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On the classification of irreducible representations of affine Hecke algebras with unequal parameters

August 2010
Representation Theory of the American Mathematical Society 16(1)

DOI:10.1090/S1088-4165-2012-00406-X

Authors:

Radboud University

Let R be a root datum with affine Weyl group

W^e

, and let

H = H (R,q)

be an affine Hecke algebra with positive, possibly unequal, parameters q. Then H is a deformation of the group algebra

\mathbb C [W^e]

, so it is natural to compare the representation theory of H and of

W^e

. We define a map from irreducible H-representations to

W^e

-representations and we show that, when extended to the Grothendieck groups of finite dimensional representations, this map becomes an isomorphism, modulo torsion. The map can be adjusted to a (nonnatural) continuous bijection from the dual space of H to that of

W^e

. We use this to prove the affine Hecke algebra version of a conjecture of Aubert, Baum and Plymen, which predicts a strong and explicit geometric similarity between the dual spaces of H and

W^e

. An important role is played by the Schwartz completion

S = S (R,q)

of H, an algebra whose representations are precisely the tempered H-representations. We construct isomorphisms

\zeta_\epsilon : S (R,q^\epsilon) \to S (R,q)

(\epsilon >0)

and injection

\zeta_0 : S (W^e) = S (R,q^0) \to S (R,q)

, depending continuously on

\epsilon

. Although

\zeta_0

is not surjective, it behaves like an algebra isomorphism in many ways. Not only does

\zeta_0

extend to a bijection on Grothendieck groups of finite dimensional representations, it also induces isomorphisms on topological K-theory and on periodic cyclic homology (the first two modulo torsion). This proves a conjecture of Higson and Plymen, which says that the K-theory of the

C^*

-completion of an affine Hecke algebra

H (R,q)

does not depend on the parameter(s) q.

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arXiv:1008.0177v2 [math.RT] 24 Sep 2010

On the classiﬁcation of irreducible

representations of aﬃne Hecke algebras

with unequal parameters

Maarten Solleveld

Mathematisches Institut, Georg-August-Universit¨at G¨ottingen

Bunsenstraße 3-5, 37073 G¨ottingen, Germany

email: Maarten.Solleveld@mathematik.uni-goettingen.de

September 2010

Abstact. Let Rbe a root datum with aﬃne Weyl group We, and let H=

H(R, q) be an aﬃne Hecke algebra with positive, possibly unequal, parameters

q. Then His a deformation of the group algebra C[We], so it is natural to

compare the representation theory of Hand of We.

We deﬁne a map from irreducible H-representations to We-representations and

we show that, when extended to the Grothendieck groups of ﬁnite dimensional

representations, this map becomes an isomorphism, modulo torsion. The map

can be adjusted to a (nonnatural) continuous bijection from the dual space of

Hto that of We. We use this to prove the aﬃne Hecke algebra version of a

conjecture of Aubert, Baum and Plymen, which predicts a strong and explicit

geometric similarity between the dual spaces of Hand We.

An important role is played by the Schwartz completion S=S(R, q) of H, an

algebra whose representations are precisely the tempered H-representations.

We construct isomorphisms ζǫ:S(R, qǫ)→ S(R, q) (ǫ > 0) and injection

ζ0:S(We) = S(R, q0)→ S(R, q), depending continuously on ǫ.

Although ζ0is not surjective, it behaves like an algebra isomorphism in many

ways. Not only does ζ0extend to a bijection on Grothendieck groups of ﬁnite

dimensional representations, it also induces isomorphisms on topological

K-theory and on periodic cyclic homology (the ﬁrst two modulo torsion).

This proves a conjecture of Higson and Plymen, which says that the K-theory

of the C∗-completion of an aﬃne Hecke algebra H(R, q) does not depend on

the parameter(s) q.

2010 Mathematics Subject Classiﬁcation.

primary: 20C08, secondary: 20G25

Contents

Introduction 3

1 Preliminaries 10

1.1 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Aﬃne Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Graded Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Analytic localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.6 The relation with reductive p-adic groups . . . . . . . . . . . . . . . 19

2 Classiﬁcation of irreducible representations 24

2.1 Two reduction theorems . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2 The Langlands classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 An aﬃne Springer correspondence . . . . . . . . . . . . . . . . . . . 35

3 Parabolically induced representations 37

3.1 Unitary representations and intertwining operators . . . . . . . . . . 38

3.2 The Schwartz algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.3 Parametrization of representations with induction data . . . . . . . . 48

3.4 The geometry of the dual space . . . . . . . . . . . . . . . . . . . . . 52

4 Parameter deformations 58

4.1 Scaling Hecke algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 Preserving unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3 Scaling intertwining operators . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Scaling Schwartz algebras . . . . . . . . . . . . . . . . . . . . . . . . 72

5 Noncommutative geometry 77

5.1 Topological K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.2 Periodic cyclic homology . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Weakly spectrum preserving morphisms . . . . . . . . . . . . . . . . 85

5.4 The Aubert–Baum–Plymen conjecture . . . . . . . . . . . . . . . . . 88

5.5 Example: type C(1)

2............................ 91

References 98

Introduction

Let R= (X, R0, Y, R∨

0, F0) be a based root datum with ﬁnite Weyl group W0and

(extended) aﬃne Weyl group We=W0⋉X. For every parameter function q:We→

C×there is an aﬃne Hecke algebra H=H(R, q).

The most important and most studied case is when qtakes the same value on all

simple (aﬃne) reﬂections in We. If this value is a prime power, then His isomor-

phic to the convolution algebra of Iwahori-biinvariant functions on a reductive p-adic

group with root datum R∨= (Y , R∨

0, X, R0, F ∨

0) [IwMa]. Moreover the generic Hecke

algebra obtained by replacing qwith a formal variable qis known [KaLu1, ChGi]

to be isomorphic to the GC×C×-equivariant K-theory of the Steinberg variety of

GC, the complex Lie group with root datum R∨. Via this geometric interpretation

Kazhdan and Lusztig [KaLu2] classiﬁed and constructed all irreducible representa-

tions of H(R, q) (when q∈C×is not a root of unity), in accordance with the local

Langlands program.

On the other hand, the deﬁnition of aﬃne Hecke algebras with generators and

relations allows one to choose the values of qon non-conjugate simple reﬂections

independently. Although this might appear to be an innocent generalization, much

less is known about aﬃne Hecke algebras with unequal parameters. The reason

is that Lusztig’s constructions in equivariant K-theory [KaLu1] allow only one de-

formation parameter. Kato [Kat2] invented a more complicated variety, called an

exotic nilpotent cone, which plays a similar role for the three parameter aﬃne Hecke

algebra of type C(1)

n. From this one can extract a classiﬁcation of the tempered dual,

for an arbitrary parameter function q[CiKa].

Like equal parameter aﬃne Hecke algebras, those with unequal parameters also

arise as intertwining algebras in the smooth representation theory of reductive p-adic

groups. One can encounter them if one looks at non-supercuspidal Bernstein com-

ponents (in the smooth dual) [Lus7, Mor]. Even for split groups unequal parameters

occur, albeit not in the principal series. It is expected that every Bernstein com-

ponent of a p-adic group can be described with an aﬃne Hecke algebra or a slight

variation on it. However, it has to be admitted that the support of this belief is not

overwhelming. In [BaMo1, BaMo2, BaCi] it is shown, in increasing generality, that

under certain conditions the equivalence between the module category of an aﬃne

Hecke algebra and a Bernstein block (in the category of smooth modules) respects

unitarity. Thus aﬃne Hecke algebras are an important tool for the classiﬁcation of

both the smooth dual and the unitary smooth dual of a reductive p-adic group.

The degenerate versions of aﬃne Hecke algebras are usually called graded Hecke

algebras. Their role in the representation theory of reductive p-adic groups [Lus7,

BaMo1, Ciu], is related to aﬃne Hecke algebras in the way that Lie algebras stand

to Lie groups. They have a very simple presentation, which makes is possible to

let them act on many function spaces. Therefore one encounters graded Hecke

algebras (with possibly unequal parameters) also independently from aﬃne Hecke

algebras, for instance in certain systems of diﬀerential equations [HeOp] and in the

classiﬁcation of the unitary dual of real reductive groups [CiTr].

In view of the above connections, it of considerable interest to classify the dual

of an aﬃne or graded Hecke algebra with unequal parameters. Since H(R, q) is

a deformation of the group algebra C[We], it is natural to expect strong similari-

ties between the duals Irr(H(R, q)) and Irr(We). Indeed, for equal parameters the

Deligne–Langlands–Kazhdan–Lusztig parametrization provides a bijection between

these duals [KaLu2, Lus5]. For unequal parameters we approach the issue via har-

monic analysis on aﬃne Hecke algebras, which forces us to consider only parameter

functions qwith values in R>0. We will assign to every irreducible H-representation

πin a natural way a representation Spr(π) of the extended aﬃne Weyl group We.

Although this construction does not always preserve irreducibility, it has a lot of

nice properties, the most important of which is:

Theorem 1. (see Theorem 2.3.1)

The collection of representations {Spr(π) : π∈Irr(H(R, q))}forms a Q-basis of the

representation ring of We.

Since the Springer correspondence for ﬁnite Weyl groups realizes Irr(W0), via

Kazhdan–Lusztig theory, as a speciﬁc subset of Irr(H(R, q)), we refer to Theorem

1 as an aﬃne Springer correspondence. It is possible to reﬁne Theorem 1 to a con-

tinuous (with respect to the Jacobson topology) bijection Irr(H(R, q)) →Irr(We).

This is related to a conjecture of Aubert, Baum and Plymen [ABP1, ABP2] (ABP-

conjecture for short) which we sketch here.

Recall that We=W0⋉Xand let Tbe the complex torus HomZ(X, C×). Cliﬀord

theory says that the irreducible We-representations with an X-weight t∈Tare in

natural bijection with the irreducible representations of the isotropy group W0,t.

The extended quotient of Tby W0is deﬁned as

T /W0=Gw∈W0{w} × Tw/W0,

with respect to the W0-action w·(w′, t) = (ww′w−1, wt). It is a model for Irr(We),

in the sense that there exist continuous bijections e

T /W0→Irr(We), which respect

the projections to T /W0.

The Bernstein presentation (included as Theorem 1.2.1) shows that C[X]∼

=O(T)

is naturally to isomorphic to a commutative subalgebra A ⊂ H(R, q), and that the

center of H(R, q) is AW0∼

=O(T/W0). Hence we have a natural map Irr(H(R, q)) →

T/W0, sending a representation to its central character. A simpliﬁed version of the

ABP-conjecture for aﬃne Hecke algebras reads:

Theorem 2. (see Theorem 5.4.2)

Let H(R, q)be an aﬃne Hecke algebra with positive, possibly unequal, parameters.

Let Q(R)be the variety of parameter functions We→C×. There exist a continuous

bijection µ:e

T /W0→Irr(H(R, q)) and a map h:e

T /W0×Q(R)→Tsuch that:

•his locally constant in the ﬁrst argument;

•for ﬁxed c∈e

T /W0, h(c, v)is a monomial in the variables v(s)±1, where s

runs through all simple aﬃne reﬂections;

•the central character of µ(W0(w, t)) is W0h(W0(w, t), q1/2)t.

The author hopes that Theorem 2 will be useful in the local Langlands program.

This could be the case for a Bernstein component sof a reductive p-adic group Gfor

which Mods(G) is equivalent to Mod(H(R, q)) (see Section 1.6 for the notations).

Recall that a Langlands parameter for Gis a certain kind of group homomorphism φ:

WF⋉C →LG, where LGis the Langlands dual group and WFis the Weil group of the

local ﬁeld Fover which Gis a variety. Let us try to extract a Langlands parameter

from W0(w, t)∈e

T /W0. The image of the distinguished Frobenius element of WF

describes the central character, so modulo conjugacy it should be h(W0(w, t), q1/2)t∈

T⊂LG.

Problematic is that the map µfrom Theorem 2 is not canonical, its construction

involves some arbitrary choices, which could lead to a diﬀerent w∈W0. Yet it is

precisely this element wthat should determine the unipotent conjugacy class that

contains the image of C\{0}under φ. Recently Lusztig [Lus9] deﬁned a map from

conjugacy classes in a Weyl group to unipotent classes in the associated complex

Lie group. Whether this map yields a suitable unipotent class for our hypothetical

φremains to be seen, for this it is probably necessary to ﬁnd a more canonical

construction of µ. The rest of such a Langlands parameter φis completely beyond

aﬃne Hecke algebras, it will have to depend on number theoretic properties of the

Bernstein component s.

Now we describe the most relevant results needed for Theorems 1 and 2. A

large step towards the determination of Irr(H(R, q)) is the Langlands classiﬁcation

(see Theorem 2.2.4), which reduces the problem to the tempered duals of parabolic

subalgebras of H(R, q). Although this is of course a well-known result, the proof for

aﬃne Hecke algebras has not been published before.

In line with Harish-Chandra’s results for reductive groups, every tempered ir-

reducible H(R, q)-representation can be obtained via induction from a discrete se-

ries representation of a parabolic subalgebra, a point of view advocated by Opdam

[Opd2]. We note that in this setting tempered and discrete series representations

can be deﬁned very easily via the A-weights of a representation. For aﬃne Hecke

algebras with irreducible root data, Opdam and the author classiﬁed the discrete

series in [OpSo2], but we emphasize that we do not use that classiﬁcation in the

present paper.

Parabolic subalgebras are given by set of simple roots P⊂F0, and induction

from HPallows an induction parameter in TP, the subtorus of Torthogonal to

P∨. So we consider induction data ξ= (P, δ, t) where P⊂F0, t ∈TPand δis

a discrete series representation of HP. Such a triple gives rise to a parabolically

induced representation

π(ξ) = π(P, δ, t) = IndH

HP(δ◦φt).

Here HP⊂ H is more or less a central extension of HPand φt:HP→ HPis a

twisted projection. The representation π(ξ) is tempered if and only if t∈TP

un, the

unitary part of TP. For the classiﬁcation of the dual it remains to decompose all

tempered parabolically induced representations and to determine when they have

constituents in common.

These phenomena are governed by intertwining operators between such represen-

tations [Opd2, Section 4]. It is already nontrivial to show that these operators are

well-deﬁned on all π(ξ) with t∈TP

un, and it is even more diﬃcult to see that they

span HomH(ξ, ξ ′). For reductive groups this is known as Harish-Chandra’s com-

pleteness theorem, and for aﬃne Hecke algebras it is a deep result due to Delorme

and Opdam [DeOp1]. The intertwining operators can be collected in a groupoid

G, which acts on the space Ξ of induction data (P, δ, t). With these tools one can

obtain a partial classiﬁcation of the dual of H(R, q):

Theorem 3. (see Theorem 3.3.2)

There exists a natural map Irr(H(R, q)) →Ξ/G, ρ 7→ Gξ+(ρ), such that:

•the map is surjective and has ﬁnite ﬁbers;

•ρis a subquotient of π(ξ+(ρ));

•ρdoes not occur in π(ξ)when ξis ”larger” than ξ+(ρ).

We note that this part of the paper is rather similar to [Sol5] for graded Hecke

algebras. The results here are somewhat stronger, and most of them can not be

derived quickly from their counterparts in [Sol5].

Yet all this does not suﬃce for Theorem 1, because we do not have much control

over the number of irreducible constituents of parabolically induced representations.

Ultimately the proof of Theorem 1 is reduced to irreducible tempered representations

with central character in HomZ(X, R>0). The author dealt with this case in [Sol6],

via the periodic cyclic homology of graded Hecke algebras.

What we discussed so far corresponds more or less to chapters 1–3 of the article.

From Theorem 1 to Theorem 2 is not a long journey, but we put Theorem 2 at the end

of the paper because we prove it together with other parts of the ABP-conjecture.

Chapters 4 and 5 are of a more analytic nature. The main object of study is the

Schwartz algebra S(R, q) of H(R, q) [Opd2, DeOp1], the analogue of the Harish-

Chandra–Schwartz algebra of a reductive p-adic group. By construction a H(R, q)-

representation extends continuously to S(R, q) if and only if it is tempered. The

Schwartz algebra is the ideal tool for the harmonic analysis of aﬃne Hecke algebras,

among others because it admits a very nice Plancherel theorem (due to Delorme

and Opdam, see [DeOp1] or Theorem 3.2.2), because the discrete series of H(R, q)

is really discrete in the dual of S(R, q), and because the inclusion H(R, q)→ S(R, q)

preserves Ext-groups of tempered representations [OpSo1].

If we vary the parameter function q, we obtain families of algebras H(R, q) and

S(R, q). It is natural to try to connect the representation theory of H(R, q) with

that of H(R, q′) for q′close to q. For general parameter deformations this is too

diﬃcult at present, but we can achieve it for deformations of the form q7→ qǫwith

ǫ∈R. On central characters of representations, this ”scaling” of qﬁts well with the

map

σǫ:T→T, t 7→ t|t|ǫ−1.

Notice that σǫ(t) = tfor t∈Tun = HomZ(X, S1). Let Modf,W0t(H(R, q)) be

the category of ﬁnite dimensional H(R, q)-representations with central character

W0t∈T/W0.

Theorem 4. (see Corollary 4.2.2)

There exists a family of additive functors

˜σǫ,t : Modf,W0t(H(R, q)) →Modf,W0σǫ(t)(H(R, qǫ)) ǫ∈[−1,1],

such that:

•for all (π, V )∈Modf,W0t(H(R, q)) and all w∈We, the map ǫ7→ ˜σǫ,t(π)(Nw)∈

EndC(V)is analytic;

•for ǫ6= 0,˜σǫ,t is an equivalence of categories;

•˜σǫ,t preserves unitarity.

For ǫ > 0 this can already be found in [Opd2], the most remarkable part is

precisely that it extends continuously to ǫ= 0, that is, to the algebra H(R, q 0) =

C[We]. In general the functors ˜σǫ,t cannot be constructed if we work only inside the

algebras H(R, qǫ), they are obtained via localizations of these algebras at certain

sets of central characters. We can do much better if we replace the algebras by their

Schwartz completions:

Theorem 5. (see Theorem 4.4.2)

For ǫ∈[0,1] there exist homomorphisms of Fr´echet *-algebras ζǫ:S(R, qǫ)→

S(R, q), such that:

•ζǫis an isomorphism for all ǫ > 0, and ζ1is the identity;

•for all w∈Wethe map ǫ7→ ζǫ(Nw)is piecewise analytic;

•for every irreducible tempered H(R, q)-representation πwith central character

W0t, the S(R, qǫ)-representations ˜σǫ,t(Nw)and π◦ζǫare equivalent.

There are some similarities with the role played by Lusztig’s asymptotic Hecke

algebra [Lus3] in [KaLu2]. In both settings an algebra is constructed, which contains

H(R, q) for a family of parameter functions q. The asymptotic Hecke algebra is of

ﬁnite type over O(T/W0), so it is only a little larger than H(R, q ). So far it has only

been constructed for equal parameter functions q, but Lusztig [Lus8] conjectures

that it also exists for unqual parameter functions. On the hand, the algebra S(R, q)

is of ﬁnite type over C∞(Tun), so it is much larger than H(R, q). Although ζǫis

an isomorphism for ǫ∈(0,1], the algebras H(R, qǫ) are embedded in S(R, q) in a

nontrivial way, in most cases ζǫ(H(R, qǫ)) is not contained in H(R, q).

Of particular interest is the homomorphism

ζ0:S(We) = S(R, q0)→ S(R, q).(1)

It cannot be an isomorphism, but it is injective and for all irreducible tempered

H(R, q)-representations πwe have π◦ζ0∼

=˜σ0(π)∼

=Spr(π). Together with Theorem

1 this results in:

Corollary 6. (see Corollary 4.4.3)

The functor Mod(S(R, q)) →Mod(S(We)) : π7→ π◦ζ0induces an isomorphism

between the Grothendieck groups of ﬁnite dimensional representations, tensored with

So Theorem 1 does not stand alone, but forms the end of a continuous family

of representations (of a family of algebras). Actually the author ﬁrst discovered the

algebra homomorphism ζ0and only later realized that the corresponding map on

representations can also be obtained in another way, thus gaining in naturality.

Apart from representation theory, the aformentioned results have some inter-

esting consequences in the noncommutative geometry of aﬃne Hecke algebras. Let

C∗(R, q) be the C∗-completion of H(R, q). It contains S(R, q) and ζǫextends to a

C∗-algebra homomorphism ζǫ:C∗(R, qǫ)→C∗(R, q ), for which Theorem 5 remains

valid. It follows quickly from this and Corollary 6 that ζ0induces an isomorphism

on topological K-theory, see Theorem 5.1.4. More precisely,

K∗(ζ0)⊗idQ:K∗(C∗(We)⋊Γ) ⊗ZQ→K∗(C∗(R, q)⋊Γ) ⊗ZQ(2)

is an isomorphism, while for equal parameters the argument also goes through with-

out ⊗ZQ. This solves a conjecture that was posed ﬁrst by Higson and Plymen

[Ply1, BCH].

Furthermore C∗(R, q) and S(R, q ) have the same topological K-theory, and via

the Chern character the complexiﬁcation of the latter is isomorphic to the periodic

cyclic homology of S(R, q). As already proved in [Sol4], H(R, q) and S(R, q) have

the same periodic cyclic homology, so we obtain a commutative diagram

HP∗(C[We]) →HP∗(S(We)) ←K∗(S(We)) ⊗ZC→K∗(C∗(We)) ⊗ZC

↓ ↓HP∗(ζ0)↓K∗(ζ0)↓K∗(ζ0)

HP∗(H(R, q)) →HP∗(S(R, q)) ←K∗(S(R, q)) ⊗ZC→K∗(C∗(R, q)) ⊗ZC,

where all the arrows are natural isomorphisms (see Corollary 5.2.2). Notice that the

Schwartz algebra S(R, q) forms a bridge between the purely algebraic H(R, q) and

the much more analytic C∗(R, q).

For the sake of clarity, the introduction is written in less generality than the

actual paper. Most notably, we can always extend our aﬃne Hecke algebras by

a group Γ of automorphisms of the Dynkin diagram of R. On the one hand this

generality is forced upon us, in particular by Lusztig’s ﬁrst reduction theorem (see

Theorem 2.1.2), which necessarily involves diagram automorphisms. On the other

hand, one advantage of having H(R, q)⋊Γ instead of just H(R, q) is that our proof of

the Aubert–Baum–Plymen conjecture applies to clearly more Bernstein components

of reductive p-adic groups.

For most of the results of this paper, the extension from H(R, q) to H(R, q)⋊Γ

is easy, mainly a matter of some extra notation. An exception is the Langlands

classiﬁcation, which hitherto was only known for commutative groups of diagram

automorphisms [BaJa1]. In our generalization (see Corollary 2.2.5) we add a new

ingredient to the Langlands data, and we show how to save the uniqueness part.

A substantial part of this article is based on the author’s PhD-thesis [Sol3],

which was written under the supervision of Opdam. We refrain from indicating all

the things that stem from [Sol3], among others because some of proofs in [Sol3] were

not worked out with the accuracy needed for research papers. Moreover, in the years

after writing this thesis many additional insights were obtained, so that in the end

actually no part of [Sol3] reached this article unscathed. The technical Chapter 4

comes closest. It should also be mentioned that the conjecture (2) formed a central

part of the author’s PhD-research. At that time it was still too diﬃcult for the

author, mainly because Theorem 1 was not available yet.

Acknowledgements. The author learned a lot about aﬃne Hecke algebras from

Eric Opdam, ﬁrst as a PhD-student a later as co-author. Without that support,

this article would not have been possible. The author also thanks Roger Plymen

for providing background information about several conjectures, and Anne-Marie

Aubert for many detailed comments, in particular concerning Section 1.6.

Chapter 1

Preliminaries

This chapter serves mainly to introduce some deﬁnitions and notations that we will

use later on. The results that we recall can be found in several other sources, like

[Lus6, Ree1, Opd2]. By default, our aﬃne Hecke algebras are endowed with unequal

parameters and may be extended with a group of automorphism of the underlying

root datum.

In the section dedicated to p-adic groups we recall what is known about the

(conjectural) relation between Bernstein components and aﬃne Hecke algebras. On

one hand this motivates the generality that we work in, on the other hand we will

use it in Section 5.4 to translate a conjecture of Aubert, Baum and Plymen to the

setting of aﬃne Hecke algebras.

1.1 Root systems

Let abe a ﬁnite dimensional real vector space and let a∗be its dual. Let Y⊂abe

a lattice and X= HomZ(Y, Z)⊂a∗the dual lattice. Let

R= (X, R0, Y, R∨

0, F0).

be a based root datum. Thus R0is a reduced root system in X, R∨

0⊂Yis the

dual root system, F0is a basis of R0and the set of positive roots is denoted R+

Furthermore we are given a bijection R0→R∨

0, α 7→ α∨such that hα , α∨i= 2 and

such that the corresponding reﬂections sα:X→X(resp. s∨

α:Y→Y) stabilize R0

(resp. R∨

0). We do not assume that R0spans a∗.

The reﬂections sαgenerate the Weyl group W0=W(R0) of R0, and S0:=

{sα:α∈F0}is the collection of simple reﬂections. We have the aﬃne Weyl group

Waﬀ =ZR0⋊W0and the extended (aﬃne) Weyl group We=X⋊W0. Both can

be considered as groups of aﬃne transformations of a∗. We denote the translation

corresponding to x∈Xby tx. As is well known, Waﬀ is a Coxeter group, and the

basis of R0gives rise to a set Saﬀ of simple (aﬃne) reﬂections. More explicitly, let

F∨

Mbe the set of maximal elements of R∨

0, with respect to the dominance ordering

coming from F0. Then

Saﬀ =S0∪ {tαsα:α∈FM}.

We write

X+:= {x∈X:hx , α∨i ≥ 0∀α∈F0},

X−:= {x∈X:hx , α∨i ≤ 0∀α∈F0}=−X+.

It is easily seen that the center of Weis the lattice

Z(We) = X+∩X−.

We say that Ris semisimple if Z(We) = 0 or equivalently if R0spans a∗. Thus a

root datum is semisimple if and only if the corresponding reductive algebraic group

is so.

The length function ℓof the Coxeter system (Waﬀ , Saﬀ ) extends naturally to

We, such that [Opd1, (1.3)]

ℓ(wtx) = ℓ(w) + Xα∈R+

0hx , α∨iw∈W0, x ∈X+.(1.1)

The elements of length zero form a subgroup Ω ⊂We, and We=Waﬀ ⋊Ω. With

Rwe also associate some other root systems. There is the non-reduced root system

Rnr := R0∪ {2α:α∨∈2Y}.

Obviously we put (2α)∨=α∨/2. Let R1be the reduced root system of long roots

in Rnr:

R1:= {α∈Rnr :α∨6∈ 2Y}.

We denote the collection of positive roots in R0by R+

0, and similarly for other root

systems.

1.2 Aﬃne Hecke algebras

There are three equivalent ways to introduce a complex parameter function for R.

(1) A map q:Saﬀ →C×such that q(s) = q(s′) if sand s′are conjugate in We.

(2) A function q:We→C×such that

q(ω) = 1 if ℓ(ω) = 0,

q(wv) = q(w)q(v) if w, v ∈Weand ℓ(wv) = ℓ(w) + ℓ(v).(1.2)

(3) A W0-invariant map q:R∨

nr →C×.

One goes from (2) to (1) by restriction, while the relation between (2) and (3) is

given by

qα∨=q(sα) = q(tαsα) if α∈R0∩R1,

qα∨=q(tαsα) if α∈R0\R1,

qα∨/2=q(sα)q(tαsα)−1if α∈R0\R1.

(1.3)

We speak of equal parameters if q(s) = q(s′)∀s, s′∈Saﬀ and of positive parameters

if q(s)∈R>0∀s∈Saﬀ .

We ﬁx a square root q1/2:Saﬀ →C×. The aﬃne Hecke algebra H=H(R, q) is

the unique associative complex algebra with basis {Nw:w∈We}and multiplication

rules NwNv=Nwv if ℓ(wv) = ℓ(w) + ℓ(v),

Ns−q(s)1/2Ns+q(s)−1/2= 0 if s∈Saﬀ .(1.4)

In the literature one also ﬁnds this algebra deﬁned in terms of the elements q(s)1/2Ns,

in which case the multiplication can be described without square roots. This explains

why q1/2does not appear in the notation H(R, q).

Notice that Nw7→ Nw−1extends to a C-linear anti-automorphism of H, so H

is isomorphic to its opposite algebra. The span of the Nwwith w∈W0is a ﬁnite

dimensional Iwahori–Hecke algebra, which we denote by H(W0, q).

Now we describe the Bernstein presentation of H. For x∈X+we put θx:=

Ntx. The corresponding semigroup morphism X+→ H(R, q)×extends to a group

homomorphism

X→ H(R, q)×:x7→ θx.

Theorem 1.2.1. (Bernstein presentation)

(a)The sets {Nwθx:w∈W0, x ∈X}and {θxNw:w∈W0, x ∈X}are bases of H.

(b)The subalgebra A:= span{θx:x∈X}is isomorphic to C[X].

(c)The center of Z(H(R, q)) of H(R, q)is AW0, where we deﬁne the action of W0

on Aby w(θx) = θwx.

(d)For f∈ A and α∈F0∩R1

fNsα−Nsαsα(f) = q(sα)1/2−q(sα)−1/2(f−sα(f))(θ0−θ−α)−1,

while for α∈F0\R1:

fNsα−Nsαsα(f) = q(sα)1/2−q(sα)−1/2+ (q1/2

α∨−q−1/2

α∨)θ−αf−sα(f)

θ0−θ−2α

Proof. These results are due to Bernstein, see [Lus6, §3]. 2

The following lemma was claimed in the proof of [Opd1, Lemma 3.1].

Lemma 1.2.2. For x∈X+

span{NuθxNv:u, v ∈W0}= span{Nw:w∈W0txW0}.(1.5)

Let Wxbe the stabilizer of xin W0and let Wxbe a set of representatives for W0/Wx.

Then the elements NuθxNvwith u∈Wxand v∈W0form a basis of (1.5).

Proof. By (1.1) ℓ(utx) = ℓ(u) + ℓ(tx), so Nuθx=Nutx. Recall the Bruhat

ordering on a Coxeter group, for example from [Hum, Sections 5.9 and 5.10]. With

induction to ℓ(v) it follows from the multiplication rules (1.4) that

NwNv−Nwv ∈span{Nw˜v: ˜v < v in the Bruhat ordering}.

Hence the sets {NuθxNv:v∈W0}and {Nw:w∈utxW0}have the same span. They

have the same cardinality and by deﬁnition the latter set is linearly independent, so

the former is linearly independent as well. Clearly W0txW0=⊔u∈WxutxW0, so

span{Nw:w∈W0txW0}=Mu∈Wxspan{Nw:w∈utxW0}

=Mu∈Wxspan{NuθxNv:v∈W0}= span{NuθxNv:u∈Wx, v ∈W0}.

The number of generators on the second line side equals the dimension of the ﬁrst

line, so they form a basis. 2

Let Tbe the complex algebraic torus

T= HomZ(X, C×)∼

=Y⊗ZC×,

so A∼

=O(T) and Z(H) = AW0∼

=O(T/W0). From Theorem 1.2.1 we see that H

is of ﬁnite rank over its center. Let t= Lie(T) and t∗be the complexiﬁcations of a

and a∗. The direct sum t=a⊕iacorresponds to the polar decomposition

T=Trs ×Tun = HomZ(X, R>0)×HomZ(X, S 1)

of Tinto a real split (or positive) part and a unitary part. The exponential map

exp : t→Tis bijective on the real parts, and we denote its inverse by log : Trs →a.

An automorphism of the Dynkin diagram of the based root system (R0, F0) is a

bijection γ:F0→F0such that

hγ(α), γ(β)∨i=hα , β∨i ∀α, β ∈F0.(1.6)

Such a γnaturally induces automorphisms of R0, R∨

0, W0and Waﬀ . It is easy to

classify all diagram automorphisms of (R0, F0): they permute the irreducible compo-

nents of R0of a given type, and the diagram automorphisms of a connected Dynkin

diagram can be seen immediately.

We will assume that the action of γon Waﬀ has been extended in some way to

We. For example, this is the case if γbelongs to the Weyl group of some larger root

system contained in X. We regard two diagram automorphisms as the same if and

only if their actions on Weare equal.

Let Γ be a ﬁnite group of diagram automorphisms of (R0, F0) and assume that

qα∨=qγ(α∨)for all γ∈Γ, α ∈Rnr . Then Γ acts on Hby algebra automorphisms

ψγthat satisfy

ψγ(Nw) = Nγ(w)w∈We,

ψγ(θx) = θγ(x)x∈X. (1.7)

Hence one can form the crossed product algebra Γ ⋉H=H⋊Γ, whose natural basis

is indexed by the group (X⋊W0)⋊Γ = X⋊(W0⋊Γ). It follows easily from (1.7)

and Theorem 1.2.1.c that Z(H⋊Γ) = AW0⋊Γ. We say that the central character of

an (irreducible) H⋊Γ-representation is positive if it lies in Trs/(W0⋊Γ).

We always assume that we have an Γ ⋉W0-invariant inner product on a. The

length function of Waﬀ also extends to X⋊W0⋊Γ, and the subgroup of elements

of length zero becomes

{w∈We⋊Γ : ℓ(w) = 0}= Γ ⋉{w∈We:ℓ(w) = 0}= Γ ⋉Ω.

More generally one can consider a ﬁnite group Γ′that acts on Rby diagram auto-

morphisms. Then the center of H⋊Γ′can be larger than AW0⋊Γ′, but apart from

that the structure is the same.

Another variation arises when Γ →Aut(H) is not a group homomorphism, but

a homomorphism twisted by a 2-cocycle κ: Γ ×Γ→C×. Instead of H⋊Γ one can

construct the algebra H ⊗ C[Γ, κ], whose multiplication is deﬁned by

NγNγ′=κ(γ, γ ′)Nγγ′,

NγhN−1

γ=γ(h),γ, γ ′∈Γ, h ∈ H.

By [Mor, Section 7] such algebras can appear in relevant examples, although it is

no explicit nontrivial are known. Let Γ∗be the Schur multiplier of Γ, also known as

representation group [CuRe]. It is a central extension of Γ that classiﬁes projective

Γ-representations, and its group algebra C[Γ∗] contains C[Γ, κ] as a direct summand.

The algebra H⋊Γ∗is well-deﬁned and contains H ⊗ C[Γ, κ] as a direct summand.

Thus we can reduce the study of aﬃne Hecke algebras with twisted group actions

the case of honest group actions.

