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Cuspidal Local Systems and Graded Hecke Algebras II
Abstract
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.
... By [Lu5] 1.15, L(s, e) is the unique irreducible quotient of a standard module X G (s, e, r 0 ) = X(s, e) associated to {(s, e)}. See [Lu1] 8 and [Lu4] 10 for the definition of the X(s, e) (and [Lu4] 10.11 for the fact that both constructed modules are isomorphic). ...
... By [Lu5] 1.15, L(s, e) is the unique irreducible quotient of a standard module X G (s, e, r 0 ) = X(s, e) associated to {(s, e)}. See [Lu1] 8 and [Lu4] 10 for the definition of the X(s, e) (and [Lu4] 10.11 for the fact that both constructed modules are isomorphic). ...
... As representations of a finite group cannot be deformed, the W -structure of X GL N (s, e, r 0 ) is independent of r 0 . Hence we can assume r 0 = 0 and we have an isomorphism of W -modules ( [Lu4] 10.13) X(s, e) ∼ = H * (B e ) ⊗ sgn. ...
In this article we calculate the signature character of certain Hermitian representations of for a p-adic field F. We further give a conjectural description for the signature character of unramified representations in terms of Kostka numbers.
... . The irreducible representations of such graded Hecke algebras were parametrized and constructed geometrically in [Lus2,Lus4]. The parameters are triples (σ, y, ρ • ) where: ...
... • u ) satisfying the analogue of (ii) on page 21. By [Lus4] G ∨,• u -conjugacy classes of such triples are naturally in bijection with Irr H(G ∨,• u , T ∨ , triv, log(q F )/2) . Let us write that as (σ, y, ρ • ) → M • y,σ,ρ • . ...
... Let us write that as (σ, y, ρ • ) → M • y,σ,ρ • . In [Lus2,Lus4] there is an extra parameter r ∈ C, but we suppress that because in this paper it will always be equal to log(q F )/2. From these parameters σ can always be chosen in Lie(T ∨ ), and then W (R u>0 )σ is the central character of M • y,σ,ρ • . Lusztig's parametrization was slightly modified in [AMS2, §3.5], essentially by composing it with the Iwahori-Matsumoto involution IM of H • s ∨ ,u . ...
Let G be a quasi-split reductive group over a non-archimedean local field. We establish a local Langlands correspondence for all irreducible smooth complex G-representations in the principal series. The parametrization map is injective, and its image is an explicitly described set of enhanced L-parameters. Our correspondence is determined by the choice of a Whittaker datum for G, and it is canonical given that choice. We show that our parametrization satisfies many expected properties, among others with respect to the enhanced L-parameters of generic representations, temperedness, cuspidal supports and central characters. Along the way we characterize genericity in terms of representations of an affine Hecke algebra.
... In the nonarchimedean setting, Kazhdan and Lusztig [KL87] and Ginzburg [CG97] established an affine counterpart to Springer theory, identifying the representation theory of the affine Hecke algebra (or equivalently unramified principal series representations of GpKq) in terms of (certain) equivariant constructible sheaves on the spaces of unipotent Langlands parameters. Lusztig [Lu95b] (following [Lu88,Lu95a]) completed this picture to an "affine generalized Springer correspondence", establishing the local Langlands correspondence between all unipotent representations of pure inner forms of GpKq for G adjoint, simple and unramified and all equivariant constructible sheaves on unipotent Langlands parameters. This work has recently been extended by Solleveld [Sol18,Sol19] first to all unipotent representations of connected G and then [Sol22a,Sol22b] to prove Vogan's padic Kazhdan-Lusztig conjecture for a much broader class of representations. ...
... The algebras Hpαq are shearings of completions of Lusztig's graded Hecke algebras [Lu88,Lu89]. We note that the equivariantization with respect to graded lifts, shearing, and de-equivariantization that appears in Koszul duality (as discussed in A.1.3) is compatible with the corresponding gradings in statements in [Sol18,Sol19], and that one can avoid such complications by working 2-periodically. ...
... For a general reductive group, there are more perverse sheaves on the nilpotent cone and its pσ, qq-fixed loci, coming from cuspidal local systems on Levi subgroups ofǦ, as classified by Lusztig (see e.g. [Lu88,Lu95a]). On the other hand there are also more unipotent representations of G and of its pure inner forms than unramified principal series. ...
