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The cocenter of graded affine Hecke algebra and the density theorem
Abstract
We determine a basis of the (twisted) cocenter of graded affine Hecke
algebras with arbitrary parameters. In this setting, we prove that the kernel
of the (twisted) trace map is the commutator subspace (Density theorem) and
that the image is the space of good forms (trace Paley-Wiener theorem).
... Proposition 4.1. [CiHe,Proposition 2.4.3] Let J ⊂ I and let w ∈ W J be an elliptic element. ...
... Proof. We follow [CiHe,Section 6]. We have V 2 = (V 2 ) W J ⊕ U as a W J -module, where U is spanned by {x 2 i |i ∈ J}. ...
... The goal of this section is to lift the spanning set constructed above for H 0 X to H c X . The proof is motivated by [CiHe,Section 6.2], with appropriate modifications. Proof. ...
We determine a basis of the cocenter (i.e., the zeroth Hochschild homology) of the degenerate affine Hecke-Clifford and spin Hecke algebras in classical types.
... The simple roots of E 6 are labeled as follows. The δ-orbits 5,6,9,6,3). Note that ρ ∨ = (8,11,15,21,15,8). ...
... The simple roots of E 6 are labeled as follows. The δ-orbits 5,6,9,6,3). Note that ρ ∨ = (8,11,15,21,15,8). ...
... Note that the different W J -conjugacy classes in O ∩ W J δ are obtained from one another by diagram automorphisms of W J . The final (and crucial) step in [9] and [6] was to use the characterization of elliptic conjugacy classes to deduce that the intersection is a single W J -conjugacy class. Now the final step may be replaced by Proposition 7.1, the proof of which is simpler than the characterization of elliptic conjugacy classes. ...
Suppose G is a connected complex semisimple group and W is its Weyl group. The lifting of an element of W to G is semisimple. This induces a well-defined map from the set of elliptic conjugacy classes of W to the set of semisimple conjugacy classes of G. In this paper, we give a uniform algorithm to compute this map. We also consider the twisted case.
... Proposition 4.1. [CiHe,Proposition 2.4.3] Let J ⊂ I and let w ∈ W J be an elliptic element. ...
... Proof. We follow [CiHe,Section 6]. We have V 2 = (V 2 ) W J ⊕ U as a W J -module, where U is spanned by {x 2 i |i ∈ J}. ...
... The goal of this section is to lift the spanning set constructed above for H 0 X to H c X . The proof is motivated by [CiHe,Section 6.2], with appropriate modifications. ...
We determine a basis of the cocenter (i.e., the zeroth Hochschild homology) of the degenerate affine Hecke-Clifford and spin Hecke algebras in classical types.
... In the same spirit as the Euler-Poincaré pairing of p-adic groups by Schneider-Stuhler [31] and others, an appropriate subspace of the virtual representations for the graded Hecke algebra is equipped with the twisted Euler-Poincaré pairing as an inner product. We shall discuss those twisted elliptic spaces defined by the twisted Euler-Poincaré pairing (based on several previous work by others [10], [11], [12], [25] and [30]). ...
... He also thanks Peter Trapa for pointing out the definition of the twisted Euler-Poincaré pairing and providing his idea on Theorem 4.11. He is also grateful for Dan Ciubotaru and Xuhua He for useful discussions on elliptic modules and their papers [11,12]. He would also like to thank Marteen Solleveld for providing many useful and detailed suggestions in an earlier version of this paper. ...
... The goal of this section is to put or recollect some results in [10], [11], [12], [25] and [30] in the framework of twisted elliptic spaces. ...
We study a twisted Euler-Poincar\'e pairing for graded affine Hecke algebras,
and give a precise connection to the twisted elliptic pairing of Weyl groups
defined by Ciubotaru-He. The Ext-groups for an interesting class of
parabolically induced modules are also studied in a connection with the twisted
Euler-Poincar\'e pairing. We also study a certain space of graded Hecke algebra
modules which equips with the twisted Euler-Poincar\'e pairing as an inner
product.
... The proof in [5] and [1] are based on a characterization of elliptic conjugacy classes using characteristic polynomials [5, Theorem 3.2.7 (P3)] and [7, Theorem 7.5 (P3)], which is proved via a case-by-case analysis. It is interesting to find a case-free proof of these results. ...