1.3 Graded Hecke algebras

Graded Hecke algebras are also known as degenerate (aﬃne) Hecke algebras. They

were introduced by Lusztig in [Lus6]. We call

R= (a∗, R0,a, R∨

0, F0) (1.8)

a degenerate root datum. We pick complex numbers kαfor α∈F0, such that

kα=kβif αand βare in the same W0-orbit. The graded Hecke algebra associated

to these data is the complex vector space

H=H(˜

R, k) = S(t∗)⊗C[W0],

with multiplication deﬁned by the following rules:

•C[W0] and S(t∗) are canonically embedded as subalgebras;

•for x∈t∗and sα∈Swe have the cross relation

x·sα−sα·sα(x) = kαhx , α∨i.(1.9)

Multiplication with any ǫ∈C×deﬁnes a bijection mǫ:t∗→t∗, which clearly

extends to an algebra automorphism of S(t∗). From the cross relation (1.9) we see

that it extends even further, to an algebra isomorphism

mǫ:H(˜

R, zk)→H(˜

R, k) (1.10)

which is the identity on C[W0].

Let Γ be a group of diagram automorphisms of ˜

Rand assume that kγ(α)=kα

for all α∈R0, γ ∈Γ. Then Γ acts on Hby the algebra automorphisms

ψγ:H→H,

ψγ(xsα) = γ(x)sγ(α)x∈t∗, α ∈Π.(1.11)

By [Sol6, Proposition 5.1.a] the center of the resulting crossed product algebra is

Z(H ⋊ Γ) = S(t∗)W0⋊Γ=O(t/(W0⋊Γ)).(1.12)

We say that the central character of an H ⋊ Γ-representation is real if it lies in

a/(W0⋊Γ).

1.4 Parabolic subalgebras

For a set of simple roots P⊂F0we introduce the notations

RP=QP∩R0R∨

P=QR∨

P∩R∨

aP=RP∨aP= (a∗

P)⊥,

a∗

P=RPaP∗= (aP)⊥,

tP=CP∨tP= (t∗

P)⊥,

t∗

P=CPtP∗= (tP)⊥,

XP=XX∩(P∨)⊥XP=X/(X∩QP),

YP=Y∩QP∨YP=Y∩P⊥,

TP= HomZ(XP,C×)TP= HomZ(XP,C×),

RP= (XP, RP, YP, R∨

P, P )RP= (X, RP, Y , R∨

P, P ),

RP= (a∗

P, RP,aP, R∨

P, P )˜

RP= (a∗, RP,a, R∨

P, P ).

(1.13)

We denote the image of x∈Xin XPby xP. Although Trs =TP,rs ×TP

rs, the product

Tun =TP,unTP

un is not direct, because the intersection

KP:= TP,un ∩TP

un =TP∩TP

can have more than one element (but only ﬁnitely many).

We deﬁne parameter functions qPand qPon the root data RPand RP, as

follows. Restrict qto a function on (RP)∨

nr and use (1.3) to extend it to W(RP)

and W(RP). Similarly the restriction of kto Pis a parameter function for the

degenerate root data ˜

RPand ˜

RP, and we denote it by kPor kP. Now we can deﬁne

the parabolic subalgebras

HP=H(RP, qP)HP=H(RP, qP),

HP=H(˜

RP, kP)HP=H(˜

RP, kP).

We notice that HP=S(tP∗)⊗HP, a tensor product of algebras. Despite our termi-

nology HPand HPare not subalgebras of H, but they are close. Namely, H(RP, qP)

is isomorphic to the subalgebra of H(R, q) generated by Aand H(W(RP), qP). We

denote the image of x∈Xin XPby xPand we let AP⊂ HPbe the commutative

subalgebra spanned by {θxP:xP∈XP}. There is natural surjective quotient map

HP→ HP:θxNw7→ θxPNw.(1.14)

Suppose that γ∈Γ⋉W0satisﬁes γ(P) = Q⊆F0. Then there are algebra isomor-

phisms

ψγ:HP→ HQ, θxPNw7→ θγ(xP)Nγwγ −1,

ψγ:HP→ HQ, θxNw7→ θγx Nγwγ−1,

ψγ:HP→HQ, fPw7→ (fP◦γ−1)w,

ψγ:HP→HQ, fw 7→ (f◦γ−1)w,

(1.15)

where fP∈ O(tP) and f∈ O(t). Sometimes we will abbreviate W0⋊Γ to W′, which

is consistent with the notation of [Sol5]. For example the group

W′

P:= {γ∈Γ⋉W0:γ(P) = P}(1.16)

acts on the algebras HPand HP. Although W′

F0= Γ, for proper subsets P(F0

the group W′

Pneed not be contained in Γ. In other words, in general W′

Pstrictly

contains the group

ΓP:= {γ∈Γ : γ(P) = P}=W′

P∩Γ.

To avoid confusion we do not use the notation WP. Instead the parabolic subgroup

of W0generated by {sα:α∈P}will be denoted W(RP). Suppose that γ∈W′

stabilizes either the root system RP, the lattice ZPor the vector space QP⊂a∗.

Then γ(P) is a basis of RP, so γ(P) = w(P) and w−1γ∈W′

Pfor a unique w∈

W(RP). Therefore

W′

ZP:= {γ∈W′:γ(ZP) = ZP}equals W(RP)⋊W′

P.(1.17)

For all x∈Xand α∈Pwe have

x−sα(x) = hx , α∨iα∈ZP,

so t(sα(x)) = t(x) for all t∈TP. Hence t(w(x)) = t(x) for all w∈W(RP), and we

can deﬁne an algebra automorphism

φt:HP→ HP, φt(θxNw) = t(x)θxNwt∈TP.(1.18)

In particular, for t∈KPthis descends to an algebra automorphism

ψt:HP→ HP, θxPNw7→ t(xP)θxPNwt∈KP.(1.19)

We can regard any representation (σ, Vσ) of H(RP, qP) as a representation of H(RP, qP)

via the quotient map (1.14). Thus we can construct the H-representation

π(P, σ, t) := IndH(R,q)

H(RP,qP)(σ◦φt).

Representations of this form are said to be parabolically induced. Similarly, for any

HP-representation (ρ, Vρ) and any λ∈tthere is an HP-representation (ρλ, Vρ⊗Cλ).

The corresponding parabolically induced representation is

π(P, ρ, λ) := IndH

HP(ρλ) = IndH

HP(Vρ⊗Cλ).

1.5 Analytic localization

A common technique in the study of Hecke algebras is localization at one or more

characters of the center. There are several ways to implement this. Lusztig [Lus6]

takes a maximal ideal Iof Z(H) and completes Hwith respect to the powers of this

ideal. This has the eﬀect of considering only those H-representations which admit

the central character corresponding to I.

For reasons that will become clear only in Chapter 4, we prefer to localize with

analytic functions on subvarieties of T/W ′. Let U⊂Tbe a nonempty W′-invariant

subset and let Can(U) (respectively Cme(U)) be the algebra of holomorphic (respec-

tively meromorphic) functions on U. There is a natural embedding

Z(H⋊Γ) = AW′∼

=O(T)W′→Can(U)W′

and isomorphisms of topological algebras

Can(U)W′⊗AW′A∼

=Can(U), Cme(U)W′⊗AW′A∼

=Cme(U).

Thus we can construct the algebras

Han(U)⋊Γ := Can(U)W′⊗Z(H⋊Γ) H⋊Γ∼

=Can(U)⊗CC[W′],

Hme(U)⋊Γ := Cme(U)W′⊗Z(H⋊Γ) H⋊Γ∼

=Cme(U)⊗CC[W′].(1.20)

The isomorphisms with the right hand side are in the category of topological vector

spaces, the algebra structure on the left hand side is determined by

(f1⊗h1)(f2⊗h2) = f1f2⊗h1h2fi∈Cme(U)W′, hi∈ H ⋊Γ.(1.21)

By [Opd2, Proposition 4.5] Z(Han(U)⋊Γ) ∼

=Can(U)W′, and similarly with mero-

morphic functions.

For any T′⊂T, let Modf,T ′(H⋊Γ) be the category of ﬁnite dimensional H⋊Γ-

modules all whose A-weights lie in T′. By [Opd2, Proposition 4.3] Modf(Han(U)⋊Γ)

is naturally equivalent to Modf ,U (H⋊Γ). (On the other hand, Hme (U)⋊Γ does

not have any nonzero ﬁnite dimensional representations over C.)

Of course graded Hecke algebras can localized in exactly the same way, and the

resulting algebras have analogous properties. By (1.12) the center of the algebra

H ⋊ Γ is isomorphic to O(t/W ′) = O(t)W′. For nonempty open W′-invariant subsets

Vof twe get the algebras

Han(V)⋊Γ := Can(V)W′⊗Z(H⋊Γ) H ⋊ Γ∼

=Can(V)⊗CC[W′],

Hme(V)⋊Γ := Cme (V)W′⊗Z(H⋊Γ) H ⋊ Γ∼

=Cme(V)⊗CC[W′].(1.22)

Let C(T /W ′) be the quotient ﬁeld of Z(H⋊Γ) ∼

=O(T /W ′) and consider

C(T /W ′)⊗Z(H⋊Γ) H⋊Γ.

As a vector space this is C(T)⊗AH⋊Γ∼

=C(T)⊗CC[W′], while its multiplication

is given by (1.21). Similarly, we let C(t/W ′) be the quotient ﬁeld of O(t/W ′) and

we construct the algebra

C(t/W ′)⊗Z(H⋊Γ) H ⋊ Γ.

Its underlying vector space is

C(t)⊗O(t)H ⋊ Γ∼

=C(t)⊗CC[W′],

and its multiplication is given by the obvious analogue of (1.21).

An important role in the harmonic analysis of H(R, q) is played by the Mac-

donald c-functions cα∈C(T) (cf. [Lus6, 3.8] and [Opd1, Section 1.7]), deﬁned

cα=





θα+q(sα)−1/2q1/2

α∨

θα+ 1

θα−q(sα)−1/2q−1/2

α∨

θα−1α∈R0\R1

(θα−q(sα)−1)(θα−1)−1α∈R0∩R1

(1.23)

Notice that cα= 1 if and only if q(sα) = q(tαsα) = 1. With this function we can

rephrase Theorem 1.2.1.d as

fNsα−Nsαsα(f) = q(sα)−1/2f−sα(f)(q(sα)cα−1).

For the graded Hecke algebra H(˜

R, k) this is much easier:

˜cα= (α+kα)α−1= 1 + kαα−1∈C(t),(1.24)

xsα−sαsα(x) = x−sα(x)(˜cα−1) α∈F0, x ∈t∗.

Given a simple root α∈F0we deﬁne elements ı0

sα∈C(T/W0)⊗Z(H)Hand ˜ısα∈

C(t/W0)⊗Z(H)Hby

q(sα)cα(1 + ıo

sα) = 1 + q(sα)1/2Nsα,

˜cα(1 + ˜ısα) = 1 + sα.(1.25)

Proposition 1.5.1. The elements ı0

sαand ˜ısαhave the following properties:

(a)The map sα7→ ı0

sα(respectively sα7→ ˜ısα) extends to a group homomorphism

from W′to the multiplicative group of C(T /W ′)⊗Z(H⋊Γ) H⋊Γ(respectively

C(t/W ′)⊗Z(H⋊Γ) H ⋊ Γ).

(b)For w∈W′and f∈C(T)∼

=C(T /W ′)⊗O(T/W ′)A(respectively ˜

f∈C(t)) we

have ı0

wfı0

w−1=w(f)(respectively ˜ıw˜

f˜ıw−1=w(˜

f)).

(c)The maps

C(T)⋊W′→C(T /W ′)⊗Z(H⋊Γ) H⋊Γ : fw 7→ fı0

C(t)⋊W′→C(t/W ′)⊗Z(H⋊Γ) H ⋊ Γ : ˜

fw 7→ ˜

f˜ıw

are algebra isomorphisms.

(d)Let P⊂F0and γ∈W′be such that γ(P)⊂F0. The automorphisms ψγfrom

(1.15) satisfy

ψγ(h) = ı0

γhı0

γ−1h∈ HPor h∈ HP,

ψγ(˜

h) = ˜ıγ˜

h˜ıγ−1˜

h∈HP.

Proof. (a), (b) and (c) with W0instead of W′can be found in [Lus6, Section 5].

Notice that Lusztig calls these elements τwand ˜τw, while we follow the notation of

[Opd2, Section 4]. We extend this to W′by deﬁning, for γ∈Γ and w∈W0:

ı0

γw := γı0

wand ˜ıγw =γ˜ıw.

For (d) see [Lus6, Section 8] or [Sol5, Lemma 3.2]. 2

We remark that by construction all the ı0

wlie in the subalgebra C(T /T F0)H(W0, q)⋊

Γ and that the ˜ıwlie in the subalgebra C(t/tF0)C[W′]. As was noticed in [Opd2,

Theorem 4.6], Proposition 1.5.1 can easily be generalized:

Corollary 1.5.2. Proposition 1.5.1 remains valid under any of the following re-

placements:

•C(T)by Cme(U)or, if all the functions cαare invertible on U, by Can(U);

•C(t)by Cme(V)or, if all the functions ˜cαare invertible on V, by Can(V).

In particular

Cme(U)⋊W′→Cme(U)W′⊗AW′H⋊Γ : fw →f ıo

w(1.26)

is an isomorphism of topological algebras.

1.6 The relation with reductive p-adic groups

Here we discuss how aﬃne Hecke algebras arise in the representation theory of

reductive p-adic groups. This section is rather sketchy, it mainly serves to provide

some motivation and to prepare for our treatment of the Aubert–Baum–Plymen

conjecture in Section 5.4 The main sources for this section are [BeDe, BeRu, Roc2,

Hei, IwMa, Mor].

Let Fbe a nonarchimedean local ﬁeld with a ﬁnite residue ﬁeld. Let Gbe a

connected reductive algebraic group deﬁned over Fand let G=G(F) be its group

of F-rational points. One brieﬂy calls Ga reductive p-adic group, even though the

characteristic of Fis allowed to be positive.

An important theme, especially in relation with the arithmetic Langlands pro-

gram, is the study of the category Mod(G) of smooth G-representations on complex

vector spaces. A ﬁrst step to simplify this problem is the Bernstein decomposition,

which we recall now. Let Pbe a parabolic subgroup of Gand Ma Levi subgroup of

P. Although Gand Mare unimodular, the modular function δPof Pis in general

not constant. Let σbe an irreducible supercuspidal representation of M. In this

situation we call (M, σ a cuspidal pair, and with parabolic induction we construct

the G-representation

P(σ) := IndG

P(δ1/2

P⊗σ).

This means that ﬁrst we inﬂate σto Pand then we apply the normalized smooth

induction functor. The normalization refers to the twist by δ1/2

P, which is useful

to preserve unitarity. Let Irr(G) be the collection of irreducible representations in

Mod(G), modulo equivalence. For every π∈Irr(G) there is a cuspidal pair (M, σ),

uniquely determined up to G-conjugacy, such that πis a subquotient of IG

P(σ).

Let G0be the normal subgroup of Ggenerated by all compact subgroups. Recall

that a character χ:G→C×is called unramiﬁed if its kernel contains G0. The group

Xur(G) of unramiﬁed characters forms a complex algebraic torus whose character

lattice is naturally isomorphic to the lattice X∗(G) of algebraic characters G→F×.

We say that two cuspidal pairs (M, σ ) and (M′, σ′) are inertially equivalent if there

exist g∈Gand χ′∈Xur(M) such that

M′=gM g−1and σ′⊗χ′∼

=σg.

With an inertial equivalence class s= [M, σ]Gone associates a full subcategory

Mods(G) of Mod(G). It objects are by deﬁnition those smooth representations π

with the property that for every irreducible subquotient ρof πthere is a (M, σ)∈

ssuch that ρis a subrepresentation of IG

P(σ). The collection B(G) of inertial

equivalence classes is countably inﬁnite (unless G={1}).

The Bernstein decomposition [BeDe, Proposition 2.10] states that

Mod(G) = Ys∈B(G)Mods(G),(1.27)

a direct product of categories. The subcategories Mods(G) (or rather their subsets of

irreducible representations) are also called the Bernstein components of the smooth

dual of G.

The Hecke algebra H(G) is the vector space of locally constant, compactly sup-

ported functions on G, endowed with the convolution product. Mod(G) is naturally

equivalent to the category Mod(H(G)) of essential H(G)-modules. (A module Vis

called essential if H(G)V=V, which is not automatic because H(G) does not have

a unit.) Corresponding to (1.27) there is a decomposition

H(G) = Ms∈B(G)H(G)s

of the Hecke algebra of Ginto two-sided ideals. In several cases H(G)sis known to

be Morita-equivalent to an aﬃne Hecke algebra.

In the classical case [IwMa, Bor] Gis split and Mods(G) is the category of

Iwahori-spherical representations. That is, those smooth G-representations Vthat

are generated by VI, where Iis an Iwahori-subgroup of G. Then H(G)sis Morita

equivalent to the algebra H(G, I) of I-biinvariant functions in H(G), and H(G, I) is

isomorphic to an aﬃne Hecke algebra H(R, q). The root datum R= (X, R0, Y , R∨

0, F0)

is dual to the root datum of (G,T), where T(F) is a split maximal torus of G=G(F).

More explicitly

•Xis the cocharacter lattice of T;

•Yis the character lattice of T;

•R∨

0is the root system of (G,T);

•R0is the set coroots of (G,T);

•F0and F∨

0are determined by I;

•qis the cardinality of the residue ﬁeld of F.

For a general inertial equivalence class s= [M, σ]Git is expected that H(G)sis

Morita equivalent to an aﬃne Hecke algebra or to a closely related kind of algebra.

We discuss some ingredients of this conjectural relation between the representation

theory of reductive p-adic groups and that of aﬃne Hecke algebras.

Let σ0be an irreducible subrepresentation of σM0and deﬁne Σ = indM

M0(σ0).

According to [BeRu, Theorem 23] IG

P(Σ) is a ﬁnitely generated projective generator

of the category Mods(G). By [BeRu, Lemma 22] Mods(G) = Mod(H(G)s) is natu-

rally equivalent to the category of right EndG(IG

P(Σ))-modules. So if EndG(IG

P(Σ))

would be isomorphic to its opposite algebra (which is likely), then it is Morita equiv-

alent to H(G)s.

Let us describe the center of EndG(IG

P(Σ)). The map

Xur(M)→Irr[M,σ]M(M) : χ7→ χ⊗σ

is surjective and its ﬁbers are cosets of a ﬁnite subgroup Stab(σ)⊂Xur (M). Let

Mσ:= \χ∈Stab(σ)ker(χ)⊂M.

Roche [Roc2, Proposition 5.3] showed that EndM(Σ) is a free O(Xur(Mσ))-module

of rank m2, where mis the multiplicity of σ0in σM0. Moreover the center of

EndM(Σ) is isomorphic to O(Xur (Mσ)), and EndM(Σ) embeds in EndG(IG

P(Σ)) by

functoriality. The group

NG(M, σ) := {g∈G:gM g−1=M , σg∼

=χ⊗σfor some χ∈Xur(M)}

acts on the family of representations {IG

P(χ⊗Σ) : χ∈Xur(M)}, and via this on

Xur(Mσ). The subgroup M⊂NG(M , σ) acts trivially, so we get an action of the

ﬁnite group

Wσ:= NG(M, σ)/M.

According to [BeDe, Th´eor`eme 2.13], the center of EndG(IG

P(Σ)) is isomorphic to

O(Xur(Mσ))Wσ=O(Xur(Mσ)/Wσ). By [Roc2, Lemma 7.3] EndG(IG

P(Σ)) is a free

EndM(Σ)-module of rank |Wσ|.

Next we indicate how to associate a root datum to EndG(IG

P(Σ)). See [Hei,

Section 6] for more details in the case of classical groups. Let Abe the maximal

split torus of Z(M), let X∗(A) be its character lattice and X∗(A) its cocharacter

lattice. There are natural isomorphisms

Xur(M)∼

=X∗(A)⊗ZC×∼

=Hom(X∗(A),C×).

In X∗(A) we have the root system R(G, A) and in X∗(A) we have the set R∨(G, A)

of coroots of (G, A). The parabolic subgroup Pdetermines positive systems R(P, A)

and R∨(P, A). Altogether we constructed a (nonreduced) based root datum

RM:= X∗(A), R∨(G, A), X∗(A), R(G, A), R∨(P, A),

from which one can of course deduce a reduced based root datum.

Yet RMis not good enough, it does not take σinto account. Put

Xσ:= Hom(Xur(Mσ),C×)∼

=Hom(Xur(Mσ∩A),C×)⊂X∗(A),

Yσ:= Hom(Xσ,Z)∼

=X∗(Mσ∩A)⊃X∗(A).

Assume for simplicity that σM0is multiplicity free, or equivalently that EndM(Σ) =

O(Xur(Mσ)). Then the above says that EndG(IG

P(Σ)) is a free module of rank |Wσ|

over C[Xσ]. We want to associate a root system to Wσ. In general Wσdoes not

contain W(G, A) = NG(M)/M, so we have to replace R(G, A) by

Rσ,nr := {α∈R(G, A) : sα∈Wσ},

and R∨(G, A) by R∨

σ,nr. Let R∨

σbe the reduced root system of R∨

σ,nr and Rσthe

dual root system, which consists of the non-multipliable roots in Rσ,nr. Let Fσbe

the unique basis of Rσcontained in R(P, A). Then W(Rσ) is a normal subgroup of

Wσand

Wσ∼

=W(Rσ)⋊Γσwhere Γσ={w∈Wσ:w(Fσ) = Fσ}.

As σis not explicit, it is diﬃcult to say which diagram automorphism groups Γσ

can occur here. A priori there does not seem to be any particular restriction.

All this suggests that, if EndG(IG

P(Σ)) is isomorphic to some aﬃne Hecke algebra,

then to

Hσ⋊Γσ:= H(Xσ, R∨

σ, Yσ, Rσ, F ∨

σ, qσ)⋊Γσ.(1.28)

In fact it also possible that the Γσ-action is twisted by a cocycle [Mor, Section 7],

but we ignore this subtlety here. We note that little would change upon replacing

Gby a disconnected group, that would only lead to a larger group of diagram

automorphisms.

We note that this description of EndG(IG

P(Σ)) is compatible with parabolic in-

duction. Every parabolic subgroup Q⊂Gcontaining Pgives rise to a subalgebra

EndQ(IQ

P(Σ)) ⊂EndG(IG

P(Σ)),

which via (1.28) and

σ,nr =R(Q, A)⊂R(P, A) = Rσ,nr

corresponds to a parabolic subalgebra HQ

σ⋊Γσ,Q ⊂ Hσ⋊Γσ.

By analogy with the Iwahori case the numbers qσ(w) are related to the aﬃne

Coxeter complex of X∗(A)⋊W(R∨(G, A)). After ﬁxing a fundamental chamber C0,

every w∈X∗(A)⋊Wσdetermines a chamber w(C0). This aﬃne Coxeter complex

can be regarded as a subset of the Bruhat–Tits building of G, so C0has a stabilizer

K⊂G. In view of [Mor, Section 6], a good candidate for qσ(w) is [KwK :K]. In

particular, for a simple reﬂection sα∈W(R∨

σ) this works out to qσ(sα) = qdα, where

qis cardinality of the residue ﬁeld of Fand dαis the dimension of the α-weight space

in the A-representation Lie(G). Hence qσis a positive parameter function and, if A

is not a split maximal torus of G,qσtends to be non-constant on the set of simple

reﬂections.

As said before, most of the above is conjectural. The problem is that in general

it is not known whether one can construct elements Nw(w∈Wσ) that satisfy the

multiplication rules of an extended aﬃne Hecke algebra. To that end one has to

study the intertwining operator between parabolically induced representations very

carefully.

Let us list the cases in which it is proven that EndG(IG

P(Σ)) is isomorphic to an

(extended) aﬃne Hecke algebra:

•Gsplit, sthe Iwahori-spherical component [IwMa, Bor];

•G=GLn(F)sarbitrary - from the work of Bushnell and Kutzko on types

[BuKu1, BuKu2, BuKu3];

•G=SLn, many s[GoRo] (for general sthe Hecke algebra is known to have a

closely related shape);

•G=SOn(F), G =Spn(F) or Gan inner form of GLn,sarbitrary [Hei];

•G=GSp4(F) or G=U(2,1),sarbitrary [Moy1, Moy2];

•Gclassical, certain s[Kim1, Kim2, Blo];

•Gsplit (with mild restrictions on the residual characteristic), sin the principal

series [Roc1];

•Garbitrary, σinduced from a level 0 cuspidal representation of a parahoric

subgroup of G[Mor, MoPr, Lus7];

Of course there is a lot of overlap in this list. For GLn, SLn, GSP4and U(2,1)

the above references do much more, they classify the smooth dual of G. In the

level 0 case, Morris [Mor] showed that the parameters qαare the same as those

for analogous Hecke algebras of ﬁnite Chevalley groups. Those parameters were

determined explicitly in [Lus1], and often they are not equal on an all simple roots.

Apart from this list, there are many inertial equivalence classes sfor which

EndG(IG

P(Σ)) is Morita-equivalent a commutative algebra. This is the case for super-

cuspidal G-representations σsuch that σG0is multiplicity-free, and more generally

it tends to happen when Rσis empty.

Chapter 2

Classiﬁcation of irreducible

representations

This chapter leads to the main result of the paper (Theorem 2.3.1). We decided to

call it an aﬃne Springer correspondence, by analogy with the classical Springer corre-

spondence. Together with Kazhdan–Lusztig-theory, the classical version parametrizes

the irreducible representations of a ﬁnite Weyl group with certain representations

of an aﬃne Hecke algebra with equal parameters. This correspondence is known

to remain valid for aﬃne or graded Hecke algebras with certain speciﬁc unequal

parameters [Ciu].

We construct a natural map from irreducible H-representations to representa-

tions of the extended aﬃne Weyl group We. Not all representations in the image

are irreducible, but the image does form a Q-basis of the representation ring of We.

The proof proceeds by reduction to a result that the author previously obtained

for graded Hecke algebras [Sol6]. To carry out this reduction, we need variations

on three well-known results in representation theory of Hecke algebras. The ﬁrst

two are due Lusztig and allow one to descend from aﬃne Hecke algebras to graded

Hecke algebras. We adjust these results to make them more suitable for aﬃne Hecke

algebras with arbitrary positive parameters.

Thirdly there is the Langlands classiﬁcation (Theorem 2.2.4), which comes from

reductive groups and reduces the classiﬁcation of irreducible representations to that

of irreducible tempered ones. For aﬃne Hecke algebras it did not appear in the

literature before, although it was of course known to experts. Because we want to

include diagram automorphisms in our aﬃne Hecke algebras, we need a more reﬁned

version of the Langlands classiﬁcation (Corollary 2.2.5). It turns out that one has

to add an extra ingredient to the Langlands parameters, and that the unicity claim

has to be changed accordingly.

However, these results do not suﬃce to complete the proof of Theorem 2.3.1 for

nontempered representations, that will be done in the next chapter.

2.1 Two reduction theorems

The study of irreducible representations of H⋊Γ is simpliﬁed by two reduction

theorems, which are essentially due to Lusztig [Lus6]. The ﬁrst one reduces to the

case of modules whose central character is positive on the lattice ZR1. The second

one relates these to modules of an associated graded Hecke algebra.

Given t∈Tand α∈R0, [Lus6, Lemma 3.15] tells us that

sα(t) = tif and only if α(t) = 1 if α∨/∈2Y

±1 if α∨∈2Y. (2.1)

We deﬁne Rt:= {α∈R0:sα(t) = t}. The collection of long roots in Rt,nr is

{β∈R1:β(t) = 1}. Let Ftbe the unique basis of Rtthat is contained in R+

0. Then

W′

Ft,t := {w∈W0⋊Γ : w(t) = t, w(Ft) = Ft}

is a group of automorphisms of the Dynkin diagram of (Rt, Ft). Moreover the

isotropy group of tin W0⋊Γ is

W′

t= (W0⋊Γ)t=W(Rt)⋊W′

Ft,t.

We can deﬁne a parameter function qtfor the based root datum

Rt:= (X, Rt, Y, R∨

t, Ft)

via restriction from R∨

nr to R∨

t,nr.

Since Ftdoes not have to be a subset of F0,Rtdoes not always ﬁt in the setting

of Subsection 1.4, but this can be ﬁxed without many problems. For u∈Tun we

deﬁne

P(u) := F0∩QRu.

Then RP(u)is a parabolic root subsystem of R0that contains Ruas a subsystem of

full rank. Although this deﬁnition would also make sense for general elements of T,

we use it only for Tun, to avoid a clash with the notation of [Opd2, Section 4.1]. We

note that the lattice

ZP(u) = ZR0∩QRu

can be strictly larger than ZRu.

To study H-representations with central character W′uc we need a well-chosen

neighborhood of uc ∈TunTrs.

Condition 2.1.1. Let B⊂tbe such that

(a)Bis an open ball centred around 0∈t;

(b)ℑ(α(b)) < π for all α∈R0, b ∈B;

(d)if cα(t)∈ {0,∞} for some α∈R0, t ∈uc exp B, then cα(uc)∈ {0,∞};

(e)if w∈W′and w(uc exp B)∩uc exp B6=∅, then w(uc) = uc.

Since W′acts isometrically on t, (a) implies that Bis W′-invariant. There

always exist balls satisﬁying these conditions, and if we have one such B, then ǫB

with ǫ∈(0,1] also works.

We will phrase our ﬁrst reduction theorem in such a way that it depends mainly

on the unitary part of the central character, it will decompose a representation in

subspaces corresponding to the points of the orbit W′u. We note that Ruc ⊂Ru

and W′

uc ⊂W′

u. Given Bsatisfying the above conditions, we deﬁne

U=W′uc exp(B), UP(u)=W′

ZP(u)uc exp(B) and Uu=W′

uuc exp(B).

We are interested in the algebras H(R, q)an(U)⋊Γ,H(RP(u), qP(u))an(UP(u))⋊W′

P(u)

and H(Ru, qu)an(Uu)⋊W′

Fu,u. Their respective centers

Can(U)W′, Can(UP(u))W′

ZP(u)and Can(Uu)W′

are naturally isomorphic, via the embeddings Uu⊂UP(u)⊂U. For any subset

⊂W′uc we deﬁne 1∈Can(U) by

1(t) = 1 if t∈exp(B)

0 if t∈U\exp(B).

Theorem 2.1.2. (First reduction theorem)

(a)There are natural isomorphisms of Can(U)W′-algebras

H(RP(u), qP(u))an(UP(u))⋊W′

P(u)∼

=1W′

ZP(u)uc(Han(U)⋊Γ) 1W′

ZP(u)uc,

H(Ru, qu)an(Uu)⋊W′

Fu,u ∼

=1W′

uuc(Han(U)⋊Γ) 1W′

uuc.

(b)These can be extended (not naturally) to isomorphisms of Can(U)W′-algebras

Han(U)⋊Γ∼

=M[W′:W′

ZP(u)]1W′

ZP(u)uc(Han(U)⋊Γ) 1W′

ZP(u)uc,

Han(U)⋊Γ∼

=M[W′:W′

u]1W′

uuc(Han(U)⋊Γ) 1W′

uuc,

where Mn(A)denotes the algebra of n×n-matrices with coeﬃcients in an alge-

bra A.

(c)The following maps are natural equivalences of categories:

Modf, U (H(R, q)⋊Γ) ↔Modf,UP(u)(HP(u)⋊W′

P(u))↔Modf,Uu(H(Ru, qu)⋊W′

Fu,u)

V7→ 1W′

ZP(u)ucV7→ 1W′

uucV

IndH⋊Γ

H(Ru,qu)⋊W′

Fu,u (π)

7→

IndHP(u)⋊W′

P(u)

H(Ru,qu)⋊W′

Fu,u (π)

7→

Proof. (a) This is a variation on [Opd2, Theorem 4.10], which itself varied on

[Lus6, Theorem 8.6]. Compared to Lusztig we replaced his Ruc by a larger root

system, we added the group Γ and we localized with analytic functions instead of

formal completions at one central character. The ﬁrst change is innocent since Ru

and RP(u)are actually easier than Lusztig’s Ruc. By [Lus6, Lemma 8.13.b] Lusztig’s

version of the isomorphisms (a) sends γ∈Γ() to 1ı0

γ1. Translated to our setting

this means that we can include the appropriate diagram automorphisms by deﬁning

W′

P(u)∋γ7→ 1W′

ZP(u)uc ı0

γ1W′

ZP(u)uc,

W′

Fu,u ∋γ7→ 1W′

uuc ı0

γ1W′

uuc.(2.2)

Finally, that Lusztig’s arguments also apply with analytic localization was already

checked by Opdam [Opd2, Section 4.1].

(b) Knowing (a), this can proved just as in [Lus6, 4.16].

of ﬁnite dimensional modules of the algebras ﬁguring in (a) and (b). Therefore the

maps in (c) are just the standard equivalences between the module categories of B

and Mn(B), translated with (a) and (b). 2

Remark 2.1.3. This reduction theorem more or less forces one to consider diagram

automorphisms: the groups W′

ZP(u)and W′

Fu,u can be nontrivial even if Γ = {id}.

The notation with induction functors in part (c) is a little sloppy, since W′

Fu,u

need not be contained in Γor in W′

ZP(u). In such cases these induction functors are

deﬁned via part (a).

The ﬁrst reduction theorem also enables us to make sense of IndH⋊Γ

H(RP,qP)⋊Γ′

Pfor

any P⊂F0and any subgroup Γ′

P⊂W′

P. Namely, ﬁrst induce from H(RP, qP)⋊Γ′

to H(RP, qP)⋊W′

ZP, then choose u∈Tun such that P(u) = Pand ﬁnally use (c).

By (2.1) we have α(u) = 1 for all α∈R1∩QRt, so α(t) = α(u)α(c)>0 for

such roots. By deﬁnition uis ﬁxed by W′

u, so Theorem 2.1.2 allows us to restrict

our attention to H⋊Γ-modules whose central character is positive on the sublattice

ZR1⊆X.

We deﬁne a parameter function kufor the degenerate root datum ˜

Ruby

ku,α =0 if α(u)26= 1

log q(sα) + α(u) log q(tαsα)/2 if α(u)2= 1.(2.3)

We note that ku,α = 0 if sαdoes not ﬁx u. Indeed, by (2.1) sα(t)6=timplies that

either α(u)26= 1 or that α(u) = −1 and α∈R0∩R1. But in the latter case sαand

tαsαare conjugate in We, so q(sα) = q(tαsα) and log q(sα) + α(u) log q(tαsα) = 0.

We will see in (4.4) that for this choice of kuthe function ˜cαcan be regarded as the

ﬁrst order approximation of q(sα)cαin a neighborhood of q= 1 and u∈T.

Now we pick u∈TW′

un , so α(u) = ±1 for all α∈R0. Then the map

expu:t→T, λ 7→ uexp(λ) (2.4)

is W′-equivariant. To ﬁnd a relation between H(R, q)⋊Γ and H(˜

Ru, ku)⋊Γ, we

ﬁrst extend these algebras with analytic localization. For every open nonempty

W′-invariant V⊂twe can deﬁne an algebra homomorphism

Φu:Hme(expu(V)) ⋊Γ→H(˜

Ru, ku)me(V)⋊Γ,

fı0

w7→ (f◦expu)˜ıw.(2.5)

Theorem 2.1.4. (Second reduction theorem)

Let Vbe as above, such that expu:V→expu(V)is bijective.

