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Homological algebra for affine Hecke algebras
Abstract
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).
... (5) [OS09] Opdam-Solleveld establishes an upper bound on the number of discrete series for affine Hecke algebras by analytic methods. (6) [OS10] Opdam-Solleveld classifies all discrete series central characters for graded Hecke algebras of crystallographic types and affine Hecke algebras, including unequal positive parameter cases in non-simply laced types. ...
... In addition to obtain a parametrization set, one also wants to study some invariants of discrete series. In particular, Opdam-Solleveld [OS09] and Ciubotaru-Kato-Kato [CKK12] studied their formal degrees. However, the formal degree is not a well-defined invariant for non-crystallographic types, and so we shall not consider that. ...
... However, several work and classical examples e.g. [OS09,Re00,Ch21,Ch24], homological properties could be a replacement for harmonic analysis in some situations. Inspired by Prasad's proposal [Pr18,Pr23], we shall study some Ext-branching laws with the hope to use homological properties reflecting some hidden harmonic analysis. ...
This article confirms the prediction that the set of discrete series central character for the graded (affine) Hecke algebra of type coincides with the set of the Heckman-Opdam central characters. Combining with previous cases of Kazhdan-Lusztig, Kriloff, Kriloff-Ram, Opdam-Solleveld, Ciubotaru-Opdam, this completes the classification of discrete series for all the graded Hecke algebras of positive parameters. Main tools include construction of calibrated modules and construction of certain minimally induced modules for discrete series. We also study the anti-sphericiity and Ext-branching laws for some discrete series.
... The important fact for the elliptic theory is that H(s ∨ ) is a deformation of an extended affine Weyl group W s ∨ = W s ∨ X * (T s ∨ ), where T s ∨ = Φ e (L ) s ∨ L for L a Levi subgroup of G that corresponds to s ∨ . This allows us to use the results of [OS09] to further reduce to R(H(s ∨ )) ∼ = R( W s ∨ ). Moreover, the latter space is equivalent to a direct sum of elliptic spaces for certain finite groups ...
... We begin by recalling several known facts about elliptic theory for affine Weyl groups and affine Hecke algebras. The main reference is [OS09] (see also [CO15]). The notation in this section is self contained and independent of the previous sections. ...
... Set R( W ) = R( W )/rad , W EP . By [OS09,Theorem 3.3], the Euler-Poincaré pairing for W can also be expressed as an elliptic integral. More precisely, define the conjugation-invariant elliptic measure μ ell on W by setting μ ell = 0 on nonelliptic conjugacy classes, and for an elliptic conjugacy class C such that v ∈ E is an isolated fixed point for some element of C, set ...
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 .
... An element wt x ∈ W is called elliptic if w ∈ W is elliptic (with respect to the action on E). For basic facts about elliptic theory for W , see [OS,sections 3.1,3.2]. There are finitely many elliptic conjugacy classes in W (and in W a ). ...
... Let R( W ) be the Grothendieck group of W -mod (the category of finite-dimensional modules), and set R( W ) = R( W )/rad , W EP . By [OS,Theorem 3.3], the Euler-Poincaré pairing for W can also be expressed as an elliptic integral. More precisely, define the conjugationinvariant elliptic measure µ ell on W by setting µ ell = 0 on nonelliptic conjugacy classes, and for an elliptic conjugacy class C such that v ∈ E is an isolated fixed point for some element of C, set ...
... By Clifford theory, the induction map Ind s : W s -mod → W -mod, Ind s (U ) := Ind W Ws⋉X (U ⊗ s) maps irreducible modules to irreducible modules. By [OS,Theorem 3.2], the map s∈T /W Ind s : ...
We define an involution on the space of compact 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 of the conjecture, including proofs for and .
... An element w ∈ W is called elliptic if w does not belong to any proper Levi subgroup L ⊂ W (see [OS1]). The following are easily seen to be equivalent: ...
... When we compose the character map with the restriction map we obtain a surjective map from R C (W ) to R C (W ). In [OS1] it was shown that the kernel of this map is spanned by the set of characters which are induced from proper Levi subgroups. We will identify R C (W ) with this quotient of R C (W ). ...
... There exists a unique conjugation invariant measure [OS1,Theorem 3.3(c)] µ ell on W , which is supported on the elliptic conjugacy classes, and which is defined by µ ell ((1 − w)(X)) = |W 0 | −1 if (1 − w)(X) has maximal rank, and µ ell ((1 − w)(X)) = 0 otherwise. This defines an integral positive semidefinite Hermitian pairing, the elliptic pairing EP W on R C (W ) by integrating f g over W with respect to the measure µ ell . ...
We give a new and independent parameterization of the set of discrete series
characters of an affine Hecke algebra , in terms of a
canonically defined basis of a certain lattice of virtual
elliptic characters of the underlying (extended) affine Weyl group. This
classification applies to all semisimple affine Hecke algebras ,
and to all , where denotes the vector
group of positive real (possibly unequal) Hecke parameters for .
By analytic Dirac induction we define for each a
continuous (in the sense of [OS2]) family
,
such that (for
some ) is an irreducible discrete series
character of . Here
is a finite union of hyperplanes in
.
In the non-simply laced cases we show that the families of virtual discrete
series characters are piecewise rational
in the parameters . Remarkably, the formal degree of
in such piecewise rational family turns
out to be rational. This implies that for each there
exists a universal rational constant determining the formal degree in the
family of discrete series characters
. We will compute
the canonical constants , and the signs . For
certain geometric parameters we will provide the comparison with the
Kazhdan-Lusztig-Langlands classification.
... There are some related studies of the extension problem in the literature [2,8,22,26,28,29,30,31,34]. While our work is motivated from some known results in the setting of p-adic groups and affine Hecke algebras, our approach is self-contained in the theory of graded affine Hecke algebras. ...
... Since discrete series are also tempered, the statement covers the case for X and Y being discrete series. The statement for affine Hecke algebra setting is proven by Opdam-Solleveld [26], and the one for p-adic group setting is proven by Meyer [22]. The method we prove Theorem 1.2 is different from theirs and essentially makes use of Theorem 1.1 nevertheless. ...
... The first application is to give an upper bound of the number irreducible discrete series. We follows the proof for affine Hecke algebra by Opdam-Solleveld [26,Proposition 3.9]. ...
In this paper, we study extensions of graded affine Hecke algebra modules. In
particular, based on an explicit projective resolution on graded affine Hecke
algebra modules, we prove a duality result for -groups. This
duality result with study on some parabolically induced modules gives a new
proof of the fact that all higher -groups between discrete series
vanish.
... Theorem 1.1 for δ = Id was established by Reeder [25] for equal parameter cases, and was independently proved by Opdam-Solleveld [28] for arbitrary parameters (in different settings). Nevertheless, our approach in proving Theorem 1.1 is independent from their work, and is self-contained. ...
... When δ = Id, the pairing defines an inner product on a subspace of the H-representation ring. This space has been known and studied in [25] and [28]. Our focus of the remaining discussion will be on the case that δ is the automorphism θ arising from the longest element in the Weyl group (see (2.2)). ...