Refined forms of the local Langlands correspondence seek to relate representations of reductive groups over local fields with sheaves on stacks of Langlands parameters. But what kind of sheaves? Conjectures in the spirit of Kazhdan-Lusztig theory (due to Vogan and Soergel) describe representations of a group and its pure inner forms with fixed central character in terms of constructible sheaves. Conjectures in the spirit of geometric Langlands (due to Fargues, Zhu and Hellmann) describe representations with varying central character of a large family of groups associated to isocrystals in terms of coherent sheaves. The latter conjectures also take place on a larger parameter space, in which Frobenius (or complex conjugation) is allowed a unipotent part. In this article we propose a general mechanism that interpolates between these two settings. This mechanism derives from the theory of cyclic homology, as interpreted through circle actions in derived algebraic geometry. We apply this perspective to categorical forms of the local Langlands conjectures for both archimedean and non-archimedean local fields. In the nonarchimedean case, we describe how circle actions relate coherent and constructible realizations of affine Hecke algebras and of all smooth representations of , and propose a mechanism to relate the two settings in general. In the archimedean case, we explain how to use circle actions to derive the constructible local Langlands correspondence (in the form due to Adams-Barbasch-Vogan and Soergel) from a coherent form (a real counterpart to Fargues' conjecture): the tamely ramified geometric Langlands conjecture on the twistor line, which we survey.
... Graded Hecke algebras H with several parameters (now typically called k) do admit a geometric interpretation [Lus2,Lus5]. (Not all combinations of parameters occur though, there are conditions on the ratios between the different k-parameters.) For this reason graded Hecke algebras, instead of affine Hecke algebras, play the main role in this paper. ...
... We will work in D b G×C × (X), the G × C × -equivariant bounded derived category of constructible sheaves on a complex variety X [BeLu]. In [Lus2,Lus5,AMS2] an important object K ∈ D b G×C × (g) was constructed from qE, by a process that bears some similarity with parabolic induction. Let g N be the variety of nilpotent elements in the Lie algebra g of G and let K N be the pullback of K to g N . ...
... Let pr 1 :ġ → g be the projection on the first coordinate. When G is connected, Lusztig [Lus5] has constructed graded Hecke algebras from K := pr 1,!q E ∈ D b G×C × (g). For our purposes the pullback K N of K to the nilpotent variety g N ⊂ g will be more suitable than K itself. ...
Graded Hecke algebras can be constructed geometrically, with constructible sheaves and equivariant cohomology. The input consists of a complex reductive group G (possibly disconnected) and a cuspidal local system on a nilpotent orbit for a Levi subgroup of G. We prove that every such "geometric" graded Hecke algebra is naturally isomorphic to the endomorphism algebra of a certain G x C*-equivariant semisimple complex of sheaves on the nilpotent cone . From there we provide an algebraic description of the G x C*-equivariant bounded derived category of constructible sheaves on . Namely, it is equivalent with the bounded derived category of finitely generated differential graded modules of a suitable direct sum of graded Hecke algebras. This can be regarded as a categorification of graded Hecke algebras.
... In the present article, we establish a generalised Springer correspondence for Z/m-graded Lie algebras and (trigonometric) degenerate double affine Hecke algebras, which was conjectured by Lusztig-Yun [19]. The main result is Theorem 9.12, which confirms the multiplicity-one conjecture proposed in [19] and can be viewed as a generalised Springer correspondence in the sense of Lusztig [15,13], for certain degenerate double affine Hecke algebras (dDAHAs) with possibly unequal parameters. Generalised Springer correspondence. ...
... The number of G λ -orbits in g η is finite. The classification of G λ -equivariant simple perverse sheaves on g η proven in [13,14,16,3] is the following: ...
... In §1, we review the sheaf-theoretic construction of graded AHAs given in [13], [3] and [14]. We prove in Proposition 1.18 that this construction is compatible with parabolic induction. ...
... In [1, Example 6.2.2] it is shown that there can be no such adjunction for the algebraic group G = SL(2, C) and the varieties X = P 1 C and Y = Spec C when we take D G (−) := D b (Per SL(2,C) (−)). As such, a more delicate approach towards the equivariant derived category was required, which lead to the competing formats that were studied in [29] (the simplicial approach); [1] and [8] (the Bernstein-Lunts approach); [65] (Lusztig's approach for use with graded Hecke algebras); and [7] (the stacky approach). We begin our examination of these categories in Part I by setting the base camp of our journey at a generalization of Lusztig's equivariant derived category as it appears in [65]. ...
... As such, a more delicate approach towards the equivariant derived category was required, which lead to the competing formats that were studied in [29] (the simplicial approach); [1] and [8] (the Bernstein-Lunts approach); [65] (Lusztig's approach for use with graded Hecke algebras); and [7] (the stacky approach). We begin our examination of these categories in Part I by setting the base camp of our journey at a generalization of Lusztig's equivariant derived category as it appears in [65]. ...
... where SfResl G (X) is a category of smooth free G-resolutions of X and Katze is the 2-category of categories. We develop and study the theory of these categories in order to have tools and methods to carefully and rigorously study the equivariant derived category of Lusztig as defined in [65]. In the beginning of this chapter we provide some basic definitions and sanity results that say, among other things, that the equivariant category over the trivial algebraic group 1, F 1 (X), is equivalent to the basic category F (X). ...