(a)The map expuinduces an isomorphism Can(expu(V))W′→Can (V)W′.

(b)Suppose that every λ∈Vsatisﬁes

hα , λi,hα , λi+ku,α /∈2πiZ\ {0}for α∈R0∩R1,

hα , λi,hα , λi+ku,α /∈πiZ\ {0}for α∈R0\R1.(2.6)

Then Φurestricts to an isomorphism of Can(U)W′-algebras

Φu:Han(expu(V)) ⋊Γ→H(˜

Ru, ku)an(V)⋊Γ.

Proof. (a) This is clear, it serves mainly to formulate (b).

(b) The case Γ = {id}is essentially [Lus6, Theorem 9.3]. The diﬀerence is that our

conditions on λreplace the conditions [Lus6, 9.1]. The general case follows easily

under the assumption that Γ ﬁxes u. 2

Given t′⊂twe denote by Modf,t′(H(˜

R, k)⋊Γ) the category of ﬁnite dimensional

H(˜

R, k)⋊Γ-modules all whose O(t)-weights lie in t′.

Corollary 2.1.5. For uc ∈TunTrs the following categories are equivalent:

(a) Modf,W ′uc (H(R, q)⋊Γ) and Modf,(W(Ru)⋊W′

Fu,u) log(c)(H(˜

Ru, ku)⋊W′

Fu,u),

(b) Modf,W ′uTr s (H(R, q)⋊Γ) and Modf ,a(H(˜

Ru, ku)⋊W′

Fu,u).

These equivalences are compatible with parabolic induction.

Proof. (a) follows from Theorems 2.1.2.b and 2.1.4.b. Notice that the conditions

(2.6) are automatically satisﬁed because qis positive and log(c)∈a, so ku,α ∈Rand

hα , log(c)i ∈ R. If we sum that equivalence over all W′c∈Trs/W ′, we ﬁnd (b). By

[BaMo2, Theorem 6.2] or [Sol5, Proposition 5.3.a] these equivalences of categories

are compatible with parabolic induction. 2

2.2 The Langlands classiﬁcation

In this section we discuss Langlands’ classiﬁcation of irreducible representations.

Basically it reduces from general representations to tempered ones, and from there

to the discrete series. Actually Langlands proved this only in the setting of real

reductive groups, but it holds just as well for p-adic reductive groups, aﬃne Hecke

algebras and graded Hecke algebras. We will only write the results for aﬃne Hecke

algebras, the graded Hecke algebra case is completely analogous and can be found

in [Eve, KrRa, Sol5].

An important tool to study H-representations is restriction to the commutative

subalgebra A∼

=O(T). We say that t∈Tis a weight of (π, V ) if there exists a

v∈V\ {0}such that π(a)v=a(t)vfor all a∈ A. It is easy to describe how the

collection of A-weights behave under parabolic induction. Recall that

WP:= {w∈W0:w(P)⊂R+

0}(2.7)

is the set of minimal length representatives of W0/W (RP).

Lemma 2.2.1. Let Γ′

Pbe a subgroup of ΓPand let σbe a representation of HP⋊Γ′

The A-weights of IndH⋊Γ

HP⋊Γ′

P(σ)are the elements γw(t)∈T, where γ∈Γ, w ∈WP

and tis an A-weight of σ.

Proof. From [Opd2, Proposition 4.20] and its proof we see that this holds in the

case Γ = Γ′

P={id}. For the general case we only have to observe that the operation

π7→ π◦ψ−1

γon H-representations has the eﬀect t7→ γ(t) on all A-weights t. 2

Temperedness of a representation is deﬁned via its A-weights. Given P⊆F0,

we have the following positive cones in aand in Trs:

a+={µ∈a:hα , µi ≥ 0∀α∈F0}, T += exp(a+),

P={µ∈aP:hα , µi ≥ 0∀α∈P}, T +

P= exp(a+

P),

aP+={µ∈aP:hα , µi ≥ 0∀α∈F0\P}, T P+= exp(aP+),

aP++ ={µ∈aP:hα , µi>0∀α∈F0\P}, T P++ = exp(aP++).

(2.8)

The antidual of a∗+:= {x∈a∗:hx , α∨i ≥ 0∀α∈F0}is

a−={λ∈a:hx , λi ≤ 0∀x∈a∗+}=Xα∈F0

λαα∨:λα≤0.(2.9)

Similarly we deﬁne

a−

P=Xα∈Pλαα∨∈aP:λα≤0.(2.10)

The interior a−− of a−equals Pα∈F0λαα∨:λα<0if F0spans a∗, and is empty

otherwise. We write T−= exp(a−) and T−− = exp(a−−).

Let t=|t|·t|t|−1∈Trs×Tun be the polar decomposition of t. An H-representation

is called tempered if |t| ∈ T−for all its A-weights t, and anti-tempered if |t|−1∈T−

for all such t. For inﬁnite dimensional representations this is not entirely satisfactory,

but we postpone a more detailed discussion to Section 3.2. Since all irreducible H-

representations have ﬁnite dimension, this vagueness does not cause any problems.

Notice that our deﬁnition mimics Harish-Chandra’s deﬁnition of admissible smooth

tempered representations of reductive p-adic groups [Wal, Section III.2]. In that

setting the crucial condition says that all exponents of such a representation must

lie in certain cone.

More restrictively we say that an irreducible H-representation belongs to the

discrete series (or simply: is discrete series) if |t| ∈ T−−, for all its A-weights t. In

particular the discrete series is empty if F0does not span a∗. This is the analogue

of Casselman’s criterium for square integrable representations of semisimple p-adic

groups [Cas, Theorem 4.4.6].

The notions tempered and discrete series apply equally well to H⋊Γ, since that

algebra contains Aand the action of Γ on Tpreserves T−. It follows more or less

directly from the deﬁnitions that the correspondence of Theorem 2.1.4 preserves

temperedness and provides a bijection between discrete series representations with

the appropriate central characters, see [Slo2, (2.11)].

It easy to detect temperedness for H(R,1) ⋊Γ = C[X⋊W0⋊Γ] = C[X⋊W′].

Lemma 2.2.2. A ﬁnite dimensional C[X⋊W′]-representation is tempered if and

only if all its A-weights lie in Tun.

This algebra has no discrete series representations, unless X= 0.

Proof. Suppose that Vis a representation of this algebra, and that t∈Tis an

A-weight with weight space Vt. For every g∈W′, gVt=Vg(t)is the g(t)-weight

space of V, which shows that every element of the orbit W′tis an A-weight of V.

But W′|t|can only be contained in T−if it equals the single element 1 ∈Trs . Hence

Vcan only be tempered if |t|= 1 for all its weights, or equivalently if all its weights

lie in Tun. By deﬁnition the latter condition also suﬃces for temperedness.

Unless X= 0, the condition |t|= 1 implies |t| 6∈ T−−, so C[X⋊W′] has no

discrete series representations. 2

The Langlands classiﬁcation looks at parabolic subalgebras of Hand irreducible

representations of those that are essentially tempered. We will describe such repre-

sentations with two data: a tempered representation and a ”complementary” part

of the central character. This is justiﬁed by the following result.

Lemma 2.2.3. Let P⊂F0, tP∈TPand tP∈TP.

(a)The map σ7→ σ◦φtPdeﬁnes an equivalence between the categories of HP-

representations with central character W(RP)tP∈TP/W (RP)and of HP-

representations with central character W(RP)tPtP∈T /W (RP).

(b)Every irreducible HP-representation is of the form σ◦φtP, where σis an ir-

reducible HP-representation and tP∈TP. Both these data are unique modulo

twists coming from KP=TP∩TP, as in (1.19).

Proof. (a) The kernel of φtPfollowed by the quotient map HP→ HPis generated

(as an ideal) by {θx−tP(x) : x∈X∩(P∨)⊥}. If ρis an HP-representation with

central character W(RP)tPtP, then the kernel of ρclearly contains these generators,

so ρfactors via φtPand this quotient map.

(b) Let ρbe an irreducible HP-representation with central character W(RP)t∈

T /W (RP). Decompose t=tPtP∈TPTP. Then part (a) yields a unique irreducible

HP-representation σsuch that ρ=σ◦φtP. The only freedom in this constuction

comes from elements u∈KP. If we replace tPby utP, then part (a) again gives a

unique σ′with ρ=σ′◦φutP, and its follows directly that σ′◦ψu=σ. 2

A Langlands datum for His a triple (P, σ, t) such that

•P⊆F0and σis an irreducible tempered HP-representation;

•t∈TPand |t| ∈ TP++.

We say that two Langlands data (P, σ, t) and (P′, σ′, t′) are equivalent if P=P′and

the HP-representations σ◦φtand σ′◦φt′are equivalent.

Theorem 2.2.4. (Langlands classiﬁcation)

(a)For every Langlands datum (P, σ, t)the H-representation π(P, σ, t) = IndH

HP(σ◦

φt)has a unique irreducible quotient L(P, σ, t).

(b)For every irreducible H-representation πthere exists a Langlands datum (P, σ, t),

unique up to equivalence, such that π∼

=L(P, σ, t).

Proof. The author learned this result from a preliminary version of [DeOp2], but

unfortunately Delorme and Opdam did not include it in the ﬁnal version. Yet the

proof in the setting of aﬃne Hecke algebras is much easier than for reductive groups.

It is basically the same as the proof of Evens [Eve] for graded Hecke algebras, see

also [KrRa, Section 2.4]. For later use we rephrase some parts of that proof in our

notation.

(a) The dominance ordering on ais deﬁned by

λ≤µif and only if hλ , αi ≤ hµ , αifor all α∈F0.(2.11)

For α∈F0we deﬁne δα∈aF0by

hβ , δαi=1 if α=β

0 if α6=β∈F0.

According to Langlands [Lan, Lemma 4.4], for every λ∈athere is a unique subset

F(λ)⊂F0such that λcan be written as

λ=λF0+X

α∈F0\F(λ)

cαδα+X

α∈F(λ)

dαα∨with λF0∈aF0, cα>0, dα≤0.(2.12)

We put λ0=Pα∈F0\F(λ)cαδα∈a+. According to [KrRa, (2.13)]

(wµ)0< µ0for all µ∈a−

P⊕aP++, w ∈WP\ {1}.(2.13)

By the deﬁnition of a Langlands datum log |s| ∈ a−

P⊕aP++ for every A-weight sof

σ◦φt. Choose ssuch that (log |s|)0is maximal with respect to the dominance order.

By Lemma 2.2.1 and (2.13) (log |s|)0is also maximal for sregarded as an A-weight

of π(P, σ, t).

Suppose that ρis an H-submodule of π(P, σ, t) of which sis an A-weight. By the

maximalty of s,ρmust contain the s-weight space of σ◦φt. The irreduciblity of σ

implies that ρcontains the HP-submodule 1 ⊗HPVσ⊂IndH

HP(σ◦φt), and therefore

ρ=π(P, σ, t). Thus the sum of all proper submodules is again proper, which means

that π(P, σ, t) has a unique maximal submodule and a unique irreducible quotient.

(b) Let sbe an A-weight of πsuch that (log |s|)0∈ais maximal and put P=

F(log |s|). Let ρbe the HP-subrepresentation ρof πgenerated by the s-weight

space. Then log |s| ∈ a−

P⊕aP++ and according to [KrRa, p. 38] (log |s′|)0= (log |s|)0

for all A-weights s′of ρ. By Lemma 2.2.3 we can write every irreducible HP-

subrepresentation of ρas σ◦φt, where σis an irreducible HP-representation and

log |t|= (log |s|)0. The AP-weights of σare of the form s′t−1and by construction

log |s′t−1|= log |s| − (log |s|)0∈a−

so σis tempered. The inclusion map σ◦φt→πinduces a nonzero H-homomorphism

π(P, σ, t)→π. Since πis irreducible, this map is surjective. Together with part (a)

this shows that πis the unique quotient of π(P, σ, t).

The proof that (P, σ ◦φt) is uniquely determined by πis easy, and exactly the same

as in the graded Hecke algebra setting, see [Eve, Theorem 2.1.iii] or [KrRa, Theorem

2.4.b]. 2

Theorem 2.2.4 can be regarded as the analogue of the Langlands classiﬁcation

for connected reductive p-adic groups. For disconnected reductive groups the clas-

siﬁcation is no longer valid as such, it has to be modiﬁed. In the case that the

component group is abelian, this is worked out in [BaJa1], via reduction to cyclic

component groups.

We work with a diagram automorphism group Γ which is more general than a

component group and does not have to be abelian. For use in Section 2.3 we have

to extend Theorem 2.2.4 to this setting.

There is a natural action of Γ on Langlands data, by

γ(P, σ, t) = (γ(P), σ ◦ψ−1

γ, γ(t)).(2.14)

Every Langlands datum yields a packet of irreducible quotients, and all data in one

Γ-orbit lead to the same packet. For γ∈ΓPthe Langlands classiﬁcation for HP

shows that the irreducible HP-representations σ◦ψγ◦φγ(t)and σ◦φtare equivalent

if and only if γ(P, σ, t) = (P, σ, t).

To get a more precise statement one needs Cliﬀord theory, as for example in

[RaRa] or [CuRe, Section 53]. Let ΓP,σ,t be the isotropy group of the Langlands

datum (P, σ, t). In [Sol5, Appendix A] a 2-cocycle κof ΓP,σ,t is constructed, giving

rise to a twisted group algebra C[ΓP,σ,t, κ]. We deﬁne a Langlands datum for H⋊Γ

as a quadruple (P, σ, t, ρ), where

•(P, σ, t) is a Langlands datum for H;

•ρis an irreducible representation of C[ΓP,σ,t, κ].

The action (2.14) extends naturally to Langlands data for H⋊Γ, since ψ−1

γinduces

an isomorphism between the relevant twisted group algebras.

From such a Langlands datum we can construct the HP⋊ΓP,σ,t-representation

σ⊗ρand the H⋊Γ-representation

πΓ(P, σ, t, ρ) := IndH⋊Γ

HP⋊ΓP,σ,t (σ◦φt)⊗ρ= IndH⋊Γ

HP⋊ΓP,σ,t (σ⊗ρ)◦φt.(2.15)

If Q⊃P, then (P, σ, t, ρ) can also be considered as a Langlands datum for HQ⋊ΓQ,

and we denote the corresponding HQ⋊ΓQ-representation by πQ,ΓQ(P, σ, t, ρ). In

particular πP,ΓP(P, σ, t, ρ) is an irreducible HP⋊ΓP-representation.

Corollary 2.2.5. (extended Langlands classiﬁcation)

(a)The H⋊Γ-representation πΓ(P, σ, t, ρ)has a unique irreducible quotient LΓ(P, σ, t, ρ).

(b)For every irreducible H⋊Γ-representation πthere exists a Langlands datum

(P, σ, t, ρ), unique modulo the action of Γ, such that π∼

=LΓ(P, σ, t, ρ).

(c)LΓ(P, σ, t, ρ)and πΓ(P, σ, t, ρ)are tempered if and only if P=F0and t∈TF0

un .

Proof. (a) and (b) By [Sol5, Theorem A.1] the H⋊Γ-representation

IndH⋊Γ

H⋊ΓP,σ,t (L(P, σ, t)⊗ρ) (2.16)

is irreducible, and every irreducible H⋊Γ-representation is of this form, for a Lang-

lands datum which is unique modulo Γ. By construction (2.16) is a quotient of

π(P, σ, t, ρ). It is the unique irreducible quotient by Theorem 2.2.4.a and because ρ

is irreducible.

|t| 6∈ T−, so |rt| 6∈ T−for any AP-weight rof σ. But the construction of L(P, σ, t)

in the proof of 2.2.4.a is precisely such that the A-weight rt of π(P, σ, t) survives

to the Langlands quotient. Since the group Γ preserves T−, its presence does not

aﬀect temperedness.

Now assume that P=F0. Since TF0++ ⊂TF0

rs and T−∩TF0

rs ={1}, this repre-

sentation can only be tempered if |t|= 1. In that case σand πΓ(P , σ, t, ρ) have the

same absolute values of A-weights, modulo Γ. But ΓT−=T−, so the temperedness

of πΓ(P, σ, t, ρ) and LΓ(P, σ, t, ρ) follows from that of σ. 2

For connected reductive p-adic groups the Langlands quotient always appears

with multiplicity one in the standard representation of which it is a quotient. Al-

though not stated explicitly in most sources, that is already part of the proof, see

[Kon] or [Sol4, Theorem 2.15]. This also holds for reductive p-adic groups with a

cyclic component group [BaJa2].

Closer examination of the proof of Theorem 2.2.4 allows us to generalize and im-

prove upon this in our setting. Let W(RP)rσ∈TP/W (RP) be the central character

of σ. Then |rσ| ∈ TP,rs = exp(aP), so we can deﬁne

ccP(σ) := W(RP) log |rσ| ∈ aP/W (RP).(2.17)

Since the inner product on ais W′-invariant, the number kccP(σ)kis well-deﬁned.

Lemma 2.2.6. Let (P, σ, t, ρ)and (P, σ′, t, ρ′)be Langlands data for H⋊Γ.

(a)The functor IndH⋊Γ

HP⋊ΓPinduces an isomorphism

HomHP⋊ΓP(πP,ΓP(P, σ, t, ρ), πP,ΓP(P, σ′, t, ρ′)) ∼

HomH⋊Γ(πΓ(P, σ, t, ρ), πΓ(P, σ′, t, ρ′)).

These spaces are one-dimensional if (σ, t, ρ)and (σ′, t, ρ′)are ΓP-conjugate, and

zero otherwise.

(b)Suppose that LΓ(Q, τ, s, ν)is a constituent of πΓ(P, σ, t, ρ), but not LΓ(P, σ, t, ρ).

Then P⊂Qand kccP(σ)k<kccQ(τ)k.

Proof. (a) We use the notation from (2.12). For any weight sof σ◦φtwe

have log |st−1| ∈ a−

Pand (log |s|)0= (log |t|)F0, where the subscript F0refers to the

decomposition of elements of twith respect to t=tF0⊕tF0. Let s′be a weight of

σ′◦φt. By (2.13)

(wlog |s′|)0<(log |s′|)0= (log |t|)F0∀w∈WP\ {id},(2.18)

with respect to the dominance order on a∗

F0. Since (γλ)0=γ(λ0) for all λ∈aand

γ∈Γ, we get



(γw log |s′|)0

<k(log |t|)F0k ∀γ∈Γ, w ∈WP\ {id}.(2.19)

In particular γw(s′) with w∈WPcan only equal the weight sof σ◦φtif w= 1.

Let vs∈Vσ⊗ρbe a nonzero weight vector. Since πP,ΓP(P, σ, t, ρ) is an irreducible

HP⋊ΓP-representation, 1 ⊗vs∈(H⋊Γ) ⊗HP⋊ΓP ,σ,t Vσ⊗ρis cyclic for πΓ(P, σ, t, ρ).

Therefore the map

HomH⋊Γ(πΓ(P, σ, t, ρ), πΓ(P, σ′, t, ρ′)) →πΓ(P, σ′, t, ρ′) : f7→ f(1 ⊗vs) (2.20)

is injective. By (2.19) the s-weight space of πΓ(P, σ′, t, ρ′) is contained in 1 ⊗Vσ′⊗ρ′.

So f(1 ⊗vs)∈1⊗Vσ′⊗ρ′and multiplying by HP⋊ΓPyields

f(C[ΓP]⊗ΓP,σ,t Vσ⊗ρ)⊂C[ΓP]⊗ΓP,σ′,t Vσ′⊗ρ′.

Thus any f∈HomH⋊Γ(πΓ(P, σ, t, ρ), πΓ(P, σ′, t, ρ′)) lies in

IndH⋊Γ

HP⋊ΓPHomHP⋊ΓP(πP,ΓP(P, σ, t, ρ), πP,ΓP(P, σ′, t, ρ′)).(2.21)

From (2.20) we see that this induction functor is injective on homomorphisms. The

modules in (2.21) are irreducible, so the dimension of (2.21) is zero or one. By

Corollary 2.2.5.b it is nonzero if and only if (σ, t, ρ) and (σ′, t, ρ′) are ΓP-conjugate.

(b) The proofs of Theorem 2.2.4.a and Corollary 2.2.5.a show that LΓ(P, σ, t, ρ)

is the unique irreducible subquotient of πΓ(P, σ, t, ρ) which has an A-weight tL

with (log |tL|)0= (log |t|)F0. Moreover of all A-weights s′of proper submodules

of πΓ(P, σ, t, ρ) satisfy (log |s′|)0<(log |t|)F0, with the notation of (2.12). In partic-

ular, for the subquotient LΓ(Q, τ, s, ν ) of πΓ(P, σ, t, ρ) we ﬁnd that

(log |s|)F0= (log |s′|)0<(log |t|)F0.

Since log |s| ∈ aQ++ and log |t| ∈ aP++, this implies P⊂Qand

k(log |s|)F0k<k(log |t|)F0k.(2.22)

According to Lemma 2.2.1 all constituents of πΓ(P, σ, t, ρ) have central character

W′(rσt)∈T /W ′. The same goes for (Q, τ , s, ν), so rσtand rτslie in the same

W0⋊Γ-orbit. Thus also

W′(log |rσt|)F0=W′(log |rτs|)F0.

By deﬁnition (log |t|)F0⊥tPand (log |s|)F0⊥tQ, so

kℜ(ccP(σ))k2+k(log |t|)F0k2=k(log |rσt|)F0k2

=k(log |rτs|)F0k2=kℜ(ccQ(τ))k2+k(log |s|)F0k2.(2.23)

Finally we use (2.22). 2

2.3 An aﬃne Springer correspondence

Given an algebra or group A, let Irr(A) be the collection of (equivalence classes

of) irreducible complex A-representations. Let GZ(A) = GZ(Modf(A)) denote the

Grothendieck group of ﬁnite length complex representations of A, and write GF(A) =

GZ(A)⊗ZFfor any ﬁeld F.

The classical Springer correspondence [Spr] realizes all irreducible representation

of a ﬁnite reﬂection group W0in the top cohomology of the associated ﬂag variety.

Kazhdan–Lusztig theory (see [KaLu2, Xi]) allows one to interpret this as a bijection

between Irr(W0) and a certain collection of irreducible representations of an aﬃne

Hecke algebra with equal parameters. As such, the ﬁnite Springer correspondence

is a specialisation of an aﬃne Springer correspondence between Irr(We) and Irr(H),

see [Lus5, Section 8] and [Ree2, Theorem 2]. We will extend this result all extended

aﬃne Hecke algebras with unequal (but positive) parameters.

For any H ⋊ Γ-representation π, let πW0⋊Γ=πW′be the restriction of πto

the subalgebra C[W′] = C[W0⋊Γ] ⊂H ⋊ Γ. Let Irr0(H ⋊ Γ) be the collection of

(equivalence classes of) irreducible tempered H ⋊ Γ-representations with real central

character. In [Sol6, Theorem 6.5.c] the author proved that the set

{πW0⋊Γ:π∈Irr0(H ⋊ Γ)}(2.24)

is a Q-basis of GQ(W0⋊Γ). When Γ is trivial, this is a kind of Springer corre-

spondence for ﬁnite Weyl groups. The only problem is that πW0may be reducible,

but that could be solved (a priori not naturally) by picking a suitable irreducible

subrepresentation of πW0.

In fact it is known from [Ciu, Corollary 3.6] that in many cases the matrix

that expresses (2.24) in terms of irreducible W0⋊Γ-representations is unipotent and

upper triangular (with respect to a suitable ordering). That would provide a natural

Springer correspondence, but unfortunately that improvement is still open in our

generality.

Theorem 2.3.1. There exists a unique system of maps

Spr : Irr(H⋊Γ) →Modf(X⋊(W0⋊Γ)),

for all extended aﬃne Hecke algebras H⋊Γ, such that:

(a)The image of Spr is a Q-basis of GQ(X⋊(W0⋊Γ)).

(b) Spr preserves the unitary part of the central character.

(d)Let u∈Tun, let ˜π∈Irr0(H(˜

Ru, ku)⋊W′

Fu,u)and let ˜π◦Φube the H(Ru, qu)⋊

W′

Fu,u-representation associated to it via Theorem 2.1.4.b. Then

SprIndH⋊Γ

H(Ru,qu)⋊W′

Fu,u (˜π◦Φu)= IndX⋊W′

X⋊W′

uCu⊗˜πW′

u,

where Cudenotes the one-dimensional X-representation with character u.

(e)If (P, σ, t, ρ)is a Langlands datum for H⋊Γ, then

Spr(LΓ(P, σ, t, ρ)) = IndX⋊(W0⋊Γ)

X⋊(W(RP)⋊ΓP,σ,t)(Spr(σ⊗ρ)◦φt).

Proof. In view of Corollary 2.1.5, properties (b) and (d) determine Spr uniquely

for all irreducible tempered representations. A glance at Lemma 2.2.2 shows that

Spr preserves temperedness.

Next Corollary 2.2.5.b and property (e) determine Spr for all irreducible repre-

sentations. By Corollary 2.2.5 every nontempered irreducible H⋊Γ-representation

πis of the form L(P, σ, t, ρ) for some Langlands datum with t6∈ Tun. By construc-

tion all A-weights of Spr(σ⊗ρ) lie in Tun, so by property (e) the absolute values of

A-weights of Spr(π) lie in W′|t|. Together with Lemma 2.2.2 this shows that Spr(π)

is not tempered.

Now we have (b)–(e), let us turn to (a). By Corollary 2.1.5 and the result

mentioned in (2.24), (a) holds if we restrict to tempered representations on both

sides. The proof that this restriction is unnecessary is more diﬃcult, we postpone

it to Section 3.4.

Remark 2.3.2. We will see in Corollary 4.4.3 that on tempered representations Spr

is given by composition with an algebra homomorphism between suitable completions

of C[X⋊(W0⋊Γ)] and H⋊Γ. That is not possible for all irreducible represen-

tations, since sometimes Spr does not preserve the dimensions of representations.

This happens if and only if π(P, σ, t, ρ)6=L(P, σ, t, ρ)in (e).

In view of Lemma 2.2.6.c it is possible to replace (e) by the condition (e’) :

Spr(πΓ(P, σ, t, ρ)) = IndX⋊(W0⋊Γ)

X⋊(W(RP)⋊ΓP,σ,t)(Spr(σ⊗ρ)◦φt).

The resulting map

Spr′:GZ(H⋊Γ) →GZ(X⋊(W0⋊Γ))

commutes with parabolic induction, but it sends some irreducible H⋊Γ-representations

to virtual We⋊Γ-representations.

Of course there also exists a version of Theorem 2.3.1 for graded Hecke algebras. It

can easily be deduced from the above using Theorem 2.1.4.b.

Chapter 3

Parabolically induced

representations

Parabolic induction is a standard tool to create a large supply of interesting rep-

resentations of reductive groups, Hecke algebras and related objects. In line with

Harish-Chandra’s philosophy of the cusp form, every irreducible tempered represen-

tation of an aﬃne Hecke algebra can be obtained via unitary induction of a discrete

series representation of a parabolic subalgebra. With the Langlands classiﬁcation

we can also reach irreducible representations that are not tempered.

Hence we consider induction data ξ= (P, δ, t), where δis a discrete series rep-

resentation of HPand t∈TPis an induction parameter. With this we associate a

representation π(ξ) = IndH

HP(δ◦φt). Among these are the principal series represen-

tations, which already exhaust the dual space of H. But that is not very satisfactory,

since a principal series representation can have many irreducible subquotients, and

it is not so easy to determine them, see [Ree1].

Instead we are mostly interested in induction data ξfor which |t|is positive (in

an appropriate sense) and in irreducible quotients of π(ξ), because the Langlands

classiﬁcation applies to these. In Theorem 3.3.2 we construct, for every irreducible

H-representation π, an essentially unique induction datum ξ+(ρ), such that ρis a

quotient of π(ξ+(ρ)). However, in general π(ξ+(ρ)) has more than one irreducible

quotient.

Another important theme in this chapter are intertwining operators between

induced representations of the form π(ξ). Their deﬁnition and most important

properties stem from the work of Opdam and Delorme [Opd2, DeOp1]. Like in the

setting of reductive groups, it is already nontrivial to show that normalized inter-

twining operators are regular on unitary induced representations. Under favorable

circumstances such intertwining operators span HomH(π(ξ), π(ξ′)). This was al-

ready known [DeOp1] for unitary induction data ξ, ξ ′, in which case π(ξ) and π(ξ′)

are tempered representations. We generalize this to pairs of positive induction data

(Theorem 3.3.1). Crucial in all these considerations is the Schwartz algebra Sof H,

the analogue of the Harish-Chandra–Schwartz algebra of a reductive p-adic group.

For the geometry of the dual space of Hit is important to understand the number

n(ξ) of irreducible H-representations ρwith ξ+(ρ) equivalent to ξ. This is governed

by a groupoid Gthat keeps track of all intertwining operators. Indeed, if t7→ ξtis a

continuous path of induction data such that all ξthave the same isotropy group in

G, then n(ξt) is constant along this path (Proposition 3.4.1).

From this we deduce that the dual of His a kind of complexiﬁcation of the

tempered dual of H. As a topological space, the tempered dual is built from certain

algebraic subvarieties of compact tori, each with a multiplicity. In this picture the

dual of His built from the corresponding complex subvarieties of complex algebraic

tori, with the same multiplicities. This geometric description is used to ﬁnish the

proof of the aﬃne Springer correspondence from Section 2.3.

3.1 Unitary representations and intertwining operators

Like for Lie groups, the classiﬁcation of the unitary dual of an aﬃne Hecke algebra

appears to be considerably more diﬃcult than the classiﬁcation of the full dual or of

the tempered dual. This is an open problem that we will not discuss in this paper,

cf. [BaMo2, BaCi]. Nevertheless we will use unitarity arguments, mainly to show

that certain representations are completely reducible. The algebra H⋊Γ is endowed

with a sesquilinear involution * and a trace τ, deﬁned by

(zNwγ)∗= ¯zγ−1Nw−1z∈C, w ∈We, γ ∈Γ,

τ(zNwγ) = zif γ=w=e,

0 otherwise.(3.1)

Since qis real-valued, this * is anti-multiplicative and τis positive. These give rise

to an Hermitian inner product on H⋊Γ:

hh , h′iτ=τ(h∗h′)h, h′∈ H ⋊Γ.(3.2)

A short calculation using the multiplication rules 1.4 shows that the basis {Nwγ:

w∈We, γ ∈Γ}of H⋊Γ is orthonormal for this inner product.

We note that Γ acts on Hby *-automorphisms, and that H,H(W0, q) and C[Γ]

are *-subalgebras of H⋊Γ. In general Ais not a *-subalgebra of H. For x∈X

[Opd2, Proposition 1.12] tells us that

θ∗

x=Nw0θ−w0(x)N−1

w0,(3.3)

where w0is the longest element of the Coxeter group W0.

Let Γ′

Pbe a subgroup of ΓPand let τbe a HP⋊Γ′

P-representation on an inner

product space Vτ. By default we will endow the vector space

H⋊Γ⊗HP⋊Γ′

PVτ∼

=C[ΓWP]⊗C[Γ′

P]Vτ

with the inner product

hh⊗v , h′⊗v′i=τ(h∗h′)hv , v′ih, h′∈C[ΓWP], v, v′∈Vτ.(3.4)

Recall that a representation πof H⋊Γ on a Hilbert space is unitary if π(h∗) = π(h)∗

for all h∈ H ⋊Γ. In particular, such representations are completely reducible.

Lemma 3.1.1. Let Γ′

Pbe a subgroup of ΓP, let σbe a ﬁnite dimensional HP⋊Γ′

P-

representation and let t∈TP.

(a)If σis unitary and t∈TP

un, then IndH⋊Γ

HP⋊Γ′

P(σ◦φt)is unitary with respect to the

inner product (3.4).

(b) IndH⋊Γ

HP⋊Γ′

P(σ◦φt)is (anti-)tempered if and only if σis (anti-)tempered and

t∈TP

un.

Proof. Since Γ acts by *-algebra automorphisms and Γ ·T−=T−, it does not

disturb the properties unitarity and temperedness. Hence it suﬃces to prove the

lemma in the case Γ = Γ′

P={id}. Then (a) and the ”if”-part of (b) are [Opd2,

Propositions 4.19 and 4.20].

For the ”only if”-part of (b), suppose that t∈TP\TP

u. Since X∩(P∨)⊥is

of ﬁnite index in XP=X/(X∩QP), there exists x∈X∩(P∨)⊥with |t(x)| 6= 1.

Possibly replacing xby −x, we may assume that |t(x)|>1. But δ(φt(θx))(v) = x(t)v

for all v∈Vδand x∈Z(X⋊WP), so the HP-representation δ◦φtis not tempered.

Hence its induction to Hcannot be tempered. Similarly, if σis not tempered, then

the restriction of σ◦φtto H(X∩QP, RP, Y /Y ∩P⊥, R∨

P, P, qP) is not tempered.

The same proof works in the anti-tempered case, we only have to replace |t(x)|>

1 by |t(x)|<1.2

Remark. It is possible that IndH⋊Γ

HP⋊Γ′

P(σ◦φt) is unitary with respect to some inner

product other than (3.4), if the conditions of part (a) are not met.

We intend to partition Irr(H⋊Γ) into ﬁnite packets, each of which is obtained

by inducing a discrete series representation of a parabolic subalgebra of H. Thus

our induction data are triples (P, δ, t), where

•P⊂F0;

•(δ, Vδ) is a discrete series representation of HP;

•t∈TP.

Let Ξ be the space of such induction data, where we regard δonly modulo equivalence

of HP-representations. We say that ξ= (P, δ, t) is unitary if t∈TP

un, and we denote

the space of unitary induction data by Ξun. Similarly we say that ξis positive if

|t| ∈ TP+, which we write as ξ∈Ξ+. Notice that, in contrast to Langlands data,

we do not require |t|to be strictly positive. We have three collections of induction

data:

Ξun ⊆Ξ+⊆Ξ.(3.5)

By default we endow these spaces with the topology for which Pand δare discrete

variables and TPcarries its natural analytic topology. We will realize every irre-

ducible H⋊Γ-representation as a quotient of a well-chosen induced representation

πΓ(ξ) := IndH⋊Γ

Hπ(P, δ, t) = IndH⋊Γ

HP(δ◦φt).

We note that for all s∈TW0⋊Γ:

πΓ(P, δ, ts) = πΓ(P, δ, t)◦φs.(3.6)

As vector space underlying πΓ(ξ) we will always take C[ΓWP]⊗Vδ. This space does

not depend on t, which will allow us to speak of maps that are continuous, smooth,

polynomial or even rational in the parameter t∈TP.

The discrete series representations of aﬃne Hecke algebras with irreducible root

data were classiﬁed in [OpSo2]. Here we recall only how their central characters can

be determined, which is related to the singularities of the elements ı0

s. Consider the

W0⋊Γ-invariant rational function

η=Yα∈R0

c−1

α∈C(T),

where cαis as in (1.23). Notice that ηdepends on the parameter function q, or

more precisely on q1/2. A coset Lof a subtorus of Tis said to be residual if the

pole order of ηalong Lequals dimC(T)−dimC(L), see [Opd4]. A residual coset

of dimension 0 is also called a residual point. Such points can exist only if Ris

semisimple, otherwise all residual cosets have dimension at least rank Z(We)>0.