... We remark that our computation of Ext-groups in Theorem 1.2 essentially uses the Extgroups for discrete series from the work of Delorme-Opdam [15] and Opdam-Solleveld [28]. Apart from the deep analytic result from [15] and [28], the main tool of our computation is the projective resolution developed in Section 3 with some careful analysis on the structure of rigid modules. ...
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.
... We also give there examples in support of the conjecture. Finally, in section 5, we rephrase the conjecture (Conjecture 5.8) in the case of representations with Iwahori fixed vectors via the homological algebra interpretation of formal degrees from [OS1]. ...
... An element wt x ∈ W e is called elliptic if w ∈ W is elliptic (with respect to the action on E), or equivalently, if wt x has an isolated fixed point in E. For basic facts about elliptic theory for W e , see [OS1,sections 3.1,3.2]. There are finitely many elliptic conjugacy classes in W e (and in W a ). ...
... Let R(W e ) be the Grothendieck group of W e -mod, and set R(W e ) = R(W e )/rad , EP W e . By [OS1,Theorem 3.3], the Euler-Poincaré pairing for W e can also be expressed as an elliptic integral. More precisely, define the conjugation-invariant elliptic measure µ ell on W e by setting µ ell = 0 on nonelliptic conjugacy classes, and for an elliptic conjugacy class C such that v ∈ E is an isolated fixed point for some element of C, set ...
We introduce a notion of elliptic fake degrees for unipotent elliptic
representations of a semisimple p-adic group. We conjecture, and verify in some
cases, that the relation between the formal degrees of unipotent discrete
series representations of a semisimple p-adic group and the elliptic fake
degrees is given by the same exotic Fourier transform matrix introduced by
Lusztig in the study of representations of finite groups of Lie type.
... Remarks. In parallel and independent work, based on methods of harmonic analysis, Opdam and Solleveld [22] have provided explicit projective resolutions of affine Hecke algebras (over the field of real numbers and with q positive) as bimodules over themselves. This also implies finiteness of global dimension, under the stronger assumptions on q made in [22]. ...
... In parallel and independent work, based on methods of harmonic analysis, Opdam and Solleveld [22] have provided explicit projective resolutions of affine Hecke algebras (over the field of real numbers and with q positive) as bimodules over themselves. This also implies finiteness of global dimension, under the stronger assumptions on q made in [22]. Derived categories and stratifications are not considered in [22]. ...
... This also implies finiteness of global dimension, under the stronger assumptions on q made in [22]. Derived categories and stratifications are not considered in [22]. In the situation of Theorem 5.8, the affine Hecke algebra H k (n, q) may be called affine quasi-hereditary, and the statement of 5.8 implies a whole set of cohomological results analogous to known results about quasihereditary algebras and highest weight categories, for example on the vanishing of certain extensions between cell lattices. ...
Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras. We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley-Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterized by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.
... The main idea in our proof is to introduce a new family of representations which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof also requires a comparison of the C * -theoretic results of Opdam, Delorme, Slooten, Solleveld [23,9,30,24,25], and the geometric construction from [12,13,7]. ...
... From [25], the formal degree of a discrete series π with central character s (not necessarily positive real) is known to equal fd(π) = C π q n 2 −n q nm+ ′ α∈Rn (α(s) − 1) , (1.4) where ′ means that the product is taken only over the nonzero factors. From Opdam-Solleveld [24], it is known that the constants C π are rational numbers, and the question is to determine them explicitly. We make use of an Euler-Poincaré type formula which expresses the formal degree as an alternating sum depending on the W -structure of the discrete series (see 4.3). ...
... We make use of an Euler-Poincaré type formula which expresses the formal degree as an alternating sum depending on the W -structure of the discrete series (see 4.3). This formula was established first in the context of p-adic groups by Schneider-Stuhler [29] and used by Reeder [26], and then for affine Hecke algebras with arbitrary real unequal parameters, it was proved in [24]. ...
We address two fundamental questions in the representation theory of affine Hecke algebras of classical types. One is an inductive
algorithm to compute characters of tempered modules, and the other is the determination of the constants in the formal degrees
of discrete series (in the form conjectured by Reeder (J.Reine Angew. Math. 520:37–93, 2000)). The former is completely different from the Lusztig-Shoji algorithm (Shoji in Invent. Math. 74:239–267, 1983; Lusztig in Ann. Math. 131:355–408, 1990), and it is more effective in a number of cases. The main idea in our proof is to introduce a new family of representations
which behave like tempered modules, but for which it is easier to analyze the effect of parameter specializations. Our proof
also requires a comparison of the C
∗-theoretic results of Opdam, Delorme, Slooten, Solleveld (J. Inst. Math. Jussieu 3:531–648, 2004; arXiv:0909.1227; Int. Math. Res. Not., 2008; Adv. Math. 220:1549–1601, 2009; Acta Math. 205:105–187, 2010), and the geometric construction from Kato (Duke Math. J. 148:305–371, 2009; Am. J. Math. 133:518–553, 2011), Ciubotaru and Kato (Adv. Math. 226:1538–1590, 2011).
... In the present paper we prove the results in a way which is intuitively more clear and which reveals their geometric origin. The methods we are using are similar to those used in [OpSo1] for the analogous statements for tempered modules over an affine Hecke algebra. The pleasant surprise is that such an explicit construction of a continuous contraction of the Schneider-Stuhler resolutions is still possible in this more complicated context, and the computations are not too unpleasant. ...
... An alternative, more explicit, description is given in [OpSo1, Section 1.1]. This works well for the Bruhat-Tits building of G/Z(G), which is the setting of [OpSo1]. ...
... An analogous result was used implicitly in [OpSo1, Section 2.3]. The proof given here also applies in the setting of [OpSo1]. ...
Let G be a reductive group over a non-archimedean local field and let S(G) be
its Schwartz algebra. We compare Ext-groups of tempered G-representations in
several module categories: smooth G-representations, algebraic S(G)-modules,
bornological S(G)-modules and an exact category of S(G)-modules on LF-spaces
which contains all admissible S(G)-modules. We simplify the proofs of known
comparison theorems for these Ext-groups, due to Meyer and Schneider-Zink. Our
method is based on the Bruhat-Tits building of G and on analytic properties of
the Schneider-Stuhler resolutions.
... The relation between the elliptic theories of H and W is given by the restriction map. Precisely, combining results of [OS1,OS2,So] for the affine Hecke algebra and Lusztig's reduction theorems [L1], one sees that the map ...
... As a consequence of the Langlands classification, one sees easily that R C (H) = R C (S). The fact that this isomorphism is an isometry with respect to the Euler-Poincaré pairings follows from [OS1,Corollary 3.7]: ...
... Combining Theorem 4.16 with the results of [OS1] on the Euler-Poincaré pairing as recalled in sections 2.5 and 2.6, one has the following result. ...