In this paper, we give explicit descriptions of the centers and cocenters of
0-Hecke algebras associated to finite Coxeter groups.
We define an involution on the elliptic space of tempered unipotent representations of inner twists of a split simple p -adic group G and investigate its behaviour with respect to restrictions to reductive quotients of maximal compact open subgroups. In particular, we formulate a precise conjecture about the relation with a version of Lusztig's nonabelian Fourier transform on the space of unipotent representations of the (possibly disconnected) reductive quotients of maximal compact subgroups. We give evidence for the conjecture, including proofs for and .
Let GG be a simple algebraic group over an algebraically closed field and let θ be a graph‐automorphism of GG. We classify the spherical unipotent conjugacy classes in the coset Gθ. As a by‐product, we show that J.‐H. Lu's characterization in characteristic zero of spherical conjugacy classes in Gθ by the dimension formula also holds for spherical unipotent conjugacy classes in Gθ in positive characteristic. If θ has order 2, we provide an alternative description of the restriction to spherical unipotent conjugacy classes in Gθ, of Lusztig's map Ψ from the set of unipotent conjugacy classes in Gθ to the set of twisted conjugacy classes of the Weyl group of GG. We also show that a twisted conjugacy class in the Weyl group has a unique maximal length element if and only if it has maximum in the Bruhat order (a result previously proved by X. He).
This paper gives a definitive solution to the problem of describing conjugacy classes in arbitrary Coxeter groups in terms of cyclic shifts. Let (W,S) be a Coxeter system. A cyclic shift of an element is a conjugate of w of the form sws for some simple reflection such that . The cyclic shift class of w is then the set of elements of W that can be obtained from w by a sequence of cyclic shifts. Given a subset such that is finite, we also call two elements K-conjugate if normalise and , where is the longest element of . Let be a conjugacy class in W, and let be the set of elements of minimal length in . Then is the disjoint union of finitely many cyclic shift classes . We define the structural conjugation graph associated to to be the graph with vertices , and with an edge between distinct vertices if they contain representatives and such that u,v are K-conjugate for some . In this paper, we compute explicitely the structural conjugation graph associated to any conjugacy class in W, and show in particular that it is connected (that is, any two conjugate elements of W differ only by a sequence of cyclic shifts and K-conjugations). Along the way, we obtain several results of independent interest, such as a description of the centraliser of an infinite order element , as well as the existence of natural decompositions of w as a product of a "torsion part" and of a "straight part", with useful properties.
In this paper, we study the relation between the cocenter and the
representation theory of affine Hecke algebras. The approach is based on the
interaction between the rigid cocenter, an important subspace of the cocenter,
and the dual object in representation theory, the rigid quotient of the
Grothendieck group of finite dimensional representations.
Affine Hecke algebras arise naturally in the study of smooth representations
of reductive p-adic groups. Finite dimensional complex representations of
affine Hecke algebras (under some restriction on the isogeny class and the
parameter function) has been studied by many mathematicians, including
Kazhdan-Lusztig \cite{KL}, Ginzburg \cite{CG}, Lusztig \cite{L1}, Reeder
\cite{Re}, Opdam-Solleveld \cite{OS}, Kato \cite{Kat}, etc. The approaches are
either geometric or analytic.
In this note, we'll discuss a different route, via the so-called
"cocenter-representation duality", to study finite dimensional representations
of affine Hecke algebras (for arbitrary isogeny class and for a generic complex
parameter). This route is more algebraic, and allows us to work with complex
parameters, instead of equal parameters or positive parameters. We also expect
that it can be eventually applied to the "modular case" (for representations
over fields of positive characteristic and for parameter equal to a root of
unity).
This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in
the study of the cocenters and class polynomials of affine Hecke algebras
and their relation to affine Deligne-Lusztig varieties. Let w be a P-alcove
element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study
the image of in the cocenter of . In the process, we obtain a
Bernstein presentation of the cocenter of . We also obtain a comparison
theorem among the class polynomials of and 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
\cite{GHKR} and \cite{GHN} on the emptiness pattern of affine Deligne-Lusztig
varieties.
We establish the Langlands classication for graded Hecke alge- bras. The proof is analogous to the proof of the classication of highest weight modules for semisimple Lie algebras.