According to [Opd2, Lemma 3.31] the collection of central characters of discrete

series representations of H(R, q) is exactly the set of W0-orbits of residual points

for (R, q). Moreover, if δis a discrete series representation of HPwith central

character W(RP)r, then rT Pis a residual coset for (R, q) [Opd2, Proposition 7.4].

Up to multiplication by an element of W0, every residual coset is of this form [Opd2,

Proposition 7.3.v].

The map that assigns to ξ∈Ξ the central character of π(ξ)∈Modf(H(R, q)) is

an algebraic morphism Ξ →T/W0. The above implies that the image of

{(P, δ, t)∈Ξ : |F0\P|=d}(3.7)

is the union of the d-dimensional residual cosets, modulo W0.

Let δ∅be the unique onedimensional representation of H∅=Cand consider

M(t) := πΓ(∅, δ∅, t) = IndH⋊Γ

A(δ∅◦φt)∼

=IndH⋊Γ

O(T)Ct, t ∈T.

The family consisting of these representations is called the principal series of H⋊Γ

and is of considerable interest. For example, by Frobenius reciprocity every irre-

ducible H⋊Γ-representation is a quotient of some principal series representation.

Lemma 3.1.2. Suppose that h∈ H ⋊Γand that M(t, h) = 0 for all tin some

Zariski-dense subset of T. Then h= 0.

Proof. Since M(t, h)∈EndC(C[ΓW0]) depends algebraically on t, it is zero for

all t∈T. Write h=Pγw∈Γ⋊W0aγ w Nγw with aγw ∈ A and suppose that h6= 0.

Then we can ﬁnd w′∈W0that aγw′6= 0 for some γ∈Γ, and such that ℓ(w′) is

maximal for this property. From Theorem 1.2.1.d we see that

M(t, h)(Ne) = X

γw∈Γ⋊W0

bγw Nγw

for some bγw with bγw′=aγw′6= 0. Therefore M(t, h) is not identically zero. This

contradiction shows that the assumption h6= 0 is untenable. 2

By [Opd2, Corollary 2.23] discrete series representations are unitary. (Although

Opdam only worked in the setting Γ = {id}, his proof also applies with general Γ.)

From this and Lemma 3.1.1 we observe:

Corollary 3.1.3. Let ξ= (P, δ, t)∈Ξ. If t∈TP

un, then πΓ(ξ)is unitary and

tempered. If t∈TP\TP

un, then πΓ(ξ)is not tempered.

For any subset Q⊂F0, let ΞQ, πQ,... denote the things Ξ,π,..., but for the

algebra HQinstead of H. For ξ= (P, δ, t)∈Ξ we deﬁne

P(ξ) := {α∈R0:|α(t)|= 1}.(3.8)

Proposition 3.1.4. Let ξ= (P, δ, t)∈Ξ+.

(a)The HP(ξ)⋊ΓP(ξ)-representation πP(ξ),ΓP(ξ)(ξ)is completely reducible.

(b)Every irreducible summand of πP(ξ),ΓP(ξ)(ξ)is of the form πP(ξ),ΓP(ξ)(P(ξ), σ, tP(ξ), ρ),

where (P(ξ), σ, tP(ξ), ρ)is a Langlands datum for H⋊Γand tP(ξ)t−1∈TP(ξ).

(c)The irreducible quotients of πΓ(ξ)are the representations LΓ(P(ξ), σ, tP(ξ), ρ),

with (P(ξ), σ, tP(ξ), ρ)coming from (b).

(d)Every irreducible H⋊Γ-representation is of the form described in (c).

(e)The functor IndH⋊Γ

HP(ξ)⋊ΓP(ξ)induces an isomorphism

EndHP(ξ)⋊ΓP(ξ)πP(ξ),ΓP(ξ)(ξ)∼

=EndH⋊Γ(π(ξ)).

Remarks. Part (a) holds for any ξ∈Ξ. In (b) tP(ξ)is uniquely determined

modulo KP(ξ).

Proof. (a) By construction there exists tP(ξ)∈TP(ξ)such that

t(tP(ξ))−1∈TP(ξ),un.(3.9)

Then πP(ξ)(P, δ, t)◦φ−1

tP(ξ)=πP(ξ)P, δ, t(tP(ξ))−1is unitary by Corollary 3.1.3. In

particular it is completely reducible, which implies that πP(ξ)P, δ, t(tP(ξ))−1is also

completely reducible. By [Sol5, Theorem A.1.c]

πP(ξ),ΓP(ξ)(ξ) = πP(ξ),ΓP(ξ)(P, δ, t) = IndHP(ξ)⋊ΓP(ξ)

HP(ξ)πP(ξ)(P, δ, t) (3.10)

remains completely irreducible.

(b) By Corollary 3.1.3 πP(ξ)P, δ, t(tP(ξ))−1is tempered and unitary, so by Lemma

2.2.6 all its irreducible summands are of the form

πP(ξ)(P(ξ), σ, k) = LP(ξ)(P(ξ), σ, k), where k∈TP(ξ)

un .

Moreover πP(ξ)P, δ, t(tP(ξ))−1C[X∩(P(ξ)∨)⊥]consists only of copies of the trivial

X∩(P(ξ)∨)⊥-representation, so k∈KP(ξ)=TP(ξ)

un ∩TP(ξ),un. Together with (3.9)

this implies ktP(ξ)t−1∈TP(ξ),un. Hence every irreducible summand of (3.10) is an

irreducible summand of some

IndHP(ξ)⋊ΓP(ξ)

HP(ξ)πP(ξ)(P(ξ), σ, ktP(ξ)).

By Cliﬀord theory (see the proof of Corollary 2.2.5) these are of the required form

πP(ξ),ΓP(ξ)(P(ξ), σ, ktP(ξ), ρ).

(d) By Corollary 2.2.5.b it suﬃces to show that every HP⋊ΓP-representation of

the form (σ◦φt)⊗ρis a direct summand of some πP,ΓP ,σ,t (ξ+). Without loss of

generality we may assume that P=F0and that ΓP,σ,t = Γ. The H-representation

σ◦φt|t|−1is irreducible and tempered, so by [DeOp1, Theorem 3.22] it is a direct

summand of π(ξ′) for some ξ′= (P′, ρ′, t′)∈Ξun. Then (P′, ρ′, t′|t|)∈Ξ+and σ◦φt

is a direct summand of π(P′, ρ′, t′|t|). By Cliﬀord theory [Sol5, Theorem A.1.b] the

H⋊Γ-representation (σ◦φt)⊗ρis a direct summand of πΓ(P′, δ′, t′|t|).

(e) Follows from (a), (b) and Lemma 2.2.6.b. 2

The parabolically induced representations πΓ(ξ) are by no means all disjoint.

The relations among them are described by certain intertwining operators, whose

construction we recall from [Opd1, Opd2].

Suppose that P, Q ⊂F0, u ∈KP, g ∈Γ⋉W0and g(P) = Q. Let δand σbe

discrete series representations of respectively HPand HQ, such that σis equivalent

with δ◦ψ−1

u◦ψ−1

g. Choose a unitary map Ig u

δ:Vδ→Vσsuch that

Igu

δ(δ(h)v) = σ(ψg◦ψu(h))(Igu

δ(v)) ∀v∈Vδ, h ∈ HP.(3.11)

Notice that any two choices of Igu

δdiﬀer only by a complex number of norm 1. In

particular Igu

δis a scalar if σ=δ.

We obtain a bijection

Igu : (C(T /W0)⊗Z(H)H)⋊Γ⊗HPVδ→(C(T /W0)⊗Z(H)H)⋊Γ⊗HQVσ,

Igu(h⊗v) = hıo

g−1⊗Igu

δ(v).(3.12)

Theorem 3.1.5. (a)The map Igu deﬁnes an intertwining operator

πΓ(gu, P, δ, t) : πΓ(P, δ, t)→πΓ(Q, σ, g(ut)).

As a map C[ΓWP]⊗CVδ→C[ΓWQ]⊗CVσit is a rational in t∈TPand

constant on TF0-cosets.

(b)This map is regular and invertible on an open neighborhood of TP

un in TP(with

respect to the analytic topology).

(c)πΓ(gu, P, δ, t)is unitary if t∈TP

un.

Remark. Due to the freedom in the choice of (3.11), for composable g1, g2∈ G

the product π(g1, g2ξ)π(g2, ξ) need not be equal to π(g1g2, ξ ). The diﬀerence is a

locally constant function whose absolute value is 1 everywhere.

Proof. If g=γw ∈Γ⋉W0, then Igu =Iγ◦Iwu, modulo this locally constant

function. It follows directly from the deﬁnitions that the theorem holds for Iγ, so

the diﬃcult part is Iwu, which is dealt with in [Opd2, Theorem 4.33 and Corollary

4.34]. 2

The intertwining operators for reﬂections acting on the unitary principal series

can be made reasonably explicit:

Lemma 3.1.6. Suppose that β∈R0and t∈Tun. Then πΓ(sβ,∅, δ∅, t)is a scalar

operator if and only if c−1

β(t) = 0.

Proof. Suppose that α∈F0, t ∈Tand c−1

α(t) = 0. Then (1.25) implies that

1 + ı0

sα(t) = 0, regarded as an element of H(W0, q ). Hence πΓ(sα,∅, δ∅, t) is a scalar

operator. Conversely, if c−1

α(t)6= 0, then (1.25) shows that 1 + ı0

sα(t) is not scalar,

because the action of 1 + q(sα)1/2Nsαon H(W0, q) has two diﬀerent eigenvalues.

With Theorem 3.1.5 we can see that this is not speciﬁc for simple reﬂections.

Find w∈W0such that w(β) = αis a simple root. Then sβ=w−1sαw, so up to a

nonzero scalar

πΓ(sβ,∅, δ∅, t) = πΓ(w−1,∅, δ∅, wt)πΓ(sα,∅, δ∅, wt)πΓ(w, ∅, δ∅, t).

Now we notice that c−1

β(t) = 0 if and only if c−1

α(wt) = 0, and that πΓ(w−1,∅, δ∅, wt) =

πΓ(w, ∅, δ∅, t)−1up to a scalar. 2

Thus it is possible to determine the H-endomorphisms for unitary principal series

representations, at least when the isotropy groups of points t∈Tun are generated

by reﬂections. The reducibility and intertwining operators for nonunitary principal

series are more complicated, and have been subjected to ample study [Kat1, Rog,

Ree1]. For other parabolically induced representations the intertwining operators are

less explicit. They can be understood better with the theory of R-groups [DeOp2].

The action of these intertwining operators on the induction data space Ξ is

described most conveniently with a groupoid Gthat includes all pairs (g, u) as above.

The base space of Gis the power set of F0, and for P, Q ⊆F0the collection of arrows

from Pto Qis

GP Q ={(g, u)∈Γ⋉W0×KP:g(P) = Q}.(3.13)

Whenever it is deﬁned, the multiplication in Gis

(g′, u′)·(g, u) = (g′g, g −1(u′)u).

Usually we will write elements of Gsimply as gu. This groupoid acts from the left

on Ξ by

(g, u)·(P, δ, t) := (g(P), δ ◦ψ−1

u◦ψ−1

g, g(ut)),

the action being deﬁned if and only if g(P)⊂F0. Since T+⊃TP+is a fundamental

domain for the action of W0on T, every element of Ξ is G-associate to an element

of Ξ+.

Although πΓ(gu(P, δ, t)) and πΓ(P, δ, t) are not always isomorphic, the existence

of rational intertwining operators has the following consequence:

Lemma 3.1.7. The H⋊Γ-representations πΓ(gu(P, δ, t)) and πΓ(P, δ, t)have the

same irreducible subquotients, counted with multiplicity.

Proof. This is not hard, the proof in the graded Hecke algebra setting [Sol5,

Lemma 3.4] also works here. 2

3.2 The Schwartz algebra

We recall the construction of various topological completions of H[Opd2]: a Hilbert

space, a C∗-algebra and a Schwartz algebra. The latter is the most relevant from

the representation theoretic point of view. All tempered representations of Hextend

to its Schwartz completion, and a close study of this Schwartz algebra reveals facts

about tempered representations for which no purely algebraic proof is known.

Let L2(R, q) be Hilbert space completion of Hwith respect to the inner product

(3.2). By means of the orthonormal basis {Nw:w∈We}we can identify L2(R, q)

with the Hilbert space L2(We) of square integrable functions W→C.

For any h∈ H the map H(R, q)→ H(R, q) : h′7→ hh′extends to a bounded

linear operator on L2(R, q). This realizes H(R, q) as a *-subalgebra of B(L2(R, q)).

Its closure C∗(R, q) is a separable unital C∗-algebra, called the C∗-algebra of H.

The Schwartz completion of Hwill, as a topological vector space, consist of

all rapidly decaying functions on We, with respect to some length function. For

this purpose the length function ℓ(w) of the Coxeter system (Waﬀ , Saﬀ ) is unsatis-

factory, because its natural extension to Weis zero on Z(We). To overcome this

inconvenience, recall that

X⊗ZR=a∗=a∗

F0⊕a∗F0=a∗

F0⊕(Z(We)⊗ZR).

Thus we can decompose any x∈X⊂a∗uniquely as x=xF0+xF0∈a∗

F0⊕a∗F0.

Now we deﬁne

N(w) = ℓ(w) + 

w(0)F0

w∈We.

Since Waﬀ ⊕Z(We) is of ﬁnite index in We, the set {w∈We:N(w) = 0}is ﬁnite.

For n∈Nwe deﬁne the following norm on H:

pnX

w∈We

hwNw= sup

w∈We|hw|(N(w) + 1)n.

The completion S=S(R, q) of Hwith respect to the family of norms {pn:n∈N}is

a nuclear Fr´echet space. It consists of all (possibly inﬁnite) sums h=Pw∈WehwNw

such that pn(h)<∞for all n∈N.

Theorem 3.2.1. There exist Cq>0, d ∈Nsuch that ∀h, h′∈ S(R, q), n ∈N

khkB(L2(R,q)) ≤Cqpd(h),

pn(h·h′)≤Cqpn+d(h)pn+d(h′).

In particular S(R, q)is a unital locally convex *-algebra, and it is contained in

C∗(R, q).

Proof. This was proven ﬁrst with representation theoretic methods in [Opd2,

Section 6.2]. Later the author found a purely analytic proof [OpSo2, Theorem A.7].

It is easily seen that the action of Γ on Hpreserves all the above norms. Hence the

crossed product S⋊Γ = S(R, q )⋊Γ (respectively C∗(R, q)⋊Γ) is a well-deﬁned

Fr´echet algebra (respectively C∗-algebra). For q= 1 we obtain the algebras

S(R,1) ⋊Γ = S(X)⋊W0⋊Γ = C∞(Tun)⋊W0⋊Γ,

C∗(R,1) ⋊Γ = C∗(X)⋊W0⋊Γ = C(Tun)⋊W0⋊Γ,(3.14)

where S(X) denotes the algebra of rapidly decreasing functions on X.

We can use these topological completions to characterize discrete series and

tempered representations. According to [Opd2, Lemma 2.22], an irreducible H⋊Γ-

representation πis discrete series if and only if it is contained in the left regular

representation of H⋊Γ on L2(R, q)⊗C[Γ], or equivalently if its character χπ:

H⋊Γ→Cextends to a continuous linear functional on L2(R, q)⊗C[Γ].

By [Opd2, Lemma 2.20] a ﬁnite dimensional H⋊Γ-representation is tempered

if and only if it extends continuously to an S⋊Γ-representation. More generally,

suppose that πis a representation of H⋊Γ on a Fr´echet space V, possibly of inﬁnite

dimension. As in [OpSo1, Proposition A.2], we deﬁne πto be tempered if it induces

a jointly continuous map (S⋊Γ) ×V→V.

A crucial role in the harmonic analysis on aﬃne Hecke algebra is played by a

particular Fourier transform, which is based on the induction data space Ξ. Let

VΓ

Ξbe the vector bundle over Ξ, whose ﬁber at (P, δ, t)∈Ξ is the representation

space C[Γ ×WP]⊗Vδof πΓ(P, δ, t). Let End(VΓ

Ξ) be the algebra bundle with ﬁbers

EndC(C[Γ ×WP]⊗Vδ). The inner product (3.4) endows EndC(C[Γ ×WP]⊗Vδ) and

End(VΓ

Ξ) with a canonical involution *. Of course these vector bundles are trivial

on every connected component of Ξ, but globally not even the dimensions need be

constant. Since Ξ has the structure of a complex algebraic variety, we can construct

the algebra of polynomial sections of End(VΓ

Ξ):

OΞ; End(VΓ

Ξ):= M

P,δ O(TP)⊗EndC(C[Γ ×WP]⊗Vδ).

Given a reasonable subset (preferably a submanifold) Ξ′⊂Ξ, we deﬁne the algebras

L2Ξ′; End(VΓ

Ξ), CΞ′; End(VΓ

Ξ)and C∞Ξ′; End(VΓ

Ξ)in similar fashion. Further-

more, if µis a suﬃciently nice measure on Ξ and Ξ′is compact, then the following

formula deﬁnes a Hermitian form on L2Ξ′; End(VΓ

Ξ):

hf1, f2iµ:= ZΞ′

tr(f1(ξ)∗f2(ξ)) dµ. (3.15)

The intertwining operators from Theorem 3.1.5 give rise to an action of the groupoid

Gon the algebra of rational sections of End(VΓ

Ξ), by

(g·f)(ξ) = πΓ(g, g−1ξ)f(g−1ξ)πΓ(g, g−1ξ)−1,(3.16)

whenever g−1ξ∈Ξ is deﬁned. This formula also deﬁnes groupoid actions of Gon

CΞ′; End(VΓ

Ξ)and on C∞Ξ′; End(VΓ

Ξ), provided that Ξ′is a G-stable submanifold

of Ξ on which all the intertwining operators are regular. Given a suitable collection

Σ of sections of (Ξ′,End(VΓ

Ξ)), we write

ΣG={f∈Σ : (g·f)(ξ) = f(ξ) for all g∈ G, ξ ∈Ξ′such that g−1ξis deﬁned}.

The Fourier transform for H⋊Γ is the algebra homomorphism

F:H⋊Γ→ OΞ; End(VΓ

Ξ),

F(h)(ξ) = π(ξ)(h).

The very deﬁnition of intertwining operators shows that the image of Fis contained

in the algebra OΞ; End(VΓ

Ξ)G. The Fourier transform also extends continuously to

various topological completions of H⋊Γ:

Theorem 3.2.2. (Plancherel theorem for aﬃne Hecke algebras)

The Fourier transform induces algebra homomorphisms

H(R, q)⋊Γ→ OΞ; End(VΓ

Ξ)G,

S(R, q)⋊Γ→C∞Ξun; End(VΓ

Ξ)G,

C∗(R, q)⋊Γ→CΞun; End(VΓ

Ξ)G.

The ﬁrst one is injective, the second is an isomorphism of Fr´echet *-algebras and

the third is an isomorphism of C∗-algebras.

Furthermore there exists a unique Plancherel measure µP l on Ξsuch that

•the support of µPl is Ξun;

•µP l is G-invariant;

•the restriction of µpl to a component (P, δ, T P)is absolutely continuous with

respect to the Haar measure of TP

un;

•the Fourier transform extends to a bijective isometry

L2(R, q)⊗C[Γ]; h,iτ→L2Ξun; End(VΓ

Ξ)G;h,iµP l .

Proof. Once again the essential case is Γ = {id}, which is a very deep result

proven by Delorme and Opdam, see Theorem 5.3 and Corollary 5.7 of [DeOp1] and

[Opd2, Theorem 4.43].

To include Γ in the picture we need a result of general nature. Let Abe any

complex Γ-algebra and endow A⊗CEnd(C[Γ]) with the Γ-action

γ·(a⊗f)(v) = γ·a⊗f(vγ)γ−1a∈A, γ ∈Γ, v ∈C[Γ], f ∈End(C[Γ]).(3.17)

There is a natural isomorphism

A⋊Γ∼

=A⊗CEnd(C[Γ])Γ.(3.18)

This is easy to show, but it appears to be one of those folklore results whose origins

are hard to retrace. In any case a proof can be found in [Sol3, Lemma A.3]. For

A=CΞun; End(VΓ

Ξ)the action (3.17) corresponds to the action of Γ on End(VΓ

Ξ)

described in (3.16). The greater part of the theorem follows from (3.18) and the case

Γ = {id}. It only remains to see how the inner products h,iτand h,iµP l behave

when Γ is included. Let us distinguish the new inner products with a subscript Γ.

On the Hecke algebra side it is easy, as the formula (3.1) does not change, so

hNγh , Nγ′h′iΓ,τ =hh , h′iτif γ=γ′,

0 if γ6=γ′.

On the spectral side the inclusion of Γ means that we replace every H-representation

π(ξ) by IndH⋊Γ

Hπ(ξ). In such an induced representation the elements of Γ permute

the H-subrepresentations γπ(ξ), while h∈ H acts by π(γ−1(h)) on γπ(ξ). The

action of Γ on Hpreserves the trace and the *, so

trπΓ(ξ, Nγh)∗πΓ(ξ, Nγ′h′)=trπ(ξ, h)∗π(ξ, h′)if γ=γ′,

0 if γ6=γ′.

In view of (3.15), this means that the L2-extension of Fis an isometry with respect

to the Plancherel measure µΓ,P l =|Γ|−1µP l.2

Corollary 3.2.3. The center of S(R, q)⋊Γ(respectively C∗(R, q)⋊Γ) is isomorphic

to C∞(Ξun)G(respectively C(Ξun)G).

Proof. This is the obvious generalization of [DeOp1, Corollary 5.5] to our setting.

Notice that Z(S⋊Γ) is larger than the closure of Z(H⋊Γ) in S⋊Γ, for example

Z(S⋊Γ) contains a nontrivial idempotent for every connected component of Ξun/G.

Varying on the notation Modf,U (H⋊Γ) we will denote by

Modf,Σ(S⋊Γ)

the category of ﬁnite dimensional S⋊Γ-modules with Z(S⋊Γ)-weights in Σ ⊂Ξun/G.

Let us compare Schwartz algebras of aﬃne Hecke algebras with those for reduc-

tive p-adic groups. Suppose that Gis reductive p-adic group and that H⋊Γ is

Morita equivalent to H(G)s, in the notation of Section 1.6. The (conjectural) iso-

morphism described in (1.28) is such that a∗=X⊗ZRcorresponds to X∗(A)⊗ZR.

The conditions for temperedness of ﬁnite length representations of H⋊Γ and H(G)s

are formulated in terms of corresponding negative cones in a∗and in X∗(A)⊗ZR.

Therefore such a Morita equivalence would preserve temperedness of representations.

Thus Modf(S⋊Γ) would be equivalent to the category of ﬁnite length modules

Modf,s(S(G)) = Modf(S(G)s),

where S(G) is the Harish-Chandra–Schwartz algebra of Gand S(G)sis its two-sided

ideal corresponding to the inertial equivalence class s∈B(G).

Moreover, is Iis an Iwahori subgroup of a split group G, it is shown in [DeOp1,

Proposition 10.2] that the isomorphism H(G, I )∼

=H(R, q) extends to an isomor-

phism S(G, I)∼

=S(R, q). Therefore it is reasonable to expect that more generally

S(G)swill be Morita equivalent to S(R, q)⋊Γ in case of an isomorphism (1.28). Fur-

ther support of this is provided by Theorem 3.2.2 in comparison with the Plancherel

theorem for S(G) [Wal] and for Cr(G) [Ply2]. These show that S(R, q)⋊Γ and

S(G)s, as well as their respective C∗-completions, have a very similar shape, which

can almost entirely be deduced from their categories of ﬁnite length modules.

3.3 Parametrization of representations with induction

data

Theorem 3.2.2 is extremely deep and useful, a large part of what follows depends on

it. It shows a clear advantage of Sover H, namely that the Fourier transform con-

sists of all smooth sections. In particular one can use any smooth section, without

knowing its preimage under F. By Corollary 3.2.3 the irreducible tempered H⋊Γ-

representations are partitioned in ﬁnite packets parametrized by Ξun/G. Moreover,

from Theorem 3.2.2 Delorme and Opdam also deduce analogues of Harish-Chandra’s

Completeness Theorem [DeOp1, Corollary 5.4] and of Langlands’ Disjointness The-

orem [DeOp1, Corollary 5.8].

We will generalize these results to all irreducible representations. For that we do

not need all induction data from Ξ, in view of Lemma 3.1.7 it suﬃces to consider

ξ∈Ξ+. At the same time this restriction to positive induction data enables us to

avoid the singularities of the intertwiners π(g, ξ).

Theorem 3.3.1. Let ξ= (P, δ, t), ξ′= (P′, δ, t′)∈Ξ+.

(a)The H⋊Γ-representations πΓ(ξ)and πΓ(ξ′)have a common irreducible quotient

if and only if there exists a g∈ G such that gξ =ξ′.

(b)The operators {πΓ(g, ξ) : g∈ G, gξ =ξ′}are regular and invertible, and they

span HomH⋊Γ(πΓ(ξ), πΓ(ξ′)).

Proof. (a) Suppose that there exists a g∈ G with gξ =ξ′. Since πΓ(γ , ξ′) is

invertible for γ∈Γ, we may replace ξ′by γξ′, which allows to assume without loss

of generality that g= (w, u)∈W0×KP.

Recall that T+is a fundamental domain for the action of W0on Trs. Since |t|and

|t′|are both in T+we must have |t|=w(|t|) = |t′|and hence P(ξ) = P(ξ′). Thus

wu(P, δ, t|t|−1) = (P′, δ′, t′|t′|−1) and by Theorem 3.1.5.b the HP(ξ)-representations

πP(ξ)(P, δ, t|t|−1) = πP(ξ)(P, δ, t)◦φ−1

|t|and

πP(ξ)(P′, δ′, t′|t|−1) = πP(ξ)(P′, δ′, t′)◦φ−1

|t|

are isomorphic. Hence πP(ξ)(P, δ, t)∼

=πP(ξ)(P′, δ′, t′), which implies that πΓ(ξ) and

πΓ(ξ′) are isomorphic. In particular πΓ(ξ) and πΓ(ξ′) have the same irreducible

quotients.

Conversely, suppose that πΓ(ξ) and πΓ(ξ′) have a common irreducible quotient.

Again we may replace ξ′by γξ′for any γ∈Γ. In view of this, Proposition 3.1.4.c

and Corollary 2.2.5.b we may assume that P(ξ) = P(ξ′) and that the HP(ξ)⋊

ΓP(ξ)-representations πP(ξ),ΓP(ξ)(ξ) and πP(ξ),ΓP(ξ)(ξ′) have a common irreducible

summand πP(ξ),ΓP(ξ)(P(ξ), σ, tP(ξ), ρ).

Pick s∈TP(ξ)

un tP(ξ)such that tP(ξ)and tP(ξ)s−1have the same isotropy group

in W0⋊Γ. This is possible because Tgis a complex algebraic subtorus of Tfor

every g∈W0⋊Γ. The HP(ξ)⋊ΓP(ξ)-representations πP(ξ),ΓP(ξ)(P, δ, ts−1) and

πP(ξ),ΓP(ξ)(P′, δ′, t′s−1) are completely reducible by Proposition 3.1.4.a, and they

have the common irreducible summand

πP(ξ),ΓP(ξ)(P(ξ), σ, tP(ξ)s−1, ρ) = IndHP(ξ)⋊ΓP(ξ)

HP(ξ)⋊ΓP(ξ),σ,tP(ξ)σ◦φtP(ξ)◦φ−1

s⊗ρ.

Moreover, because every irreducible summand is of this form,

HomHP(ξ)⋊ΓP(ξ)πP(ξ),ΓP(ξ)(P, δ, ts−1), πP(ξ),ΓP(ξ)(P′, δ′, t′s−1)∼

HomHP(ξ)⋊ΓP(ξ)πP(ξ),ΓP(ξ)(P, δ, t), πP(ξ),ΓP(ξ)(P′, δ′, t′)6= 0.(3.19)

Since tP(ξ)s−1∈Tun, we have ts−1, t′s−1∈Tun. So |t|=|t′|and the repre-

sentations πP(ξ),ΓP(ξ)(P, δ, ts−1) and πP(ξ),ΓP(ξ)(P′, δ′, t′s−1) extend continuously to

S(RP(ξ), qP(ξ))⋊ΓP(ξ). Now Theorem 3.2.2 for this algebra shows that the left

hand side of (3.19) is spanned by the intertwiners πP(ξ),ΓP(ξ)(g, P, δ, ts−1) with

g(P, δ, ts−1) = (P′, δ′, t′s−1). Since (3.19) is nonzero, there exists at least one such

g∈ G. The choice of sguarantees that g(P, δ, t) = (P′, δ′, t′) as well.

(b) By Theorem 3.1.5 the πP(ξ),ΓP(ξ)(g, P, δ, ts−1) are invertible and constant on

TP(ξ)-cosets. Hence the πP(ξ),ΓP(ξ)(g, P, δ, t) span the right hand side of (3.19), and

they are invertible. 2

It is interesting to compare Theorem 3.3.1 with [Ree1], which describes the H-

endomorphisms of principal series representations M(t). It transpires that the re-

sults of [Ree1] simplify considerably when |t|is in the positive Weyl chamber: then

EndH(M(t)) is semisimple and all its irreducible quotients occur with multiplicity

one.

Now we can prove the desired partition of Irr(H⋊Γ) in packets:

Theorem 3.3.2. Let πbe an irreducible H⋊Γ-representation. There exists a unique

association class G(P, δ, t)∈Ξ/Gsuch that the following equivalent properties hold:

(a)πis isomorphic to an irreducible quotient of πΓ(ξ+), for some ξ+∈Ξ+∩

G(P, δ, t);

(b)πis a constituent of πΓ(P, δ, t), and kccP(δ)kis maximal for this property.

Proof. Proposition 3.1.4.d says that there exists ξ+= (P′, δ′, t′)∈Ξ+satisfying

(a), and by Theorem 3.3.1 its G-association class is unique.

Let ξ= (P, δ, t)∈Ξ such that πis a constituent of πΓ(ξ) and kccP(δ)kis maximal

under this condition. By Lemma 3.1.7 we may assume that ξ∈Ξ+. Suppose that π

is not isomorphic to a quotient of πΓ(ξ). In view of Proposition 3.1.4 this means that

there exist Langlands data (P(ξ+), σ′, t′P(ξ+), ρ′) and (P(ξ), σ, tP(ξ), ρ) such that π∼

LΓ(P(ξ+), σ′, t′P(ξ+), ρ′) is a constituent but not a quotient of πΓ(P(ξ), σ, tP(ξ), ρ).

Now Lemma 2.2.6.b tells us that



ccP(ξ)(σ)

<

ccP(ξ+)(σ′)

.

But ccP(ξ)(σ) = WP(ξ)ccP(δ) and ccP(ξ+)(σ′) = WP(ξ+)ccP′(δ′), so

kccP(δ)k<

ccP′(δ′)

,

contradicting the maximality of kccP(δ)k. Therefore πmust be a quotient of πΓ(ξ).

Thus the association class Gξsatisﬁes not only (b) but also (a), which at the

same time shows that is unique. In particular conditions (a) and (b) turn out to be

equivalent. 2

All these constructs with induction data have direct analogues in the setting of

graded Hecke algebras [Sol5, Sections 6 and 8]. Concretely, ˜

Ξ is the space of all

triples ˜

ξ= (Q, σ, λ), where Q⊂F0, σ is a discrete series representation of HQand

λ∈tQ. The subsets of unitary (respectively positive) induction data are obtained

by imposing the restriction λ∈iaQ(respectively λ∈aQ++iaQ). The corresponding

induced representation is

πΓ(˜

ξ) = IndH⋊Γ

Hπ(Q, σ, λ) = IndH⋊Γ

HQ(σλ).

The groupoid ˜

Gand its action on ˜

Ξ are deﬁned like G, but without the parts KP.

We would like to understand the relation between induction data for H⋊Γ and for

H ⋊ Γ. We consider, for every u∈Tun, the induction data for ( ˜

Ru, ku) with λ∈a.

Thus we arrive at the space b

Ξ of quadruples ˆ

ξ= (u, ˜

P , ˜

δ, λ) such that:

•u∈Tun;

•˜

P⊂Fu;

•˜

δis a discrete series representation of H(˜

Ru, ˜

P, ku, ˜

P);

•λ∈a˜

The H⋊Γ-representation associated to ˆ

ξis

πΓ(u, ˜

P , ˜

δ, λ) = IndH⋊Γ

H(Ru,qu)π(˜

P , ˜

δ, λ),

where the H(˜

Ru, ku)-representation π(˜

P , ˜

δ, λ) is considered as a representation of

H(˜

Ru, qu), via Theorem 2.1.4.

For g∈ G the map ψgfrom (1.19) and (1.15) induces an algebra isomorphism

H(˜

Ru, ku)→H(˜

Rg(u), kg(u)), and the stabilizer in Gof u∈Tun is the groupoid ˜

associated to ( ˜

Ru, ku). This leads to an action of Gon b

Ξ.

The collections of H⋊Γ-representations corresponding to Ξ and to b

Ξ are almost

the same, but not entirely:

Lemma 3.3.3. There exists a natural ﬁnite-to-one surjection

Ξ/G → b

Ξ/G,Gξ7→ Gˆ

ξ,

with the following property. Given ˆ

ξ∈b

Ξone can ﬁnd ξi∈Ξ(not necessarily all

diﬀerent) such that SiGξiis the preimage of Gˆ

ξand πΓ(ˆ

ξ) = LiπΓ(ξi).

Proof. Given ξ= (P, δ, t)∈Ξ, let t=uPcP∈TP

u×TP

rs be the polar decomposi-

tion of tand let uPcP∈TP,u ×TP,rs be an AP-weight of δ. Put

W+

P,uP={w∈W(RP) : w(uP) = uP, w(R+

P,uP) = R+

P,uP}.(3.20)

By Theorem 2.1.2 there exists a unique discrete series representation

δ1of H(RP,uP, qP,uP)⋊W+

P,uPsuch that δ∼

=IndHP

H(RP,uP,qP,uP)⋊W+

P,uP

(δ1).

Then automatically

δ◦φt∼

=IndHP

H(RP

uP,qP

uP)⋊W+

P,uP

(δ1◦φt).