We define the algebraic Dirac induction map \Ind_D for graded affine Hecke
algebras. The map \Ind_D is a Hecke algebra analog of the explicit
realization of the Baum-Connes assembly map in the K-theory of the reduced
-algebra of a real reductive group using Dirac operators. The definition
of \Ind_D is uniform over the parameter space of the graded affine Hecke
algebra. We show that the map \Ind_D defines an isometric isomorphism from
the space of elliptic characters of the Weyl group (relative to its reflection
representation) to the space of elliptic characters of the graded affine Hecke
algebra. We also study a related analytically defined global elliptic Dirac
operator between unitary representations of the graded affine Hecke algebra
which are realized in the spaces of sections of vector bundles associated to
certain representations of the pin cover of the Weyl group. In this way we
realize all irreducible discrete series modules of the Hecke algebra in the
kernels (and indices) of such analytic Dirac operators. This can be viewed as a
graded Hecke algebra analogue of the construction of discrete series
representations for semisimple Lie groups due to Parthasarathy and
Atiyah-Schmid.
... All ingredients necessary for the above line of arguments have been developed in detail in the context of abstract affine Hecke algebras as well [DeOp1,DeOp2,OpSo1] . The analytic R-groups are defined in terms of the Weyl group and the Plancherel density, all of which allow for explicit determination [Slo]. ...
... Since the algebra A ξ is a twisted crossed product of a formal power series ring with R ξ , it is not hard to compute the Ext-groups between modules in Mod Wξ,tor bor (S) using Theorem 1. But we are more interested in computing their Ext-groups in the category of H-modules or equivalently [Mey1,OpSo1] in Mod bor (S). To go from there to Mod Wξ,tor bor (S) boils down to applying the formal completion functor at a central character. ...
... The definition of EP also applies naturally to the Grothendieck group G C (H) of finite dimensional representations of an abstract affine Hecke algebra H with positive parameters. Here one uses that the category of finitely generated H-modules has finite cohomological dimension by [OpSo1]. The form EP H on G C (H) is also Hermitian [OpSo1,Theorem 3.5.a]. ...
Let be irreducible tempered representations of an affine Hecke
algebra H with positive parameters. We compute the higher extension groups
explicitly in terms of the representations of analytic
R-groups corresponding to and . The result has immediate
applications to the computation of the Euler-Poincar\'e pairing ,
the alternating sum of the dimensions of the Ext-groups. The resulting formula
for is equal to Arthur's formula for the elliptic pairing of
tempered characters in the setting of reductive p-adic groups. Our proof
applies equally well to affine Hecke algebras and to reductive groups over
non-archimedean local fields of arbitrary characteristic. This sheds new light
on the formula of Arthur and gives a new proof of Kazhdan's orthogonality
conjecture for the Euler-Poincar\'e pairing of admissible characters.
... The definition of EP also applies naturally to the Grothendieck group G C (H) of finite dimensional representations of an affine Hecke algebra H with positive parameters. Here one uses that the category of finitely generated H-modules has finite cohomological dimension by [OpSo1]. The form EP H on G C (H) is Hermitian ([OpSo1, Theorem 3.5 a)]). ...
... (2) and (3) does not seem to have an obvious counterpart in the setting of affine Hecke algebras. Analogues are known for the Euler-Poincaré pairing of representations of either a finite Weyl group [Ree] or of an (extended) affine Weyl group W e [OpSo1,Theorem 3.3]. To relate these to EP H one must compare EP H and EP W e , which is done in [Ree,Section 5.6] (for affine Hecke algebras with equal parameters) and in [OpSo1,Chapter 3]. ...
... Analogues are known for the Euler-Poincaré pairing of representations of either a finite Weyl group [Ree] or of an (extended) affine Weyl group W e [OpSo1,Theorem 3.3]. To relate these to EP H one must compare EP H and EP W e , which is done in [Ree,Section 5.6] (for affine Hecke algebras with equal parameters) and in [OpSo1,Chapter 3]. Recent results from [Sol3] allow us to conclude that G C (H) modulo the radical of EP H equals the vector space Ell(H) of "elliptic characters", and that this space does not depend on the parameters q (Theorem 2.3). ...
Let be tempered representations of an affine Hecke algebra with positive parameters. We study their Euler--Poincar\'e pairing , the alternating sum of the dimensions of the Ext-groups. We show that can be expressed in a simple formula involving an analytic R-group, analogous to a formula of Arthur in the setting of reductive p-adic groups. Our proof applies equally well to affine Hecke algebras and to reductive groups over nonarchimedean local fields of arbitrary characteristic. Comment: 22 pages
... There exists a scaling map σ 0 : G(H) → G(W ) (see [OS,Theorem 1.7]) which descends to a mapσ 0 : Ell(H) → Ell (W ). The finite dimensional Z-module Ell(W ) can be described completely explicitly in terms of the elliptic characters of the isotropy groups W t (with t ∈ T ) for the action of W 0 on T . ...
... The finite dimensional Z-module Ell(W ) can be described completely explicitly in terms of the elliptic characters of the isotropy groups W t (with t ∈ T ) for the action of W 0 on T . The pairing EP on Ell(W ) can be described in these terms as well, and it turns out that EP is positive definite on Ell(W ) (for all these results, consult [OS,Chapter 3]). It turns out that Ell (W ) is nonzero only if R is semisimple, and that the support of Ell(W ) as a Z-module is contained in the the set of orbits W 0 s such that R s,1 ⊂ R 1 is of maximal rank. ...
... Proposition 3.11. A continuous family of irreducible discrete series characters U ∋ q → δ(q) is compatible with the scaling mapsσ ǫ (with ǫ > 0) of [OS,Theorem 1.7] in the sense thatσ ǫ (δ(q)) = δ(q ǫ ). ...
We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases). Comment: 68 pages, 5 figures. In the second version an appendix was added
... In the Iwahori case, Reeder [Re1] described the elements x such that R u (G) x ell = 0, but this description can be extended to the setting of unipotent representations using the results of Opdam-Solleveld [OS1] for Hecke algebras and the transfer from Hecke algebras modules to representations of the group due to Lusztig [Lu1] and Opdam [Op2]. To state this, we need one more definition. ...
... An analogous result was proved in [OS1,Section 3] in the setting of the affine Hecke algebra H of an affine Weyl group W and with arbitrary positive parameters. In that case, one has an isometric isomorphism R(H) ell → R(W ) ell with respect to the Euler-Poincaré pairings on both spaces. ...
In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig's nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G_2.
... Elliptic representation theory, introduced by Arthur [Ar93], studies the Grothendieck group of certain representations of a Lie-theoretic group modulo those induced from proper parabolic subgroups. The elliptic theory of representations of semisimple p-adic groups and Iwahori-Hecke algebras is further studied intensively, e.g., Schneider-Stuhler [SS97], Bezrukavnikov [Be98], Reeder [Re01], Opdam-Solleveld [OS09]. ...
... Opdam and Solleveld in [OS09] studied the affine Hecke algebras for positive parameters and showed that Theorem 3.10. Let q be a positive parameter function onH. ...
This survey article, is written as an extended note and supplement of my
lectures in the current developments in mathematics conference in 2015. We
discuss some recent developments on the conjugacy classes of affine Weyl groups
and p-adic groups, and some applications to Shimura varieties and to
representations of affine Hecke algebras.