This paper studies affine Deligne-Lusztig varieties X_{\tw}(b) in the
affine flag variety of a quasi-split tamely ramified group. We describe the
geometric structure of X_{\tw}(b) for a minimal length element \tw in the
conjugacy class of an extended affine Weyl group, generalizing one of the main
results in \cite{HL} to the affine case. We then provide a reduction method
that relates the structure of X_{\tw}(b) for arbitrary elements \tw in the
extended affine Weyl group to those associated with minimal length elements.
Based on this reduction, we establish a connection between the dimension of
affine Deligne-Lusztig varieties and the degree of the class polynomial of
affine Hecke algebras. As a consequence, we prove a conjecture of G\"ortz,
Haines, Kottwitz and Reuman in \cite{GHKR}.
The Kazhdan-Lusztig parameters are important parameters in the representation
theory of p-adic groups and affine Hecke algebras. We show that the
Kazhdan-Lusztig parameters have a definite geometric structure, namely that of
the extended quotient T//W of a complex torus T by a finite Weyl group W.
More generally, we show that the corresponding parameters, in the principal
series of a reductive p-adic group with connected centre, admit such a
geometric structure. This confirms, in a special case, our recently formulated
geometric conjecture. In the course of this study, we provide a unified
framework for Kazhdan-Lusztig parameters on the one hand, and Springer
parameters on the other hand. Our framework contains a complex parameter s,
and allows us to interpolate between and . When ,
we recover the parameters which occur in the Springer correspondence; when , we recover the Kazhdan-Lusztig parameters.
Let R be a root datum with affine Weyl group , and let
be an affine Hecke algebra with positive, possibly unequal, parameters q.
Then H is a deformation of the group algebra , so it is
natural to compare the representation theory of H and of .
We define a map from irreducible H-representations to -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 . 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 .
An important role is played by the Schwartz completion of H,
an algebra whose representations are precisely the tempered
H-representations. We construct isomorphisms and injection , depending continuously on .
Although is not surjective, it behaves like an algebra isomorphism
in many ways. Not only does 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 -completion of an affine Hecke algebra does
not depend on the parameter(s) q.
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 H r;p;n , can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to ane 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.
F72.42> , where G # is the semi-direct product Go ! oe ? of G with the group ! oe ? generated by oe . When ß is irreducible then S is uniquely determined up to an ` th root i of unity in C . Let M (G) be the category of G -modules. An element E of M (G) is called finitely generated if for any filtered system of proper subobjects E i in E , the subobject Sigma i E i is proper in E . Let K(G) be the Grothendieck group of finitely generated G -modules, and R(G) the Grothendieck group of G -modules of finite length.
An affine Hecke algebra contains a large abelian subalgebra spanned by the Bernstein–Zelevinski–Lusztig basis elements , where x runs over (an extension of) the root lattice. The centre of is the subalgebra of Weyl group invariant elements in . The natural trace (‘evaluation at the identity’) of the affine Hecke algebra can be written as integral of a certain rational n -form (with values in the linear dual of ) over a cycle in the algebraic torus . This cycle is homologous to a union of ‘local cycles’. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum of . From this result we derive the Plancherel formula of the affine Hecke algebra.
AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32
We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer's reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple p-adic group.
Suppose G is a simple reductive p-adic group with Weyl group W . We give a classification of the irreducible representations of W which can be extended to real hermitian representations of the associated graded Hecke algebra H. Such representations correspond to unitary representations of G which have a small spectrum when restricted to an Iwahori subgroup.
Representations of affine and graded Hecke algebras associated to Weyl groups play an important role in the Langlands correspondence for the admissible representations of a reductive p-adic group. We work in the general setting of a graded Hecke algebra associated to any real reflection group with arbitrary parameters. In this setting we provide a classification of all irreducible representations of graded Hecke algebras associated to dihedral groups. Dimensions of generalized weight spaces, Langlands parameters, and a Springer-type correspondence are included in the classification. We also give an explicit construction of all irreducible calibrated representations (those pos-sessing a simultaneous eigenbasis for the commutative subalgebra) of a general graded Hecke algebra. While most of the techniques used have appeared pre-viously in various contexts, we include a complete and streamlined exposition of all necessary results, including the Langlands classification of irreducible representations and the irreducibility criterion for principal series representa-tions.
