Content uploaded by Xuhua He
Author content
All content in this area was uploaded by Xuhua He on Jan 31, 2015
Content may be subject to copyright.
arXiv:1310.3940v1 [math.RT] 15 Oct 2013
P-ALCOVES, PARABOLIC SUBALGEBRAS AND
COCENTERS OF AFFINE HECKE ALGEBRAS
XUHUA HE AND SIAN NIE
Abstract. This is a continuation of the sequence of papers [8], [6]
in the study of the cocenters and class polynomials of affine Hecke
algebras Hand their relation to affine Deligne-Lusztig varieties.
Let wbe a P-alcove element, as introduced in [3] and [4]. In this
paper, we study the image of Twin the cocenter of H. In the
process, we obtain a Bernstein presentation of the cocenter of H.
We also obtain a comparison theorem among the class polynomials
of Hand of its parabolic subalgebras, which is analogous to the
Hodge-Newton decomposition theorem for affine Deligne-Lusztig
varieties. As a consequence, we present a new proof of [3] and [4]
on the emptiness pattern of affine Deligne-Lusztig varieties.
Introduction
0.1. The purpose of this paper is twofold. We use some ideas arising
from affine Deligne-Lusztig varieties to study affine Hecke algebras, and
we apply the results on affine Hecke algebras to affine Deligne-Lusztig
varieties.
For simplicity, we only discuss the equal-parameter case in the in-
troduction. The case of unequal parameters and the twisted cocenters
will also be presented in this paper.
Let R= (X, R, Y, R∨, F0) be a based root datum and let ˜
Wbe the
associated extended affine Weyl group. An affine Hecke algebra His
a deformation of the group algebra of ˜
W. It is a free Z[v, v−1]-algebra
with basis {Tw}, where w∈˜
W. The relations among the Tware given
in §1.3. This is the Iwahori-Matsumoto presentation of H.
The cocenter ¯
H=H/[H,H] of His a useful tool in the study of the
representation theory and structure of p-adic groups. We will discuss
some applications of the cocenter as they serve as the motivation for
this paper.
Let R(H) be the Grothendieck group of representations of H. Then
the trace map T r :¯
H→R(H)∗relates the cocenter ¯
Hto the repre-
sentations of H. This map was studied in [1], [9].
In [8], we provide a standard basis of the cocenter ¯
H, which is con-
structed as follows. For each conjugacy class Oof ˜
W, we choose a
X.H. was partially supported by HKRGC grant 602011.
1
2 XUHUA HE AND SIAN NIE
minimal length representative wO. Then the image of TwOin ¯
His in-
dependent of the choice of wOand the set {TwO}, where Oranges over
all the conjugacy classes of ˜
W, is a basis of ¯
H. This is the Iwahori-
Matsumoto presentation of ¯
H.
Moreover, for any w∈˜
W,
Tw≡X
O
fw,OTwOmod [H,H]
for some fw,O∈N[v−v−1]. The coefficients fw,Oare called the class
polynomials.
In [6], the first-named author proved the “dimension=degree” the-
orem which relates the degrees of the class polynomials of Hto the
dimensions of the affine Deligne-Lusztig varieties of the corresponding
p-adic group G.
0.2. Let J⊂S0and let HJbe the corresponding parabolic subalgebra
of H. For a given w∈˜
W, we would like to express Twas an element
in HJ+ [H,H] for some J.
This is useful for the representation theory because a large number
of the representations of Hare built on the parabolically induced rep-
resentations IndH
HJ(−) for some J. It is also useful for the study of
affine Deligne-Lusztig varieties as one would like to compare the affine
Deligne-Lusztig varieties for Gand for the Levi subgroups of G.
We prove that
Theorem A. Let Pbe a (semistandard) parabolic subgroup of Gand
let wbe a P-alcove element of type J. Then Tw∈HJ+ [H,H].
The notion of P-alcove elements was introduced by G¨ortz, Haines,
Kottwitz, and Reuman in [3] and generalized in [4]. Roughly speaking,
wis a P-alcove element if the finite part of wlies in the finite Weyl
group of Pand it sends the fundamental alcove to a certain region of
the apartment. See [3, Section 3] for a visualization.
0.3. Let Obe a conjugacy class of ˜
Wand wObe a minimal length
element of O. We may regard wOas a P-alcove element for some P.
In this case, we have a sharper result:
Theorem B. Let Obe a conjugacy class of ˜
Wand let J⊂S0be such
that O∩˜
WJcontains an elliptic element of ˜
WJ. Then
TwO≡TJ
ymod [H,H]
for some y∈O∩˜
WJof minimal length (with respect to the length func-
tion on ˜
WJ) in its ˜
WJ-conjugacy class. Here TJ
yis the corresponding
Iwahori-Matsumoto element in HJ.
The description of the element TJ
yin Huses Bernstein presentation.
Thus Theorem B gives a Bernstein presentation of the cocenter ¯
H.
3
Notice that in the Bernstein presentation of the basis of ¯
H, there are
exactly Nelements that are not represented by elements in a proper
parabolic subalgebra of H, where Nis the number of elliptic conjugacy
classes of ˜
W. On the other hand, Opdam and Solleveld showed in [15,
Proposition 3.9] and [16, Theorem 7.1] that the dimension of the space
of “elliptic trace functions” on Halso equals N. It would be interesting
to relate these results via the trace map.
0.4. We may also compare the class polynomials of Hand of HJas
follows:
Theorem C. Let P=z−1PJzwith z∈JW0be a semistandard para-
bolic subgroup of Gand let ˜wbe a P-alcove element. Suppose that
T˜w≡X
O
f˜w,OTwOmod [H,H],
TJ
z˜wz−1≡X
O′
fJ
z˜wz−1,O′TJ
w′
Omod [HJ,HJ].
Then f˜w,O=PO′⊂OfJ
z˜wz−1,O′.
The Hodge-Newton decomposition theorem, which is proved in [3,
Theorem 1.1.4], says that if P=M N is a semistandard parabolic
subgroup of Gand ˜wis a P-alcove element, then the corresponding
affine Deligne-Lusztig varieties for the group Gand for the group M
are locally isomorphic.
Recall that there is a close relation between the class polynomials
and the affine Deligne-Lusztig varieties. Thus Theorem C above can be
regarded as an algebraic analog of the Hodge-Newton decomposition
theorem in [3].
Combining Theorem C with the “degree=dimension” Theorem, we
can derive an algebraic proof of [3, Theorem 1.1.2] and [4, Corollary
3.6.1] on the emptiness pattern of affine Deligne-Lusztig varieties.
1. Affine Hecke algebras
1.1. Let R= (X, R, Y, R∨, F0) be a based root datum, where R⊂X
is the set of roots, R∨⊂Yis the set of coroots and F0⊂Ris the set
of simple roots. By definition, there exist a bijection α7→ α∨from R
to R∨and a perfect pairing h,i:X×Y→Zsuch that hα, α∨i= 2
and the corresponding reflections sα:X→Xstabilizes Rand s∨
α:
Y→Ystabilizes R∨. We denote by R+⊂Rthe set of positive roots
determined by F0. Let X+={λ∈X;hλ, α∨i>0,∀α∈R+}.
The reflections sαgenerate the Weyl group W0=W(R) of Rand
S0={sα;α∈F0}is the set of simple reflections.
An automorphism of Ris an automorphism δof Xsuch that δ(F0) =
F0. Let Γ be a subgroup of automorphisms of R.
4 XUHUA HE AND SIAN NIE
1.2. Let V=X⊗ZR. For α∈Rand k∈Z, set
Hα,k ={v∈V;hv, α∨i=k}.
Let H={Hα,k;α∈R, k ∈Z}. Connected components of V− ∪H∈HH
are called alcoves. Let
C0={v∈V; 0 <hv, α∨i<1,∀α∈R+}
be the fundamental alcove.
Let W=ZR⋊W0be the affine Weyl group and S⊃S0be the
set of simple reflections in W. Then (W, S) is a Coxeter group. Set
˜
W= (X⋊W0)⋊Γ = X⋊(W0⋊Γ). Then Wis a subgroup of ˜
W.
Both Wand ˜
Wcan be regarded as groups of affine transformations of
V, which send alcoves to alcoves. For λ∈X, we denote by tλ∈W
the corresponding translation. For any hyperplane H=Hα,k ∈Hwith
α∈Rand k∈Z, we denote by sH=tkαsα∈Wthe reflection of V
along H.