... Affine Hecke algebras appear naturally in the representation theory of reductive p-adic groups as convolution algebras of compactly supported functions, such as Iwahori-Hecke algebras e.g., [Bo], [IM], and their generalizations in the theory of types, e.g., [HM], [BK], or as endomorphism algebras of certain projective generators [Be], [Hei]. The representation theory of affine Hecke algebras with (equal and unequal) parameters that are not roots of unity was extensively studied, e.g., [KL], [CG], [Lu1,Lu2], [BM1,BM2], [Xi1], [Op,OS1,OS2,So], [Re], [Kat]. The representation theory of affine Hecke algebras for parameter equal to a root of unity plays an important role in the study of modular representations of p-adic groups [Vi] and the simple modular representations for affine Hecke algebras associated to general linear groups were classified in [AM]. ...
... A familiar object in the literature is the elliptic quotientR 0 (H) obtained by taking the quotient of R(H) C by the span of all proper parabolically induced modules [BDK]. The elliptic representation theory of reductive p-adic groups and associated Hecke algebras has been an area of active research, e.g., [Ar], [BDK], [Be], [Kaz1], [SS], [Re], [OS2]. The dual object toR 0 (H) is the elliptic cocenterH ell , the subspace ofH on which all proper parabolically induced modules vanish. ...
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.
... One important ingredient is analogous properties in the affine highest weight category introduced by Kleshchev [32] (also see [29]), see the proofs in Section 5. Roughly speaking such ingredient reduces to the computations of Ext-groups for tempered modules. Such Ext-groups are now better understood due to the work on discrete series by Silberger, Meyer, Opdam-Solleveld [39,44,49] using analytic methods and by [11] using algebraic methods; and more general case [45] via R-groups. We also refer the reader to [12] for more discussions. ...
Let G n be an inner form of a general linear group over a non-Archimedean local field. We fix an arbitrary irreducible representation σ of G n . Building on the work of Lapid-Mínguez on the irreducibility of parabolic inductions, we show how to define a full subcategory of the category of smooth representations of some G m , on which the parabolic induction functor τ ↦ τ × σ is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-Hölder sequence of the big derivative.
... We give self-contained treatments in Section 5. For the explicit Ext-groups in tempered case, it is better understood due to the work in the discrete series case by Silberger, Meyer, Opdam-Solleveld [Si79,Me06,OS09] using analytic methods and by [Ch16] using algebraic methods; and more general case [OS13] via R-groups, although we do not use them directly. ...
Let be an inner form of a general linear group over a non-Archimedean field. We fix an arbitrary irreducible representation of . Lapid-M\'inguez give a combinatorial criteria for the irreducibility of parabolic induction when the inducing data is of the form when is a segment representation. We show that their criteria can be used to define a full subcategory of the category of smooth representation of some , on which the parabolic induction functor is fully-faithful. A key ingredient of our proof for the fully-faithfulness is constructions of indecomposable representations of length 2. Such result for a special situation has been previously applied in proving the local non-tempered Gan-Gross-Prasad conjecture for non-Archimedean general linear groups. In this article, we apply the fully-faithful result to prove a certain big derivative arising from Jacquet functor satisfies the property that its socle is irreducible and has multiplicity one in the Jordan-H\"older sequence of the big derivative.
... Of course the selection of topics in any survey is to a considerable extent the taste of the author. To preserve a reasonable size, we felt forced to omit many interesting aspects of affine Hecke algebras: the Kazhdan-Lusztig basis [KaLu1], asymptotic Hecke algebras [Lus3,Lus9], unitary representations [BaMo1,Ciu], the Schwartz and C * -completions of affine Hecke algebras [Opd2,DeOp1], homological algebra [OpSo1,Sol5], formal degrees of representations [OpSo2,CKK], spectral transfer morphisms [Opd3,Opd4] and so on. We apologize for these and other omissions and refer the reader to the literature. ...
This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations.
Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with parameters in R≥1, and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.
... Of course the selection of topics in any survey is to a considerable extent the taste of the author. To preserve a reasonable size, we felt forced to omit many interesting aspects of affine Hecke algebras: the Kazhdan-Lusztig basis [KaLu1], asymptotic Hecke algebras [Lus7], unitary representations [BaMo1,Ciu], the Schwartz and C *completions of affine Hecke algebras [Opd2,DeOp1], homological algebra [OpSo1,Sol5], formal degrees of representations [OpSo2,CKK], spectral transfer morphisms [Opd3,Opd4] and so on. We apologize for these and other omissions and refer the reader to the literature. ...
This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible representations. Only at the end we establish a new result: a natural bijection between the set of irreducible representations of an affine Hecke algebra with real parameters , and the set of irreducible representations of the affine Weyl group underlying the algebra. This can be regarded as a generalized Springer correspondence with affine Hecke algebras.
... We remark that in type A case, Theorem A follows through a Morita equivalence with an affine Hecke algebra of type A (see e.g. Opdam-Solleveld [OS09]). ...
We generalize Lusztig's geometric construction of the Poincaré-Birkhoff-Witt (PBW) bases of finite quantum groups of type ADE under the framework of Varagnolo and Vasserot. In particular, every PBW basis of such quantum groups is proven to yield a semi-orthogonal collection in the module category of the Khovanov-Lauda-Rouquier (KLR) algebras. This enables us to prove Lusztig's conjecture on the positivity of the canonical (lower global) bases in terms of the (lower) PBW bases. In addition, we verify Kashiwara's problem on the finiteness of the global dimensions of the KLR algebras of type ADE.
... The notion is based on the special properties of the μ-function of an affine Hecke algebra [52,53] which are intimitely related to its basic role in the derivation of the Plancherel formula for affine Hecke algebras via residues [21,52,55,56]. This approach to the computation of formal degrees has its origin in the theory of spherical functions for p-adic reductive groups [47], and was further inspired by early observations of Lusztig [35,38] and Reeder [59,60] on the behaviour of formal degrees within unipotent L-packets. ...
In this paper we will give a complete classification of the spectral transfer
morphisms between the unipotent affine Hecke algebras of the various inner
forms of a given quasi-split absolutely simple algebraic group, defined over a
non-archimidean local field and split over an unramified extension
of . As an application of these results, the results of [O4] on the
spectral correspondences associated with such morphisms and some results of
Ciubotaru, Kato and Kato [CKK] we prove a conjecture of Hiraga, Ichino and
Ikeda [HII] on the formal degrees and adjoint gamma factors for all unipotent
discrete series characters of unramified simple groups of adjoint type defined
over .
... The continuous families of discrete series characters described in the previous paragraph arise from underlying families of "scaling isomorphisms" which can be defined on the level of formal completions of affine Hecke algebras at a fixed central character. This construction yields (see [OS1,Theorem 1.7]) a family bijections for ∈ (0, 1] ...
We introduce the notion of spectral transfer morphisms between normalized
affine Hecke algebras, and show that such morphisms induce spectral measure
preserving correspondences on the level of the tempered spectra of the affine
Hecke algebras involved.
... Notice that in the Bernstein presentation of the basis ofH, there are exactly N elements that are not represented by elements in a proper parabolic subalgebra of H, where N is the number of elliptic conjugacy classes ofW . 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 H also equals N. It would be interesting to relate these results via the trace map. ...
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.
... A similar identity involving combinatorial data and cohomological data is known for a Koszul algebra A • over a field F; .3) is defined and which satisfy the identity p A · χ A = 1. helpful discussions, and M. Solleveld for pointing out that in a slightly different context projective resolutions of affine Hecke algebras were already constructed in [11]. In an earlier version of this paper the complex C and its properties were discussed in detail. ...