This article deals with various topics related with Grothendieck groups, invariant distributions, parabolic and compact inductions... for a p-adic group G. The main result is a description of the K
0 of the Hecke algebra ℋ of G in terms of discrete series of Levi subgroups, which has an interesting behavior with regard to parabolic restriction and induction. A similar description – but no more compatible with these parabolic functors – is obtained for =ℋ/[ℋ,ℋ] and the Hattori rank map gets an easy description in this dictionary.¶We follow a beautiful idea of J. Bernstein consisting in comparing two natural filtrations on these objects, one of combinatorial nature and one of topological nature. The combinatorial filtrations are related to the structure of Levi subgroupsin G and have counterparts concerning many classical objects of interest as the Grothendieck group of finite length G-modules R(G), the set Ωsr
of regular semi-simple conjugacy classes, and the variety Θ(G) of infinitesimal characters. These filtrations will turn out to be “compatible”, in a sense to be specified, with regard to all the classical operations or morphisms between these objects.
For the affine Hecke algebras of type A, Grojnowski has developed a combinatorial labelling of irreducible modules using formal characters and crystal operators, also obtaining certain special branching rules.
In this thesis, the same approach is applied to affine Hecke algebras of type B, where it does not lead to full results in all cases. For certain eigenvalues of lattice operators only partial results can be achieved.
One main result is the irreducibility of certain modules that are induced from type A to type B, giving a one-to-one corresondance of irreducible objects in certain full subcategories of the module categories in type A and type B.
This yields an analogous combinatorial description in type B as in type A, including branching rules.
The space of elliptic virtual representations of a p-adic group is endowed with a natural inner product EP( , ), defined analytically by Kazhdan and homologically by Schneider–Stuhler. Arthur has computed EP in terms of analytic R-groups. For Iwahori spherical representations, we show that EP can also be expressed in terms of a corresponding inner product on space of elliptic virtual representations of Weyl groups. This leads to an explicit description of both elliptic representation theories, in terms of the Kazhdan–Lusztig and Springer correspondences
We prove a strong induction theorem for graded Hecke algebras and we classify
the tempered and square integrable representations of such algebras using
methods of equivariant homology.
Let W be a finite Coxeter group and let F be an automorphism of W that leaves the set of generators of W invariant. We establish certain properties of elements of minimal length in the F-conjugacy classes of W that allow us to define character tables for the corresponding twisted Iwahori–Hecke algebras. These results are extensions of results obtained by Geck and Pfeiffer in the case where F is trivial.
We present a geometric description for tempered modules of the affine Hecke algebra of type Cn with arbitrary (non-root of unity) unequal parameters, using the exotic Deligne–Langlands correspondence (Kato (2009) [18]). Our description has several applications to the structure of the tempered modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sgn representation of the Weyl group, equivalently, via the Iwahori–Matsumoto involution, of spherical cuspidal modules. This latter combinatorial classification was expected from Heckman and Opdam (1997) [15], and determines the L2-solutions for the Lieb–McGuire system.
Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H-modules with real central character forms a Q-basis of the representation ring of W.Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety.
In this paper we study homological properties of modules over an affine Hecke algebra H. In particular we prove a comparison result for higher extensions of tempered modules when passing to the Schwartz algebra S, a certain topological completion of the affine Hecke algebra. The proof is self-contained and based on a direct construction of a bounded contraction of certain standard resolutions of H-modules.This construction applies for all positive parameters of the affine Hecke algebra. This is an important feature, since it is an ingredient to analyse how the irreducible discrete series representations of H arise in generic families over the parameter space of H. For irreducible non-simply laced affine Hecke algebras this will enable us to give a complete classification of the discrete series characters, for all positive parameters (we will report on this application in a separate article).
We study the minimal length elements in some double cosets of Coxeter groups and use them to study Lusztig's G-stable pieces and the generalization of G-stable pieces introduced by Lu and Yakimov. We also use them to study the minimal length elements in a conjugacy class of a finite Coxeter group and prove a conjecture in [M. Geck, S. Kim, G. Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2) (2000) 570–600].