For any ˜w∈˜
W, we denote by ℓ( ˜w) the number of hyperplanes in H
separating C0from ˜w(C0). By [10], the length function is given by the
following formula
ℓ(tχwτ ) = X
α,w−1(α)∈R+
|hχ, α∨i| +X
α∈R+,w−1(α)∈R−
|hχ, α∨i − 1|.
Here χ∈X,w∈W0and τ∈Γ.
If ˜w∈W, then ℓ( ˜w) is just the word length in the Coxeter system
(W, S). Let Ω = {˜w∈˜
W;ℓ( ˜w) = 0}. Then ˜
W=W⋊Ω.
1.3. Let q
1
2
s, s ∈Sbe indeterminates. We assume that q
1
2
s=q
1
2
tif s, t
are conjugate in ˜
W. Let A=Z[q1
2
s, q−1
2
s]s∈Sbe the ring of Laurant
polynomials in q
1
2
s, s ∈Swith integer coefficients.
The (generic) Hecke algebra Hassociated to the extend affine Weyl
group ˜
Wis an associative A-algebra with basis {T˜w; ˜w∈˜
W}subject
to the following relations
T˜xT˜y=T˜x˜y,if ℓ(˜x) + ℓ(˜y) = ℓ(˜x˜y);
(Ts−q
1
2
s)(Ts+q−1
2
s) = 0,for s∈S.
If q
1
2
s=q
1
2
tfor all s, t ∈S, then we call Hthe (generic) Hecke algebra
with equal parameter.
This is the Iwahori-Matsumoto presentation of H. It reflects the
structure of (quasi) Coxeter group ˜
W.
1.4. In this section, we recall the Bernstein presentation of H. It is
used to construct a basis of the center of Hand is useful in the study
of representations of H.
5
For any λ∈X, we may write λas λ=χ−χ′for χ, χ′∈X+. Now
set θλ=TχT−1
χ′. It is easy to see that θλis independent of the choice
of χ, χ′. The following results can be found in [12].
(1) θλθλ′=θλ+λ′for λ, λ′∈X.
(2) The set {θλTw;λ∈X, w ∈W0}and {Twθλ;λ∈X, w ∈W0}are
A-basis of H.
(3) For λ∈X+, set zλ=Pλ′∈W·λθλ′. Then zλ, λ ∈X+is an A-basis
of the center of H.
(4) θχTsα−Tsαθsα(χ)= (q1
2
sα−q−1
2
sα)θχ−θsα(χ)
1−θ−αfor α∈F0such that
α∨/∈2Yand χ∈X.
The following special cases will be used a lot in this paper.
(5) Let α∈F0and χ∈X. If hχ, α∨i= 0, then θχTsα=Tsαθχ.
(6) Let α∈F0and χ∈X. If hχ, α∨i= 1, then θsα(χ)=T−1
sαθχT−1
sα.
1.5. For any J⊂S0, let RJbe the set of roots spanned by αfor
α∈Jand R∨
Jbe the set of coroots spanned by α∨for α∈J. Let
RJ= (X, RJ, Y, R∨
J, J) be the based root datum corresponding to J.
Let WJ⊂W0be the subgroup generated by sαfor α∈Jand set
˜
WJ= (X⋊WJ)⋊ΓJ. Here ΓJ={δ∈Γ; δ(RJ) = RJ}. As in §1.2, we
set HJ={Hα,k ∈H;α∈RJ, k ∈Z}and CJ={v∈V; 0 <hv, α∨i<
1, α ∈R+
J}. For any ˜w∈˜
WJ, we denote by ℓJ( ˜w) the number of
hyperplanes in HJseparating CJfrom ˜wCJ.
We denote by ˜
WJ(resp. J˜
W) the set of minimal coset representatives
in ˜
W /WJ(resp. WJ\˜
W). For J, K ⊂S0, we simply write ˜
WJ∩K˜
W
as K˜
WJ.
Let HJ⊂Hbe the subalgebra generated by θλfor λ∈Xand Tw
for w∈WJ⋊ΓJ. We call HJa parabolic subalgebra of H.
It is known that HJis the Hecke algebra associated to the extend
affine Weyl group ˜
WJand the parameter function p
1
2
t, where tranges
over simple reflections in ˜
WJ. The parameter function p
1
2
tis determined
by q1
2
s(see [15, 1.2]). We denote by {TJ
˜w}˜w∈˜
WJthe Iwahori-Matsumoto
basis of HJ.
2. The Iwahori-Matsumoto presentation of ¯
H
2.1. We follow [8].
For w, w′∈˜
Wand s∈S, we write ws
−→ w′if w′=sws and ℓ(w′)6
ℓ(w). We write w→w′if there is a sequence w=w0, w1,··· , wn=w′
of elements in ˜
Wsuch that for any k,wk−1
s
−→ wkfor some s∈S.
We write w≈w′if w→w′and w′→w. It is easy to see that
w≈w′if w→w′and ℓ(w) = ℓ(w′).
We call ˜w, ˜w′∈˜
Welementarily strongly conjugate if ℓ( ˜w) = ℓ( ˜w′)
and there exists x∈Wsuch that ˜w′=x˜wx−1and ℓ(x˜w) = ℓ(x) + ℓ( ˜w)
or ℓ( ˜wx−1) = ℓ(x) + ℓ( ˜w). We call ˜w, ˜w′strongly conjugate if there
6 XUHUA HE AND SIAN NIE
is a sequence ˜w= ˜w0,˜w1,··· ,˜wn= ˜w′such that for each i, ˜wi−1is
elementarily strongly conjugate to ˜wi. We write ˜w∼˜w′if ˜wand ˜w′
are strongly conjugate. We write ˜w˜∼˜w′if ˜w∼δ˜w′δ−1for some δ∈Ω.
Now we recall one of the main results in [8].
Theorem 2.1. Let Obe a conjugacy class of ˜
Wand Omin be the set of
minimal length elements in O. Then
(1) For any ˜w′∈O, there exists ˜w′′ ∈Omin such that ˜w′→˜w′′.
(2) Let ˜w′,˜w′′ ∈Omin, then ˜w′˜∼˜w′′ .
2.2. Let h, h′∈H, we call [h, h′] = hh′−h′hthe commutator of hand
h′. Let [H,H] be the A-submodule of Hgenerated by all commutators.
We call the quotient H/[H,H] the cocenter of Hand denote it by ¯
H.
It follows easily from definition that T˜w≡T˜w′mod [H,H] if ˜w˜∼˜w′.
Hence by Theorem 2.1 (2), for any conjugacy class Oof ˜
Wand ˜w, ˜w′∈
Omin,T˜w≡T˜w′mod [H,H]. We denote by TOthe image of T˜win ¯
H
for any ˜w∈Omin.
Theorem 2.2. (1) The elements {TO}, where Oranges over all the
conjugacy classes of ˜
W, span ¯
Has an A-module.
(2) If q
1
2
s=q
1
2
tfor all s, t ∈S, then {TO}is a basis of ¯
H.
We call {TO}the Iwahori-Matsumoto presentation of the cocenter ¯
H
of affine Hecke algebra H.
The equal parameter case was proved in [8, Theorem 5.3 & Theorem
6.7]. Part (1) for the unequal parameter case can be proved in the same
way as in loc. cit. We expect that Part (2) remains valid for unequal
parameter case. One possible approach is to use the classification of
irreducible representations and a generalization of density theorem. We
do not go into details in this paper.
3. Some length formulas
3.1. The strategy to prove Theorem B in this paper is as follows. For
a given conjugacy class O, we
•construct a minimal length element in O, which is used for the
Iwahori-Matsumoto presentation of ¯
H;
•construct a suitable J, and an element in O∩˜
WJ, of minimal
length in its ˜
WJ-conjugacy class, which is used for the Bernstein
presentation of ¯
H;
•find the explicit relation between the two different elements.
To do this, we need to relate the length function on ˜
Wwith the
length function on ˜
WJfor some J⊂S0. This is what we will do in
this section. Another important technique is the “partial conjugation”
method introduced in [5], which will be discussed in the next section.
7
3.2. Let n=♯(W0⋊Γ). For any ˜w∈˜
W, ˜wn=tλfor some λ∈X.
We set ν˜w=λ/n ∈Vand call it the Newton point of ˜w. Let ¯ν˜wbe
the unique dominant element in the W0-orbit of ν˜w. Then the map
˜
W→V, ˜w7→ ¯ν˜wis constant on the conjugacy class of ˜
W. For any
conjugacy class O, we set νO= ¯ν˜wfor any ˜w∈Oand call it the
Newton point of O.