For any Hecke algebra associated to a Coxeter group (W,S) and a distinguished element of a commutative ring with unit R we introduce a finite chain complex of left H-modules which reflects many properties of the Coxeter complex of (W,S), i.e., it is acyclic if (W,S) is non-spherical (cf. Thm. A), and H is of type FP under suitable conditions on the distinguished element (cf. Prop. B). There exists a canonical trace function (cf. Prop. 5.1). This trace function evaluated on the Hattori-Stallings rank of can be considered as the Euler characteristic of H. It will be shown that for generic values of q the Euler characteristic coincides with the reciprocal of the Poincar\'e series of evaluated in q (cf. Thm. C).
... The starting point is a reinterpretation of the Lusztig-Shoji algorithm in terms of certain q-elliptic pairings for W with respect to the reflection representation V (see section 2.1); here q is an indeterminate. This is a generalization of the elliptic pairing of W introduced in [26], see also [24,8,7]. The q-elliptic pairing is also related to Kato's notion of (graded) Kostka systems for the semidirect product A W = C[W ] ⋉ S(V ) and the graded Euler-Poincaré pairing. ...
We provide a direct connection between Springer theory, via Green
polynomials, the irreducible representations of the pin cover \wti W, a
certain double cover of the Weyl group W, and an extended Dirac operator for
graded Hecke algebras. Our approach leads to a new and uniform construction of
the irreducible genuine \wti W-characters. In the process, we give a
construction of the action by an outer automorphism of the Dynkin diagram on
the cohomology groups of Springer theory, and we also introduce a q-elliptic
pairing for W with respect to the reflection representation V. These
constructions are of independent interest. The q-elliptic pairing is a
generalization of the elliptic pairing of W introduced by Reeder, and it is
also related to S. Kato's notion of (graded) Kostka systems for the semidirect
product A_W=\bC[W]\ltimes S(V).
... , r with q being a formal parameter. For more details about this algebra, see [Opd03,Opd04,DO08,OS09,DDF12]. ...
We categorify the extended affine Hecke algebra and the affine quantum Schur
algebra S(n,r) for 2 < r < n, using Elias-Khovanov and Khovanov-Lauda type
diagrams. We also define the affine analogue of the Elias-Khovanov and the
Khovanov-Lauda 2-representations of these categorifications into an extension
of the 2-category of affine (singular) Soergel bimodules.
... It was already proved in Lemma 4.6.1 and it is a consequence of the explicit description of the basis of H δ . The case δ = 1 was proved in the setting of affine Hecke algebras in [OS,Proposition 3.9] via different methods. Proof. ...
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).
... [NO82] D.VII). In this case, an explicit bi-resolution of A is constructed by Opdam-Solleveld [OS09]. ...
We present simple conditions which guarantee a geometric convolution algebra
to behave like a variant of the quasi-hereditary algebra. In particular,
standard modules of the affine Hecke algebras of type , and the
quiver Schur algebras are shown to satisfy the Brauer-Humphreys type
reciprocity and the semi-orthogonality property. In addition, we present a new
criterion of purity of weights in the geometric side. This yields a proof of
Shoji's conjecture on limit symbols of type [Shoji, Adv. Stud.
Pure Math. 40 (2004)], and the purity of the exotic Springer fibers [K, Duke
Math. 148 (2009)]. Using this, we describe the leading terms of the
-realization of a solution of the Lieb-McGuire system in the
appendix. In [K, arXiv:1203.5254], we apply the results of this paper to the
KLR algebras of type to establish Kashwara's problem and
Lusztig's conjecture.
... In the case where V = X, we have obtained an exact resolution (C or c (X (•) , X)) I → X I of H = X I as an (H, H)-bimodule: it is the resolution (0.1). Note that, if the field k has characteristic 0, then for H ′ such a result is already contained (with a different proof) in [37]. 0.2.2. ...
Let F be a locally compact nonarchimedean field with residue characteristic p
and G the group of F-rational points of a connected split reductive group over
F. For k an arbitrary field, we study the homological properties of the
Iwahori-Hecke k-algebra H' and of the pro-p Iwahori-Hecke k-algebra H of G. We
prove that both of these algebras are Gorenstein rings with self-injective
dimension bounded above by the rank of G. If G is semisimple, we also show that
this upper bound is sharp, that both H and H' are Auslander-Gorenstein and that
there is a duality functor on the finitely generated modules of H (respectively
H'). We obtain the analogous Gorenstein and Auslander-Gorenstein properties for
the graded rings associated to H and H'.
When k has characteristic p, we prove that in most cases H and H' have
infinite global dimension. In particular, we deduce that the category of smooth
k-representations of G=PGL(2,Q_p) generated by their invariant vectors under
the pro-p-Iwahori subgroup has infinite global dimension (at least if k is
algebraically closed).
... This problem is raised by Kashiwara several times in his lectures on KLR algebras. We remark that in type A case, Theorem A follows from the work of Opdam-Solleveld [OS09] through Morita equivalences with affine Hecke algebras of type A, but their proof is completely different from that of ours. ...
We generalize Lusztig's geometric construction of the PBW bases of finite
quantum groups of type under the framework of
[Varagnolo-Vasserot, J. reine angew. Math. 659 (2011)]. In particular, every
PBW basis of such quantum groups is proven to yield a semi-orthogonal
collection in the module category of the KLR-algebras. This enables us to prove
Lusztig's conjecture on the positivity of the canonical (lower global) bases in
terms of the (lower) PBW bases in the case. In addition, we
verify Kashiwara's problem on the finiteness of the global dimensions of the
KLR-algebras of type .
... In particular, one has the identity p (H .3) is defined and which satisfy the identity p (A,B,λ) · χ (A,B,λ) = 1. helpful discussions, and M. Solleveld for pointing out that in a slightly different context projective resolutions of affine Hecke algebras were already constructed in [12]. Our gratitude goes also to A. Mathas and S. Schroll for informing us about the Deodhar complex (cf. ...
It is shown that the Euler characteristic χ(H,B,εq) of a Z〚[q〛]-Hecke algebra H associated with a finitely generated Coxeter group (W,S) coincides with (p_{(W,S)}(q))^{−1}, where p_{(W,S)}(t) is the Poincaré series of (W,S).
To an associative R-algebra A equipped with an R-linear involution_^#: A ->
A^{op}, a linear representation , and a free R-basis B satisfying
certain conditions one can associate an Euler characteristic (cf.
{\S}4.5). These algebras will be called {\it Euler algebras}.
Under a suitable condition on the distinguished parameter , we show
that the R-Hecke algebra H_q associated with a finitely generated Coxeter group
(W,S) is an R-Euler algebra, and its Euler characteristic coincides with
, where is the Poincar\'e series associated
with (W,S).
... The Schwartz algebra is the ideal tool for the harmonic analysis of affine Hecke algebras, among others because it admits a very nice Plancherel theorem (due to Delorme and Opdam, see [DeOp1] or Theorem 3.2.2), because the discrete series of H(R, q) is really discrete in the dual of S(R, q), and because the inclusion H(R, q) → S(R, q) preserves Ext-groups of tempered representations [OpSo1]. ...