Let δ′be an irreducible direct summand of the restriction of δ1to H(RP,uP, qP,uP),

such that uPcPis a weight of δ′. Then δ1◦φtis a direct summand of

IndH(RP

uP,qP

uP)⋊W+

P,uP

H(RP

uP,qP

uP)(δ′◦φt),(3.21)

so πΓ(ξ) is a direct summand of

IndH⋊Γ

H(RP

uP,qP

uP)(δ′◦φt).(3.22)

By Theorem 2.1.4 δ′can also be regarded as a discrete series representation ˜

δof

H(˜

RP,uP, kP,uP) with central character W(RP,uP)cP. Then δ′◦φtcorresponds to

the representation ˜

δlog(cP)of H(˜

uP, kP

uP). Let ˜

Pbe the unique basis of RP,uP

contained in R+

All in all (P, δ, t) gives rise to the induction datum ˜

ξ= ( ˜

P , ˜

δ, log(cP)) for the

graded Hecke algebra H(˜

RP,uP, kP,uP). Since RP,uPis a parabolic root subsystem

of Ru,˜

ξcan also be regarded as an induction datum for H(˜

Ru, ku). Let us check

the possible freedom in the above construction. All AP-weights of δare in the same

WP-orbit, so another choice of uPcPdiﬀers only by an element of WP. All possible

choices of δ′above are conjugated by the action of the group W+

P,uP, and (W0⋊Γ)u

is the unitary part of the central character of πΓ(ξ). Therefore ξdetermines the

quadruple ˆ

ξ:= (u, ˜

P , ˜

δ, log(cP)) ∈b

uniquely modulo the action of G. That yields a map Ξ →b

Ξ/G, ξ → Gˆ

ξ, and since

the actions of Gare deﬁned in the same way on both sides, this map factors via Ξ/G.

By reversing the above steps one can reconstruct the representations δ′◦φtand

(3.22) from ˜

ξ , uPand uP. In fact it suﬃces to know ˜

ξand the product u=uPuP∈

Tun. Namely, the only additional ambiguity comes from the group KP, but this is

inessential since

(δ′◦ψk−1)◦φkt ∼

=δ′◦φtfor k∈KP.

By construction δ1is a direct summand of IndH(RP,uP,qP ,uP)⋊W+

P,uP

H(RP,uP,qP,uP)(δ′), and the other

constituents δj(1 < j ≤n) are also discrete series representations. Hence (3.22) is

a direct sum of ﬁnitely many parabolically induced representations

πΓ(P, IndHP

H(RP,uP,qP,uP)⋊W+

P,uP

(δj), t).

Now Corollary 2.1.5 assures that our map Ξ/G → b

Ξ/Gis surjective and that the

preimage of G(u, ˜

P , ˜

δ, log(cP)) consists precisely of the association classes

G(P, IndHP

H(RP,uP,qP,uP)⋊W+

P,uP

(δj), t) (1 ≤j≤n).2

Remark. Things simplify considerably if the group W+

P,uP={id}in (3.20), then the

map Ξ/G → b

Ξ/Gis bijective on (P, δ, T P)/G. In many cases this group is indeed

trivial, but not always. See [OpSo2, Section 8], where W+

P,uPis denoted Γs(e).

3.4 The geometry of the dual space

For any algebra Athe set Irr(A) has a natural topology, the Jacobson topology.

This is the noncommutative generalization of the Zariski topology, by deﬁnition all

its closed sets are of the form

V(S) := {π∈Irr(A) : π(s) = 0 ∀s∈S}S⊂A.

In this section we discuss the topology and the geometry of Irr(H⋊Γ), and we

compare it with the dual of S⋊Γ. This will be useful for the proof of Theorem 2.3.1

and for our discussion of periodic cyclic homology in Section 5.2.

Parabolic induction gives, for every discrete series representation δof a parabolic

subalgebra HP, a family of H⋊Γ-representations πΓ(P, δ, t), parametrized by t∈TP.

The group

GP,δ := {g∈ G :g(P) = P, δ ◦ψ−1

g∼

=δ}

acts algebraically on TP, and by Lemma 3.1.7 points in the same orbit lead to

representations with the same irreducible subquotients.

Theorem 3.3.2 allows us to associate to every π∈Irr(H⋊Γ) an induction datum

ξ+(π)∈Ξ+, unique modulo G, such that πis a quotient of πΓ(ξ+(π)). For any

subset U⊂TPwe deﬁne

IrrP,δ,U (H⋊Γ) = {π∈Irr(H⋊Γ) : Gξ+(π)∩(P, δ, U )6=∅}.

For U=TPor U={t}we abbreviate this to IrrP,δ(H⋊Γ) or IrrP,δ,t(H⋊Γ).

Proposition 3.4.1. Let Ube a subset of TP+TP

un such that every g∈ GP,δ with

gU ∩U6=∅ﬁxes Upointwise. For arbitrary t∈Uthere are canonical bijections

IrrP,δ (HP⋊ΓP,δ,t )×U→IrrP,δ,U (HP⋊ΓP,δ,t)→IrrP,δ,U (H⋊Γ).

Remark. It is not unreasonable to expect that the Jacobson topology of H⋊Γ

induces the Zariski topology on IrrP,δ (HP⋊ΓP,δ,t)×U, where IrrP,δ (HP⋊ΓP,δ,t) is

regarded as a discrete space. However, while it is easy to see that all V(h) become

Zariski-closed in IrrP,δ (HP⋊ΓP,δ,t)×U, it is not clear that one can obtain all

Zariski-closed subsets in this way. That might require some extra conditions on U.

Proof. By assumption every t∈Uhas the same stabilizer GP,δ,t ⊂ GP,δ. Accord-

ing to Theorem 3.3.1 the operators {πΓ(g, P, δ, t) : g∈ GP,δ,t}span EndH⋊Γ(πΓ(P, δ, t)).

By deﬁnition all elements of IrrP,δ,t(H⋊Γ) occur as a quotient of πΓ(P, δ, t), but

the latter representation also has other constituents if it is not completely re-

ducible. We have to avoid that situation if we want to ﬁnd a direct relation between

EndH⋊Γ(πΓ(P, δ, t)) and IrrP,δ,t(H⋊Γ).

Since πΓ(P, δ, t) and πΓ(g, P, δ, t) are unitary for t∈TP

un, there exists an open

GP,δ -stable tubular neighborhood TP

ǫof TP

un in TP, such that πΓ(P, δ, t) is completely

reducible and πΓ(g, P, δ, t) is regular and invertible, for all t∈TP

ǫand g∈ GP,δ . For

every t∈Uwe can ﬁnd r∈R>0such that t|t|r−1∈TP

ǫ. Let Uǫ⊂TP

ǫbe the

resulting collection. For every t∈Uǫthe algebras

{πΓ(P, δ, t)(h) : h∈ H ⋊Γ}and span{πΓ(g, P, δ, t) : g∈ GP,δ,t}

are each others’ commutant in

EndC(πΓ(P, δ, t)) = EndCC[ΓΓP]⊗CVδ.

Hence there is a natural bijection between

•isotypical components of πΓ(P, δ, t) as a H⋊Γ-representation;

•isotypical components of πΓ(P, δ, t) as a GP,δ,t-representation.

The operators πΓ(P, δ, t) are rational in t∈TP, and regular on TP

ǫ. As there are

only ﬁnitely many inequivalent GP,δ,t-representations of a ﬁxed ﬁnite dimension, the

isomorphism class of πΓ(P, δ, t) as a GP,δ,t-representation does not depend on t∈Uǫ.

This provides a natural bijection

IrrP,δ,Uǫ(H⋊Γ) ←→ IrrP,δ,t(H⋊Γ) ×Uǫt∈Uǫ.(3.23)

The extended Langlands classiﬁcation (Corollary 2.2.5) shows that there is a canon-

ical bijection IrrP,δ,t|t|r−1(H⋊Γ) ↔IrrP,δ,t(H⋊Γ) for every r∈R>0, which allows

us to extend (3.23) uniquely to

IrrP,δ,U (H⋊Γ) ←→ IrrP,δ,t0(H⋊Γ) ×U t0∈U. (3.24)

The above also holds with the algebras H⋊ΓP,δ,t or HP⋊ΓP,δ,t in the role of H⋊Γ.

Since the construction of the intertwiners corresponding to g∈ GP,δ,t is the same in

all three cases, we obtain natural isomorphisms

EndHP⋊ΓP,δ,t IndHP⋊ΓP,δ,t

HPδ∼

=EndHP⋊ΓP,δ,t (πP,ΓP,δ,t (P, δ, t)) ∼

=EndH⋊Γ(πΓ(P, δ, t)).

Now the above bijection between isotypical components shows that the maps

IrrP,δ (HP⋊ΓP,δ,t)→IrrP,δ,t(HP⋊ΓP,δ,t)→IrrP,δ,t(H⋊Γ)

ρ7→ ρ◦φt7→ IndH⋊Γ

HP⋊ΓP,δ,t (ρ◦φt)

are bijective, and (3.24) allows us to extend this from one tto U. 2

Theorem 3.3.2 shows that IrrP,δ (H⋊Γ) and IrrQ,σ (H⋊Γ) are either equal or

disjoint, depending on whether or not (P, δ) and (Q, σ) are G-associate. The sets

IrrP,δ (H⋊Γ) are usually not closed in Irr(H⋊Γ), because we did not include all

constituents of the representations πΓ(P, δ, t). We can use their closures to deﬁne

a stratiﬁcation of Irr(H⋊Γ), and a corresponding stratiﬁcation of Irr(S⋊Γ). By

Corollary 3.1.3 we may identify IrrP,δ (S⋊Γ) with the tempered part IrrP,δ,T P

un (H⋊Γ)

of IrrP,δ(H⋊Γ).

Let ∆ be a set of representatives for the action of Gon pairs (P, δ). Then

the cardinality of ∆ equals the number of connected components of Ξun/Gand by

Theorem 3.2.2

S⋊Γ∼

=M(P,δ)∈∆C∞(TP

un)⊗CEnd(C[ΓWP]⊗CVδ)GP ,δ .(3.25)

Lemma 3.4.2. There exist ﬁltrations by two-sided ideals

H⋊Γ = F0(H⋊Γ) ⊃F1(H⋊Γ) ⊃ ··· ⊃ F|∆|(H⋊Γ) = 0,

S⋊Γ = F0(S⋊Γ) ⊃F1(S⋊Γ) ⊃ ··· ⊃ F|∆|(S⋊Γ) = 0,

with Fi(H⋊Γ) ⊂Fi(S⋊Γ), such that for all i > 0:

(a) IrrFi−1(S⋊Γ)/Fi(S⋊Γ)∼

=IrrPi,δi(S⋊Γ),

(b) IrrFi−1(H⋊Γ)/Fi(H⋊Γ)∼

=IrrPi,δi(H⋊Γ).

Remark. Analogous ﬁltrations of Hecke algebras of reductive p-adic groups are

described in [Sol4, Lemma 2.17]. The proof in our setting is basically the same.

Proof. We number the elements of ∆ such that

kccPi(δi)k ≥ 

ccPj(δj)

if j≤i, (3.26)

and we deﬁne

Fi(H⋊Γ) = {h∈ H ⋊Γ : π(h) = 0 for all π∈IrrPj,δj(H⋊Γ), j ≤i},

Fi(S⋊Γ) = {h∈ S ⋊Γ : π(h) = 0 for all π∈IrrPj,δj(S⋊Γ), j ≤i}.

Clearly Fi(H⋊Γ) ⊂Fi(S⋊Γ) and

Fi−1(S⋊Γ)/Fi(S⋊Γ) ∼

=C∞(TP

un)⊗CEnd(C[Γ ×WP]⊗CVδ)GPi,δi,

which establishes (a). For (b), we ﬁrst show that

[j≤iIrrPj,δj(H⋊Γ) (3.27)

is closed in the Jacobson topology of Irr(H⋊Γ). Its Jacobson-closure consists of

all irreducible subquotients πof πΓ(Pj, δj, t), for j≤iand t∈TPj. Suppose that

π /∈IrrPj,δj(H⋊Γ). By Theorem 3.3.2 π∈IrrP,δ (H⋊Γ) for some discrete series

representation δof HPwith kccP(δ)k>

ccPj(δj)

. Select (Pn, δn) and g∈ G such

that g(P, δ) = (Pn, δn). Then π∈IrrPn,δn(H⋊Γ) and

kccPn(δn)k=kccP(δ)k>

ccPj(δj)

,

so n < j ≤iby (3.26). Therefore (3.27) is indeed Jacobson-closed and

Irr(Fi(H⋊Γ)) ∼

=[j<i IrrPj,δj(H⋊Γ),

which implies (b). 2

The ﬁltrations from Lemma 3.4.2 help us to compare the dual of H⋊Γ with its

tempered dual, which can be identiﬁed with the dual of S⋊Γ. The space IrrP,δ (H⋊Γ)

comes with a ﬁnite-to-one projection onto

TP+TP

un/GP,δ ∼

=TP/(W(RP)⋊GP,δ).(3.28)

The subspace IrrP,δ (S⋊Γ) ⊂IrrP,δ(H⋊Γ) is the inverse image of TP

un/(W(RP)⋊GP,δ )

under this projection. By Proposition 3.4.1 the ﬁber at t∈TPessentially depends

only on the stabilizer GP,δ,t. Since GP,δ acts algebraically on TP, the collection of

points t∈TPsuch that the ﬁber at (WP⋊GP,δ )thas exactly mpoints (for some

ﬁxed m∈N) is a complex aﬃne variety, say TP,m. As the action of GP,δ stabilizes

un, the variety TP,m is already determined by its intersection with TP

un. Hence one

can reconstruct the set IrrP,δ (H⋊Γ) from IrrP,δ(S⋊Γ). With these insights we can

ﬁnally complete the proof of property (a) of our aﬃne Springer correspondence.

Continuation of the proof of Theorem 2.3.1.

Extend Spr to a Q-linear map

SprQ:GQ(H⋊Γ) →GQ(X⋊(W0⋊Γ)).(3.29)

We have to show that SprQis a bijection. To formulate the proof we introduce

some new terminology, partly taken from [Opd3]. We know from Lemma 3.1.1 that

representations of the form

πP,ΓP,δ,u (P, δ, u) = IndHP⋊ΓP ,δ,u

HP(δ◦φu) with (P, δ, u)∈Ξun

are tempered and unitary. Let σbe an irreducible summand of such an HP⋊ΓP,δ,u-

representation and let TW(RP)⋊ΓP,δ,u

1be the connected component of TW(RP)⋊ΓP,δ,u

that contains 1 ∈T. We call

IndH⋊Γ

HP⋊ΓP,δ,u (σ◦φt) : t∈TW(RP)⋊ΓP,δ,u

1.(3.30)

a smooth d-dimensional family of representations, where dis the dimension of the

complex algebraic variety TW(RP)⋊ΓP,δ,u . If we restrict the parameter tto Tun, then

we add the adjective tempered to this description.

We note that these representations are irreducible for tin a Zariski-open subset

of TW(RP)⋊ΓP,δ,u

1. The proof of Proposition 3.1.4.b shows that

IndH⋊Γ

HP⋊ΓP,δ,u (σ◦φt)

is always a direct sum of representations of the form πΓ(P′, σ′, t′, ρ′), where (P′, σ′, t′, ρ′)

is almost a Langlands datum for H⋊Γ, the only diﬀerence being that |t′| ∈ TP′

need not be positive. Nevertheless we can specify a unique ”Langlands constituent”

of πΓ(P′, σ′, t′, ρ′), by the following procedure. Choose g∈ G such that g(P′, δ′, t′)∈

Ξ+. By Proposition 3.1.4.b g(P′, σ′, t′, ρ′) is a Langlands datum for H⋊Γ and, as in

Lemma 3.1.7, πΓ(g(P′, σ′, t′, ρ′)) has the same trace and the same semisimplication

as πΓ(P′, σ′, t′, ρ′). We deﬁne

LπΓ(P′, σ′, t′, ρ′)=LΓ(g(P′, σ′, t′, ρ′)).(3.31)

In view of Corollary 2.2.5 the Langlands constituent appears only once in πΓ(P′, σ′, t′, ρ′)

and Lemma 2.2.6.b characterizes it as the constituent which is minimal in the ap-

propriate sense.

Let LIndH⋊Γ

HP⋊ΓP,δ,u (σ◦φt)be the direct sum of the Langlands constituents of

the irreducible summands of IndH⋊Γ

HP⋊ΓP,δ,u (σ◦φt). The family

LIndH⋊Γ

HP⋊ΓP,δ,u (σ◦φt):t∈TW(RP)⋊ΓP,δ,u

0(3.32)

cannot be called smooth, because the traces of these representations do not depend

continuously on t. Let us call it an L-smooth d-dimensional family of representations.

By properties (d) and (e) of Theorem 2.3.1

SprQLIndH⋊Γ

HP⋊ΓP,δ,u (σ◦φt)= IndX⋊(W0⋊Γ)

X⋊(W(RP)⋊ΓP,δ,u)(SprQ(σ)◦φt).(3.33)

The right hand side is almost a smooth d-dimensional family of X⋊(W0⋊Γ)-

representations. Not entirely, because SprQ(σ) is in general reducible and because a

priori this family could be only a part of a higher dimensional smooth family.

Let Gd

Q(H⋊Γ) be the Q-submodule of GQ(H⋊Γ) spanned by the representations

(3.32), for all L-smooth families of dimension at least d∈Z≥0. This is a decreasing

sequence of Q-submodules of GQ(H⋊Γ), by convention

Q(H⋊Γ) = GQ(H⋊Γ)

and Gd

Q(H⋊Γ) = 0 when d > dimC(T) = rank(X). We deﬁne Gd

Q(S⋊Γ) analogously,

with tempered smooth families of dimension at least d. Now (3.33) says that

SprQGd

Q(H⋊Γ)⊂Gd

Q(X⋊W0⋊Γ),

SprQGd

Q(S⋊Γ)⊂Gd

Q(S(X)⋊W0⋊Γ).(3.34)

Let us consider the graded vector spaces associated to these ﬁltrations. For tempered

representations SprQinduces Q-linear maps

Q(S⋊Γ)/Gd+1

Q(S⋊Γ) →Gd

Q(S(We)⋊Γ)/Gd+1

Q(S(We)⋊Γ).(3.35)

We proved in Section 2.3 that

SprQ:GQ(S⋊Γ) →GQ(S(We)⋊Γ) (3.36)

is bijective, so (3.35) is bijective for all d∈Z≥0. We will show that

Q(H⋊Γ)/Gd+1

Q(H⋊Γ) →Gd

Q(X⋊W0⋊Γ)/Gd+1

Q(X⋊W0⋊Γ) (3.37)

is bijective as well. For d > dimC(T) there is nothing to prove, so take d≤dimC(T).

Pick smooth d-dimensional families {πi,t :i∈I, t ∈Vi}such that

πi,t :i∈I, t ∈(Vi∩TP

un)/GP,δ(3.38)

is a basis of Gd

Q(S⋊Γ)/Gd+1

Q(S⋊Γ). The bijectivity of (3.35) implies that

SprQ(πi,t) : i∈I , t ∈(Vi∩TP

un)/GP,δ

is a basis of Gd

Q(S(We)⋊Γ)/Gd+1

Q(S(We)⋊Γ). The discussion following (3.28)

shows that L-smooth d-dimensional families and tempered smooth d-dimensional

families are in natural bijection. Hence

L(πi,t) : i∈I , t ∈Vi/GP,δ 

is a basis of Gd

Q(H⋊Γ)/Gd+1

Q(H⋊Γ) and

SprQ(πi,t) : i∈I , t ∈Vi/GP,δ 

is a basis of Gd

Q(X⋊W0⋊Γ)/Gd+1

Q(X⋊W0⋊Γ). Therefore (3.37) is indeed bijective.

From this we deduce, with some standard applications of the ﬁve lemma, that (3.29)

is a bijection. 2.

Chapter 4

Parameter deformations

Let H=H(R, q) be an aﬃne Hecke algebra associated to an equal parameter func-

tion q. Varying the parameter q∈C×yields a family a algebras, whose members are

specializations of an aﬃne Hecke algebra with a formal variable q. The Kazhdan–

Lusztig-parametrization of Irr(H(R, q)) [KaLu2, Ree2] provides a bijection between

the irreducible representations of H(R, q) and H(R, q′), as long as q, q′∈C×are

not roots of unity. Moreover, every πq∈Irr(H(R, q)) is part of a family of repre-

sentations {πq:q∈C×}which depends algebraically on q.

It is our conviction that a similar structure underlies the representation theory of

aﬃne Hecke algebras with unequal parameters. However, at present a proof seems

to be out of reach. We have more control when we restrict ourselves to positive

parameter functions and to parameter deformations of the form q7→ qǫwith ǫ∈R.

We call this scaling of the parameter function, because it corresponds to multiplying

the parameters of a graded Hecke algebra with ǫ. Notice that H(R, q0) = C[We].

We can relate representations of H(R, q) to H(R, qǫ)-representations by applying

a similar scaling operation on suitable subsets of the space T∼

=Irr(A). We construct

a family of functors

˜σǫ: Modf(H(R, q)) →Modf(H(R, qǫ))

which is an equivalence of categories for ǫ6= 0, and which preserves many prop-

erties of representations (Corollary 4.2.2). In particular this provides families of

representations {˜σǫ(π) : ǫ∈R}that depend analytically on ǫ.

The Schwartz algebra S(R, q) behaves even better under scaling of the parameter

function q. As qcan be varied in several directions, we have a higher dimensional

family of Fr´echet algebras S(R, q), which is known to depend continuously on q

[OpSo2, Appendix A]. This was exploited for the main results of [OpSo2], but the

techniques used there to study general deformations of qare speciﬁc to discrete series

representations.

To get going at the other series, we only scale q. Via a detailed study of the

Fourier transform of S(R, q) (see Theorem 3.2.2) we construct homomorphisms of

Fr´echet *-algebras

ζǫ:S(R, qǫ)→ S(R, q)ǫ∈[0,1],

which depend piecewise analytically on ǫand are isomorphisms for ǫ > 0 (Theorem

4.4.2). It is not known whether this is possible with H(R, q) instead of S(R, q),

when qis not an equal parameter function.

The most remarkable part is that these maps extend continuously to ǫ= 0, that

is, to a map ζ0:S(We)→ S(R, q). Of course ζ0cannot be an isomorphism, but it

is injective and and some ways behaves like an isomorphism. In fact, we show that

for irreducible tempered H(R, q )-representations the aﬃne Springer correspondence

from Section 2.3 and the functors ˜σ0and π7→ π◦ζ0agree (Corollary 4.4.3).

4.1 Scaling Hecke algebras

As we saw in Section 2.3, there is a correspondence between tempered representations

of H⋊Γ and of We⋊Γ. On central characters this correspondence has the eﬀect

of forgetting the real split part and retaining the unitary part. These elements of

T/(W0⋊Γ) are connected by the path (W0⋊Γ)cǫu, with ǫ∈[0,1]. Opdam [Opd2,

Section 5] was the ﬁrst to realize that one can interpolate not only the central

characters, but also the representations themselves. In this section we will recall

and generalize the relevant results of Opdam. In contrast to the previous sections

we will not include an extra diagram automorphism group Γ in our considerations,

as the notation is already heavy enough. However, it can be checked easily that all

the results admit obvious generalizations with such a Γ.

First we discuss the situation for graded Hecke algebras, which is considerably

easier. Let V⊂tbe a nonempty open W0-invariant subset. Recall the elements

˜ıw∈Cme(V)W0⊗Z(H(˜

R,k)) H(˜

R, k) from Proposition 1.5.1. Given any ǫ∈Cwe

deﬁne a scaling map

λǫ:V→ǫV, v 7→ ǫv.

For ǫ6= 0 it induces an algebra isomorphism

mǫ:Cme(ǫV )W0⊗Z(H(˜

R,ǫk)) H(˜

R, k)→Cme(V)W0⊗Z(H(˜

R,k)) H(˜

R, k),

f˜ıw,ǫ 7→ (f◦λǫ)˜ıw.(4.1)

Let us calculate the image of a simple reﬂection sα∈S0:

mǫ(1 + sα) = mǫ˜c−1

α,ǫ(1 + ˜ısα,ǫ)=mǫα

ǫkα+α(1 + ˜ısα,ǫ)=ǫα

ǫkα+ǫα(1 + ˜ısα)

= ˜c−1

α(1 + ˜ısα) = 1 + sα.

That is, mǫ(w) = wfor all w∈W0⊂H(˜

R, ǫk), so mǫis indeed the same as (1.10).

In particular we see that mǫcan be deﬁned without using meromorphic functions.

These maps have a limit homomorphism

m0:H(˜

R,0) = O(t)⋊W0→H(˜

R, k),

fw 7→ f(0)w, (4.2)

with the property that

C→Cme(V)W0⊗Z(H(˜

R,k)) H(˜

R, k) : ǫ7→ mǫ(f w)

is analytic for all f∈ O(t) and w∈W0.

From now we assume that ǫis real. Let qǫbe the parameter function qǫ(w) =

q(w)ǫ, which is well-deﬁned because q(w)∈R>0for all w∈We. We obtain a family

of algebras Hǫ=H(R, qǫ) with H(R, q0) = C[We]. To relate representations of

these algebras we use their analytic localizations, as in Section 1.5.

Let cα,ǫ be the c-function with respect to (R, qǫ), as in (1.23). For ǫ∈[−1,1]\{0}

the ball ǫB ⊂tsatisﬁes the conditions 2.1.1 with respect to ucǫ∈Tand the

parameter function qǫ(which enters via the function cα,ǫ). For ǫ= 0 the point

ǫB ={0}trivially fulﬁlls all the conditions, except that it is not open. We write

Uǫ=W0ucǫexp(ǫB)ǫ∈[−1,1],

and we deﬁne a W0-equivariant scaling map

σǫ:U1→Uǫ, w(uc exp(b)) 7→ w(ucǫexp(ǫb)).

As was noted in [Opd2, Lemma 5.1], σǫis an analytic diﬀeomorphism for ǫ6= 0,

while σ0is a locally constant map with range U0=W0u. Let ı0

w,ǫ be the element

constructed in Proposition 1.5.1. By (1.26) σǫinduces an algebra isomorphism

ρǫ:Hme

ǫ(Uǫ)→ Hme(U)ǫ∈[−1,1] \ {0},

fı0

w,ǫ 7→ (f◦σǫ)ı0

w,(4.3)

where f∈Cme(U) and w∈W0. We will show that these maps depend continuously

on ǫand have a well-deﬁned limit as ǫ→0.

Lemma 4.1.1. For ǫ∈[−1,1]\{0}and α∈R0write dα,ǫ = (cα,ǫ ◦σǫ)c−1

α∈Cme(U).

This deﬁnes a bounded invertible analytic function on U×([−1,1] \ {0}), which

extends to a function on U×[−1,1] with the same properties.

Proof. This extends [Opd2, Lemma 5.2] to ǫ= 0. Let us write

dα,ǫ(t) = f1f2f3f4

g1g2g3g4

(t) = 1 + θ−α(t)

1 + θ−α(σǫ(t)) ×

1 + q(sα)−ǫ/2q(tαsα)ǫ/2θ−α(σǫ(t))

1 + q(sα)−1/2q(tαsα)1/2θ−α(t)

1−θ−α(t)

1−θ−α(σǫ(t))

1−q(sα)−ǫ/2q(tαsα)−ǫ/2θ−α(σǫ(t))

1−q(sα)−1/2q(tαsα)−1/2θ−α(t)

We see that dα,ǫ(t) extends to an invertible analytic function on U×[−1,1] if none

of the quotients fn/gnhas a zero or a pole on this domain. By Condition 2.1.1.c

there is a unique b∈w(log(c) + B) such that t=w(u) exp(b). This deﬁnes a

coordinate system on w(uc) exp(B), and σǫ(t) = w(u) exp(ǫb). By Condition 2.1.1.d,

if either fn(t) = 0 or gn(t) = 0 for some t∈w(uc exp(B)) ⊂U, then fn(w(uc)) =

gn(w(uc)) = 0. One can easily check that in this situation

fn(t)

gn(t)= 1−e−α(b)ǫ

1−e−α(b)!(−1)n

Again by Condition 2.1.1.d the only critical points of this function are those for

which α(b) = 0. For ǫ6= 0 both the numerator and the denominator have a zero of

order 1 at such points, so the singularity is removable. For the case ǫ= 0 we need

to have a closer look. In our new coordinate system we can write

cα,ǫ(σǫ(t)) = f2(t)f4(t)

g1(t)g3(t)=

u(w−1α) + q(sα)−1/2q(tαsα)1/2e−α(b)ǫ

u(w−1α) + e−α(b)ǫ

u(w−1α)−q(sα)−1/2q(tαsα)−1/2e−α(b)ǫ

u(w−1α)−e−α(b)ǫ.

Standard calculations using L’Hospital’s rule show that

lim

ǫ→0cα,ǫ(σǫ(t)) = 









1 if u(w−1α)26= 1

2α(b) + log q(sα)−log q(tαsα)

2α(b)if u(w−1α) = −1

2α(b) + log q(sα) + log q(tαsα)

2α(b)if u(w−1α) = 1.

In other words,

lim

ǫ→0cα,ǫ(σǫ(t)) = (ku,α +α(b))/α(b) = ˜cα(b) if α(w−1u)2= 1.(4.4)

Thus at least dα,0= limǫ→0dα,ǫ exists as a meromorphic function on U. For

u(w−1α)26= 1, dα,0=c−1

αis invertible by Condition 2.1.1.d. For u(w−1α) = −1

we have

dα,0(t) = 1−e−α(b)

α(b)

α(b) + log q(sα)/q(tαsα)/2

1−q(sα)−1/2q(tαsα)1/2e−α(b)

1 + e−α(b)

1 + q(sα)−1/2q(tαsα)−1/2e−α(b)

while for u(w−1α) = 1

dα,0(t) = 1−e−α(b)

α(b)

1 + e−α(b)

1 + q(sα)−1/2q(tαsα)1/2e−α(b)

α(b) + log q(sα)q(tαsα)/2

1−q(sα)−1/2q(tαsα)−1/2e−α(b)

These expressions deﬁne invertible functions by Condition 2.1.1.c. We conclude that

dα,ǫ(t) and d−1

α,ǫ(t) are indeed analytic functions on U×[−1,1]. Since this domain is

compact, they are bounded. 2

We can use Lemma 4.1.1 to show that the maps ρǫpreserve analyticity:

Proposition 4.1.2. The maps (4.3) restrict to isomorphisms of topological algebras

ρǫ:Han

ǫ(Uǫ)∼

−−→ Han(U)

There is a well-deﬁned limit homomorphism

ρ0= lim

ǫ→0ρǫ:C[We]→ Han(U)

such that for every w∈Wethe map

[−1,1] → Han(U) : ǫ7→ ρǫ(Nw)

is analytic.

Proof. The ﬁrst statement is [Opd2, Theorem 5.3], but for the remainder we

need to prove this anyway. It is clear that ρǫrestricts to an isomorphism between

Can(Uǫ) and Can (U). For a simple reﬂection sα∈S0corresponding to α∈F0we

have

Ns+q(s)−ǫ/2=q(s)ǫ/2cα,ǫ(ı0

s,ǫ + 1)

ρǫ(Ns) = q(s)ǫ/2(cα,ǫ ◦σǫ)(ı0

s+ 1) −q(s)−ǫ/2

=q(s)(ǫ−1)/2(cα,ǫ ◦σǫ)c−1

αNs+q(s)−1/2−q(s)−ǫ/2

=q(s)(ǫ−1)/2dα,ǫNs+q(s)−1/2−q(s)−ǫ/2

(4.5)

By Lemma 4.1.1 such elements are analytic in ǫ∈[−1,1] and t∈U, so in particular

they belong to Han(U). Moreover, since every dα,ǫ is invertible and by (1.26), the

set {ρǫ(Nw) : w∈W0}is a basis for Han(U) as a Can(U)-module. Therefore ρǫ

restricts to an isomorphism between the topological algebras Han

ǫ(Uǫ) and Han(U)

for ǫ6= 0.

Given any w∈We, there is a unique x∈X+such that w∈W0xW0. By Lemma

1.2.2 there exist unique coeﬃcients cw,u,v(q)∈Csuch that

Nw=X

u∈Wx,v∈W0

cw,u,v (q)NuθxNv.

From (1.4) we see that in fact cw,u,v (q)∈Q{q(s)1/2:s∈Saﬀ }, so in particular

these coeﬃcients depend analytically on q. Moreover ρǫ(θx) = θx◦σǫdepends

analytically on ǫ, as a function on U, so

ρǫ(Nw) = X

u∈Wx,v∈W0

cw,u,v (qǫ)ρǫ(Nu)(θx◦ρǫ)ρǫ(Nv)

is analytic in ǫ∈[−1,1]. Thus ρ0exists as a linear map. But, being a limit of

algebra homomorphisms, it must also be multiplicative. 2

Suppose that u∈Tun is W0-invariant, so that

expu:W0(log(c) + B)→U=uW0cexp(B)

is a W0-equivariant bijection. Then clearly

σǫ◦expu= expu◦λǫ:ǫW0(log(c) + B)→Uǫ.(4.6)

Let Φube as in (2.5), with V=W0log(c) + B. It follows from (4.6), (4.1) and (4.3)

that

mǫ◦Φu,ǫ = Φu◦ρǫǫ∈[−1,1] \ {0}

as maps Hme

ǫ(Uǫ)→Cme(W0log(c) + B)W0⊗Z(H(˜

Ru,ku)) H(˜

Ru, ku). The maps Φu,ǫ

can also be deﬁned for ǫ→0, simply by

Φu,0(fNw) = (f◦expu)w∈Can(t)W0⊗Z(H(˜

Ru,ku)) H(˜

Ru, ku),

where f∈Can(T) and w∈W0. By (4.5) for α∈F0

mǫ◦Φu,ǫ(Nsα) = mǫ◦Φu,ǫq(sα)ǫ/2cα,ǫ(ı0

s,ǫ + 1) −q(sα)ǫ/2

=mǫq(sα)ǫ/2(cα,ǫ ◦expu)(˜ısα,ǫ + 1) −q(sα)ǫ/2

=q(sα)ǫ/2(cα,ǫ ◦expu◦λǫ)(˜ısα+ 1) −q(sα)ǫ/2

=q(sα)ǫ/2(cα,ǫ ◦expu◦λǫ)˜c−1

α(sα+ 1) −q(sα)ǫ/2

The W0-invariance of uimplies that α(u)2= 1, so by (4.4) lim

ǫ→0(cα,ǫ◦expu◦λǫ)˜c−1

α= 1.

Hence

lim

ǫ→0mǫ◦Φu,ǫ(Nsα) = sα=m0◦Φu,0(Nsα).

On the other hand, it is clear that for f∈C[X]∼

=O(T)

lim

ǫ→0mǫ◦Φu,ǫ(f) = lim

ǫ→0f◦expu◦λǫ=f(u) = m0◦Φu,0(f).

Since ρ0= limǫ→0ρǫexists, we can conclude that

m0◦Φu,0= Φu◦ρ0:C[X⋊W0]→H(˜

Ru, ku).(4.7)

4.2 Preserving unitarity

Proposition 4.1.2 shows that for ǫ∈[−1,1] \{0}there is an equivalence between the

categories Modf,U (H) and Modf ,Uǫ(Hǫ). It would be nice if this equivalence would

preserve unitarity, but that is not automatic. In fact these categories are not even

closed under taking duals of H-representations.

From (3.3) we see that an H-representation with central character W0tcan only

be unitary if t−1∈W0t, where t−1(x) = t(−x) for x∈X. To deﬁne a * on Han(U)

we must thus replace Uby

U±1:= U∪ {t−1:t∈U}.