For ˜w∈˜
W, set
V˜w={v∈V; ˜w(v) = v+ν˜w}.
By [8, Lemma 2.2], V˜w⊂Vis a nonempty affine subspace and ˜w V ˜w=
V˜w+ν˜w=V˜w. Let p:˜
W=X⋊(W0⋊Γ) →W0⋊Γ be the projection
map. Let ube an element in V˜w. By the definition of V˜w,Vp( ˜w)=
{v−u;v∈V˜w}. In particular, ν˜w∈Vp( ˜w).
Let E⊂Vbe a convex subset. Set HE={H∈H;E⊂H}and
WE⊂Wto be the subgroup generated by sHwith H∈HE. We say
a point p∈Eis regular in Eif for any H∈H,v∈Himplies that
E⊂H. Then regular points of Eform an open dense subset of V˜w.
For any λ∈V, set Jλ={s∈S0;s(λ) = λ}.
Proposition 3.1. Let ˜w∈˜
Wsuch that ¯
C0contains a regular point e
of V˜w. Then ˜wis of minimal length in its conjugacy class if and only
if it is of minimal length in its WV˜w-conjugacy class;
Proof. Note that for any x∈WV˜w,¯
C0contains a regular point of
V˜w=x−1V˜w=Vx−1˜wx, hence by [8, Proposition 2.5 & Proposition 2.8],
the minimal length of elements in the conjugacy class of ˜wequals
h¯ν˜w, ρ∨i+ min
C♯HV˜w(C, ˜wC) = h¯ν˜w, ρ∨i+ min
x∈WV˜w
♯HV˜w(xC0,˜wxC0)
=h¯ν˜w, ρ∨i+ min
x∈WV˜w
♯HV˜w(C0, x−1˜wxC0)
= min
x∈WV˜w
ℓ(x−1˜wx),
where Cranges over all connected components of V− ∪H∈H˜wHand
ρ∨=1
2Pα∈R+α∨.
Proposition 3.2. Let ˜w∈˜
Wsuch that ¯
C0contains a regular point of
V˜w. Let J⊂S0. Assume there exists z∈JW0such that z˜wz−1∈˜
WJ.
Then
ℓ( ˜w) = ℓJ(z˜wz−1) + h¯ν˜w,2ρ∨i − h¯νJ
z˜wz−1,2ρ∨
Ji,
where ¯νJ
z˜wz−1denotes the unique J-dominant element in the WJ-orbit
of νz˜wz−1and ρ∨
J=1
2Pα∈R+
Jα∨. In particular, if J⊂Jνz˜wz−1, we have
ℓ( ˜w) = ℓJ(z˜wz−1) + h¯ν˜w,2ρ∨i.
Proof. By [8, Proposition 2.8] we have
ℓ( ˜w) = h¯ν˜w,2ρ∨i+♯HV˜w(C, ˜wC ),
8 XUHUA HE AND SIAN NIE
where Cis the connected component of V− ∪H∈HV˜wHcontaining C0.
Since z∈JW0,zC0⊂CJand hence ¯
CJcontains a regular point of
zV ˜w=Vz˜wz −1. Applying [8, Proposition 2.8] to ℓJinstead of ℓwe
obtain
ℓJ(z˜wz−1) = h¯νJ
˜w,2ρ∨
Ji+♯HJ∩HVz˜wz−1(C′, z ˜wz−1C′)
=h¯νJ
˜w,2ρ∨
Ji+♯HVz˜wz−1(C′, z ˜wz−1C′),
where C′is the connected component of V−∪H∈HVz˜wz−1Hcontaining CJ
and the second equality follows from the fact that HVz˜wz−1⊂HJ. Since
zC =C′, the map H7→ zH induces a bijection between HV˜w(C, ˜wC)
and HVz˜wz−1(C′, z ˜wz−1C′).
Lemma 3.3. Let J⊂S0and z∈JW0. Let s∈Sand t=zsz−1.
(1) If t∈˜
WJ, then ℓJ(t) = 1.
(2) If t /∈˜
WJ, then zs =xz′for some x∈˜
WJwith ℓJ(x) = 0 and
z′∈JW0.
Proof. Assume s=sHis the reflection along some hyperplane H∈H.
Since s∈S,¯
C0contains some regular point of H. Since z∈JW0,
zC0⊂CJ. If t∈˜
WJ, then ¯
CJcontains some regular point of H′=zH
and hence t=sH′is of length one with respect to ℓJ.
If t∈W0−WJ, then s∈S0and zs ∈JW0. In this case, z′=zs and
x= 1. If t /∈˜
WJ∪W0, then s=tθsθfor some maximal coroot θ∨with
z(θ)/∈RJ. Then zs =tz(θ)uz′for some u∈WJand z′∈JW0. Let
α∈R+
J. Since z′, z ∈JW0and that θ∨is a maximal coroot, we have
hz(θ), α∨i=1,if u−1(α)<0;
0,Otherwise.
In other words, ℓJ(tz(θ)u) = 0.
Corollary 3.4. Let ˜w′∈˜
Wand z′∈JW0such that z′˜wz′−1∈˜
WJ.
Let s∈Ssuch that ˜w′and ˜w=s˜w′sare of the same length. Let zbe
the unique minimal element of the coset WJz′p(s). Then z˜wz−1and
z′˜w′z′−1belong to the same ˜
WJ-conjugacy class and
ℓJ(z˜wz−1) = ℓJ(z′˜w′z′−1).
Proof. The first statement follow form the construction of z.
Without loss of generality, we may assume that ˜w′s > ˜w′> s ˜w′.
Let t=z′sz′−1. If t∈˜
WJ, then ℓJ(t) = 1 by Lemma 3.3. Since
˜w′> s ˜w′, the reflection hyperplane H∈Hof sseparates C0from ˜w′C0.
Hence z′Hseparates CJfrom z′˜w′z′−1CJsince z′C0⊂CJ, which means
that z′˜w′z′−1> tz′˜w′z′−1. Similarly, z′˜w′z′−1t > z′˜w′z′−1. Therefore
ℓJ(z˜wz−1) = ℓJ(tz′˜w′z′−1t) = ℓJ(z′˜w′z′−1).
If t /∈˜
WJ, then zs =x−1z′for some x∈˜
WJwith ℓJ(x) = 0. Hence
z˜wz−1=x−1z′˜w′z′−1xand ℓJ(z˜wz−1) = ℓJ(z′˜w′z′−1).
9
Proposition 3.5. Let Obe a conjugacy class of ˜
Wand J⊂S0such
that O∩˜
WJ6=∅. Let ˜w∈Omin and z∈JW0, such that z˜wz−1∈
˜
WJ. Then z˜wz−1is of minimal length (with respect to ℓJ) in its ˜
WJ-
conjugayc class.
Proof. By [8, Proposition 2.5 & Lemma 2.7], there exists ˜w→˜w′∈
Omin such that ¯
C0contains a regular point of V˜w′. By Corollary 3.4, it
suffices to consider the case that ¯
C0contains a regular point of V˜w. By
Proposition 3.1 and Proposition 3.2,
ℓJ(z˜wz−1) = min
x∈WV˜w
ℓJ(zx ˜wx−1z−1) = min
y∈WVz˜wz−1
ℓJ(yz ˜wz−1y−1).
Note that ¯
CJcontains a regular point of Vz˜wz−1. Applying Proposition
3.1 to ℓJand z˜wz−1we obtain the desired result.
4. A family of partial conjugacy classes
4.1. In this section, we consider an arbitrary Coxeter group (W, S).
Let T=∪w∈WwSw−1⊂Wbe the set of reflections in W. Let R=
{±1} × T. For s∈S, define Us:R→Rby Us(ǫ, t) = (ǫ(−1)δs,t , sts).
Let Γ be a subgroup of automorphisms of Wsuch that δ(S) = Sfor
all δ∈Γ. Let ˜
W=W⋊Γ. For any δ∈Γ, define Uδ:R→Rby
Uδ(ǫ, t) = (ǫ, δ(t)). Then UδUsUδ−1=Uδ(s)for s∈Sand δ∈Γ.
We have the following result.
Proposition 4.1. (1) There is a unique homomorphism Uof ˜
Winto
the group of permutations of Rsuch that U(s) = Usfor all s∈Sand
U(δ) = Uδfor all δ∈Γ.
(2) For any w∈˜
Wand t∈T,tw < w if and only if for ǫ=±1,
U(w−1)(ǫ, t) = (−ǫ, w−1tw).
The case Γ = {1}is in [13, Proposition 1.5 & Lemma 2.2]. The
general case can be reduced to that case easily.