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.
... 1 Note: After the original version of this paper is circulated (in 2006, with different argument and weaker conclusion in Theorem D, and consequently give a classification of Hmodules only with a help of Lusztig's results [Lu88,Lu89,Lu95b]), there appeared two kinds of related works. One is the study of geometry which is connected to our nilcone by Achar-Henderson [AH08], Enomoto [En08], Finkelberg-Ginzburg-Travkin [FGT08], Springer [Sp07], Travkin [Tr08], and the other is the classification of tempered dual by Opdam and Solleveld [OS07,OS08,So07]. For the former, I have included explanations about the situation as much as I could in order to avoid potential problems. ...
Let G be a complex symplectic group. We introduce a G x (C ^x) ^{l + 1}-variety N_{l}, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type C_n ^(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N_2. This enables us to establish a Deligne-Langlands type classification of "non-critical" simple H-modules. As applications, we present a character formula and multiplicity formulas of H-modules. Comment: v7, 52pages. Corrected typos and errors in the proofs of Lemma 4.1 and Theorem 6.2 modulo Proposition 6.7, final version, accepted for publication in Duke Math
We introduce the local and global indices of Dirac operators for the rational Cherednik algebra , where G is a complex reflection group acting on a finite-dimensional vector space . We investigate precise relations between the (local) Dirac index of a simple module in the category of , the graded G-character of the module, the Euler-Poincar\'e pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for constructed from finite-dimensional G-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function c. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl-Opdam operators.
Kazhdan and Lusztig identified the affine Hecke algebra ℋ with an equivariant K K -group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields F F with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from K K -theory to Hochschild homology and thereby identify ℋ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf . As a result the derived category of ℋ-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu).
In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of GL n ( F ) into coherent sheaves on the stack of Langlands parameters.
Hiraga, Ichino and Ikeda have conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F, in terms of local Langlands parameters. In these lectures we shall present a proof of these conjectures for Lusztig's class of representations of unipotent reduction if F is p-adic and G is of adjoint type and splits over an unramified extension of F. This is based on the author's paper [Spectral transfer morphisms for unipotent affine Hecke algebras, Selecta Math. (N.S.) {\bf 22} (2016), no. 4, 2143--2207]. More generally for G connected reductive (still assumed to be split over an unramified extension of F), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction. We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors. The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in \it{op. cit.}
In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen filtrations. Our approach involves the use of information from the Langlands classification as well as some deeper understanding on some structure of some modules. Such module arises from the image of a Knapp-Stein type intertwining operator and is a quotient of a generalized standard module. As an application, we compute the Ext-groups for irreducible modules in a block for the graded Hecke algebra of type C3, assuming the truth of a version of Jantzen conjecture.
In this paper, we consider the relation between two nonabelian Fourier transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig parameters for unipotent elliptic representations of a split p-adic group and the second is defined in terms of the pseudocoefficients of these representations and Lusztig’s nonabelian Fourier transform for characters of finite groups of Lie type. We exemplify this relation in the case of the p-adic group of type G2.
We introduce the local and global indices of Dirac operators for the rational Cherednik algebra , where G is a complex reflection group acting on a finite-dimensional vector space . We investigate precise relations between the (local) Dirac index of a simple module in the category of , the graded G-character of the module, the Euler-Poincar\'e pairing, and the composition series polynomials for standard modules. In the global theory, we introduce integral-reflection modules for constructed from finite-dimensional G-modules. We define and compute the index of a Dirac operator on the integral-reflection module and show that the index is, in a sense, independent of the parameter function c. The study of the kernel of these global Dirac operators leads naturally to a notion of dualised generalised Dunkl-Opdam operators.
In this paper, we establish connections between the first extensions of
simple modules and certain filtrations of of standard modules in the setting of
graded Hecke algebras. The filtrations involved are radical filtrations and
Jantzen filtrations. Our approach involves the use of information from the
Langlands classification as well as some deeper analysis on structure of some
modules. Such modules arise from the image of a Knapp-Stein type intertwining
operator and is a quotient of a generalized standard module.
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).
Cerebral ischemia/reperfusion leads to selective neuronal death in specific brain areas. The effects of forebrain ischemia produced by four-vessel occlussion were investigated in rats. Immunohistochemical localization of ubiquitin in the brain cortex and hippocampus was studied. Using antibodies against ubiquitin we have found that ubiquitin immunoreactivity (UIR) was normally present in all neurons of hippocampus and in the cerebral cortex. After 20 min of ischemia and 24 h of reperfusion the number of ubiquitin positive cells was decreased in CA2 and CA3 regions of hippocampus and ubiquitin-negative neurons were found in the most vulnerable region of CA1 subfield. Cerebral cortex showed significant changes of ubiquitin immunoreactivity in the nuclei and perikarya of pyramidal neurons. Seven days of reperfusion after 20 min of ischemia showed recovery of UIR in the hippocampal CA2, CA3 regions and in granule cells layer of the dentate gyrus. The most vulnerable CA1 pyramidal cells showed morphological changes of the cell bodies and irregularities of the cell membrane. Interestingly, number of those cells display intensive UIR in their perikarya and in the nuclei as well. Comparison of UIR in CA1 pyramidal cells after ischemia/reperfusion with those in the control sections showed increase in ubiquitin reaction. Remarkable UIR was also found in all layers in the cerebral cortex. These data suggest that recovery of ubiquitin immunoreactivity in some regions of hippocampus and cerebral cortex may be a prerequisite to neuronal survival after ischemia/reperfusion. In CA1 region the accumulation of ubiquitinated proteins in pyramidal cells was visualized without their degradation. The stress protein ubiquitin is one of the regulatory proteins playing a role in ischemic tolerance.
This is a brief introduction to the quiver Hecke algebras of Khovanov, Lauda
and Rouquier, emphasizing their application to the categorification of quantum
groups. The text is based on lectures given by the author at the ICRA workshop
in Bielefeld in August, 2012.
A uniform parametrization for the irreducible spin representations of Weyl
groups in terms of nilpotent orbits is recently achieved by Ciubotaru (2011).
This paper is a generalization of this result to other real reflection groups.
Let be a root system with the real reflection
group W. We define a special subset of points in which will be
called solvable points. Those solvable points, in the case R
crystallographic, correspond to the nilpotent orbits whose elements have a
solvable centralizer in the corresponding Lie algebra. Then a connection
between the irreducible spin representations of W and those solvable points
in is established.
For a Weyl group W, we give a simple closed formula (valid on elliptic
conjugacy classes) for the character of the representation of W in each
A-isotypic component of the full homology of a Springer fiber. We also give a
formula (valid again on elliptic conjugacy classes) of the W-character of an
irreducible discrete series representation with real central character of a
graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin
double cover of W and the Dirac operator for graded affine Hecke algebras play
key roles.
We show that affine Hecke algebras of rank two with generic parameters are
affine cellular in the sense of Koenig-Xi.
We give a common framework for the classification of projective spin
irreducible representations of a Weyl group, modeled after the Springer
correspondence for ordinary representations.
