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# The cocenter of graded affine Hecke algebra and the density theorem

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*in*Journal of Pure and Applied Algebra 220(1) · August 2012

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DOI: 10.1016/j.jpaa.2015.06.018 · Source: arXiv

Cite this publicationAbstract

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).

- ... The Proposition 7.1 is used in an essential way to prove that in finite Weyl groups, elliptic conjugacy classes never fuse. [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. ...Preprint
- Mar 2019

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. - ... 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. ...Article
- Feb 2015

In this paper, we give explicit descriptions of the centers and cocenters of $0$-Hecke algebras associated to finite Coxeter groups. - Article
- May 2016
- J ALGEBRA

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 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.
- ArticleFull-text available
- Jun 2014

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 $\ch$ 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 $T_w$ in the cocenter of $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ 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.

- Article
- Nov 2001
- COMPOS MATH

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 - Bernstein's isomorphism and good forms, K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras
- Jan 1992

- 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; - Cuspidal local systems and graded algebras II, Representations of groups
- Jan 1994
- 217-275

- 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
- Jan 1986
- 47-180

- [ 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) - 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.
- Article
- Feb 1983
- T AM MATH SOC

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. - Article
- Jul 1999
- T AM MATH SOC

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. - Article
- Apr 2002
- Represent Theor

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. - Article
- Jan 2000
- INVENT MATH

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 \(\overline{\mathcal{H}}\)=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. - Article
- Aug 2008
- ALGEBR REPRESENT TH

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. - Kazhdan-Lusztig parameters and extended quotients, arXiv:1102.4172 Reduction to real infinitesimal character in affine Hecke algebras
- Jan 1993
- 611-635

- 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. - Article
- Jan 1988
- PUBL MATH-PARIS

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. - Article
- Jul 2000
- J Algebra

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. - Article
- Jan 2011
- ADV MATH

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. - Article
- Mar 2010
- J ALGEBRA

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. - Article
- Mar 2009
- ADV MATH

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). - Article
- Nov 2007
- ADV MATH

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]. - 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}.
- Article
- Aug 2011
- J ALGEBRA

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. - Article
- Aug 2011
- DUKE MATH J

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 $W$. - 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 $s = 1$ and $s = \sqrt q$. When $s = 1$, we recover the parameters which occur in the Springer correspondence; when $s = \sqrt q$, we recover the Kazhdan-Lusztig parameters.
- 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) $q$.
- Article
- Sep 2009
- Am J Math

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. - ArticleFull-text available
- Sep 2002

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. - ArticleFull-text available
- Aug 1997

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. - ArticleFull-text available
- Feb 2001

An affine Hecke algebra H contains a large abelian subalgebra A spanned by the Bernstein-Zelevinski-Lusztig basis elements theta(x), where x runs over (an extension of) the root lattice. The centre Z of H is the subalgebra of Weyl group invariant elements in A. 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 H) over a cycle in the algebraic torus T = Spec(A). 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 W0 \ T of Z. From this result we derive the Plancherel formula of the affine Hecke algebra.