4.2. Let J⊂S. We consider the action of WJon ˜
Wby w·w′=
ww′w−1for w∈WJand w∈˜
W. Each orbit is called a WJ-conjugacy
class or a partial conjugacy class of ˜
W(with respect to WJ). We set
ΓJ={δ∈Γ; δ(J) = J}.
Lemma 4.2. Let I⊂Sand w∈WI⋊ΓI. Then wis of minimal
length in its WI-conjugacy class if and only if wis of minimal in its
W-conjugacy class.
Proof. The “if” part is trivial.
Now we show the “only if” part. Suppose that wis a minimal length
element in its WI-conjugacy class. An element in the W-conjugacy
class of wis of the form xwx−1for some x∈W. Write x=x1y, where
10 XUHUA HE AND SIAN NIE
x1∈WIand y∈WI. Then xwx−1=x1(ywy−1)x−1
1. Here ywy−1∈WI
is in the WI-conjugacy class of w. Hence ℓ(ywy−1)>ℓ(y). Now
ℓ(xwx−1)>ℓ(x1(ywy−1)) −ℓ(x1) = ℓ(x1) + ℓ(ywy−1)−ℓ(x1)
=ℓ(ywy−1)>ℓ(w).
Thus wis a minimal length element in its W-conjugacy class.
4.3. In general, a WJ-conjugacy class in ˜
Wmay not contain any ele-
ment in WJ⋊ΓJ. To study the minimal length elements in this partial
conjugacy class, we introduce the notation I(J, w).
For any w∈J˜
W, set
I(J, w) = max{K⊂J;wK w−1=K}.
Since w(K1∪K2)w−1=wK1w−1∪wK2w−1,I(J, w) is well-defined.
We have that
(a) I(J, w) = ∩i>0w−iJwi.
Set K=∩i>0w−iJwi. Let s∈I(J, w). Then wisw−i∈I(J, w)⊂J
for all i. Thus s∈K. On the other hand, wKw−1⊂K. Since Kis a
finite set, wKw−1=K. Thus K⊂I(J, w).
(a) is proved.
Lemma 4.3. Let w∈J˜
Wand x∈WJ. Then x∈WI(J,w)if and only
if w−ixwi∈WJfor all i∈Z.
Proof. If x∈WI(J,w), then w−1xw ∈Ww−1(I(J,w)) =WI(J,w). So
w−ixwi∈WJfor all i∈Z.
Suppose that w−ixwi∈WJfor all i∈Z. We write xas x=ux1,
where u∈WJ∩w−1Jw and x1∈J∩w−1J w W. By [14, 2.1(a)], wx1∈JW
and wx ∈WJ(wx1). Since wxw−1∈WJ,wx ∈WJw. Therefore
WJ(wx1)∩WJw6=∅and wx1=w. So x1= 1 and x∈WJ∩w−1J w .
Applying the same argument, x∈W∩i>0w−iJwi=WI(J,w).
4.4. Similar to §2.1, for w, w′∈˜
W, we write w→Jw′if there is a
sequence w=w0, w1,··· , wn=w′of elements in ˜
Wsuch that for any
k,wk−1
s
−→ wkfor some s∈J. The notations ∼Jand ≈Jare defined
in a similar way.
The following result is proved in [5].
Theorem 4.4. Let Obe a WJ-conjugacy class of ˜
W. Then there exists
a unique element ˜w∈J˜
Wand a WI(J, ˜w)-conjugacy class Cof WI(J, ˜w)˜w
such that O∩WI(J, ˜w)˜w=C. In this case,
(1) for any ˜w′∈O, there exists x∈WI(J, ˜w)such that ˜w′→Jx˜w.
(2) for any two minimal length elements x, x′of C,x∼I(J, ˜w)x′.
Now we prove the following result.
Theorem 4.5. Let I⊂J⊂Sand let w, u ∈˜
Wbe of minimal length in
the same WJ-conjugacy class such that u∈J˜
W,w∈I˜
Wand wIw−1=
11
I. Then there exists h∈I(J,u)WJIsuch that hIh−1⊂I(J, u)and
hwh−1=u.
Remark. This result can be interpreted as a conjugation on a family
of partial conjugacy classes in the following sense. Let I1be the set
of WI-conjugacy classes that intersects WIwand I2be the set of WJ-
conjugacy classes that intersects WI(J,u)u. Then
(1) There is an injective map I1→I2which sends a WI-conjugacy
class O1in I1to the unique WJ-conjugacy class O2in I2that contains
O1.
(2) Conjugating by hsends O1∩WIwinto O2∩WI(J,u)u.
Proof. Let b=fǫ ∈WJwith f∈WJIand ǫ∈WIsuch that bwb−1=u.
We show that
(a) fwf −1=u.
Set x=ǫwǫ−1w−1. Then x∈WIand u=fxwf −1. Suppose that
x6= 1. Then sx < x for some s∈I. Set t=fsf −1. Since f∈WI,
tf =fs > f . Thus U(f−1)(ǫ, t) = (ǫ, f −1tf) = (ǫ, s). Since sx < x,
U(x−1)(ǫ, s) = (−ǫ, x−1sx). Notice that w∈IWwith wIw−1=I
and f−1∈IW. Thus wf −1∈IW. Hence U(f w−1)(−ǫ, x−1sx) =
(−ǫ, fw−1x−1sxwf −1).
Therefore U(fw−1x−1f−1)(ǫ, t) = (−ǫ, f w−1x−1sxwf−1) and
tfxwf−1=f(sx)wf −1< fxwf−1=u.
Applying this argument successively, we have fwf −1< u. This
contradicts our assumption that uis of minimal length in the WK-
conjugacy class containing fwf−1. Hence x= 1 and f wf −1=u.
(a) is proved.
Now we write fas f=πh with π∈WI(J,u)and h∈I(J,u)WJI.
Then w=h−1(π−1uπu−1)uh. Similar to the proof of (a), we have that
w>h−1uh. By our assumption, wis a minimal length element in the
WJ-conjugacy class of h−1uh. Thus w=h−1uh.
For any x∈WIand i∈Z,hwi=uihand
ui(hxh−1)u−i= (hwi)x(hwi)−1=h(wixw−i)h−1∈hWIh−1⊂WJ.
By Lemma 4.3, hxh−1∈WI(J,u). Thus hWIh−1⊂WI(J,u). Since
h∈I(J,u)WJI, we have hIh−1⊂I(J, u).
5. Bernstein presentation of the cocenter of H
5.1. We fix a conjugacy class Oof ˜
Wand will construct a subset Jof
S0, as small as possible, such that TwO∈HJ+ [H,H].
By [8, Proposition 2.5 & Lemma 2.7], there exists ˜w′∈Omin such
that ¯
C0contains a regular point e′of V˜w′. We choose v∈Vsuch that
V˜w′=V˜w′+vand v, ν ˜w′∈¯
Cfor some Weyl Chamber C. We write Jfor
JνO∩J¯v. Let z∈JW0with z(ν˜w′) = νOand z(v) = ¯v. Set ˜w0=z˜w′z−1.
12 XUHUA HE AND SIAN NIE
By Proposition 3.5, ˜w0is of minimal length (with respect to ℓJ) in its
˜
WJ-conjugacy class.
Unless otherwise stated, we keep the notations in the rest of this
section. The main result of this section is
Theorem 5.1. We keep the notations in §5.1. Then
TwO≡TJ
˜w0mod [H,H].
5.2. The idea of the proof is as follows.
Suppose ˜w0=tλ0w0and ˜w′=tλ′w′. Then we need to compare
TJ
tλ0and Ttλ′. Although λ0and λare in the same W0-orbit, the relate
between TJ
tλ0and Ttλ′is complicated. Roughly speaking, we write λ0
as λ0=µ1−µ2for J-dominant coweights, i.e., hµ1, αii,hµ2, αii>0 for
i∈J. Then
TJ
tλ0=TJ
tµ1(TJ
tµ2)−1=θµ1θ−1
µ2.
The right hand side is not easy to compute.
To overcome the difficulty, we replace ˜w0by another minimal length
element ˜w1in its ˜
WJ-conjugacy class whose translation part is J-
dominant and replace ˜w′by another minimal length element ˜w2in
Oand study the relation between ˜w1and ˜w2instead. The construction
of ˜w1and ˜w2uses “partial conjugation action”.
5.3. Recall that e′∈¯
C0is a regular element of V˜w′. Set e=z(e′).
Since e∈z(¯
C0) and is a regular point of V˜w0, we have
(1) 0 6|he, α∨i| 61 for any α∈R.