Let ±W0be the group {±1} ×W0, which acts on Tby −w(t) = w(t)−1. The above

means that we need the following strengthening of Condition 2.1.1.e:

(e’) As Condition 2.1.1.e, but with ±W0⋊Γ instead of W′.

Lemma 4.1.1 and Proposition 4.1.2 remain valid under this condition, with the same

proof. Equations (3.3) and (1.20) show that the involution from Hextends naturally

to Hme(U±1)⋊Γ by

(Nγf)∗=Nw0(f◦ −w0)N−1

w0Nγ−1γ∈W′, f ∈Cme(U±1),(4.8)

where w0is the longest element of W0. According to [Opd2, (4.56)]

(ı0

w)∗=Nw0Y

α∈R+

0∩w′R−

cα

c−α

ı0

w′N−1

w0,(4.9)

where w∈W0and w′=w0w−1w0. We extend the map from Proposition 4.1.2 to

ρǫ:Han

ǫ(Uǫ)⋊Γ→ Han(U)⋊Γ by deﬁning ρǫ(Nγ) = Nγfor γ∈Γ. Usually the

maps ρǫdo not preserve the *, but this can be ﬁxed. For ǫ∈[−1,1] consider the

element

Mǫ=ρǫ(N−1

w0,ǫ)∗Nw0Yα∈R+

dα,ǫ ∈ Han(U)

We will use Mǫto extend [Opd2, Corollary 5.7]. However, this result contained a

small mistake: the element Aǫin [Opd2] is not entirely correct, we replace it by Mǫ.

Theorem 4.2.1. For all ǫ∈[−1,1] the element Mǫ∈ Han (U±1)is invertible,

positive and bounded. It has a positive square root M1/2

ǫand the map ǫ7→ M1/2

ǫis

analytic. The map

˜ρǫ:Han

ǫ(Uǫ)⋊Γ→ Han(U±1)⋊Γ, h 7→ M1/2

ǫρǫ(h)M−1/2

is a homomorphism of topological *-algebras, and an isomorphism if ǫ6= 0. For any

w∈We⋊Γthe map

[−1,1] → Han(U±1)⋊Γ : ǫ7→ ˜ρǫ(Nw)

is analytic.

Proof. By Lemma 4.1.1 and Proposition 4.1.2 the Mǫare invertible, bounded

and analytic in ǫ. Consider, for ǫ6= 0, the automorphism µǫof Hme(U±1) given by

µǫ(h) = ρǫ(ρ−1

ǫ(h)∗)∗.

We will discuss its eﬀect on three kinds of elements. Firstly, for f∈Cme(U±1) we

have, by (4.8) and the W0-equivariance of σǫ:

µǫ(f) = ρǫ((f◦σǫ)∗)∗

=ρǫNw0(f◦(−w0)◦σǫ)N−1

w0,ǫ∗

=ρǫ(N−1

w0,ǫ)∗f◦ −w0∗ρǫ(Nw0,ǫ )∗

=ρǫ(N−1

w0,ǫ)∗Nw0f N −1

w0ρǫ(Nw0,ǫ)∗

=ρǫ(N−1

w0,ǫ)∗Nw0Q

α∈R+

dα,ǫfQ

α∈R+

d−1

α,ǫN−1

w0ρǫ(Nw0,ǫ)∗

=MǫfM −1

ǫ.

(4.10)

Secondly, suppose that the simple reﬂections sand s′=w0sw0∈S0correspond to

αand put −w0α∈F0. Using Propostion 1.5.1.b for Hme

ǫ(U±1) and (4.9) we ﬁnd

Mǫı0

sM−1

ǫ=ρǫ(N−1

w0,ǫ)∗Nw0Q

α∈R+

dα,ǫı0

α∈R+

d−1

α,ǫN−1

w0ρǫ(Nw0,ǫ)∗

=ρǫ(N−1

w0,ǫ)∗Nw0ı0

sd−1

α,ǫd−α,ǫ N−1

w0ρǫ(Nw0,ǫ)∗

=ρǫ(N−1

w0,ǫ)∗Nw0ı0

c−α

cα

N−1

w0Nw0

cα,ǫ ◦σǫ

c−α,ǫ ◦σǫ

N−1

w0ρǫ(Nw0,ǫ)∗

=ρǫ(N−1

w0,ǫ)∗(ı0

s′)∗cα′,ǫ ◦σǫ

c−α′,ǫ ◦σǫ∗

ρǫ(Nw0,ǫ)∗

=ρǫ(Nw0,ǫ)cα′,ǫ ◦σǫ

c−α′,ǫ ◦σǫ

ı0

s′ρǫ(N−1

w0,ǫ)∗

=ρǫNw0

cα′

c−α′

ı0

s′,ǫN−1

w0,ǫ∗

=ρǫ(ı0

s,ǫ)∗∗=µǫ(ı0

s).

(4.11)

Thirdly, for γ∈Γ by deﬁnition

µǫ(Nγ) = ρǫ(ρ−1

ǫ(Nγ)∗)∗=ρǫ(N−1

γ)∗= (N−1

γ)∗=Nγ.

Since elements of the above three types generate Hme(U±1)⋊Γ, we conclude that

µǫ(h) = MǫhM−1

ǫfor all h∈ Hme(U±1)⋊Γ.

Now we can see that

ρǫN−1

w0,ǫ∗=ρǫ(N∗

w0,ǫ)−1∗=ρǫ(N−1

w0,ǫ)∗∗

=µǫρǫ(N−1

w0,ǫ),=Mǫρǫ(N−1

w0,ǫ)M−1

Ne=M−1

ǫρǫ(N−1

w0,ǫ)∗Nw0Q

α∈R+

dα,ǫ =ρǫ(N−1

w0,ǫ)M−1

ǫNw0Q

α∈R+

dα,ǫ,

Mǫ=Nw0Q

α∈R+

dα,ǫρǫ(N−1

w0,ǫ) = ρǫ(N−1

w0,ǫ)∗Nw0Q

α∈R+

dα,ǫ∗∗

=ρǫ(N−1

w0,ǫ)∗Nw0Qα∈R+

1dα,ǫ∗=M∗

Thus the elements Mǫare Hermitian ∀ǫ6= 0. By continuity in ǫ, M0is also Her-

mitian. Moreover they are all invertible, and M1=Ne, so they are in fact strictly

positive. We already knew that the element ǫ7→ Mǫof

Can([−1,1]; Han(U±1)) ∼

=Can([−1,1] ×U±1)⊗AH

is bounded, so we can construct its square root using holomorphic functional calculus

in the Fr´echet algebra Can

b([−1,1] ×U±1)⊗AH, where the subscript bdenotes

bounded functions. This ensures that ǫ→M1/2

ǫis still analytic. Finally, for ǫ6= 0

˜ρǫ(h)∗=M1/2

ǫρǫ(h)M−1/2

ǫ∗

=M−1/2

ǫρǫ(h)∗M1/2

=M−1/2

ǫµǫ(ρǫ(h∗))M1/2

=M1/2

ǫρǫ(h∗)M−1/2

ǫ= ˜ρǫ(h∗)

(4.12)

Again this extends to ǫ= 0 by continuity. 2

Corollary 4.2.2. For uc ∈Uand ǫ∈[−1,1] there is a family of additive functors

˜σǫ,uc : Modf ,W ′uc(H⋊Γ) →Modf, W ′σǫ(uc)(Hǫ⋊Γ),

(π, V )7→ (π◦˜ρǫ, V )

with the following properties:

(a)for all w∈We⋊Γand (π, V )the map [−1,1] →EndC(V) : ǫ7→ ˜σǫ,uc(π)(Nw)

is analytic;

(b)for ǫ6= 0 , σǫ,uc is an equivalence of categories;

(d)for ǫ < 0,˜σǫ,uc exchanges tempered and anti-tempered modules, where anti-

tempered means that |s|−1∈T−for all A-weights s∈T;

(e)for ǫ≥0,˜σǫ,uc preserves temperedness;

(f)for ǫ > 0,˜σǫ,uc preserves the discrete series.

Proof. Parts (a), (b) and (c) follow immediately from Theorem 4.2.1. Let (π, V )

be a ﬁnite dimensional Han (U±1)⋊Γ-representation. Conjugation by M1/2

ǫdoes

not change the isomorphism class of π, so ˜σǫ,uc(π) has the same Aǫ-weights as π◦ρǫ,

which by construction are σǫof the A-weights of π. Now parts (d), (e) and (f) are

obvious consequences of |σǫ(t)|=|t|ǫ.2

As the notation indicates, ˜σǫ,uc depends on the previously chosen base point uc.

For one t∈Tthere can exist several possible base points such that t∈U, and these

could in principle give rise to diﬀerent functors ˜σǫ,t. This ambiguity disappears if

we restrict to t=uc in Corollary 4.2.2. Then

˜σǫ:= Mt∈T/W0

˜σǫ,t : Modf(H⋊Γ) →Modf(Hǫ⋊Γ)

is an additive functor which also has the properties described in Corollary 4.2.2.

The functor ˜σǫwas already used in [OpSo1, Theorem 1.7].

The image of ˜σ0is contained in ModTu n (C[We]), so this map is certainly not

bijective, not even after passing to the associated Grothendieck groups of modules.

Nevertheless ˜σ0clearly is related to the map Spr from Section 2.3, in fact these maps

agree on irreducible tempered H⋊Γ-modules:

Lemma 4.2.3. Suppose that ǫ∈[−1,1], uc ∈TunTrs and t∈TP(u). Let Γ′

ube a

subgroup of W′

Fu,u and π∈Modf,(W(Ru)⋊Γ′

u)uc(H(˜

Ru, qu)⋊Γ′

u).

(a)The following Hǫ⋊Γ-representations are canonically isomorphic:

˜σǫIndH⋊Γ

H(Ru,qu)⋊Γ′

u(π◦φt),IndHǫ⋊Γ

H(Ru,qǫ

u)⋊Γ′

u˜σǫ(π)◦φt|t|ǫ−1),

IndH⋊Γ

H(Ru,qu)⋊Γ′

u(π◦φt)◦ρǫ,IndHǫ⋊Γ

H(Ru,qǫ

u)⋊Γ′

u(π◦φt)◦ρǫ.

(b) ˜σ0= Spr on Irr(S⋊Γ).

Proof. (a) By deﬁnition ˜σǫis given by composition with ˜ρǫ, and the diﬀerence

between ρǫand ˜ρǫis only an inner automorphism. Hence the map

˜σǫIndH⋊Γ

H(Ru,qu)⋊Γ′

u(π◦φt)→IndH⋊Γ

H(Ru,qu)⋊Γ′

u(π◦φt)◦ρǫ:v7→ M−1/2

ǫv

is an invertible intertwiner.

The two representations in the right hand column are naturally isomorphic if

and only if the H(Ru, qǫ

u)⋊Γ′

u-representations

˜σǫ(π)◦φt|t|ǫ−1and π◦φt◦ρǫ(4.13)

are so. Notice that φtand φt|t|ǫ−1are well-deﬁned because t∈TP(u)⊆TW(Ru). As

we just showed, the left hand side of (4.13) is naturally isomorphic to π◦ρǫ◦φt|t|ǫ−1.

Applying σǫonce with center uct and once with center uc results in

σǫ,uct(uct) = ut|t|−1cǫ|t|ǫ=σǫ,uc (uc)t|t|ǫ−1.

This implies that ρǫ,uc ◦φt|t|ǫ−1=φt◦ρǫ,uct, which shows that the representations

(4.13) can indeed by identiﬁed.

Now we turn to the most diﬃcult case, the two representations in the bottom

row. In view of Theorem 2.1.2.b

IndH⋊Γ

H(Ru,qu)⋊Γ′

u(π◦φt)◦ρǫand IndHǫ⋊Γ

H(Ru,qǫ

u)⋊Γ′

uπ◦φt◦ρǫ

are isomorphic if only if the

H(Ru, qǫ

u)an(Uǫ,u)⋊W′

Fu,u-representation IndH(Ru,qǫ

u)⋊W′

Fu,u

H(Ru,qǫ

u)⋊Γ′

uπ◦φt◦ρǫ

corresponds to the

1W′

uuc(H(R, qǫ)an(Uǫ)⋊Γ)1W′

uuc-representation 1W′

uucIndH⋊Γ

H(Ru,qu)⋊Γ′

u(π◦φt)◦ρǫ

via the isomorphism from Theorem 2.1.2.a. It is clear from the deﬁnition (4.3) that

IndH(Ru,qǫ

u)⋊W′

Fu,u

H(Ru,qǫ

u)⋊Γ′

u(π◦φt)◦ρǫ∼

=IndH(Ru,qǫ

u)⋊W′

Fu,u

H(Ru,qǫ

u)⋊Γ′

u(π◦φt)◦ρǫ,

so it suﬃces to show that the following diagram commutes:

H(Ru, qǫ

u)an(Uǫ,u )⋊W′

Fu,u →1W′

uuc(H(R, qǫ)an(Uǫ)⋊Γ)1W′

uuc

↓ρǫ↓ρǫ

H(Ru, qu)an(Uu)⋊W′

Fu,u →1W′

uuc(H(R, q)an(U)⋊Γ)1W′

uuc.

For elements of H(Ru, qǫ

u)an(Uǫ,u ) this is easy, since the eﬀect of the vertical arrows

is only extension of functions from Uǫ,u (resp. Uu) to Uǫ(resp. U) by 0. For elements

of W′

Fu,u the commutativity follows from (2.2) and (4.3).

(b) By Corollary 2.1.5.b every irreducible tempered H⋊Γ-representation πis of the

form IndH⋊Γ

H(Ru,qu)⋊W′

Fu,u (˜π◦Φu) for some ˜π∈Irr0H(Ru, qu)an(Uu)⋊W′

Fu,u. By

Theorem 2.3.1.d

Spr(π) = IndX⋊W′

X⋊W′

uCu⊗˜πW′

u.

Using (4.7) we can rewrite

Cu⊗˜πW′

u=Cu⊗(˜π◦m0)W′

u= ˜π◦m0◦Φu,0= ˜π◦Φu◦ρ0∼

=˜π◦Φu◦˜ρ0= ˜σ0(˜π◦Φu).

Now we can apply part (a):

Spr(π)∼

=IndX⋊W′

X⋊W′

u˜σ0(˜π◦Φu)∼

=˜σ0IndH⋊Γ

H(Ru,qu)⋊W′

Fu,u (˜π◦Φu)= ˜σ0(π).2

4.3 Scaling intertwining operators

We will show that the scaling maps ˜σǫgive rise to scaled versions of the intertwining

operators πΓ(g, ξ ). We will use this to study the behaviour of the components of the

Fourier transform of S⋊Γ under scaling of q.

As we remarked at the start of Section 4.1, the results of that section can easily

be extended to H⋊Γ, and we will use that generality here. Recall that the groupoid

Gfrom (3.13) includes Γ and is deﬁned independently of q. Let us realize the

representation

πΓ

ǫ(P, ˜σǫ(δ), t) on C[ΓWP]⊗Vδas IndHǫ⋊Γ

ǫ(δ◦˜ρǫ◦φt,ǫ).

For all ǫ∈[−1,1] we obtain algebra homomorphisms

Fǫ:H(R, qǫ)⋊Γ→MP,δ O(TP)⊗EndC(C[ΓWP]⊗Vδ),

Fǫ(h)(P, δ, t) = πΓ

ǫ(P, ˜σǫ(δ), t, h).

(4.14)

Rational intertwining operators πΓ

ǫ(g, P, δ′, t) can be deﬁned as in (3.12) for all Hǫ-

representations of the form πΓ

ǫ(P, δ′, t), where δ′is irreducible but not necessarily

discrete series. In particular, for ǫ6= 0 the (P, δ)-component of the image of Fǫis

invariant under an action of the group

GP,˜σǫ(δ):= {g∈ G :g(P) = P, ˜σǫ(δ)◦ψ−1

g∼

=˜σǫ(δ)}

via such intertwiners. As in (3.16), the action is not on polynomial but on rational

sections.

Proposition 4.3.1. Let ǫ∈[−1,1] \ {0}, let g∈ G with g(P)⊂F0and let δ′be a

discrete series representation of Hg(P)that is equivalent to δ◦ψ−1

(a)The Hg(P),ǫ-representations ˜σǫ(δ′)and ˜σǫ(δ)◦ψ−1

g,ǫ are unitarily equivalent.

(b)GP,˜σǫ(δ)=GP,δ and GP,˜σǫ(δ),t =GP,δ,t for all t∈TP.

(c)The intertwiners πΓ

ǫ(g, P, ˜σǫ(δ), t)∈HomCC[ΓWP]⊗Vδ,C[ΓWg(P)]⊗Vδ′de-

pend rationally on t∈TPand analytically on ǫ, whenever they are regular.

(d)For t∈TP

un the πΓ

ǫ(g, P, ˜σǫ(δ), t)are regular and unitary, and πΓ

0(g, P, ˜σ0(δ), t) :=

limǫ→0πΓ

ǫ(g, P, ˜σǫ(δ), t)exists.

Proof. (a) First we show that

ψg◦ρǫ=ρǫ◦ψg,ǫ.(4.15)

Write g=γ−1uwith u∈KPand γ∈W0⋊Γ. The automorphism ψufrom (1.19)

reﬂects the translation of TPby u∈TP

un: that changes Uto uU , but apart from

that it commutes with σǫ. Hence ψu◦ρǫ=ρǫ◦ψu,ǫ .

The isomorphism ψγ:HP→ Hγ(P)from (1.15) is more diﬃcult to deal with,

because it acts nontrivially on the Nwwith w∈WP. However, by Propostion 1.5.1.d

this is the restriction to HPof the automorphism

ψγ:h7→ ı0

γhı0

γ−1of the algebra C(T/W0)⊗Z(H)H⋊Γ.

Similarly ψγ,ǫ (h) = ı0

γ,ǫ hı0

γ−1,ǫ. From these formula it is clear that ψγ◦ρǫ=ρǫ◦ψγ,ǫ

on Hme

ǫ(Uǫ). This establishes (4.15).

Let Ig

δ:Vδ→Vδ′be as in (3.11). We claim that

v7→ Ig

δδ(ψ−1

g(M1/2

ǫ)M−1/2

ǫ)v(4.16)

is an intertwiner between ˜σǫ(δ)◦ψ−1

g,ǫ and ˜σǫ(δ′). Indeed, for v∈Vδand h∈ Han

ǫ(Uǫ):

δδ(ψ−1

g(M1/2

ǫ)M−1/2

ǫ)˜σǫ(δ)◦ψg,ǫ(h)v=

δδ(ψ−1

g(M1/2

ǫM−1/2

ǫ)δ(M1/2

ǫρǫ(ψ−1

g,ǫ h)M−1/2

ǫ)v=

δδ◦ψ−1

g(M1/2

ǫρǫ(h)M−1/2

ǫ)δ(ψ−1

g(M1/2

ǫ)M−1/2

ǫ)v=

δ′(M1/2

ǫρǫ(h)M−1/2

ǫ)Ig

δδ(ψ−1

g(M1/2

ǫ)M−1/2

ǫ)v=

˜σǫ(δ′)(h)Ig

δδ(ψ−1

g(M1/2

ǫ)M−1/2

ǫ)v.

Obviously (4.16) is invertible, so it is an equivalence between the irreducible rep-

resentations ˜σǫ(δ)◦ψ−1

g,ǫ and ˜σǫ(δ′). Since both are unitary, there exists a unitary

intertwiner between, which by the irreducibility must be a scalar multiple of (4.16).

We deﬁne Ig

δ,ǫ as the unique positive multiple of (4.16) that is unitary.

(b) By part (a)

g(P, ˜σǫ(δ), t) = (g(P),˜σǫ(δ)◦ψ−1

g,ǫ , g(t)) ∼

=(g(P),˜σǫ(δ′), g(t)) ∼

=(g(P),˜σǫ(δ◦ψ−1

g), g(t)),

so the stabilizer of (P, ˜σǫ(δ), t) does not depend on ǫ∈[−1,1].

δ,ǫ depends analytically on ǫ∈[−1,1]. By deﬁnition ı0

γis

rational in t∈Tand analytic in ǫ, away from the poles. By deﬁnition (3.12)

πΓ

ǫ(g, P, ˜σǫ(δ), t)(h⊗HP

ǫv) = hı0

γ⊗Hg(P)

ǫIg

δ,ǫ(v),

so πΓ

ǫ(g, P, ˜σǫ(δ), t) has the required properties.

(d) All possible singularities of the intertwining operators come from poles and

zero of the c-functions from (1.23). By Theorem 3.1.5.c πΓ

ǫ(g, P, ˜σǫ(δ), t) is unitary

for all t∈TP

un and ǫ∈(0,1]. In particular all the singularities on this domain are

removable. On the other hand, the explicit formula for cαshows that the singularities

for qǫ(ǫ6= 0) and t∈Tun are the same as those for qand t∈Tun. Therefore

πΓ

ǫ(g, P, ˜σǫ(δ), t) is also regular for ǫ∈[−1,0) and t∈TP

un.

Since these linear maps are analytic in ǫ∈[−1,1]\{0}and unitary for ǫ > 0, they

are also unitary for ǫ < 0. Hence all the matrix coeﬃcients of πΓ

ǫ(g, P, ˜σǫ(δ), t) are

uniformly bounded on TP

un×[−1,1]\{0}, which implies that every possible singularity

at ǫ= 0 is removable. In particular limǫ→0πΓ

ǫ(g, P, ˜σǫ(δ), t) exists. 2

We ﬁx a discrete series representation δof HPand we abbreviate

AP,δ := C∞(TP

un)⊗C(C[ΓWP]⊗Vδ).

Proposition 4.3.1 says among others that for g∈ GP,δ

[−1,1] →A×

P,δ :ǫ7→ πΓ

ǫ(g, P, ˜σǫ(δ),·)

is an analytic map. The group GP,˜σǫ(δ)=GP,δ acts on AP,δ by

(g·ǫf)(t) = πΓ

ǫ(g, P, ˜σǫ(δ), g−1t)f(g−1t)πΓ

ǫ(g, P, ˜σǫ(δ), g−1t)−1.(4.17)

By construction the δ-component of the image of Fǫconsists of GP, ˜σǫ(δ)-invariant

sections for ǫ6= 0, and by Proposition 4.3.1 this also goes for ǫ= 0. We intend

to show that the algebras AGP, ˜σǫ(δ)

P,δ for ǫ∈[−1,1] are all isomorphic. (Although

GP,˜σǫ(δ)=GP,δ we prefer the longer notation here, because it indicates which action

on AP,δ we consider.) We must be careful when taking invariants, because

GP,δ →A×

P,δ :g7→ πΓ

ǫ(g, P, ˜σǫ(δ),·) (4.18)

is not necessarily a group homomorphism. However, the lack of multiplicativity is

small, it is only caused by the freedom in the choice of a scalar in (3.11). In other

words, (4.18) deﬁnes a projective representation of GP,δ on AP,δ . Recall [CuRe,

Section 53] that the Schur multiplier G∗

P,δ is a ﬁnite central extension of GP,δ , with

the property that every projective representation of GP,δ lifts to a unique linear

representation of G∗

P,δ . This means that for every lift g∗∈ G∗

P,δ of g∈ GP,δ there is

a unique scalar multiple πΓ

ǫ(g∗, P, ˜σǫ(δ),·) of πΓ

ǫ(g, P, ˜σǫ(δ),·) such that

G∗

P,δ →A×

P,δ :g∗7→ πΓ

ǫ(g∗, P, ˜σǫ(δ),·)

becomes multiplicative. Since πΓ

ǫ(g, P, ˜σǫ(δ),·) is unitary, so is πΓ

ǫ(g∗, P, ˜σǫ(δ),·).

Notice that GP,δ and G∗

P,δ ﬁx the same elements of AP,δ, because the action (4.17) is

deﬁned via conjugation with πΓ

ǫ(g, P, ˜σǫ(δ),·). According to [CuRe, Section 53] the

way lift (4.18) from GP,δ to G∗

P,δ is completely determined by the cohomology class

of the 2-cocycle

GP,δ × GP,δ →C×: (g1, g2)7→ Ig1

δ,ǫIg2

δ,ǫIg−1

2g−1

δ,ǫ .(4.19)

This cocycle depends analytically on ǫand GP,δ is a ﬁnite group, so the class of

(4.19) in H2(GP,δ ,C) does not depend on ǫ. (In most cases this cohomology class

is trivial, but examples are known in which it is nontrivial, see [DeOp2, Section

6.2].) Hence the ratio between πΓ

ǫ(g∗, P, ˜σǫ(δ),·) and πΓ

ǫ(g, P, ˜σǫ(δ),·) also depends

analytically on ǫ.

For g∗∈ G∗

P,δ we deﬁne λg∗:TP→TPby λg∗(t) = g(t). In the remainder of this

section we will work mostly with G∗

P,δ , and to simplify the notation we will denote

its typical elements by ginstead of g∗. For g∈ G∗

P,δ and t∈TP

un we write

ug,ǫ(t) := πΓ

ǫ(g, λ−1

g(t)),

so that the multiplicativity translates into

ugg′,ǫ =ug,ǫ(ug′,ǫ ◦λ−1

g).(4.20)

From the above we know that ug,ǫ ∈AP,δ is unitary and analytic in ǫ. These

elements can be used to identify AGP, ˜σǫ(δ)

P,δ with a corner in a larger algebra. Consider

the crossed product AP,δ ⋊λG∗

P,δ , where the action of G∗

P,δ on AP,δ comes only from

the action on C(TP

un) induced by the λg. In particular this action is independent of

ǫ. On AP,δ ⋊λG∗

P,δ we can deﬁne actions of G∗

P,δ by

g·ǫa=ug,ǫgag−1u−1

g,ǫ .(4.21)

For a∈AP,δ this recovers the action (4.17). An advantage of introducing the Schur

multiplier is that, by (4.20), g7→ ug,ǫgis a homomorphism from G∗

P,δ to the unitary

group of AP,δ ⋊λG∗

P,δ . Hence

[−1,1] →AP,δ ⋊λG∗

P,δ :ǫ7→ pδ,ǫ := |G∗

P,δ |−1Xg∈G∗

P,δ

ug,ǫg(4.22)

is a family of projections, depending analytically on ǫ. This was ﬁrst observed in

[Ros] and worked out in [Sol3, Lemma A.2],

AGP,˜σǫ(δ)

P,δ →pδ,ǫ(AP,δ ⋊λG∗

P,δ )pδ,ǫ :a7→ pδ,ǫapδ,ǫ (4.23)

is an isomorphism of Fr´echet *-algebras. Its inverse is the map

Xg∈G∗

P,δ

agg7→ |G∗

P,δ |ae.

Lemma 4.3.2. The Fr´echet *-algebras AGP,˜σǫ(δ)

P,δ ∼

=pδ,ǫ(AP,δ ⋊λG∗

P,δ )pδ,ǫ are all iso-

morphic, by C∞(TP

un)GP ,δ -linear isomorphisms that are piecewise analytic in ǫ.

Proof. According to [Phi, Lemma 1.15] the projections pδ(uǫ) are all conjugate,

by elements depending continuously on ǫ. That already proves that all the Fr´echet

algebras AGP,˜σǫ(δ)

P,δ are isomorphic. Since C∞(TP

un)GP ,δ is the center of both AGP, ˜σǫ(δ)

P,δ

and pδ,ǫ(AP,δ ⋊λG∗

P,δ )pδ,ǫ, these isomorphisms are C∞(TP

un)GP ,δ -linear.

To show that the isomorphisms can be made piecewise analytic we construct

the conjugating elements explicitly, using the recipe of [Bla, Proposition 4.32]. For

ǫ, ǫ′∈[−1,1] consider

z(δ, ǫ, ǫ′) := (2pδ,ǫ′)−1)(2pδ,ǫ)−1) + 1 ∈AP,δ ⋊λG∗

P,δ .

Clearly this is analytic in ǫand ǫ′and

pδ,ǫ′z(δ, ǫ, ǫ′) = 2pδ,ǫ′pδ,ǫ =z(δ, ǫ, ǫ′)pδ,ǫ.

The enveloping C∗-algebra of AP,δ ⋊λG∗

P,δ is

Cδ:= EndCC[ΓWP]⊗Vδ⊗C(TP

un)⋊λG∗

P,δ .

Let k·k be its norm and suppose that kpδ(uǫ)−pδ(u′

ǫ)k<1. Then

kz(δ, ǫ, ǫ′)−2k=

4pδ,ǫ′pδ,ǫ −2pδ,ǫ −2pδ,ǫ′



=

−2(pδ,ǫ −pδ,ǫ′)2



≤2

pδ,ǫ −pδ,ǫ′

2<2

so z(δ, ǫ, ǫ′) is invertible in Cδ. But AP,δ ⋊λG∗

P,δ is closed under the holomorphic

functional calculus of Cδ, so z(δ, ǫ, ǫ′) is also invertible in this Fr´echet algebra. More-

over, because the Fr´echet topology on AP,δ ⋊λG∗

P,δ is ﬁner than the induced topology

from Cδ, there exists an open interval Iǫcontaining ǫsuch that z(δ, ǫ, ǫ′) is invertible

for all ǫ′∈Iǫ. For such ǫ, ǫ′we construct the unitary element

u(δ, ǫ, ǫ′) := z(δ, ǫ, ǫ′)|z(δ, ǫ, ǫ′)|−1∈AP,δ ⋊λG∗

P,δ .

By construction the map

pδ,ǫ(AP,δ ⋊λG∗

P,δ )pδ,ǫ →pδ,ǫ′(AP,δ ⋊λG∗

P,δ )pδ,ǫ′:x7→ u(δ, ǫ, ǫ′)xu(δ, ǫ, ǫ′)−1(4.24)

is an isomorphism of Fr´echet *-algebras. Notice that (4.24) also deﬁnes an isomor-

phism between the respective C∗-completions.

Let us pick a ﬁnite cover {Iǫi}m

i=1 of [−1,1]. Then for every ǫ, ǫ′∈[−1,1] an iso-

morphism between pδ,ǫ(AP,δ ⋊λG∗

P,δ )pδ,ǫ and pδ,ǫ′(AP,δ ⋊λG∗

P,δ )pδ,ǫ′can be obtained

by composing at most misomorphisms of the form (4.24). 2

4.4 Scaling Schwartz algebras

It follows from Corollary 4.2.2 that, for ǫ∈(0,1], the functor ˜σǫis an equivalence

between the categories of ﬁnite dimensional tempered modules of Hand of Hǫ=

H(R, qǫ). We will combine this with the explicit description of the Fourier transform

of S(R, q) from Theorem 3.2.2 to construct ”scaling isomorphisms”

ζǫ:S(R, qǫ)⋊Γ→ S(R, q)⋊Γ.

These algebra homomorphisms depend continuously on ǫand they turn out to have

a well-deﬁned limit

ζ0:S(We)⋊Γ = S(R, q0)⋊Γ→ S(R, q)⋊Γ.

Although ζ0is no longer surjective, it provides the connection between the repre-

sentation theories of S(R, q) and S(We) that we already discussed in Section 2.3.

Recall from (3.25) that ∆ is a set of representatives for G-association classes of

discrete series representations of parabolic subalgebras of H, and that

F:S(R, q)⋊Γ→M(P,δ)∈∆C∞(TP

un)⊗EndC(C[ΓWP]⊗Vδ)GP ,δ (4.25)

is an isomorphism of Fr´echet *-algebras. Together with (4.14) and Lemma 4.3.2,

this implies the existence of a continuous family of algebra homomorphisms, with

some nice properties:

Proposition 4.4.1. There exists a collection of injective *-homomorphisms

ζǫ:H(R, qǫ)⋊Γ→ S(R, q)⋊Γǫ∈[−1,1],

such that:

(a)For ǫ < 0(respectively ǫ > 0) the map π7→ π◦ζǫis an equivalence between

Modf(S(R, q)⋊Γ) and the category of ﬁnite dimensional anti-tempered (respec-

tively tempered) H(R, qǫ)⋊Γ-modules.

(b)ζ1:H⋊Γ→ S ⋊Γis the canonical embedding.

(c)ζǫ(Nw) = Nwfor all w∈Z(We).

(d)For all w∈Wethe map

[−1,1] → S(R, q )⋊Γ : ǫ7→ ζǫ(Nw)

is piecewise analytic, and in particular analytic at 0.

(e)For all π∈Modf(S(R, q)⋊Γ) the representations π◦ζǫand ˜σǫ(π)are equivalent.

In particular πΓ(P, δ, t)◦ζǫ∼

=πΓ

ǫ(P, ˜σǫ(δ), t)for all (P, δ, t)∈Ξun.

(f)ζǫpreserves the trace τ.

Proof. Let ζP,δ,ǫ :AGP,˜σǫ(δ)

P,δ →AGP,δ

P,δ be the isomorphism from Lemma 4.3.2. We

already observed that δ-component of the image of Fǫis invariant under GP,˜σǫ(δ), so

we can deﬁne

ζǫ:= F−1◦M(P,δ)∈∆ζP,δ,ǫ◦ Fǫ.

Now (b) holds by construction and (c) follows from the C∞(TP

un)GP ,δ -linearity in

Lemma 4.3.2. For (d) we use Theorem 4.2.1 and Lemma 4.3.2. From the last lines

of the proof of Lemma 4.3.2 we see how we can arrange that ζǫis analytic at 0: it

suﬃces to take ǫi= 0 and to use the elements u(δ, 0, ǫ′) for ǫ′in a neighborhood of

ǫ= 0.

Any ﬁnite dimensional module decomposes canonically as a direct sum of parts

corresponding to diﬀerent central characters. Hence in (e) it suﬃces to consider

(π, V )∈Modf,G(P,δ,t)(S(R, q)⋊Γ).That is, π:S(R, q)⋊Γ→EndC(V) factors via

evP,δ,t :S(R, q)⋊Γ→EndC(C[ΓWP]⊗Vδ), h 7→ πΓ(P, δ, t, h).

As ζǫ,P,δ is C∞(TP

un)GP ,δ -linear, π◦ζǫ:S(R, qǫ)⋊Γ→EndC(V) factors via

evP, ˜σǫ(δ),t :S(R, qǫ)⋊Γ→EndC(C[ΓWP]⊗Vδ).

Moreover, by Lemma 4.3.2 the ﬁnite dimensional C∗-algebras evP,δ,t(S(R, q)⋊Γ)

and evP, ˜σǫ(δ),t (S(R, qǫ)⋊Γ) are isomorphic, by isomorphisms depending continuously

on ǫ∈[−1,1]. We have two families of representations on V, π ◦ζǫand ˜σǫ(π), which

agree at ǫ= 1 and all whose matrix coeﬃcients are continuous in ǫ. Since a ﬁnite

dimensional semisimple algebra has only ﬁnitely many equivalence classes of repre-

sentations on V, such equivalence classes are rigid under continuous deformations.

Therefore π◦ζǫ∼

=˜σǫ(π) for all ǫ∈[−1,1]. Now Lemma 4.2.3 (or a simpler version

of the above argument) shows that πΓ(P, δ, t)◦ζǫ∼

=πΓ

ǫ(P, ˜σǫ(δ), t), concluding the

proof of (e).