Let H = H (R,q) be an affine Hecke algebra with complex, possibly unequal
parameters q, which are not roots of unity. We compute the Hochschild and the
cyclic homology of H. It turns out that these are independent of q and that
they admit an easy description in terms of the extended quotient of a torus by
a Weyl group, both of which are canonically associated to the root datum R. For
q positive we also prove that the representations of the family of algebras H
(R,q^e) come in families which depend analytically on the complex number e.
Analogous results are obtained for graded Hecke algebras and for Schwartz
completions of affine Hecke algebras.
We give a geometric proof that minimal length elements in a (twisted)
conjugacy class of a finite Coxeter group W have remarkable properties with
respect to conjugation, taking powers in the associated Braid group and taking
centralizer in W.
We establish a transfer of unitarity for a Bernstein component of the
category of smooth representations of a reductive p-adic group to the
associated Hecke algebra, in the framework of the theory of types, whenever the
Hecke algebra is an affine Hecke algebra with geometric parameters, in the
sense of Lusztig (possibly extended by a group of automorphisms of the root
datum). It is known that there is a large class of such examples (detailed in
the paper). As a consequence, we establish relations between the unitary duals
of different groups, in the spirit of endoscopy.
Bernstein's isomorphism and good forms, K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras
- Y Flicker
Y. Flicker, Bernstein's isomorphism and good forms, K-Theory and Algebraic Geometry:
Connections with Quadratic Forms and Division Algebras, 1992 Summer Research Institute;
Kazhdan-Lusztig parameters and extended quotients, arXiv:1102.4172 Reduction to real infinitesimal character in affine Hecke algebras
- Abp ] A.-M Aubert
- P Baum
- R Plymen
ABP] A.-M. Aubert, P. Baum, R. Plymen, Kazhdan-Lusztig parameters and extended quotients, arXiv:1102.4172. [BM1] D. Barbasch, A. Moy, Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6 (1993), no. 3, 611–635.
Cuspidal local systems and graded algebras II, Representations of groups
- G Lusztig
G. Lusztig, Cuspidal local systems and graded algebras II, Representations of groups (Banff, AB, 1994), Amer. Math. Soc., Providence, 1995, 217–275.
Unitary equivalences for reductive p-adic groups, to appear in Amer Trace Paley-Wiener theorem for reductive p-adic groups
- [ Bc
- D Barbasch
- D Ciubotaru
[BC] D. Barbasch, D. Ciubotaru, Unitary equivalences for reductive p-adic groups, to appear in Amer. J. Math. [BDK] J. Bernstein, P. Deligne, D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, J. d'Analyse Math. 47 (1986), 180–192.
in the sense of Bala-Carter [Ca1]), one allows every φ of Springer type " , while for e quasi-distinguished, but not distinguished, one allows only φ = 1. For example, for E 6 , a basis is labeled by the five pairs
- E Concretely
Concretely, for type E, for every distinguished e (in the sense of Bala-Carter [Ca1]),
one allows every φ of " Springer type ", while for e quasi-distinguished, but not
distinguished, one allows only φ = 1. For example, for E 6, a basis is labeled by the
five pairs
(E 6, 1), (E 6 (a 1 ), 1), (E 6 (a 3 ), 1), (E 6 (a 3 ), sgn), (D 4 (a 1 ), 1).
(8.9.2)
- A.-M Aubert
- P Baum
- R Plymen
A.-M. Aubert, P. Baum, R. Plymen, Kazhdan-Lusztig parameters and extended quotients,
arXiv:1102.4172.
- X He
X. He Geometric and homological properties of affine Deligne-Lusztig varieties,
arXiv:1201.4901.
- X He
- S Nie
X. He and S. Nie, Minimal length elements of finite Coxeter groups, arXiv:1108.0282v1, to
appear in Duke Math. J.
- A Ram
- J Ramagge
A. Ram, J. Ramagge, Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory,
Birkhäuser, Trends in Mathematics (2003), 428-466.
UT 84112 E-mail address: ciubo@math
- M Solleveld
M. Solleveld, Hochschild homology of affine Hecke algebras, arXiv:1108.5286.
(D. Ciubotaru) Dept. of Mathematics, University of Utah, Salt Lake City, UT 84112
E-mail address: ciubo@math.utah.edu
(X. He) Dept. of Mathematics, Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong
E-mail address: maxhhe@ust.hk