(2) he, α∨i>0 for any α∈R+
J.
(3) If e∈Hα,k for some α∈Rand k∈Z, then V˜w0⊂Hα,k and
α∈RJ. In particular, Je⊂Jand by (2), RJe={α∈R;he, α∨i= 0}.
By Theorem 4.4, the WJe-conjugacy class of ˜w0contains a minimal
length element ˜w1of the form ˜w1=tλw1x1with λ∈X,w1∈WJ⋊ΓJ
and x1∈WI(Je,tλw1)such that tλw1∈Je˜
Wand x1is of minimal length
in its Ad(w1)-twisted conjugacy class of WI(Je,tλw1).
By Theorem 4.4, there exists a minimal length element ˜w1in the
W0-conjugacy of ˜w0and ˜w′, which has the form ˜w2=t¯
λw2x2such that
t¯
λw2∈S0˜
Wand x2∈WI(S0,t¯
λw2). Since ˜w′∈Omin, ˜w1∈Omin.
We have the following results on λand econstructed above.
Lemma 5.2. Keep notations in §5.3. Then we have
(1) For any α∈R+,h˜w1(e), α∨i>−1.
(2) hλ, α∨i>−1for any α∈R+.
(3) hλ, α∨i>0for any α∈R+
J.
(4) For any α∈R+, if hλ, α∨i=−1, then he, α∨i<0.
Proof. For any α∈R+,
hλ, α∨i+hw1(e), α∨i=hλ+w1x1(e), α∨i=h˜w1(e), α∨i=he+νO, α∨i.
13
(1) By §5.3 (1), he, α∨i>−1. So he+νO, α∨i>−1. If he+
νO, α∨i=−1, then he, α∨i=−1. Therefore α∈R+
Jby §5.3 (3), which
contradicts §5.3 (2).
(2) By §5.3 (1), hw1(e), α∨i61. By (1), hλ, α∨i=h˜w1(e), α∨i −
hw1(e), α∨i>−2. Since hλ, α∨i ∈ Z,hλ, α∨i>−1.
(3) By §5.3 (2), 0 6he, α∨i=hνO+e, α∨i=hλ, α∨i+hw1(e), α∨i.
If hλ, α∨i<0, then by §5.3 (1), he, α∨i= 0 and hence α∈R+
Jeby §5.3
(4). Since tλw1∈Je˜
W,hλ, α∨i>0, which is a contradiction. Therefore
hλ, α∨i>0 for all α∈R+
J.
(4) Suppose that he, α∨i>0. Since hλ, α∨i=−1 and hw1(e), α∨i6
1, he+νO, α∨i60. Thus he, α∨i=hνO, α∨i= 0. Therefore α∈R+
Jby
§5.3 (3), which contradicts (3).
Lemma 5.3. We keep the notations in §5.3. Then
(1) ˜w1∈˜
WJis of minimal length (with respect to ℓJ) in its ˜
WJ-
conjugacy class.
(2) tλw1∈J˜
W.
Proof. Since WJe⊂WJfixes V˜w0, we have ˜w1∈˜
WJand V˜w0=V˜w1.
(1) Since ℓ( ˜w1)6ℓ( ˜w0), we have ℓJ( ˜w1)6ℓJ( ˜w0) by Proposition
3.2. By Proposition 3.5, ˜w0is a minimal length (with respect to ℓJ) in
its conjugacy class of ˜
WJ. So is ˜w1.
(2) It suffices to show that hλ, α∨i>1 for any α∈R+
Jwith w−1
1(α)<
0. Suppose that hλ, α∨i<1. By Lemma 5.2 (3), hλ, α∨i= 0. Hence
by §5.3 (2),
0>he, w−1
1(α∨)i=hw1(e), α∨i=h˜w1(e), α∨i=he+νO, α∨i>0.
Thus by §5.3 (2) again, he, α∨i= 0 and α∈R+
Je. However, tλw1∈Je˜
W
by our construction. Hence hλ, α∨i>1, which is a contradiction.
Combining Proposition 3.2 with Lemma 5.3, we obtain
Corollary 5.4. Keep notations in §5.3. Then
ℓ(z−1tλw1z) = hνO,2ρ∨i+hλ, 2ρ∨
Ji − ℓ(w1),
ℓ(z−1tλw1x1z) = hνO,2ρ∨i+hλ, 2ρ∨
Ji − ℓ(w1) + ℓ(x1).
Lemma 5.5. Keep the notations in §5.3. Let y∈J¯
λW0be the unique
element such that y(λ) = ¯
λ. Then
(1) ℓ(yw1y−1) = 2ℓ(y) + ℓ(w1).
(2) h¯
λ, 2ρ∨i=hνO,2ρ∨i+hλ, 2ρ∨
Ji+ 2ℓ(y).
(3) yJλy−1⊂J¯
λ.
Proof. By definition, for any α∈R+,y(α)∈R+if and only if hλ, α∨i>
0. By Lemma 5.2 (2), ℓ(y) = ♯{α∈R+;hλ, α∨i=−1}.
(1) Let α∈R+such that w−1
1(α)<0. Then α∈R+
Jsince w1∈
WJ⋊ΓJ. Hence hλ, α∨i>0 by Lemma 5.2 (3). Therefore y(α)>0.
Hence ℓ(yw1) = ℓ(y) + ℓ(w1).
14 XUHUA HE AND SIAN NIE
To show ℓ(yw1y−1) = ℓ(yw1) + ℓ(y−1), we have to prove that for any
β∈R+with y(β)<0, we have yw1(β)∈R+.
Assume y(β)<0. Then hλ, β∨i<0. Thus hλ, β∨i=−1 by Lemma
5.2(2). Moreover, we have β /∈R+
Jand he, β∨i<0 by Lemma 5.2 (3)
and (4). Since w1∈WJ⋊ΓJ,w1(β)>0. By Lemma 5.2 (1),
−1<h˜w1(e), w(β∨)i=hλ, w1(β∨)i+hw1(e), w1(β∨)i=hλ, w1(β∨)i+he, β∨i
<hλ, w1(β∨)i.
Therefore hλ, w1(β∨)i>0 and yw1(β)∈R+.
(2) By Lemma 2.1 and Lemma 5.2(2), we have that
h¯
λ, 2ρ∨i=X
α∈R+
|hλ, α∨i| =X
α∈R+
hλ, α∨i+ 2♯{α∈R+;hλ, α∨i=−1}
=hλ, 2ρ∨i+ 2ℓ(y).
Since w1∈WJ⋊ΓJ,wk
1(ρ∨−ρ∨
J) = ρ∨−ρ∨
Jfor all i∈Z. Let
m=|WJ⋊ΓJ|. Then Pm
k=1 wk
1(λ) = mνOand
h¯
λ, 2ρ∨i − hλ, 2ρ∨
Ji= 2ℓ(y) + hλ, 2(ρ∨−ρ∨
J)i
= 2ℓ(y) + 1
m
m
X
k=1
hλ, 2w−k
1(ρ∨−ρ∨
J)i= 2ℓ(y) + 1
m
m
X
k=1
hwk
1(λ),2(ρ∨−ρ∨
J)i
= 2ℓ(y) + hνO,2(ρ∨−ρ∨
J)i= 2ℓ(y) + hνO,2ρ∨i.
(3) Notice that WJ¯
λyWJλ=WJ¯
λ(yWJλy−1)y=WJ¯
λyand y∈J¯
λW0,
we see that yis the unique minimal element of the double coset WJ¯
λyWJλ,
that is, y∈J¯
λW0Jλ. Moreover yWJλy−1⊂WJ¯
λ. Thus ysends simple
roots of Jλto simple roots of J¯
λ.
Proposition 5.6. Keep the notations in §5.3. Set I=yI (Je, tλw1)y−1.
Then there exists h∈I(J¯
λ,w2)WJ¯
λ
Isuch that
(1) hIh−1⊂I(J¯
λ, w2).
(2) w2=hyw1y−1h−1.
(3) Both w2and yw1y−1are of minimal lengths in their common
WJ¯
λ-conjugacy class.
(4) hy ˜w1y−1h−1∈Omin.
Remark. By Lemma 5.5 (3), I⊂y(Jλ)⊂J¯
λ. Moreover, we have
yw1y−1∈IW0and yw1y−1Iyw−1
1y−1=Iby the construction of w1.