Property (a) is a consequence of (e), Corollary 4.2.2 and Lemma 3.1.1.b. Ac-

cording to [Opd2, Lemma 5.5] the scaling maps σǫpreserve the Plancherel measure

µP l for ǫ > 0. By part (e) and Theorem 4.2.1, so do the maps π7→ π◦ζǫ. Now the

last part of Theorem 3.2.2 shows that

τ(ζǫ(h)) = τ(h) for all ǫ∈(0,1], h ∈ Hǫ⋊Γ.(4.26)

By part (d) there is a small ǫ′>0 such that (4.26) also holds for all ǫ∈[−ǫ′,0].

Moreover, from the end of the proof of Lemma 4.3.2 we see that we can make

the maps from (d) analytic at any given ǫ∈[−1,1]. Of course this involves some

choices, but they do not inﬂuence τbecause they all come from conjugation in

certain algebras. Therefore (4.26) extends to all ǫ∈[−1,1].

As concerns the injectivity of ζǫ, suppose that h∈ker(ζǫ). Then M(t, h) = 0

for all unitary principal series representations M(t). Since Tun is Zariski-dense in T,

Lemma 3.1.2 for Hǫ⋊Γ shows that h= 0.2

Notice that for ǫ < 0 composition with ζǫdoes not preserve the local (in the

dual space) traces µP l(ξ)tr πΓ(ξ), because by Theorem 3.2.2 and Proposition 4.4.1

˜σǫdoes not preserve the support of the Plancherel measure.

In general ζǫ(Hǫ⋊Γ) is not contained in H⋊Γ, for two reasons: ζδ,ǫ usually does

not preserve polynomiality, and not every polynomial section is in the image of F.

For ǫ≥0 the ζǫextend continuously to S(R, qǫ)⋊Γ:

Theorem 4.4.2. For ǫ∈[0,1] there exist homomorphisms of Fr´echet *-algebras

ζǫ:S(R, qǫ)⋊Γ→ S(R, q)⋊Γ

ζǫ:C∗(R, qǫ)⋊Γ→C∗(R, q)⋊Γ

with the following properties:

(a)ζǫis an isomorphism for ǫ > 0, and ζ0is injective.

(b)ζ1is the identity.

(c)ζǫ(h) = hfor all h∈ S(Z(W)).

(d)Let x∈C∗(We⋊Γ) and let h=Pw∈We⋊ΓhwNwwith pn(h)<∞for all n∈N.

Then the following maps are continuous:

[0,1] → S(R, q )⋊Γ, ǫ 7→ ζǫ(h),

[0,1] →B(L2(R, q)), ǫ 7→ ζ−1

ǫζ0(x).

(e)For all π∈Modf(S(R, q)⋊Γ) the representations π◦ζǫand ˜σǫ(π)are equivalent.

(f)ζǫpreserves the trace τ.

Proof. For any (P, δ)∈∆ the representation ˜σǫ(δ), although not necessarily

irreducible if ǫ= 0, is certainly completely reducible, being unitary. Hence by

Theorem 3.2.2 every irreducible constituent π1of ˜σǫ(δ) is a direct summand of

IndHǫ,P

HP1

ǫ,P

(δ1◦φt1,ǫ)

for a P1⊂P, a discrete series representation δ1of H(RP1, qǫ) and a

t1∈HomZ(XP)P1, S1= HomZX/(X∩(P∨)⊥+QP1), S1⊂Tu

Consequently, πǫ(P, π1, t) is a direct summand of

IndHǫ

ǫIndHǫ,P

HP1

ǫ,P

(δ1◦φt1,ǫ)◦φt,ǫ=πǫ(P1, δ1, tt1)

In particular for t∈TP

un every matrix coeﬃcient of πΓ

ǫ(P, ˜σǫ(δ), t) appears in the

Fourier transform of S(R, qǫ)⋊Γ, so (4.14) extends to the respective Schwartz

and C∗-completions, as required. By Corollary 4.2.2.b and (4.25) these maps are

isomorphisms if ǫ > 0. Since (4.24) extends continuously to the appropriate C∗-

completions, so does the algebra homomorphism ζǫfrom Proposition 4.4.1.

Properties (b), (c), (e) and (f) are direct consequences of the corresponding parts

of Proposition 4.4.1.

To see that ζ0remains injective we vary on the proof of Proposition 4.4.1. By

(e) the family of representations

It◦ζǫ∼

=πΓ

ǫ(∅,˜σǫ(δ∅), t) = πΓ

ǫ(∅, δ∅, t)

with t∈Tun becomes precisely the unitary principal series of We⋊Γ when ǫ→0. By

Lemma 2.2.2 and Frobenius reciprocity every irreducible tempered representation of

H(R, q0)⋊Γ = C[We⋊Γ] is a quotient of a unitary principal series representation.

Hence every element of C∗(R, q0)⋊Γ = C∗(We⋊Γ) that lies in the kernel of ζ0

annihilates all irreducible tempered We⋊Γ-representations, and must be 0.

The assumptions in (d) mean that we can consider has an element of S(R, qǫ)⋊

Γ for every ǫ. Moreover the sum Pw∈We⋊ΓhwNwconverges uniformly to hin

S(R, qǫ)⋊Γ. For every ﬁnite partial sum h′the map ǫ7→ φǫ(h′) is continuous by

Proposition 4.4.1.e, so this also holds for hitself.

For ǫ∈(0,1] we consider

ζ−1

ǫζ0(x)−x=F−1

ǫM

P,δ∈∆

ζ−1

P,δ,ǫζP,δ,0(F0(x)) − Fǫ(x).(4.27)

Since ζP,δ,0is invertible, both ζ−1

P,δ,ǫζP,δ,0(F0(x)) and Fǫ(x) are continuous in ǫand

converge to F0(x) as ǫ↓0. The continuity of Fǫwith respect to ǫimplies that the

F−1

ǫare also continuous with respect to ǫ, so ǫ7→ ζ−1

ǫζ0(x) is continuous.

The expression between the large brackets in (4.27) also depends continuously

on ǫand converges to 0 as ǫ↓0. Furthermore

F−1

ǫ:M(P,δ)∈∆C(TP

un)⊗EndC(C[ΓWP]⊗Vδ)GP ,˜σǫ(δ)→B(L2(R, q))

is a homomorphism of C∗-algebras, so it cannot increase the norms. Therefore

lim

ǫ↓0(ζ−1

ǫζ0(x)−x) = 0 in B(L2(R, q)).2

The homomorphisms from Theorem 4.4.2 are by no means natural, the construc-

tion involves a lot of arbitrary choices. Nevertheless one can prove [Sol3, Lemma

5.22] that it is possible to interpolate continuously between two such homomor-

phisms, obtained from diﬀerent choices. In other words, the homotopy class of

ζǫ:S(R, qǫ)⋊Γ→ S(R, q)⋊Γ is canonical.

It is quite remarkable that ζ0preserves the trace τ, because the measure space

(Ξun, µP l ) diﬀers substantially from Tun with the Haar measure (which corresponds

to the algebra S(X)⋊W′). For example, the former is disconnected and can have

isolated points, while the latter is a connected manifold. So the preservation of the

trace already indicates that it is not possible to separate all components of Ξun/G

using only elements from the image of ζ0.

Corollary 4.4.3. For π∈Irr(S(R, q)⋊Γ) we have Spr(π)∼

=˜σ0(π)∼

=π◦ζ0, where

Spr is as in Section 2.3. The map ζ0induces a linear bijection

GQ(ζ0) : GQ(S(R, q)⋊Γ) →GQ(S(We)⋊Γ).

Proof. The ﬁrst claim follows from Theorem 4.4.2.e and Lemma 4.2.3.b. The

second statement follows from the ﬁrst and Theorem 2.3.1.a 2

Chapter 5

Noncommutative geometry

Aﬃne Hecke algebras have some clear connections with noncommutative geometry.

Already classical is the isomorphism between an aﬃne Hecke algebra (with one

formal parameter q) and the equivariant K-theory of a Steinberg variety, see [Lus2,

KaLu2, ChGi]. Of a more analytic nature is K-theory of the C∗-completion C∗(R, q)

of H(R, q). It is relevant for the representation theory of aﬃne Hecke algebras

because topological K-theory is built from ﬁnitely generated projective modules.

Since K-theory tends to be invariant under small perturbations, it is expected [Ply1,

BCH] that K∗(C∗(R, q)) does not depend on q. We prove this modulo torsion

(Theorem 5.1.4).

For the algebra H(R, q) periodic cyclic homology is more suitable than K-theory.

Although periodic cyclic homology is not obviously related to representation theory,

there is a link for certain classes of algebras [Sol6]. From [BaNi] it is known that

HP∗(H(R, q)) ∼

=HP∗(C[We]) when qis an equal parameter function, but the proof

is by no means straightforward.

We connect these two theories via the Schwartz completion of H(R, q). For

this algebra both topological K-theory and periodic cyclic homology are meaning-

ful invariants. Notwithstanding the diﬀerent nature of the algebras H(R, q) and

S(R, q), they have the same periodic cyclic homology (Theorem 5.2.1). We deduce

the existence of natural isomorphisms

HP∗(H(R, q)) ∼

=HP∗(S(R, q)) ∼

=K∗(S(R, q)) ⊗ZC∼

=K∗(C∗(R, q)) ⊗ZC.

Moreover the scaling maps from Chapter 4 provide isomorphisms from these vector

spaces to the corresponding invariants of group algebras of We(Corollary 5.2.2).

Notice the similarity with the ideas of [BHP, Sol4].

Our method of proof actually shows that S(R, q) and S(We) are geometrically

equivalent (Lemma 5.3.1), a term coined by Aubert, Baum and Plymen [ABP1] to

formulate a conjecture for Hecke algebras of p-adic groups. This conjecture (which

we call the ABP-conjecture) describes the structure of Bernstein components in the

smooth dual of a reductive p-adic group. Translated to aﬃne Hecke algebras this

conjectures says among others that the dual of Hcan be parametrized with the ex-

tended quotient e

T /W0. The topological space Irr(H(R, q)), with its central character

map to T/W0, should then by obtained from e

T /W0, with its canonical projection

onto T/W0, via translating the components of e

T /W0in algebraic dependence on q.

We verify the ABP-conjecture for all aﬃne Hecke algebras with positive parameters,

possibly extended with a group of diagram automorphisms (Theorem 5.4.2). Hence

the ABP-conjecture holds for all Bernstein components of p-adic groups, whose

Hecke algebras are known to be Morita equivalent to aﬃne Hecke algebras.

In the ﬁnal section we calculate in detail what happens for root data with R0of

type B2/C2. Interestingly, this also shows that the representation theory of H(R, q)

seems to behave very well under general deformations of the parameter function q.

5.1 Topological K-theory

By means of the canonical basis {Nw:w∈We⋊Γ}we can identify the topological

vector spaces underlying the algebras S(R, q)⋊Γ for all positive parameter functions

q. It is clear from the rules (1.4) that the multiplication in aﬃne Hecke algebras

depends continuously on q, in some sense. To make that precise, one endows ﬁnite

dimensional subspaces of H(R, q)⋊Γ with their standard topology, and one deﬁnes a

topology on the space of positive parameter functions by identifying them with tupels

of positive real numbers. This can be extended to the Schwartz completions: by

[OpSo2, Lemma A.8] the multiplication in S(R, q)⋊Γ is continuous in q, with respect

to the Fr´echet topology on S(R, q)⋊Γ. This opens the possibility to investigate

this ﬁeld of Fr´echet algebras with analytic techniques that relate the algebras for

diﬀerent q’s, a strategy that was used at some crucial points in [OpSo2].

We denote the topological K-theory of a Fr´echet algebra Aby K∗(A) = K0(A)⊕

K1(A). Since S(R, q)⋊Γ is closed under the holomorphic functional calculus of

C∗(R, q)⋊Γ [OpSo2, Theorem A.7], the density theorem [Bos, Th´eor`eme A.2.1]

tells us that the inclusion S(R, q)⋊Γ→C∗(R, q )⋊Γ induces an isomorphism

K∗(S(R, q)⋊Γ) ∼

=K∗(C∗(R, q)⋊Γ).(5.1)

K-theory for C∗-algebras is homotopy-invariant, so it is natural to expect the fol-

lowing:

Conjecture 5.1.1. For any positive parameter function qthe abelian groups

K∗(C∗(We)⋊Γ) and K∗(C∗(R, q)⋊Γ) are naturally isomorphic.

This conjecture stems from Higson and Plymen (see [Ply1, 6.4] and [BCH, 6.21]),

at least when Γ = {id}and qis constant on Saﬀ. It is similar to the Connes–

Kasparov conjecture for Lie groups, see [BCH, Sections 4–6] for more background.

Independently Opdam [Opd2, Section 1.0.1] formulated Conjecture 5.1.1 for unequal

parameters. We will discuss its relevance for the representation theory of aﬃne

Hecke algebras, and we will prove a slightly weaker version, obtained by applying

the functor ⊗ZQ.

Recall [Phi] that for any unital Fr´echet algebra A, K0(A) (respectively K1(A))

is generated by idempotents (respectively invertible elements) in matrix algebras

Mn(A). The K-groups are obtained by taking equivalence classes with respect to

the relation generated by stabilization and homotopy equivalence.

Lemma 5.1.2. Suppose that

κǫ:K∗(C∗(We)⋊Γ) →K∗(C∗(R, q)⋊Γ) ǫ∈[0,1]

is a family of group homomorphisms with the following property.

For every idempotent e0∈Mn(S(R, q0)⋊Γ) = Mn(S(We)⋊Γ) (resp. invertible

element x0∈Mn(S(R, q0)⋊Γ)) there exists a C∗-norm-continuous path ǫ7→ eǫ

(resp. ǫ7→ xǫ) in the Fr´echet space underlying Mn(S(R, q)⋊Γ), such that κǫ[e0] =

[eǫ](resp. κǫ[x0] = [xǫ]).

Then κǫ=K∗(ζ−1

ǫζ0), with ζǫ:C∗(R, qǫ)⋊Γ→C∗(R, q)⋊Γas in Theorem 4.4.2.

By ”C∗-norm-continuous” we mean that the path in B(L2(R, q)) deﬁned by

mapping ǫto the operator of left multiplication by eǫ(with respect to qǫ), is contin-

uous. It follows from [OpSo2, Proposition A.5] that every Fr´echet-continuous path

is C∗-norm-continuous. By Theorem 4.4.2.d the maps K∗(ζ−1

ǫζ0) have the property

that the κǫare supposed to possess, so at least the statement is meaningful.

Proof. By deﬁnition K∗(ζ−1

ǫζ0)[e0] = [ζ−1

ǫζ0(e0)], where we extend the ζǫto

matrix algebras over C∗(R, qǫ) in the obvious way. The paths ǫ→eǫand ǫ→

ζ−1

ǫζ0(e0) are both C∗-norm-continuous, so we can ﬁnd ǫ′>0 such that



ζ−1

ǫζ0(e0)−eǫ

<k2e0−1k−1=

2ζ−1

ǫζ0(e0)−1

−1for all ǫ≤ǫ′.

Then by [Bla, Proposition 4.3.2] eǫand ζ−1

ǫζ0(e0) are connected by a path of idem-

potents in Mn(C∗(R, qǫ)⋊Γ), so

κǫ[e0] = [eǫ] = [ζ−1

ǫζ0(e0)] = K∗(ζ−1

ǫζ0)[e0] for all ǫ≤ǫ′.

For ǫ≥ǫ′

K∗(ζ−1

ǫζ0)[e0] = K∗(ζ−1

ǫζǫ′)K∗(ζ−1

ǫ′ζ0)[e0] = K∗(ζ−1

ǫζǫ′)[eǫ′].

By parts (a) and (d) of Theorem 4.4.2, K∗(ζ−1

ǫζǫ′) (ǫ≥ǫ′) is the only family of

maps K0(C∗(R, qǫ′)) ⋊Γ) →K0(C∗(R, q)⋊Γ) that comes from continuous paths of

idempotents.

Now K1. Choose ǫ′>0 such that



ζ−1

ǫζ0(x0)x−1

ǫ−1

<1 for all ǫ≤ǫ′.

Then ζ−1

ǫζ0(x0)x−1

ǫis homotopic to 1 in GLn(C(R, qǫ)⋊Γ), so

K1(ζ−1

ǫζ0)[x0] = [ζ−1

ǫζ0(x0)] = [xǫ] = κǫ[x0] for all ǫ≤ǫ′.

The argument for ǫ≥ǫ′is just as for K0.2

This lemma says that the map

K∗(ζ0) : K∗(C∗(We)⋊Γ) →K∗(C∗(R, q)⋊Γ

is natural: it does not really depend on ζ0, only the topological properties of idem-

potents and invertible elements in matrix algebras over S(R, qǫ)⋊Γ with ǫ∈[0,1].

To prove that K∗(ζ0) becomes an isomorphism after tensoring with Q, we need some

preparations of a more general nature.

For topological spaces Y⊂Xand a topological algebra Awe write

C0(X, Y ;A) = {f:X→A|fis continuous and fY= 0}.

We omit Y(resp. A) from the notation if Y=∅(resp. A=C). A C(X)-algebra

Bis a C-algebra endowed with a unital algebra homomorphism from C(X) to the

center of the multiplier algebra of B. A morphism of C(X)-algebras is an algebra

homomorphism that is also a C(X)-module map.

Lemma 5.1.3. Let Σbe a ﬁnite simplicial complex, let Aand Bbe C(Σ)-Banach-

algebras and let φ:A→Ba morphism of C(Σ)-Banach-algebras. Suppose that

(a)for every simplex σof Σthere are ﬁnite dimensional C-algebras Aσand Bσsuch

that

C0(σ, δσ)A∼

=C0(σ, δσ;Aσ) and C0(σ, δσ)B∼

=C0(σ, δσ;Bσ);

(b)for every x∈σ\δσ the localized map φ(x) : Aσ→Bσinduces an isomorphism

on K-theory.

Then K∗(φ) : K∗(A)∼

−−→ K∗(B)is an isomorphism.

Proof. Let Σnbe the n-skeleton of Σ and consider the ideals

I0=C(Σ) ⊃I1=C(Σ; Σ0)⊃ ··· ⊃ In=C0(Σ,Σn−1)⊃ · ·· (5.2)

They give rise to ideals InAand InB. Because Σ is ﬁnite, all these ideals are 0 for

large n. We can identify

InA/In+1A∼

=C0(Σn,Σn−1)A∼

σ∈Σ : dim σ=n

AC0(σ, δσ) := M

σ∈Σ : dim σ=n

C0(σ, δσ;Aσ),(5.3)

and similarly for B. Because φis C(Σ)-linear, it induces homomorphisms

φ(σ) : C0(σ, δσ;Aσ)→C0(σ, δσ;Bσ).

By the additivity of and the excision property of the K-functor (see e.g. [Bla, Theo-

rem 9.3.1]), it suﬃces to show that every φ(σ) induces an isomorphism on K-theory.

Let xbe any interior point of σ. Because σ\δσ is contractible, φσis homotopic

to idC0(σ,δσ)⊗φ(xσ). By assumption the latter map induces an isomorphism on

K-theory. With the homotopy invariance of K-theory it follows that

K∗(φ(σ)) = K∗idC0(σ,δσ)⊗φ(xσ)

is an isomorphism. 2

Obviously this lemma is in no way optimal: one can generalize it to larger classes

of algebras and one can relax the ﬁniteness assumption on Σ. Because we do not

need it, we will not bother to write down such generalizations. What will need

however, is that Lemma 5.1.3 is also valid for similar functors, in particular for

A7→ K∗(A)⊗ZC.

Theorem 5.1.4. The map

K∗(ζ0)⊗idQ:K∗(C∗(We)⋊Γ) ⊗ZQ→K∗(C∗(R, q)⋊Γ) ⊗ZQ

is a Q-linear bijection.

Proof. Consider the projection

pr : Ξun/G → Tun/W ′,G(P, δ, t)7→ W′r|r|−t,

where W(RP)r∈TP/W (RP) is the central character of δ. With this projection and

Theorem 3.2.2 we make C∗(R, q) into a C(Tun/W ′)-algebra. By Theorem 4.4.2.e

ζ0:C∗(We)⋊Γ→C∗(R, q)⋊Γ is a morphism of C(Tun/W ′)−C∗-algebras. Choose

a triangulation Σ of Tun, such that:

•w(σ)∈Σ for every simplex σ∈Σ and every w∈W′;

•TG

un is a subcomplex of Σ, for every subgroup G⊂W′;

•the star of any simplex σis W′

σ-equivariantly contractible.

Then Σ/W ′is a triangulation of Tun/W ′. From Theorem 3.2.2 and the proof of

[Sol1, Lemma 7] we see that A=C∗(We)⋊Γ and B=C∗(R, q)⋊Γ and are of the

form required in condition (a) of Lemma 5.1.3. For any u∈Tun we write

Au:= C∗(We)⋊Γ/ker Iu,

Bu:= LGξ∈Ξun/G,pr(ξ)=W′uC∗(R, q)⋊Γ/ker πΓ(ξ).

Condition (b) of Lemma 5.1.3 for K∗(?)⊗ZQmeans that the map ζ0(W′u) : Au→Bu

should induce an isomorphism

K∗(ζ0(W′u)) : K∗(Au)⊗ZQ→K∗(Bu)⊗ZQ.(5.4)

As for all ﬁnite dimensional semisimple algebras,

K∗(Au) = K0(Au) = GZ(A) and K∗(Bu) = K0(Bu) = GZ(Bu).

With these identiﬁcations K0(ζ0(W′u)) sends a projective module eMn(Au) to the

projective module ζ0(W′u)(e)Mn(Bu). The free abelian groups GZ(Au) and GZ(Bu)

have natural bases consisting of irreducible modules. With respect to these bases

the matrix of K0(ζ0(W′u)) is the transpose of the matrix of

˜σ0:GZ(Bu)→GZ(Au), π 7→ π◦ζ0(W′u).

By Theorem 2.3.1.a ˜σ0⊗idQ:GQ(Au)→GQ(Bu) is a bijection, so (5.4) is also a

bijection. Now we can apply Lemma 5.1.3, which ﬁnishes the proof. 2

So we proved Conjecture 5.1.1 modulo torsion, which raises the question what

information is still contained in the torsion part. It is known that K∗(C∗(R, q)⋊Γ)

is a ﬁnitely group. Indeed, by [Sol1, Theorem 6] this is the case for all Fr´echet

algebras if the type described in Theorem 3.2.2. Hence the torsion subgroup of

K∗(C∗(R, q)⋊Γ) is ﬁnite. In fact the author does not know any examples of

nontrivial torsion elements in such K-groups, but it is conceivable that they exist.

It turns out that this is related to the multiplicities of W′-representations in the

aﬃne Springer correspondence from Section 2.3.

Lemma 5.1.5. The following are equivalent:

(a)K∗(ζ0) : K∗(C∗(We)⋊Γ) →K∗(C∗(R, q)⋊Γ) is an isomorphism.

(b)K0(ζ0) : K0(C∗(We)⋊Γ) →K0(C∗(R, q)⋊Γ) is surjective.

(c)For every u∈Tun the map Spr induces a surjection from the Grothendieck group

of Modf,W ′uTr s (S(R, q)⋊Γ) to that of Modf,W ′u(We⋊Γ).

(d)The map Spr induces a bijection SprZ:GZ(S(R, q)⋊Γ) →GZ(S(We)⋊Γ).

Proof.

(a) ⇒(b) Obvious.

(b) ⇒(c) We use the notation from the proof of Theorem 5.1.4, in particular K0(Au)

and K0(Bu) are the Grothendieck groups referred to in (c). We claim that the

canonical map

K0(C∗(R, q)⋊Γ) →K0(Bu) (5.5)

is surjective. Recall that K0(Bu) is built from idempotents. Given any idempotent

eu∈Mn(Bu) we want to ﬁnd an idempotent e∈Mn(C∗(R, q)⋊Γ) that maps to it.

By Theorem 3.2.2 this means that on every connected component (P, δ, T P

un)/GP,δ of

Ξun/Gwe have to ﬁnd an idempotent eP,δ in

MnC(TP

un)⊗EndC(C[ΓWP]⊗Vδ)GP ,δ ,

which in every point of pr−1(W′u)∩(P, δ, T P

un)/GP,δ takes the value prescribed by

eu. Recall that the groupoid Gwas built from elements of W′and from the groups

KP=TP∩TP. The latter elements permute the components of Ξun freely, so

pr−1(W′u) intersects every component of Ξun in at most one G-association class.

Therefore we can always ﬁnd such a eP,δ, proving the claim (5.5).

Together with assumption (b) this implies that

K0(C∗(We)⋊Γ) K0(ζ0)

−−−−→ K0(C∗(R, q)⋊Γ) →K0(Bu)

is surjective. The underlying C∗-algebra homomorphisms factors via C∗(We)⋊Γ→

Au, so

K0(ζ0(W′u)) : K0(Au)→K0(Bu).(5.6)

is also surjective.

the notation of (5.4) SprZis the direct sum, over all W′u∈Tun, of the maps

GZ(Bu)→GZ(Au) : π7→ π◦ζ0(W′u).(5.7)

As we noticed in the proof of Theorem 5.1.4, the matrix of this map is the trans-

pose of the matrix of (5.6). We showed in the aforementioned proof that the latter

map becomes an isomorphism after applying ⊗ZQ. As K0(Au) and K0(Bu) are free

abelian groups, this implies that K0(ζ0(W′u)) is injective. So under assumption (c)

(5.6) is in fact an isomorphism. Hence, with respect to the natural bases it is given

by an integral matrix with determinant ±1. Then the same goes for (5.7), so that

map is also bijective. Therefore SprZis bijective.

(d) ⇒(a) The above shows that under assumption (d) the maps (5.7) and (5.6) are

bijections. Since K1(Au) = K1(Bu) = 0, we may apply Lemma 5.1.2. 2

By Corollary 2.1.5.b and property (d) of Theorem 2.3.1, condition (c) of Lemma

5.1.5 can be reformulated as follows: for all u∈Tun the map π7→ πW′

uinduces a

bijection from the Grothendieck group of Modf,a(H(˜

Ru, ku)⋊W′

Fu,u) to GZ(W′

u).

According to [Ciu, Corollary 3.6] this statement is valid for all graded Hecke

algebras of ”geometric type”. Hence Conjecture 5.1.1 holds, including torsion, for

many important examples of aﬃne Hecke algebras.

In particular, let Ibe an Iwahori subgroup of a split reductive p-adic group G

with root datum R, as in Section 1.6. By [Ply2] the completion C∗

r(G, I) of H(G, I)

is isomorphic to C∗(R, q), where qis some prime power. It is interesting to combine

Conjecture 5.1.1 with the Baum–Connes conjecture. Let βG be the aﬃne Bruhat–

Tits building of Gand identify a∗with an apartment. The Baum–Connes conjecture

for groups like Gand Wewas proven by V. Laﬀorgue[Laf], see also [Sol4] (For We

it can of course be done more elementarily.) We obtain a diagram

KWe

∗(a∗)→K∗(C∗

r(We)) →K∗(C∗(R, q)) →K∗(C∗

r(G, I))

↓ ↑

∗(βG)→K∗(C∗

r(G)) →Ls∈B(G)K∗(C∗

r(G)s)

(5.8)

in which all the horizontal maps are natural isomorphisms, while the vertical maps

pick the factor of K∗(C∗

r(G)) corresponding to the Iwahori-spherical component in

B(G). For the group G=GLn(F) this goes back to [Ply1]. Notice that (5.8)

realizes KWe

∗(a∗) as a direct summand of KG

∗(βG), which is by no means obvious in

equivariant K-homology.

5.2 Periodic cyclic homology

Periodic cyclic homology is rather similar to topological K-theory, but the former

functor is deﬁned on larger classes of algebras. For example one can take the peri-

odic cyclic homology of nontopological algebras like H(R, q), while it is much more

diﬃcult to make sense of the topological K-theory of aﬃne Hecke algebras without

completing them. By deﬁnition the periodic cyclic homology of an algebra over a

ﬁeld Fis an F-vector space. Whereas topological K-theory for C∗-algebras is the

generalization of K-theory for topological spaces, periodic cyclic homology for non-

commutative algebras can be regarded as the analogue of De Rham cohomology for

manifolds.

In [Sol6, Theorem 3.3] the author proved with homological-algebraic techniques

that the periodic cyclic homology of an (extended) graded Hecke algebra H(˜

R, k)⋊Γ

does not depend on the parameter function k. Subsequently he translated this into

a representation-theoretic statement, which we already used in (2.24): the collection

of irreducible tempered H(˜

R, k)⋊Γ-representations with real central character forms

a basis of GQ(W0⋊Γ).

We will devise a reversed chain of arguments for aﬃne Hecke algebras. Via

topological K-theory we will use the aﬃne Springer correspondence to show that

H(R, q)⋊Γ and S(R, q)⋊Γ have the same periodic cyclic homology, and that it

does not depend on the (positive) parameter function q. The material in this section

can be compared with [BHP, Sol4].

Recall that the Chern character is a natural transformation K∗→HP∗, where

we write HP∗(A) = H P0(A)⊕HP1(A). By (5.1) and [Sol1, Theorem 6] there are

natural isomorphisms

K∗(C∗(R, q)⋊Γ) ⊗ZC←K∗(S(R, q)⋊Γ) ⊗ZC→H P∗(S(R, q)),(5.9)

the ﬁrst one is induced by the embedding S(R, q)⋊Γ→C∗(R, q)⋊Γ, one the second

one by the Chern character. Here and elsewhere in this paper the periodic cyclic

homology of topological algebras is always meant with respect to the completed

projective tensor product. (One needs a tensor product to build the diﬀerential

complex whose homology is HP∗.) By contrast, in the deﬁnition of the periodic cyclic

homology of nontopological algebras we simply use the algebraic tensor product

over C.

Theorem 5.2.1. The inclusion H(R, q)⋊Γ→ S(R, q)⋊Γinduces an isomorphism

on periodic cyclic homology.

Proof. In [Sol4, Theorem 3.3] the author proved the corresponding result for

Hecke algebras of reductive p-adic groups. The proof from [Sol4] also applies in

our setting, the important representation-theoretic ingredients being Theorem 3.2.2,

Proposition 3.4.1 and Lemma 3.4.2. A sketch of this proof already appeared in [Sol2].

Corollary 5.2.2. There exists a natural commutative diagram

HP∗(C[We]⋊Γ) →HP∗(S(We)⋊Γ) ←K∗(S(We)⋊Γ) →K∗(C∗(We)⋊Γ)

↓ ↓HP∗(ζ0)↓K∗(ζ0)↓K∗(ζ0)

HP∗(H(R, q)⋊Γ) →H P∗(S(R, q)⋊Γ) ←K∗(S(R, q)⋊Γ) →K∗(C∗(R, q)⋊Γ)

After applying ⊗ZCto the K-groups, all these maps are isomorphisms.

Proof. The horizontal maps are induced the inclusion maps

H(R, q)⋊Γ→ S(R, q)⋊Γ→C∗(R, q)⋊Γ

and by the Chern character K∗→HP∗. The vertical maps (expect the leftmost

one) are induced by the Fr´echet algebra homomorphisms ζ0from Theorem 4.4.2.

According to (5.9) and Theorem 5.2.1 all the horizontal maps become isomorphisms

after tensoring the K-groups with C. By Lemma 5.1.2 the maps K∗(ζ0) are natural,

and by Theorem 5.1.4 they become isomorphisms after applying ⊗ZC. The diagram

commutes by functoriality, so H P∗(ζ0) is also a natural isomorphism. Finally, we

deﬁne HP∗(C[We]⋊Γ) →HP∗(H(R, q)⋊Γ) as the unique map that makes the

entire diagram commute. 2

Remark 5.2.3. Whether the leftmost vertical map comes from a suitable algebra

homomorphism C[We]⋊Γ→ H(R, q)⋊Γis doubtful, no such homomorphism is

known if q6= 1.

Suppose that Xis the weight lattice of R∨

0, that q∈C\ {0}is any complex

number which is not a root of unity, and that q(s) = qfor all s∈Saﬀ . In this setting

an isomorphism HP∗(C[We]) ∼

=HP∗(H(R, q)) was already constructed by Baum

and Nistor [BaNi, Theorem 11]. Their proof makes essential use of the Kazhdan–

Lusztig classiﬁcation [KaLu2, Theorem 7.12] of irreducible H(R, q)-representations,

and of Lusztig’s asymptotic Hecke algebra [Lus3, Lus4].

For graded Hecke algebras things are even better than in Corollary 5.2.2: in

[Sol6, Theorem 3.4] it was proven that not only HP∗(H(˜

R, k)⋊Γ), but also the

cyclic homology and the Hochschild homology of H(˜

R, k)⋊Γ are independent of k.

Whether or not this can be transferred to H(R, q) is unclear to the author. The

point is that the comparison of H(˜

R, k)⋊Γ with H(R, q)⋊Γ goes only via analytic

localizations of these algebras. Since the eﬀect of localization on the dual space is

very easy, we can translate the comparison between localized Hecke algebras to a

comparison between their dual spaces. By [Sol6, Theorem 4.5] the periodic cyclic

homology of a ﬁnite type algebra essentially depends only on its dual space, so it

is not surprising that the parameter independence of HP∗can be transferred from

graded Hecke algebras to aﬃne Hecke algebras,

On the other hand, the Hochschild homology of an algebra changes in a nontrivial

way under localization. Therefore one would in ﬁrst instance only ﬁnd a comparison

between the Hochschild homology of two localized aﬃne Hecke algebras with the

same root datum but diﬀerent parameters q. Possibly, provided that one would

know enough about HH∗(H(R, q)⋊Γ), one could deduce that also this vector space

is independent of q. We remark that most certainly the Z(H(R, q)⋊Γ)-module

structure of HH∗(H(R, q)⋊Γ) will depend on q, because that is already the case

for graded Hecke algebras, see the remark to Theorem 3.4 in [Sol6].

5.3 Weakly spectrum preserving morphisms

For the statement and the proof of the Aubert–Baum–Plymen conjecture we need

spectrum preserving morphisms and relaxed versions of those. These notions were

developed in [BaNi, Nis]. Baum and Nistor work in the category of ﬁnite type k-

algebras, where kis the coordinate ring of some complex aﬃne variety. Since we are

also interested in certain Fr´echet algebras, we work in a larger class of algebras.

We cannot do without some ﬁniteness assumptions, but it suﬃces to impose them

on representations. So, throughout this section we assume that for all our complex

algebras Athere exists a N∈Nsuch that the dimensions of irreducible A-modules

are uniformly bounded by N. In particular π7→ ker πis a bijection from Irr(A) to

the collection of primitive ideals of A. A homomorphism φ:A→Bbetween two

such algebras is called spectrum preserving if

•for every primitive ideal J⊂B, there is exactly one primitive ideal I⊂A

containing φ−1(J);

•the map J7→ Iinduces a bijection Irr(φ) : Irr(B)→Irr(A).