We define an analogue of the Casimir element for a graded affine Hecke algebra , and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H
D
(X) of an -module X, and show that H
D
(X) carries a representation of a canonical double cover of the Weyl group . Our main result shows that the -structure on the Dirac cohomology of an irreducible -module X determines the central character of X in a precise way. This can be interpreted as p-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of .
Recently the classification of unipotent representations of a simple p-adic group was obtained by Lusztig based on his ideas of character sheaves. An alternative strategy had been proposed by Reeder based on a comparison of formal degrees. We show how the formal degrees can be computed from a residue calculation and thereby turn the approach of Reeder into an efficient route.
Iwahori�Hecke algebras are ubiquitous. One encounters these algebras in subjects as diverse as harmonic analysis, equivariant K-theory, orthogonal polynomials, quantum groups, knot theory, algebraic combinatorics, and integrable models in statistical physics. In this exposition we will mostly concentrate on the analytic aspects of affine Hecke algebras and study them from the perspective of operator algebras. We will discuss the Plancherel theorem for these type of algebras, and based on a conjectural invariance property of their (operator algebraic) K-theory, study the structure of the tempered dual.
Proefschrift Universiteit van Amsterdam. Met samenvatting in het Nederlands.
For a general affine Hecke algebra H we study its Schwartz completion S. The main theorem is an exact description of the image of S under the Fourier isomorphism. An important ingredient in the proof of this result is the definition and computation of the constant terms of a coefficient of a generalized principal series representation. Finally we discuss some consequences of the main theorem for the theory of tempered representations of H.
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 introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases). Comment: 68 pages, 5 figures. In the second version an appendix was added
In this paper, F F is a non-Archimedean local field and G G is the group of F F -points of a connected reductive algebraic group defined over F F . Also, τ \tau is an irreducible representation of a compact open subgroup J J of G G , the pair ( J , τ ) (J,\tau ) being a type in G G . The pair ( J , τ ) (J,\tau ) is assumed to be a cover of a type ( J L , τ L ) (J_{L},\tau _{L}) in a Levi subgroup L L of G G . We give conditions, generalizing those of earlier work, under which the Hecke algebra H ( G , τ ) \scr H(G,\tau ) is the tensor product of a canonical image of H ( L , τ L ) \scr H(L,\tau _{L}) and a sub-algebra H ( K , τ ) \scr H(K,\tau ) , for a compact open subgroup K K of G G containing J J .
We describe a duality operator for representations of the Hecke algebra of a Weyl group or of an affine Weyl group in terms of a certain involution on this algebra.
Rust. Although we made hundreds of changes, we did not alter any of the methods used or the numbering of any results. Casselman has neither approved nor disavowed the end product. Several mathematical and stylistic questions have been put aside for another day, which has not yet arrived. These are all indicated in footnotes.
In this paper, F is a non-Archimedean local field and G is the group of F-points of a connected reductive algebraic group defined over F. Also, τ is an irreducible representation of a compact open subgroup J of G, the pair (J, τ) being a type in G. The pair (J, τ) is assumed to be a cover of a type (JL, τL) in a Levi subgroup L of G. We give conditions, generalizing those of earlier work, under which the Hecke algebra H(G, τ) is the tensor product of a canonical image of H(L, τL) and a sub-algebra H(K, τ), for a compact open subgroup K of G containing J.
This graduate textbook presents a concrete and up-to-date introduction to the theory of Coxeter groups. The book is self-contained, making it suitable either for courses and seminars or for self-study. The first part is devoted to establishing concrete examples. Finite reflection groups acting on Euclidean spaces are discussed, and the first part ends with the construction of the affine Weyl groups, a class of Coxeter groups that plays a major role in Lie theory. The second part (which is logically independent of, but motivated by, the first) develops from scratch the properties of Coxeter groups in general, including the Bruhat ordering and the seminal work of Kazhdan and Lusztig on representations of Hecke algebras associated with Coxeter groups is introduced. Finally a number of interesting complementary topics as well as connections with Lie theory are sketched. The book concludes with an extensive bibliography on Coxeter groups and their applications.
The complex representation theory of a finite Lie group G G is related to that of certain "generic algebras". As a consequence, formulae are derived ("the Comparison Theorem"), relating multiplicities in G G to multiplicities in the Weyl group W W of G G . Applications include an explicit description of the dual (see below) of an arbitrary irreducible complex representation of G G .
We describe a duality operator for representations of the Hecke algebra of a Weyl group or of an affine Weyl group in terms of a certain involution on this algebra.
Here, F denotes a non-Archimedean local field (with finite residue field) and G the group of F-points of a connected reductive algebraic group defined over F. Let R(G) denote the category of smooth, complex representations of G. Let B(G) be the set of pairs (L, σ), where L is an F-Levi subgroup of G and σ is an irreducible supercuspidal representation of L, taken modulo the equivalence relation generated by twisting with unramified quasi characters and G-conjugacy. To s ∈ B(G), one can attach a full (abelian) sub-category Rs(G) of R(G); the theory of the Bernstein centre shows that R(G) is the direct product of these Rs(G). The object of the paper is to give a general method for describing these factor categories via representations of compact open subgroups within a uniform framework. Fix s ∈ B(G). Let K be a compact open subgroup of G and ρ an irreducible smooth representation of K. The pair K, ρ is an s-type if it has the following property: an irreducible representation π of G contains ρ if and only if π in Rs(G). Let H(G, ρ) be the Hecke algebra of compactly supported ρ-spherical functions on G; if (K,ρ) is an s-type, then the category Rs(G) is canonically equivalent to the category H(G, ρ)-Mod of H(G, ρ-modules. Let M be a Levi subgroup of G; there is a canonical map B(M) to B(G). Take t ∈ B(M) with image s ∈ B(G). The choice of a parabolic subgroup of G with Levi component M gives functors of parabolic induction and Jacquet restriction connecting Rt(M) with Rs(G). We assume given a t-type (KM,ρM) in M; the paper concerns a general method of constructing from this data an s-type (K,ρ) in G. One thus obtains a description of these induction and restriction functors in terms of an injective ring homomorphism H(M,ρM) → H(G,ρ). The method applies in a wide variety of cases, and subsumes much previous work. Under further conditions, observed in certain particularly interesting cases, one can go some distance to describing H(G,ρ) explicitly. This enables one to isolate cases in which the map on Hecke algebras is an isomorphism, and this in turn implies powerful intertwining theorems for the types. 1991 Mathematics Subject Classification: 22E50.
Suppose that W is a Weyl group, let C(W) be a space of functions on W, with complex values, invariant under conjugation. We can define an “elliptic scalar product” on C(W). It is a natural ingredient to the representation theory of p-adic reductive groups. Let G be a reductive group over the algebraic closure of a finite field. The generalized Springer correspondence gives a bijection between two sets:–the set of pairs (U,E), where U is an unipotent orbit of G and E is a G-equivariant irreducible local system on U;–the disjoint union of the sets of irreducible representations of certain Weyl groups related to G. Using Kazhdan–Lusztig polynomials, we modify the generalized Springer correspondence. By the modified correspondence, a pair (U,E) as above maps to a representation of a certain Weyl group, and this representation is, in general, reducible. There is no simple formula that relates the elliptic scalar product and the generalized Springer correspondence. But a simple formula does exist, and we prove it, that relates the elliptic scalar product and the modified generalized Springer correspondence. Our result is, in fact, a corollary of a theorem of Lusztig on the restriction of character-sheaves to the unipotent variety.