Proof. By Theorem 4.4, there exists a minimal length element in the
W0-conjugacy class of tλw1of the form t¯
λuc, where u∈J¯
λ(W0⋊Γ) and
c∈WI(J¯
λ,u). Again by Theorem 4.4, there exists c′∈WI(J¯
λ,u)such that
uc′is of minimal length in the WJ¯
λ-conjugacy class of uc. Note that
t¯
λuc and t¯
λuc′are in the same WJ¯
λ-conjugacy class. So by the choice
of t¯
λuc, we have
ℓ(tλ)−ℓ(u) + ℓ(c) = ℓ(t¯
λuc)6ℓ(t¯
λuc′) = ℓ(tλ)−ℓ(u) + ℓ(c′),
15
that is, ℓ(c)6ℓ(c′). Hence ℓ(uc) = ℓ(u) + ℓ(c)6ℓ(u) + ℓ(c′) = ℓ(uc′).
Therefore
(a) uc is of minimal length in its WJ¯
λ-conjugacy class.
By Corollary 5.4, ℓ(z−1tλw1z) = hνO,2ρ∨i+hλ, 2ρ∨
Ji − ℓ(w1). Apply-
ing Lemma 5.5, we have
ℓ(yw1y−1) = 2ℓ(y) + hνO,2ρ∨i+hλ, 2ρ∨
Ji − ℓ(z−1tλw1z).
On the other hand, ℓ(t¯
λuc) = h¯
λ, 2ρ∨i − ℓ(u) + ℓ(c). Hence
ℓ(uc) = h¯
λ, 2ρ∨i+ 2ℓ(c)−ℓ(t¯
λuc).
Since t¯
λuc and t¯
λyw1y−1are in the same W0-conjugacy class, then uc
and yw1y−1are in the same WJ¯
λ-conjugacy class. By (a) and Lemma
5.5, we see that
06ℓ(yw1y−1)−ℓ(uc) = ℓ(t¯
λuc)−ℓ(z−1tλw1z)−2ℓ(c).
Notice that by our construction, t¯
λuc is of minimal length in the W0-
conjugacy class of z−1tλw1z. Hence c= 1 and ℓ(yw1y−1) = ℓ(u). By
(a), both uand yw1y−1are of minimal lengths in the their common
WJ¯
λ-conjugacy class.
By Proposition 4.5, there exists h∈I(J¯
λ,u)WJ¯
λ
Isuch that u=
hyw1y−1h−1and hI h−1⊂I(J¯
λ, u). Thus hy ˜w1y−1h−1∈t¯
λuWI(J¯
λ,u)=
t¯
λuWI(S0,t¯
λu). The W0-conjugacy class of ˜w1intersects both t¯
λuWI(S0,t¯
λu)
and t¯
λw2WI(S0,t¯
λw2). By Theorem 4.4, w2=u.
By definition, x1is a minimal length element in the Ad(w1)-twisted
conjugacy class by WI(Je,tλw1). Thus hyx1y−1h−1is of minimal length in
its Ad(w2)-twisted onjugacy class by WhIh−1. By Lemma 4.2, hyx1y−1h−1
is of minimal length in its Ad(w2)-twisted conjugacy class by WI(J¯
λ,w2).
Thus by Theorem 4.4, hy ˜w1y−1h−1=t¯
λw2(hyx1y−1h−1) is of minimal
length in the W0-conjugacy class of ˜w′. So hy ˜w1y−1h−1∈Omin.
5.4. Now we prove Theorem 5.1.
By Lemma 5.3, ˜w0and ˜w1are of minimal length (with respect to ℓJ)
in their ˜
WJ-conjugacy class. Hence by §2.2,
(a) TJ
˜w0≡TJ
˜w1=θλT−1
w−1
1
Tx1mod [HJ,HJ].
Let x′=yx1y−1∈WI⊂WJ¯
λand x′′ =hx′h−1∈WhIh−1. We show
that
(b) θλT−1
w−1
1
Tx1≡θ¯
λT−1
yw−1
1y−1Tx′mod [H,H].
Let y=sr···s1be a reduced expression. For each k, let αkbe the
positive simple root corresponding to skand let λk=sk···s1(λ). Since
ys1···sk−1(αk)<0, then
hλk−1, α∨
ki=hλ, s1···sk−1(α∨
k)i<0.
16 XUHUA HE AND SIAN NIE
By Lemma 5.2(2), hλk−1, α∨
ki=−1. By §1.4 (6), Tskθλk−1=θλkT−1
sk.
Applying it successively, we have that
Tyθλ=Tsr···Ts1θλ=θy(λ)T−1
sr···T−1
s1=θ¯
λT−1
y−1.
Since y∈J¯
λW0, we have ℓ(x′y) = ℓ(yx1) = ℓ(x′)+ℓ(y) = ℓ(y)+ℓ(x1).
By Lemma 5.5, ℓ(yw−1
1y−1) = 2ℓ(y) + ℓ(w1). Hence TyTx1T−1
y=Tx′
and TyTw−1
1Ty−1=Tyw−1
1y−1. Therefore
TyθλT−1
w−1
1
Tx1T−1
y=θ¯
λT−1
y−1T−1
w−1
1
Tx1T−1
y=θ¯
λ(T−1
y−1T−1
w−1
1
T−1
y)(TyTx1T−1
y)
=θ¯
λT−1
yw−1
1y−1Tx′.
(b) is proved.
Notice that h∈WJ¯
λ. By §1.4 (5), Thθ¯
λT−1
h=θ¯
λ. By Proposition
5.6,
ℓ(yw1y−1h−1) = ℓ(h−1w2) = ℓ(h−1) + ℓ(w2) = ℓ(yw1y−1) + ℓ(h−1).
Thus ThT−1
yw−1
1y−1T−1
h=T−1
w−1
2
. Since h∈I(J¯
λ,w2)WJ¯
λ
Iand h(I)⊂
I(J¯
λ, w2), we have that ℓ(x′′h) = ℓ(hx′) = ℓ(h) + ℓ(x′) = ℓ(x′′ ) + ℓ(h)
and ThTx′T−1
h=Tx′′ . So
Thθ¯
λTyw−1
1y−1Tx′T−1
h=θ¯
λ(ThT−1
yw−1
1y−1T−1
h)(ThTx′T−1
h)
=θ¯
λT−1
w−1
2
Tx′′ .
By Proposition 5.6, hy ˜w1y−1h−1=t¯
λw2x′′ and ˜w′are both of mini-
mal lengths in O. By Theorem 2.1 and §2.2,
(c) T˜w′≡Tt¯
λw2x′′ =θ¯
λT−1
w−1
2
Tx′′ ≡θ¯
λT−1
yw−1
1y−1Tx′mod [H,H].
Combining (a), (b) and (c),
TwO≡TJ
˜w0mod [H,H].
Example 5.7. Let’s consider the extended affine Weyl group ˜
Was-
sociated to GL8. Here ˜
W∼
=Z8⋊S8, where the permutation group
S8of {1,2,··· ,8}acts on Z8in a natural way. Let ˜u=tχσwith
χ= [χ1,··· , χ8] and σ∈S8. Then
ℓ(˜u) = X
i<j,σ(i)<σ(j)
|λi−λj|+X
i<j,σ(i)>σ(j)
|λi−λj−1|.
Take χ= [1,1,1,1,1,0,0,0] ∈Z8and x= (6,3,1)(7,4,8,5,2) ∈S8.
Then ˜w′=tχx∈S0˜
Wis an minimal length element in its conjugacy
class.
Let J={(1,2),(2,3),(4,5),(6,7),(7,8)} ⊂ S0and ˜w∈˜
WJ=Z8⋊
WJwith λ= [1,1,0,1,1,1,0,0] and w= (3,2,1)(7,5,8,6,4). Then
ℓJ( ˜w) = 0. In particular, ˜wis of minimal length (in the sense of ℓJ) in
its conjugacy class of ˜
WJ.
17
By Theorem 5.1,
T˜w′≡TJ
˜w=θλT−1
w−1mod [H,H].
5.5. We call an element w∈W0⋊Γelliptic if Vw⊂VW0and an
element ˜w∈˜
Welliptic if p( ˜w) is elliptic in W0⋊Γ. By definition, if ˜w
is elliptic, then ν˜w∈VW0.
A conjugacy class Oin W0⋊Γ or ˜
Wis called elliptic if ˜wis elliptic
for some (or, equivalently any) ˜w∈O.
Now we discuss the choice of vin §5.1. If we assume furthermore
that vis a regular point of Vp( ˜w′), then ¯v=z(v) is a regular point of
Vp( ˜w0). Thus Vp( ˜w0)⊂ ∩α∈RJHα,0=VWJ. Hence ˜w0is elliptic in ˜
WJ.