We can relax these conditions in the following way. Suppose that there exists ﬁltra-

tions A=A0⊃A1⊃ ··· ⊃ An= 0,

B=B0⊃B1⊃ ··· ⊃ Bn= 0 (5.10)

by two sided ideals, such that φ(Ai)⊂Bifor all i. We call φ:A→Bweakly

spectrum preserving if all the induced maps φi:Ai/Ai+1 →Bi/Bi+1 are spectrum

preserving. In this case there are bijections

⊔iIrr(Ai/Ai+1)→Irr(A),

⊔iIrr(Bi/Bi+1)→Irr(B),

Irr(φ) := ⊔iIrr(φi) : Irr(B)→Irr(A).

Notice that Irr(φ) depends not only on φ, but also on the ﬁltrations of Aand B.

Lemma 5.3.1. Let φ:A→Bbe a weakly spectrum preserving morphism, and

suppose that the dimensions of irreducible B-modules are uniformly bounded by N∈

N. Then Irr(φ)−1(V(I)) = V(φ(I)N)for every two-sided ideal I⊂A. In particular

the bijection Irr(φ)is continuous with respect to the Jacobson topology (cf. Section

3.4).

Proof. We proceed with induction to the length nof the ﬁltration. For n= 1

the morphism φis spectrum preserving, so the statement reduces to [BaNi, Lemma

9]. For n > 1 the induction hypothesis applies to the homomorphisms φ:A1→B1

and φ1:A/A1→B/B1. So for π∈Irr(B/B1)⊂Irr(B) we have

π∈Irr(φ)−1(V(I)) ⊂Irr(B)⇐⇒

π∈Irr(φ1)−1(V(I+A1/A1)) ⊂Irr(B/B1)⇐⇒

π∈V(φ(I)N+B1/B1)) ⊂Irr(B/B1)⇐⇒

π∈V(φ(I)N)) ⊂Irr(B).

A similar argument applies to π∈Irr(B1)⊂Irr(B).2

The automatic continuity of Irr(φ) enables us to extract a useful map from the

Fr´echet algebra morphism ζ0:

Lemma 5.3.2. The morphism ζ0:S(We)⋊Γ→ S(R, q)⋊Γis weakly spectrum

preserving.

Proof. We wil make use of the proofs of Lemma 5.1.3 and Theorem 5.1.4. There

we constructed a W′-equivariant triangulation of Tun, which lead to two-sided ideals

In=C∞

0(Σ,Σn−1)W′⊂C∞(TW′

un ,

InS(R, q)⋊Γ⊂ S(R, q)⋊Γ,

InS(We)⋊Γ⊂ S(We)⋊Γ.

(Here and below we regard the n-skeleton Σnboth as a simplicial complex and as

a subset of Tun). It suﬃces to show that the induced map

ζ0,n :InS(We)⋊Γ/In+1S(We)⋊Γ→InS(R, q)⋊Γ/In+1S(R, q)⋊Γ (5.11)

is spectrum preserving, for every n. Fortunately the dual spaces of these quotient

algebras are rather simple, by (5.3)

IrrInS(R, q)⋊Γ/In+1S(R, q)⋊Γ∼

σ∈Σ/W ′,dim σ=n

(σ\δσ)×Irrxσ(S(R, q)⋊Γ),

(5.12)

where xσ∈σ\δσ and Irrxσ(S(R, q)⋊Γ) denotes the dual space of the algebra

Gξ∈Ξun/G,pr(ξ)=W′xσS(R, q)⋊Γ/ker πΓ(ξ).(5.13)

By construction ζ0,n is C∞

0-linear, so in particular it is linear over

C∞

0(Σn,Σn−1) := In/In+1.

We know from Theorem 2.3.1.a and Corollary 4.4.3 that GQ(ζ0,n) is a bijection, so

in particular

GQ(ζ0,n) : GQ(Modxσ(S(R, q)⋊Γ)) →GQ(Modxσ(S(We)⋊Γ)) (5.14)

is a bijection. Any ordering (π1, π2,...,πk) of Irrxσ(S(R, q)⋊Γ) gives rise to a

ﬁltration of (5.13) by ideals

Bi:= \i

j=1 ker πji= 0,1,...,k.

Since we are dealing with two ﬁnite dimensional semisimple algebras of the same

rank k, (5.14) can be described completely with a matrix M∈GLk(Z). Order

Irrxσ(S(R, q)⋊Γ) and Irrxσ(S(We)⋊Γ) such that all the principal minors of Mare

nonsingular. Then the corresponding ideals Biof (5.13) and Ai⊂ S(We)⋊Γ/ker Ixσ

are such that ζ0,n(xσ) induces spectrum preserving morphisms Ai/Ai+1 →Bi/Bi+1.

Hence ζ0,n(xσ) is weakly spectrum preserving.

It follows from this and (5.12) that for any n-dimensional simplex σ∈Σ/W ′we

can construct ﬁltrations by two-sided ideals in

C∞

0(Σ, δσ)W′S(R, q )⋊Γ/C∞

0(Σ, σ)W′S(R, q)⋊Γ

and in

C∞

0(Σ, δσ)W′S(We)⋊Γ/C ∞

0(Σ, σ)W′S(We)⋊Γ,

with respect to which the map induced by ζ0,n is weakly spectrum preserving. We

do this for all such simplices σ, and then (5.3) show that (5.11) is weakly spectrum

preserving. 2

A related notion that we will use in the next section is geometric equivalence of

algebras, as deﬁned in [ABP1, Section 4]. The basic idea is to call Aand Bgeomet-

rically equivalent if they are Morita-equivalent or if there exists a weakly spectrum

preserving morphism φ:A→B. Furthermore two ﬁnite type k-algebras are geo-

metrically equivalent if they only diﬀer by an algebraic deformation of the k-module

structure. Now one deﬁnes geometric equivalence to be the equivalence relation (on

the category of ﬁnite type k-algebras) generated by these three elementary moves.

So whenever two algebras Aand Bare geometrically equivalent, they are so by

various sequences of elementary moves. Every such sequence induces a bijection

between the dual spaces of Aand B, which however need not be continuous, since

the map Irr(φ) from Lemma 5.3.1 is usually not a homeomorphism. Nevertheless,

by [BaNi, Theorem 8] every weakly spectrum preserving morphism of ﬁnite type

algebras φ:A→Binduces an isomorphism HP∗(φ) : HP∗(A)→HP∗(B). The

other two moves are easily seen to respect periodic cyclic homology, so geometric

equivalence implies HP -equivalence.

5.4 The Aubert–Baum–Plymen conjecture

In a series of papers [ABP1, ABP2, ABP3, ABP4] Aubert, Baum and Plymen de-

veloped a conjecture that describes the structure of Bernstein components in the

smooth dual of a reductive p-adic group. We will rephrase this conjecture for aﬃne

Hecke algebras, and prove it in that setting.

A central role is played by extended quotients. Let Gbe a ﬁnite group acting

continuously on a topological space T. We endow

T:= {(g, t)∈G×T:g·t=t}

with the subspace topology from G×T. Then Galso acts continuously on e

T, by

g·(g′, t) = (gg′g−1, g ·t).

The extended quotient of Tby Gis deﬁned as e

T /G. It comes with a projection onto

the normal quotient: e

T /G →T /G :G(g, t)7→ Gt.

The ﬁber over Gt ∈T/G can be identiﬁed with the collection hGtiof conjugacy

classes in the isotropy group Gt.

The relevance of the extended quotient for representation theory comes from

crossed product algebras. Suppose that F(T) is an algebra of continuous complex

valued functions on T, which separates the points of Tand is stable under the action

of Gon C(T). These conditions ensure that the crossed product F(T)⋊Gis well-

deﬁned. The dual space of this algebra was determined in classical results that go

back to Frobenius and Cliﬀord (see [CuRe, Section 49]). The collection of irreducible

representations with a F(T)-weight t∈Tis in natural bijection with Irr(Gt), via

the map

π7→ IndF(T)⋊G

F(T)⋊GxCx⊗π.

Since |Irr(Gx)|=|hGxi|, there exists a bijection

T /G →Irr(F(T)⋊G) (5.15)

which maps G(g, t) to representation with a F(T)-weight t. With a little more

work one can ﬁnd a continuous bijection. However, it is not natural and a not a

homeomorphism, except in very simple cases.

We return to an extended aﬃne Hecke algebra H(R, q)⋊Γ. As described in

Section 1.2, the parameter function qis completely determined by its values on the

quotient set R∨

nr/W0⋊Γ. Let Q(R) be the complex variety of all maps R∨

nr/W0⋊Γ→

C×. To every v∈ Q(R) we associate the parameter function qα∨=v(α∨)2.

Conjecture 5.4.1. (ABP-conjecture for aﬃne Hecke algebras)

(a)The algebras C[We]⋊Γand H(R, q)⋊Γare geometrically equivalent.

(b)There exists a canonical isomorphism HP∗(C[We]⋊Γ) ∼

=HP∗(H(R, q)⋊Γ).

(c)There exists a continuous bijection µ:e

T /W ′→Irr(H(R, q)⋊Γ) such that

µ(g

Tun/W ′) = Irr(S(R, q)⋊Γ).

(d)For every connected component cof e

T /W ′there exists a smooth morphism of

algebraic varieties hc:Q(R)→Twith the following properties.

For all components cwe have hc(1) = 1, and

prq1/2(e

T /W ′−T /W ′) = {t∈T:Itis reducible }/W ′,

where prv:e

T /W ′→T /W ′is deﬁned by

prv(W′(w, t)) = W′hc(v)tfor v∈ Q(R), W ′(w, t)∈c.

Moreover µcan be chosen such that the central character of µ(W′(w, t)) is

W′hc(q1/2)tfor W′(w, t)∈c.

We will now discuss the diﬀerent parts of the ABP-conjecture. As we mentioned

at the end of Section 5.3, every explicit geometric equivalence gives rise to an isomor-

phism on periodic cyclic homology. However, this isomorphism need not be natural,

so (b) does not yet follow from (a). In [ABP1, ABP3] we see that Aubert, Baum

and Plymen have a geometric equivalence via Lusztig’s asymptotic Hecke algebra

[Lus4] in mind. The corresponding isomorphism [BaNi, Theorem 11]

HP∗(H(R, q)) ∼

=HP∗(C[We])

can be regarded as canonical, albeit in a rather weak sense.

Unfortunately, for unequal parameter functions this asymptotic Hecke algebra

exists only as a conjecture. Neither does the author know any other way to construct

a geometric equivalence between H(R, q)⋊Γ and C[We]⋊Γ, so this part of the

conjecture remains open for unequal parameter functions. As a substitute we oﬀer

Lemma 5.3.2, which has approximately the same strength. It is weaker because it

concerns only topologically completed versions of the algebras, but it is stronger

in the sense that the geometric equivalence consists of only one weakly spectrum

preserving morphism.

Part (b) of Conjecture 5.4.1 was already dealt with in Corollary 5.2.2.

Let us construct a map µas in part (c). Recall from (3.38) that there exist

tempered smooth families {πi,t :t∈Vi}which together form a basis of GQ(S(R, q)⋊

Γ). The parameter space of such a family is of the form Vi=uiexp(agi) for some

ui∈Tun, gi∈W′. By (3.35) the number of families with parameter space of the

form gVifor some g∈W′, is precisely the number of components of g

Tun/W ′whose

projection onto Tun/W ′is W′Vi. Hence we can ﬁnd a continuous bijection

Tun/W ′→ {πi,t :t∈Vi, i ∈I}.

This will be our map µwhenever the image πi,t is irreducible, which is the case

on a Zariski-dense subset of g

Tun/W ′. To extend µcontinuously to all nongeneric

points, we need to ﬁnd irreducible subrepresentations π′

i,t ⊂πi,t, such that {πi,t :

t∈Vi}= Irr(S(R, q)⋊Γ). For 0-dimensional components all point are generic, so

there is nothing to do. If we have already deﬁned µon all components of g

Tun/W ′

of dimension smaller than d, and (i, t) corresponds to a nongeneric point W′(w, t)

in a component of dimension d, than we choose for µ(W′(w, t)) any irreducible

subrepresentation of πi,t that we did not have yet in the image of the previously

handled components. This process can be carried out completely can (3.35), and

yields a continuous bijection

µ:g

Tun/W ′→Irr(S(R, q)⋊Γ).(5.16)

From this and Lemmas 5.3.1 and 5.3.2 we obtain continuous bijections

Tun/W ′→Irr(S(R, q)⋊Γ) →Irr(S(We)⋊Γ).

As explained after (3.28) and (3.38), these can be extended canonically to continuous

bijections e

T /W ′→Irr(H(R, q)⋊Γ) →Irr(We⋊Γ).(5.17)

Now we show that part (d) of the conjecture is valid for this µ. Suppose that

W′(w, t0)∈g

Tun/W ′is such that the corresponding representation πi,t is irreducible.

With Theorem 3.3.2 we can ﬁnd an induction datum ξ+(πi,t) = (P, δ, t1)∈Ξun such

that πi,t is a subquotient of πΓ(P, δ, t1). Then µ(W′(w, t0t2)) is a subquotient of

πΓ(P, δ, t1t2) for all t2∈exp(tw), so its central character is W′rt1t2, where W(RP)r∈

TP/W (RP) is the central character of the HP-representation δ. According to [Opd2,

Lemma 3.31] r∈TPis a residual point for RP, which by Proposition 2.63 and

Theorem 2.58 of [OpSo2] means that the coordinates of rcan be expressed as an

element of TP,un times a monomial in the variables {q(s)±1/2:s∈Saﬀ }. Hence

we can write |r|=hc(q1/2), where cis the component of e

T /W ′containing W′(w, t)

and hc:Q(R)→TP⊂Tis a smooth algebraic morphism with hc(1) = 1. Now

W′hc(q1/2)t0t2is by construction the central character of µ(W′(w, t0t2)). We note

that the discrete series representation δ∅of H∅=Chas central character 1 ∈T∅=

{1}, so hc= 1 when chas dimension rank(X).

Let prq1/2be as in part (d) of Conjecture 5.4.1 and temporarily denote the

diﬀerence of two sets by −. Then prq1/2(e

T /W ′−T /W ′) is the set of central characters

of µ(e

T /W ′−T /W ′). Since µparametrizes irreducible representations, and since

every π∈Irr(H(R, q)⋊Γ) with central character tis a quotient of the principal

series representation It, no element of prq1/2(e

T /W ′−T /W ′) can be the parameter

of an irreducible principal series representation.

Conversely, suppose that t∈Tis not in the aforementioned set. In view of

Lemma 3.1.7 we may assume that t∈T+. Then there is, up to isomorphism, only

one πt∈Irr(H(R, q)⋊Γ) with central character W′t. Hence the induction datum

ξ+(πt) is (∅, δ∅, t)∈Ξ+. By Theorem 3.3.1 the intertwining operators

{π(g, ∅, δ∅, t) : g∈ G, g(∅, δ∅, t) = (∅, δ∅, t)}

span EndH⋊Γ(It). If any of these intertwiners were nonscalar, then Itwould contain

nonisomorphic irreducible constituents. The latter is impossible, so EndH⋊Γ(It) =

Cid. This in turn implies that πtcannot be both a subrepresentation and a quotient

of It, unless πt=It. Therefore prq1/2(e

T /W ′−T /W ′) is precisely the subset of t∈T

for which the principal series representation Itis reducible.

By (3.7) this set contains all residual cosets of dimension smaller than dimC(T).

However, in general prq1/2(e

T /W ′−T /W ′) is larger, because a unitary principal series

representation can be reducible.

We note that by Proposition 4.1.2 the same map prvalso makes part (d) valid

for the scaled parameter functions qǫwith ǫ∈R. However, for other parameter

functions changes can occur.

Summarizing, we showed that:

Theorem 5.4.2. Parts (b), (c) and (d) of Conjecture 5.4.1 hold for every extended

aﬃne Hecke algebra with a positive parameter function q. Part (a) holds for the

Schwartz completions of the algebras in question.

Hence the Aubert–Baum–Plymen conjecture [ABP1, ABP2] for a Bernstein com-

ponent sof a reductive p-adic group Gholds whenever the algebra H(G)sis Morita-

equivalent to an extended aﬃne Hecke algebra in the way described in Section 1.6. In

particular this applies to all Bernstein components listed at the end of that section.

5.5 Example: type C(1)

In the ﬁnal section we illustrate what the Aubert–Baum–Plymen conjecture looks

like for an aﬃne Hecke algebra with R0of type B2/C2and Xthe root lattice. More

general results for type C(1)

naﬃne Hecke algebras can be found in [Kat2, CiKa]. For

other examples we refer to [Sol3, Chapter 6].

Consider the based root datum Rwith

X=Y=Z,

R0={x∈X:kxk= 1 or kxk=√2},

R∨

0={y∈Y:kxk= 2 or kxk=√2},

F0={α1=1

−1, α2= ( 0

1)}

Then α4= ( 1

1) is the longest root and α∨

3= ( 2

0) is the longest coroot, so

Saﬀ ={sα1, sα2, tα3sα3}.

We write si=sαifor 1 ≤i≤4 and s0=tα3sα3. The Weyl group W0is isomorphic

to D4and consists of the elements

W0={e, ρπ/2, ρπ, ρ−π/2} ∪ {s1, s2, s3, s4},

where ρθdenotes the rotation with angle θ. The aﬃne Weyl group of Ris the

Coxeter group

Waﬀ =We=X⋊W0=hs0, s1, s2|s2

i= (s0s2)2= (s1s2)4= (s0s1)4=ei.

Furthermore Rnr =R0∪ {±α2,±α3}and X+={(m

n)∈X:m≥n≥0}.

We note that Ris the root datum of the algebraic group SO5, while R∨corre-

sponds to Sp4. Let Fbe a p-adic ﬁeld whose residue ﬁeld has qelements, and let s

be the Iwahori-spherical component of Sp4(F). Then Mods(Sp4(F)∼

=Mod(H(R, q))

and Kazhdan–Lusztig theory describes the irreducible representations in this cate-

gory with data from Sp4(F).

But there are many more parameter functions for R. Since s0, s1and s2are not

conjugate in We, we can independently choose three parameters

q0=q(s0) = qα∨

2/2, q1=q(s1) = qα1, q2=q(s2) = qα∨

Several combinations of these parameters occur in Hecke algebras associated to non-

split p-adic groups, see [Lus7]. The c-functions are

cα1=θα1−q−1

θα1−1and cα2=θα2+q−1/2

2q1/2

θα2+ 1

θα2−q−1/2

1q−1/2

θα2−1.

For q0=q2the relations from Theorem 1.2.1.d simplify to

fNsi=Nsisi(f) = (q1/2

i−q−1/2

i)(f−si(f))(1 −θ−αi)−1i= 1,2.(5.18)

In contrast with graded Hecke algebras, H(R=R(SO5), q1, q2=q0) is not isomor-

phic to H(R∨=R(Sp4), q2, q1). The reason is that in H(R∨, q2, q1) the relation

fNs(0

2)=Ns(0

2)s(0

2)(f) = (q1/2

2−q−1/2

2)(f−s(0

2)(f))(1 −θ0

−2)−1f∈ A

holds, which really diﬀers from (5.18) because the root lattice Z−1

1+Z(0

2) does

not equal Xfor the root datum R∨.

We will work out the tempered dual of H(R, q) for almost all positive parameter

functions q. To this end we discuss for each parabolic subalgebra HPwith P⊂

{α1, α2}separately. Its contribution to Irr(S(R, q)) will of course depend on q, and

be even be empty in some cases.

•P=∅

XP={0}, XP=X, YP={0}, Y P=Y, RP=∅, R∨

P=∅, W (RP) = {e},

TP={1}, T P=T, GP=W0,HP=C,HP=A∼

=C[X].

We must determine the reducibility of the unitary principal series representations

It= IndH

ACt=π(∅, δ∅, t)t∈Tun.

By Theorem 3.3.1 EndH(It) is spanned by the intertwining operators π(w, ∅, δ∅, t)

with w∈W0and w(t) = t. For a root α∈R0with sα(t) = t, Lemma 3.1.6 tells us

that π(sα,∅, δ∅, t) is a scalar if and only if c−1

α(t) = 0.

Let us write t= (t3, t2) with ti=t(αi). A fundamental domain

for the action of W0on Tun is {t= (eiφ, eiψ ) : 0 ≤ψ≤φ≤π}:

(−1,1)

(1,1)

The isotropy groups are trivial for all interior points, so Itis irreducible for such

t. Below we list the necessary data for all boundary points:

t W0,t conditions EndH(It) # irreducibles

(eiφ,1), φ ∈(0, π)hs2iq0q26= 1 C1

q0q2= 1 C[hs2i] 2

(−1, eiψ), ψ ∈(0, π)hs3iq26=q0C1

q2=q0C[hs3i] 2

(eiφ, eiφ ), φ ∈(0, π)hs1iq16= 1 C1

q1= 1 C[hs1i] 2

(−1,1) hs2, s3iq06=q2, q0q26= 1 C1

q0=q26= 1 C[hs3i] 2

q0=q−1

26= 1 C[hs2i] 2

q0=q2= 1 C[hs2, s3i] 4

(−1,−1) W0q06=q2, q16= 1 C1

q0=q2, q16= 1 C[hs2i] 2

q1= 1, q06=q2C[hs1i] 2

q0=q2, q1= 1 C[W0] 5

(1,1) W0q0q26= 1, q16= 1 C1

q0=q−1

2, q16= 1 C[hs2i] 2

q1= 1, q06=q−1

2C[hs3i] 2

q0=q−1

2, q1= 1 C[W0] 5

•P={α1}

XP=X/Zα4∼

=Zα1/2, XP=X/Zα1, YP=Zα∨

1, Y P=Zα∨

4, RP={±α1},

R∨

P={±α∨

1}, W (RP) = {e, s1}, T P={t∈T:t3=t2}, TP={t∈T:t2t3= 1},

TP∩TP={(1,1),(−1,−1)},GP={e, s4} × TP∩TP,

HP=H(RP, q(s1) = q1=qα∨

1,HP=HP⋉ C[XP].

The root datum RPis of type C(1)

1, which means that R∨

0is of type C1=A1and

generates the lattice YP. For q1= 1 there are no residual points, for q16= 1 there

are two orbits, namely W(RP)(q1/2

1, q−1/2

1) and W(RP)(−q1/2

1,−q−1/2

1). Both orbits

carry a unique discrete series representation, which has dimension one. The formulas

for these representations are not diﬃcult, but they depend on whether q2>1 or

q2<1. So we obtain two families of H(R, q)-representations:

π{α1}, δ1,(t2, t2)= IndH

HP(δ1◦φ(t2,t2))q16= 1, t2∈S1,

π{α1}, δ′

1,(t2, t2)= IndH

HP(δ′

1◦φ(t2,t2))q16= 1, t2∈S1.

The action of GPon these families is such that s4(t2, t2) = (t−1

2, t−1

2), while (−1,−1) ∈

GPsimultaneously exchanges δ1with δ′

1and (t2, t2) with (−t2,−t2). A fundamental

domain for this action is {({α1}, δ1,(eiφ, eiφ )) : φ∈[0, π]}. For φ∈(0, π) these

points have a trivial stabilizer in GP, so the corresponding H-representations are

irreducible. On the other hand, the element s4∈ GPﬁxes the points with φ= 0

or φ=π, so the representations π({α1}, δ1,(1,1)) and π({α1}, δ1,(−1,−1)) can be

reducible. Whether or not this happens depends on more subtle relations between

q0, q1and q2.

•P={α2}

XP=X/Zα3∼

=Zα2, XP=X/Zα2∼

=Zα3, YP=Zα∨

1, Y P=Zα∨

3/2,

R0={±α2}, R∨

0={±α∨

2}, W (RP) = {e, s2},

TP={t∈T:t2= 1}, TP={t∈T:t3= 1},GP={e, s3},

HP∼

=H(RP, q(s2) = q2, qα∨

2=q0),HP∼

=HP⊗C[XP].

The root datum RPis of type A(1)

1, which diﬀers from C(1)

1in the sense that XP

is the root lattice. There are two orbits of residual points: W(RP)(1, q1/2

0q1/2

2) and

W(RP)(1,−q1/2

0q−1/2

2). That is, these points are residual unless they equal (1,1) or

(1,−1). Both orbits admit a unique discrete series representation, of dimension one,

which denote by δ+or δ−. Like for P={α1}, the explicit formulas depend on which

of the AP-characters are in T−−

P. Again we ﬁnd two families of H-representations:

π{α2}, δ+,(t3,1)= IndH

HP(δ+◦φ(t3,1))q0q26= 1, t3∈S1,

π{α2}, δ−,(t3,1)= IndH

HP(δ−◦φ(t3,1))q0/q26= 1, t3∈S1.

The group GPacts on these families by s3(t3,1) = (t−1

3,1). A fundamental domain is,

for both families, given by t∈ {eiφ :φ∈[0, π]}. For φ∈(0,1π) the representations

π{α2}, δ+,(eiφ,1), because the isotropy group in GPis trivial. For φ∈ {0, π}

the intertwining operator associated to s3∈ GPis not necessarily scalar, so we ﬁnd

either one or two irreducible constituents. Remarkably enough, this depends not

only on the parameters q0and q2of HP, but also on q1, as we will see later.

•P={α2}=F0

Here simply HP=HP=H(R, q). We have to determine all residual points, and

how many inequivalent discrete series they carry. The former can be done by hand,

but that is quite elaborate. It is more convenient to use [Opd2, Theorem 7.7] and

[HeOp, Proposition 4.2], which say that for generic qthere are 40 residual points.

Representatives for the 5 W0-orbits are

r1(q) = (q1/2

0q1/2

2q−1

1, q1/2

0q1/2

2),

r2(q) = (q−1/2

0q−1/2

2q−1

1, q−1/2

0q−1/2

2),

r3(q) = (−q1/2

0q−1/2

2q−1

1,−q1/2

0q−1/2

2),

r4(q) = (−q−1/2

0q1/2

2q−1

1,−q1/2

0q−1/2

2),

r5(q) = (−q1/2

0q−1/2

2, q−1/2

0q−1/2

2).

Since every W0-orbit of residual points carries at least one discrete series represen-

tation,

dimQG0

Q(H(R, q))/G1

Q(H(R, q))= dimQG0

Q(S(R, q))/G1

Q(S(R, q))≥5.

On the other hand one can easily check, for example with the calculations for

P=∅, q = 1, that dimQG0

Q(We)/G1

Q(We)= 5. With (3.37) we deduce that

every W ri(q) carries a unique discrete series representation δ(ri). So far for generic

parameter functions.

For nongeneric qthe W ri(q) are still the only possible residual points, but some

of them may cease to be residual for certain q. In such cases ri(q) is absorbed by

the tempered part r T P

un of some onedimensional residual coset r T P. (If ri(q) is

absorbed by Tun, which is the tempered part of the twodimensional residual coset

T, then ri(q) is also absorbed by a onedimensional residual coset.) This happens in

the following cases:

residual point qabsorbed by

r1(q)q0q2=q1W0(q1/2

1, q−1/2

1)Tα1

q0q2=q2

1W0(1, q1/2

0q1/2

2)Tα2

r2(q)q0q2=q−1

1W0(q1/2

1, q−1/2

1)Tα1

q0q2=q−2

1W0(1, q1/2

0q1/2

2)Tα2

r3(q)q0/q2=q1W0(q1/2

1, q−1/2

1)Tα1

q0/q2=q2

1W0(1,−q1/2

0q−1/2

2)Tα2

r4(q)q0/q2=q−1

1W0(q1/2

1, q−1/2

1)Tα1

q0/q2=q−2

1W0(1,−q1/2

0q−1/2

2)Tα2

r5(q)q0=q2W0(1, q1/2

0q1/2

2)Tα2

q0q2= 1 W0(1,−q1/2

0q−1/2

2)Tα2

It is also possible that two orbits of residual points conﬂuence, but stay residual.

The deep result [OpSo2, Theorem 3.4] says that in general situations of this type

the discrete series representations with conﬂuencing central character do not merge

and remain irreducible.

The geometric content of the Aubert–Baum–Plymen conjecture is best illustrated

with some pictures of the tempered dual of H(R, q), for various q. Of course Thas

real dimension four, so we cannot draw it. But the unitary principal series can be

parametrized by Tun/W0, which is simply a 45-45-90 triangle.

The other components of Irr(S(R, q)) will lie close to Tun/W0if qis close to 1,

which we will assume in our pictures. We indicate what conﬂuence occurs when

qis scaled to 1 by drawing any π∈Irr(S(R, q)) close to the unitary part of its

central character. To distinguish the three onedimensional components, we denote

the series obtained from inducing δ1/δ+/δ−by L1/L+/L−.

Finally, we have to represent graphically how many inequivalent irreducible rep-

resentations a given parabolically induced representation π(ξ) contains. By default,

π(ξ) is itself irreducible. When π(ξ) contains two diﬀerent irreducibles, we draw the

corresponding point fatter. When there are more than two, we write the number of

irreducibles next to it.

q = 1

1q q = 1

0 2

q = q = 1

q = q = q

2 1

q = q = q

1/2

q q = q = 1

2 1 0

q = q , q = 1

2 1 q = 1

−

q generic

Almost everything in this picture can be deduced from the above calculations,

the ABP-conjecture (or rather Theorem 5.4.2) and [OpSo2, Theorem 3.4]. The

only thing that cannot be detected with these methods is what happens at the

conﬂuences r1(q)→(q1/2

1, q−1/2

1)∈L1for q0=q2=q1/2

1and r1(q)→(1, q2)∈L+

for q0=q1=q2. For these qthere are four unitary induction data that give rise to

representations with central character in Trs/W0. So three of them are irreducible,

and one contains two diﬀerent irreducibles. We can see that the reducibility does not

occur in the unitary principal series or in the discrete series, which leaves the two

intermediate series. Fortunately one can explicitly determine all subrepresentations

of π{α1}, δ1,(1,1)(for q0=q2=q1/2

1) and of π{α2}, δ+,(1,1)(for q0=q1=q2),

see [Slo1, Section 8.1.2]. In fact the graded Hecke algebras corresponding to these

parameter functions are precisely the ones assciated to Sp4and to SO5. Hence

one may also determine the reducibilty of the aforementioned representations via

the Deligne–Langlands–Kazhdan–Lusztig parametrization. Thirdly, it is possible to

analyse these parabolically induced representations with R-groups, as in [DeOp2].

The comparison of the tempered duals for diﬀerent parameter functions clearly

shows that this aﬃne Hecke algebra behaves well with respect to general parameter

deformations, not necessarily of the form q7→ qǫ. We see that for small pertubations

of qit is always possible to ﬁnd regions U/W0⊂T /W0such that the number of

tempered irreducibles with central character in U/W0remains stable. (The W0-

types of these representations can change however.) It is reasonable to expect that

something similar holds for general aﬃne Hecke algebras, probably that would follow

from the existence of an appropriate asymptotic Hecke algebra [Lus8].

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Topological K-theory of affine Hecke algebras

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Maarten Solleveld

Let H(R,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of H(R,q), that is, the K-theory of its C*-completion C*_r (R,q). We will prove that

K_* (C*_r (R,q))

does not depend on the parameter q. For this we use representation theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras. Thus, for the computation of these K-groups it suffices to work out the case q=1. These algebras are considerably simpler than for q not 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine

K_* (C*_r (R,q))

for all classical root data R, and for some others as well. This will be useful to analyse the K-theory of the reduced C*-algebra of any classical p-adic group. For the computations in the case q=1 we study the more general situation of a finite group \Gamma acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product

C(M) \rtimes \Gamma

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Article

Full-text available

Feb 2024

Maarten Solleveld

Consider a reductive p -adic group G , its (complex-valued) Hecke algebra \mathcal H (G) , and the Harish-Chandra–Schwartz algebra \mathcal S (G) . We compute the Hochschild homology groups of \mathcal H (G) and of \mathcal S (G) , and we describe the outcomes in several ways. Our main tools are algebraic families of smooth G -representations. With those we construct maps from HH_{n} (\mathcal H (G)) and HH_{n} (\mathcal S(G)) to modules of differential n -forms on affine varieties. For n = 0 , this provides a description of the cocentres of these algebras in terms of nice linear functions on the Grothendieck group of finite length (tempered) G -representations. It is known from [J. Algebra 606 (2022), 371–470] that every Bernstein ideal \mathcal H (G)^{\mathfrak s} of \mathcal H (G) is closely related to a crossed product algebra of the form \mathcal O (T)\rtimes W . Here \mathcal O (T) denotes the regular functions on the variety T of unramified characters of a Levi subgroup L of G , and W is a finite group acting on T . We make this relation even stronger by establishing an isomorphism between HH_{*} (\mathcal H (G)^{\mathfrak s}) and HH_{*} (\mathcal O (T)\rtimes W) , although we have to say that in some cases it is necessary to twist \mathbb{C} [W] by a 2-cocycle. Similarly, we prove that the Hochschild homology of the two-sided ideal \mathcal S (G)^{\mathfrak s} of \mathcal S (G) is isomorphic to HH_{*} (C^{\infty} (T_{u})\rtimes W) , where T_{u} denotes the Lie group of unitary unramified characters of L . In these pictures of HH_{*} (\mathcal H (G)) and HH_{*} (\mathcal S (G)) , we also show how the Bernstein centre of \mathcal H (G) acts. Finally, we derive similar expressions for the (periodic) cyclic homology groups of \mathcal H (G) and of \mathcal S (G) and we relate that to topological K-theory.

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We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r,p,n) and the corresponding cyclotomic Hecke algebras Hr,p,n, can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of simply connected affine Hecke algebras by using an extended form of Clifford theory. This extension of Clifford theory is given in the Appendix.

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Hecke algebras.- Affine Weyl groups and affine Hecke algebras.- A generalized two-sided cell of an affine Weyl group.- qs-analogue of weight multiplicity.- Kazhdan-Lusztig classification on simple modules of affine Hecke algebras.- An equivalence relation in T x ?*.- The lowest two-sided cell.- Principal series representations and induced modules.- Isogenous affine Hecke algebras.- Quotient algebras.- The based rings of cells in affine Weyl groups of type .- Simple modules attached to c 1.

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This expository note will state the ABP (Aubert-Baum-Plymen) conjecture. The conjecture can be stated at four levels: 1. K-theory of C*-algebras 2. Periodic cyclic homology of finite type algebras 3. Geometric equivalence of finite type algebras 4. Representation theory. The emphasis in this note will be on representation theory.

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Je suis parvenu dans ma these a construire une k-theorie bivariante pour les algebres de banach. Cela m'a permis de demontrer la conjecture de baum-connes pour les groupes de lie semi-simples et les groupes reductifs p-adiques, ainsi que pour les sous-groupes discrets de type fini de sl#3(f), avec f une extension finie de q#p et pour les sous-groupes discrets cocompacts de sp(n, 1), sl#3(r) et sl#3(c). J'ai aussi demontre une conjecture analogue a la conjecture de baum-connes pour les algebres l#1 (et plus generalement toutes les bonnes completions) des sous-groupes fermes des groupes de lie semi-simples et des groupes reductifs p-adiques.

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