The group G acts on D(G) by conjugation x:tt--)l,t x, x E G, ~ ED(G). We denote by J(G)CD(G) the subspace spanned by differences /x-/x x, /x ED(G), x E G, and define D(G) = D(G)/J(G). Let R(G) be the space of locally constant functions f on G invariant under conjugation. We denote by ( ):R(G) [)(G)-oC the natural pairing (f,~) = d~'fGf*lx where/x E D(G) is a representative of /.Z and f*(g)= f(g-'). It is clear that ( , ) is a perfect pairing. Let ;/CD(G) be the space of measures /.L such that /~" =/x for all x E G. The convolution defines an algebra structure on D(G) and ,~ is the center of D(G), ,~ acts naturally on D(G) and R(G). It is easy to check the following results. Proposition lc. For any fE R(G), t~ Ef)(G), z E,~ we have (zf, g)= . We choose a Haar measure dg on G. Then any /~ ~D(G) can be written in the form /x = F~,dg where F~ is a locally constant function on G. For any /z E D(G) we define a function ~ on G by
Let G be a connected reductive group defined over a local non Archimedean field F with residue field F; let P be a parahoric subgroup with associated reductive quotient M. If σ is an irreducible cuspidal representation of M(F) it provides an irreducible representation of P by inflation. We show that the pair (P,σ) is an -type as defined by Bushnell and Kutzko. The cardinality of can be bigger than one; we show that if one replaces P by the full centraliser of the associated facet in the enlarged affine building of G, and σ by any irreducible smooth representation of which contains σ on restriction then (,) is an -type for a singleton set . Our methods employ invertible elements in the associated Hecke algebra (σ) and they imply that the appropriate parabolic induction functor and its left adjoint can be realised algebraically via pullbacks from ring homomorphisms.
The Bruhat-Tits building X of a connected reductive group G over a nonarchimedean local field K is a rather intriguing G-space. It displays in a geometric way the inner structure of the locally compact group G like the classification of maximal compact subgroups or the theory of the Iwahori subgroup. One might consider X not quite as a full analogue of a real symmetric space but as a kind of skeleton of such an analogue. As such it immediately turned out to be an important technical device in the smooth representation theory of the group G. As a reminder let us mention that the irreducible smooth representations of G lie at the core of the local Langlands program which aims at understanding the absolute Galois group of the local field K. In this paper we develop a systematic and conceptional theory which allows to pass in a functorial way from smooth representations of G to equivariant objects on X. There actually will be two such constructions--a homological and a cohomological one. Since the building carries a natural C W-structure the notion of a coefficient system (or cosheaf) on X makes sense. In the homological theory we will construct functors from smooth representations to G-equivariant coefficient systems on X. It should be stressed that the definition of the coefficient system only involves the original G-representation as far as the action of certain compact open subgroups of G is concerned. One therefore might consider the whole construction as a kind of localization process. Our main result will be that the cellular chain complex naturally associated with a coefficient system provides (under mild assumptions) a functorial projective resolution of the G-representation we started with. In the cohomological theory we will associate, again functorially, G-equivariant sheaves on X with smooth G-representations. The main task which we will achieve then is the computation of the cohomology with compact support of the sheaves coming from an irreducible smooth G-representation. The result can best be formulated in terms of a certain duality functor on the category of finite length smooth G-representations. As a major application we will prove Zelevinsky's conjecture in [Zel] that his duality map on the level of Grothendieck groups preserves irreducibility. For carrying
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
Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered
representations of G to the derived category of all smooth representations of G is fully faithful. Here we consider representations on bornological vector spaces. As a consequence, if G is semi-simple, V and W are tempered irreducible representations of G, and V or W is square-integrable, then Ext
G
n
(V, W) ≅ 0 for all n ≥ 1. We use this to prove in full generality a formula for the formal dimension of square-integrable representations due
to Schneider and Stuhler.
49> 0 ) = Ext i C (V 0 ; V ) = 0 for all integers i ? 0 ? This question is motivated by the orthogonal decomposition of the Schwartz algebra of H given by the Plancherel formula ([Sil, Th.3, page 4679] for example). I tried to prove without success that the answer was yes, some years ago while writing [Vig1]. The answer (yes) is an exercise for GL(n; F ) for any integer n ? 1. It can be worth to publish it. Let H = G := GL(n; F ). Let V 2 C irreducible essentially square integrable. We can describe all the irreducible V 0 2 C such that Ext i C (V 0 ; V ) 6= 0 for at least one integer i 0.
Let A be an algebra with a countable basis and let B be, say, a Frechet algebra that contains A as a dense subalgebra. This embedding induces a functor from the derived category of B-modules to the derived category of A-modules. In many important examples, this functor is fully faithful. We study this property in some detail, giving several equivalent conditions, examples, and applications. To prepare for this, we explain carefully how to do homological algebra with modules over bornological algebras. We construct the derived category of bornological left A-modules and some standard derived functors, with special emphasis on the adjoint associativity between the tensor product and the internal Hom functor. We also discuss the category of essential modules over a non-unital algebra and its functoriality.
Geometry of buildings is used to prove some homological properties of the category of smooth representations of a reductive p-adic group (Kazhdan's "pairing conjecture", Bernstein's description of homological duality in terms of Deligne-Lusztig duality). A different proof had been obtained a little earlier by Schneider and Stuhler.
Given a metrizable topological vector space, we can also use its von Neumann
bornology or its bornology of precompact subsets to do analysis. We show that
the bornological and topological approaches are equivalent for many problems.
For instance, they yield the same concepts of convergence for sequences of
points or linear operators, of continuity of functions, of completeness and
completion. We also show that the bornological and topological versions of
Grothendieck's approximation property are equivalent for Frechet spaces. These
results are important for applications in noncommutative geometry. Finally, we
investigate the class of ``smooth'' subalgebras appropriate for local cyclic
homology and apply some of our results in this context.
Let G(K) be the group of K-rational points of a connected adjoint simple algebraic group defined over a non-archimedean local field K. In this paper we classify the unipotent representations of G(K) in terms of the geometry of the Langlands dual group. (This was known earlier in the special case where G(K) is an inner form of a split group.) We also determine which representations are tempered or square integrable.
Harmonic analysis for affine Hecke algebras, in: Current Developments in Mathemat-ics
- G J Heckman
- E M Opdam
G.J. Heckman, E.M. Opdam, Harmonic analysis for affine Hecke algebras, in: Current Developments in Mathemat-ics, International Press, 1997, pp. 37–60.
Homological algebra for Schwartz algebras of reductive p-adic groups, in: Noncommutative Geometry and Number Theory
- R Meyer
R. Meyer, Homological algebra for Schwartz algebras of reductive p-adic groups, in: Noncommutative Geometry and Number Theory, in: Aspects Math., vol. E37, Vieweg Verlag, 2006, pp. 263–300.
The discrete series of affine Hecke algebras
- E M Opdam
- M S Solleveld
E.M. Opdam, M.S. Solleveld, "The discrete series of affine Hecke algebras"
(in preparation)