5.6. Let Obe a conjugacy class and ˜w, ˜w′∈Owith ν˜w=ν˜w′=νO.
Let x∈˜
Wsuch that x˜wx−1= ˜w′. Then x∈˜
WJO. In particular, the
set {˜w∈O∩˜
WJνO;ν˜w=νO}is a single ˜
WJνO-conjugacy class.
Let Abe the set of pairs (J, C), where J⊂S0,Cis an elliptic
conjugacy class of ˜
WJand ν˜wis dominant for some (or, equivalently
any) ˜w∈C. For any (J, C),(J′, C′)∈A, we write (J, C)∼(J′, C′) if
ν˜w=ν˜w′for ˜w∈Cand ˜w′∈C′and there exists x∈J′(WJν˜w
⋊ΓJν˜w)J
such that xJx−1=J′and xCx−1=C′.
Lemma 5.8. The map from Ato the set of conjugacy classes of ˜
W
sending (J, C)to the unique conjugacy class Oof ˜
Wwith C⊂Ogives
a bijection from A/∼to the set of conjugacy classes of ˜
W.
Proof. If (J, C)∼(J′, C ′), then Cand C′are in the same conjugacy
class of ˜
W. On the other hand, suppose that Cand C′are in the
same conjugacy class O. Let ˜w∈Cand Jν˜w. Then ν˜w∈VWJand
J⊂Jν˜w. Similarly, J′⊂Jν˜w. Then C, C′⊂ { ˜w1∈O∩˜
WJν˜w;ν˜w1=νO}
is in the same ˜
WK-conjugacy class. In particular, there exists x∈
WJν˜w
⋊ΓJν˜wsuch that x˜wx−1∈C′. Hence p(x˜wx−1) is an elliptic
element in WJ′⋊ΓJ′. By [2, Proposition 5.2], x=x′x1for some
x′∈WJ′,x1∈J′(WJν˜w
⋊ΓJν˜w)Jsuch that x1Jx−1
1=J′. Hence
x1˜
WJx−1
1=˜
WJ′and x1Cx−1
1=C′.
Now combining Theorem 5.1 and Theorem 2.2, we have
Theorem 5.9. (1) The elements {TJ
O}(J,O)∈A/∼span ¯
Has an A-module.
(2) If q
1
2
s=q
1
2
tfor all s, t ∈S, then {TJ
O}(J,O)∈A/∼is a basis of ¯
H.
This gives Bernstein presentation of the cocenter ¯
H.
6. P-alcove elements and the cocenter of H
6.1. For any α∈Rand an alcove C, let k(α, C) be the unique integer k
such that Clies in the region between the hyperplanes Hα,k and Hα,k−1.
For any alcoves Cand C′, we say that C>αC′if k(α, C)>k(α, C ′).
18 XUHUA HE AND SIAN NIE
Let J⊂S0and z∈W0. Following [4, §4.1], we say an element
˜w∈˜
Wis a (J, z)-alcove element1if
(1) z˜wz−1∈˜
WJand
(2) ˜wC0>αC0for all α∈z−1(R+−R+
J).
Note that if ˜wis a (J, z)-alcove element, then it is also a (J, uz)-alcove
element for any u∈WJ.
If ˜wis a (J, z)-alcove element, we may also call ˜waP-alcove ele-
ment, where P=z−1PJzis a semistandard parabolic subgroup of the
connected reductive group Gassociated to the root datum R.
Lemma 6.1. Let ˜w∈˜
Wbe a (J, z)-alcove and let s∈S.
(1) If ℓ( ˜w) = ℓ(s˜ws), then s˜ws is a (J, zp(s))-alcove element;
(2) If ˜w > s ˜ws, then zp(s)z−1∈WJ. Moreover, both s˜wand s˜ws
are (J, z)-alcove elements.
Remark. In part (2), s˜wand s˜ws are also (J, zp(s))-alcove elements.
Proof. Part (1) is proved in [4, Lemma 4.4.3].
Assume ˜w > s ˜ws and s=sHis the reflection along H=Hα,k ∈H
for some α∈Rand k∈Z. By replacing αby −αif necessary, we can
assume that z(α)∈R+. If z(α)/∈RJ, then α, p( ˜w)(α)∈z−1(R+−R+
J).
Note that ˜w > s ˜ws, so H, ˜wH ∈H(C0,˜wC0). Hence ˜wC0>αC0and
˜wC0>p( ˜w)(α)C0since ˜wis a (J, z)-alcove. Applying ˜wto the first
inequality we have ˜w2C0>p( ˜w)(α)˜wC0. Hence both C0and ˜w2C0are
separated from ˜wC0by ˜wH. In other words, C0and ˜w2C0are on the
same side of ˜wH . So ˜wC0>αC0and ˜w2C0>p( ˜w)(α)˜wC0can’t happen
at the same time. That is a contradiction. The “moreover” part follows
from [4, Lemma 4.4.2].
Theorem 6.2. Let ˜w∈˜
W,J⊂S0and z∈JW0such that ˜wis a
(J, z)-alcove. Then
T˜w∈HJ+ [H,H].
Proof. We argue by induction on the length of ˜w. Suppose that ˜wis of
minimal length in its conjugacy class. By [8, Proposition 2.5 & Lemma
2.7] and Lemma 6.1, we may assume further that ¯
C0contains a regular
point of V˜w.
Let µ∈Vbe a dominant vector such that J=Jµ. Since ˜wis
a (J, z)-alcove, then zp( ˜w)z−1(µ) = µ, that is, z−1(µ) + V˜w=V˜w.
Moreover
(a) R+−R+
J⊂ {α∈R+;hz(ν˜w), α∨i>0}.
Let v=ν˜w+ǫz−1(µ) with ǫa sufficiently small positive real number.
We have V˜w=V˜w+v. Let z1=uz with u∈WJsuch that hz1(v), α∨i>
0 for each α∈R+
J. Let β∈R+−R+
J. By (a), hz1(ν˜w), β∨i=
1In fact, for ˜w∈X⋊W0and δ∈Γ, ˜wδ is a (J, z)-alcove element if and only
if ˜wC0is a (J, z−1, δ )-alcove in [4, §4.1]. This is a generalization of the P-alcove
introduced in [3].
19
hz(ν˜w), u−1(β∨)i>0. Moreover hz1z−1(µ), β∨i=hµ, u−1(β∨)i>0.
Hence hz1(v), β∨i>0. So z1(v) is dominant. Since vlies in a suffi-
ciently small neighborhood of ν˜w,z1(ν˜w) is also dominant. Now apply-
ing Proposition 5.1 (2), T˜w∈HJ¯ν˜w∩J¯v+ [H,H].
Let α∈RJ¯ν˜w∩J¯v. Then hz1(v), α∨i=hz1(ν˜w), α∨i= 0. Hence
hz1z−1(µ), α∨i=hµ, α∨i= 0. Thus J¯ν˜w∩J¯v⊂J. The statement holds
for ˜w.
Now assume that ˜wis not of minimal length in its conjugacy class
and the statement holds for all ˜w′∈˜
Wwith ℓ( ˜w′)< ℓ( ˜w).
By Theorem 2.1, there exist ˜w1∼
=˜wand s∈Ssuch that ℓ(s˜w1s)<
ℓ( ˜w1) = ℓ( ˜w). Then
T˜w≡T˜w1≡Ts˜w1s+ (q1
2
s−q−1
2
s)Ts˜w1mod [H,H].
Here ℓ(s˜w1s), ℓ(s˜w1)< ℓ( ˜w). By Lemma 6.1, ˜w1, s ˜w1s, s ˜w1are (J, z1)-
alcove elements for some z1∈JW0. The statement follows from induc-
tion hypothesis.
6.2. We introduce the class polynomials, following [8, Theorem 5.3].
Suppose that q1
2
s=q1
2
tfor all s, t ∈S. We simply write vfor q1
2
s. In
this case, the parameter function p
1
2
tin §1.5 also equals to v.
Let ˜w∈˜
W. Then for any conjugacy class Oof ˜
W, there exists a
polynomial f˜w,O∈Z[v−v−1] with nonnegative coefficient such that
f˜w,Ois nonzero only for finitely many Oand
(a) T˜w≡X
O
f˜w,OTOmod [ ˜
H, ˜
H].
The polynomials can be constructed explicitly as follows.
If ˜wis a minimal element in a conjugacy class of ˜
W, then we set
f˜w,O=(1,if ˜w∈O
0,if ˜w /∈O. Suppose that ˜wis not a minimal element in
its conjugacy class and that for any ˜w′∈˜
Wwith ℓ( ˜w′)< ℓ( ˜w), f˜w′,Ois
already defined. By Theorem 2.1, there exist ˜w1≈˜wand s∈Ssuch
that ℓ(s˜w1s)< ℓ( ˜w1) = ℓ( ˜w). In this case, ℓ(s˜w)< ℓ( ˜w) and we define
f˜w,Oas
f˜w,O= (vs−v−1
s)fs˜w1,O+fs˜w1s,O.
Theorem 6.3. Let ˜w∈˜
W,J⊂S0and z∈JW0such that ˜wis a
(J, z)-alcove. Let
T˜w≡X
O
f˜w,OTwOmod [H,H];
TJ
z˜wz−1≡X
O′
fJ
z˜wz−1,O′TJ
wO′mod [HJ,HJ],
20 XUHUA HE AND SIAN NIE
where Oand O′run over all the conjugacy classes of ˜
Wand ˜
WJre-
spectively in the above summations. Then
f˜w,O=X
O′⊂O
fJ
z˜wz−1,O′.
Proof. We argue by induction on the length of ˜w. If ˜wis of minimal
length in its conjugacy class, then by Proposition 3.5, z˜wz−1is also a
minimal length element (with respect to ℓJ) in its ˜
WJ-conjugacy class.
The statement holds in this case.
Now assume that ˜wis not of minimal length in its conjugacy class
and the statement holds for all ˜w′∈˜
Wwith ℓ( ˜w′)< ℓ( ˜w).
By Theorem 2.1, there exist ˜w1∼
=˜wand s∈Ssuch that ℓ(s˜w1s)<
ℓ( ˜w1) = ℓ( ˜w). By Corollary 3.4 and Lemma 6.1, there exists z1∈
JW0such that ˜w1, s ˜w1s, s ˜w1are (J, z1)-alcove elements and z˜wz−1∼
=
z1˜w1z−1
1with respect to ˜
WJ.
Let t=zsz−1. Then by Lemma 6.1 and Lemma 3.3, t∈˜
WJ
and ℓJ(t) = 1. By the proof of Corollary 3.4, ℓJ(tz1˜w1z−1
1t−1)<
ℓJ(z1˜w1z−1
1). So by the construction of class polynomials,
f˜w,O=f˜w1,O= (v−v−1)fs˜w1,O+fs˜w1s,O;
fJ
z˜wz−1,O′=fJ
z1˜w1z−1
1,O′= (v−v−1)fJ
tz1˜w1z−1
1,O′+fJ
tz1˜w1z−1
1t−1,O′.
The statement follows from induction hypothesis.
6.3. In the rest of this section, we discuss some application to affine
Deligne-Lusztig varieties.
Let Fqbe the finite field with qelements. Let kbe an algebraic
closure of Fq. Let F=Fq((ǫ)), the field of Laurent series over Fq, and
L=k((ǫ)), the field of Laurent series over k.
Let Gbe a quasi-split connected reductive group over Fwhich splits
over a tamely ramified extension of F. Let σbe the Frobenius auto-
morphism of L/F . We denote the induced automorphism on G(L) also
by σ.
Let Ibe a σ-invariant Iwahori subgroup of G(L). The I-double cosets
in G(L) are parameterized by the extended affine Weyl group WG. The
automorphism on WGinduced by σis denoted by δ. Set ˜
W=WG⋊hδi.
For ˜w∈WGand b∈G(L), set
X˜w(b) = {gI∈G(L)/I;g−1bσ(g)∈I˜wI}.
This is the affine Deligne-Lusztig variety attached to ˜wand b. It plays
an important role in arithmetic geometry. We refer to [3], [4] and [6]
for further information.
The relation between the affine Deligne-Lusztig varieties and the
class polynomials of the associated affine Hecke algebra is found in [6,
Theorem 6.1].
21
Theorem 6.4. Let b∈G(L)and ˜w∈˜
W. Then
dim(X˜w(b)) = max
O
1
2(ℓ( ˜w) + ℓ(wO) + deg(f˜wδ,O)) − h¯νb,2ρi,
where Oranges over the ˜
W-conjugacy class of WGδ⊂˜
Wsuch that νO
equals the Newton point of band κG(x) = κG(b)for some (or equiva-
lently, any) x∈WGwith xδ ∈O. Here κGis the Kottwitz map [11].
6.4. For J⊂S0, let MJbe the corresponding Levi subgroup of Gde-
fined in [4, 3.2] and κJthe Kottwitz map for MJ(L). As a consequence
of Theorem 6.3, we have
Theorem 6.5. Let ˜w∈WGand z∈W0. Suppose ˜wδ is a (J, z)-alcove
element. Then for any b∈MJ(L),X˜w(b) = ∅unless κJ(z˜wδ(z)−1) =
κJ(b).
Remark. This result was first proved in [3, Theorem 1.1.2] for split
groups and then generalized to tamely ramified groups in [4, Corollary
3.6.1]. The approach there is geometric, using Moy-Prasad filtration.
The approach here is more algebraic.
Proof. Assume X˜w(b)6=∅. By Theorem 6.4, there exists a conjugacy
class Oof WGδsuch that f˜wδ,O6= 0, νO= ¯νband κG(b) = κG(x) for
some (or equivalently, any) x∈WGwith xδ ∈O. By Theorem 6.3,
there exists a ˜
WJ-conjugacy class O′⊂Osuch that fJ
z˜wδz−1,O′6= 0.
Choose b′∈MJ(L) such that νb′=νO′and κJ(b′) = κJ(x′) for some
(or equivalently, any) x′∈WMJwith x′δ∈O′. By [4, Proposition
3.5.1], band b′belong to the same σ-conjugacy class of MJ(L). Since
the affine Deligne-Lusztig variety XMJ
z˜wδ(z)−1(b′) for MJis nonempty, we
have κJ(z˜wδ(z)−1) = κJ(b′) = κJ(b).
References
[1] J. Bernstein, P. Deligne, D. Kazhdan, Trace Paley-Wiener theorem for reduc-
tive p-adic groups, J. d’Analyse Math. 47 (1986), 180–192.
[2] D. Ciubotaru and X. He, The cocenter of graded affine Hecke algebra and the
density theorem, arXiv:1208.0914.
[3] U. G¨ortz, T. Haines, R. Kottwitz, D. Reuman, Affine Deligne-Lusztig varieties
in affine flag varieties, Compos. Math. 146 (2010), no. 5, 1339–1382.
[4] U. G¨ortz, X. He and S. Nie, P-alcoves and nonemptiness of affine Deligne-
Lusztig varieties, arXiv:1211.3784.
[5] X. He, Minimal length elements in some double cosets of Coxeter groups, Adv.
Math. 215 (2007), no. 2, 469–503.
[6] X. He, Geometric and homological properties of affine Deligne-Lusztig vari-
eties, arXiv:1201.4901, to appear in Ann. of Math.
[7] X. He and S. Nie, Minimal length elments of finite Coxeter groups, Duke Math.
J., 161 (2012), 2945–2967.
[8] X. He and S. Nie, Minimal length elements of extended affine Weyl groups, II,
arXiv:1112.0824.
[9] D. Kazhdan, Representations groups over close local fields, J. d’Analyse Math.
47 (1986), 175–179.
22 XUHUA HE AND SIAN NIE
[10] N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the struc-
ture of the Hecke rings of p-adic Chevalley groups, Inst. Hautes ´
Etudes Sci.
Publ. Math. (1965), no. 25, 5–48.
[11] R. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109
(1997), 255–339.
[12] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math.
Soc. 2(1989), 599–635.
[13] G. Lusztig, Hecke algebras with unequal parameters, CRM Monograph Series,
18. American Mathematical Society, Providence, RI, 2003.
[14] G. Lusztig, Parabolic character sheaves. I, Mosc. Math. J. 4(2004), no. 1,
153–179.
[15] E. Opdam and M. Solleveld, Homological algebra for affine Hecke algebras,
Adv. Math. 220 (2009), no. 5, 1549–1601.
[16] E. Opdam and M. Solleveld, Discrete series characters for affine Hecke algebras
and their formal degrees, Acta Math. 205(2010), 105–187.
Department of Mathematics and Institute for advanced Study, The
Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong
E-mail address:xuhuahe@gmail.com
Max Planck Institute for mathematics, Vivatsgasse 7, 53111, Bonn,
Germany
E-mail address:niesian@amss.ac.cn