In this thesis we study two main topics which culminate in a proof that four distinct definitions of the equivariant derived category of a smooth algebraic group G acting on a variety X are in fact equivalent. In the first part of this thesis we introduce and study equivariant categories on a quasi-projective variety X. These are a generalization of the equivariant derived category of Lusztig and are indexed by certain pseudofunctors that take values in the 2-category of categories. This 2-categorical generalization allow us to prove rigorously and carefully when such categories are additive, monoidal, triangulated, admit t-structures, among and more. We also define equivariant functors and natural transformations before using these to prove how to lift adjoints to the equivariant setting. We also give a careful foundation of how to manipulate t-structures on these equivariant categories for future use and with an eye towards future applications. In the final part of this thesis we prove a four-way equivalence between the different formulations of the equivariant derived category of -adic sheaves on a quasi-projective variety X. We show that the equivariant derived category of Lusztig is equivalent to the equivariant derived category of Bernstein-Lunts and the simplicial equivariant derived category. We then show that these equivariant derived categories are equivalent to the derived -adic category of Behrend on the algebraic stack . We also provide an isomorphism of the simplicial equivariant derived category on the variety X with the simplicial equivariant derived category on the simplicial presentation of , as well as prove explicit equivalences between the categories of equivariant -adic sheaves, local systems, and perverse sheaves with the classical incarnations of such categories of equivariant sheaves.
... Graded Hecke algebras H with several parameters (now typically called k) do admit a geometric interpretation [Lus1,Lus3]. (Not all combinations of parameters occur though, there are conditions on the ratios between the different k-parameters.) For this reason graded Hecke algebras, instead of affine Hecke algebras, play the main role in this paper. ...
... The appropriate geometric objects are still equivariant sheaves on a variety associated to a complex reductive group, but now the sheaves are constructible and one considers their equivariant cohomology instead of their K-theory. Results in [Lus3] strongly suggest that the module category of H is equivalent with some category of equivariant constructible sheaves. We will make this precise by involving derived categories. ...
... We will work in D G×C × (X), the equivariant (bounded) derived category of constructible sheaves on a complex variety X [BeLu]. In [Lus1,Lus3,AMS2] an important object K ∈ D G×C × (g) was constructed from qE, by a process that bears some similarity with parabolic induction. Let g N be the set of nilpotent elements in the Lie algebra g of G and let K N be the pullback of K to g N . ...
The local Langlands correspondence matches irreducible representations of a reductive p-adic group G(F) with enhanced L-parameters. It is conjectured by Hellmann and Zhu that it can be categorified. Then it should become a fully faithful functor from a derived category of representations to a derived category of equivariant sheaves on some variety of L-parameters. We approach this conjecture in the case of finite length G(F)-representations. Then it runs via graded Hecke algebras associated to Bernstein components or to enhanced L-parameters. Here we work with graded Hecke algebras H on the Galois side, those can be constructed entirely in terms of a complex reductive group, endowed with data from L-parameters. We fix an arbitrary central character (\sigma,r) of H (which encodes the image of Frobenius by an L-parameter). That leads to a variety g_N^{\sigma,r} of nilpotent elements in the Lie algebra of G (possibilities for the monodromy operator from an L-parameter) and to a complex of equivariant constructible sheaves K_{N,\sigma,r} on g_N^{\sigma,r}. We relate the (derived) endomorphism algebra of (g_N^{\sigma,r}, K_{N,\sigma,r}) to a localization of H, which yields an equivalence between the appropriate categories of finite length modules of these algebras. From there we construct a fully faithful functor between: - the bounded derived category of finite length H-modules specified by the central character (\sigma,r), - the equivariant bounded derived category of constructible sheaves on g_N^{\sigma,r}. Also, we explicitly determine the images of standard modules under this functor. We expect that these results pave the way for more general instances of the aforementioned conjectural extension of the local Langlands correspondence.
... Fix a maximal torus T , for a choice of Borel B containing T . By [Lusa,Prop 3.2], we have an isomorphism: ...
... (3.4.1) End 0, * (Ind gr Q (L)) ⊗ End * ,gr (L) → End * ,gr (Ind gr Q (L)) By [Lusa,Prop 3.2], (3.4.1) is an isomorphism and can be identified as maps of graded vector spaces: ...
We give a combinatorial description of the dg category of character sheaves on a complex reductive group G, extending results of [Li] for G simply-connected. We also explicitly identify the parabolic induction/restriction functors.
... Recall that the degenerate affine Hecke algebra H is a C[ ]-algebra generated by Sym(t * ) and the group algebra C[W ], subject to interacting relations between them. As will be reviewed in Section 2.1, H acts on the T-equivariant cohomology H * T (T * B), through an isomorphism of convolution algebras [CG10,Lus88]. The operators corresponding to w ∈ W ⊂ H are usually referred to as the Demazure-Lusztig operators [Gin98]. ...
... Let H G * (St) denote the G-equivariant Borel-Moore homology of the Steinberg variety, which has an associative algebra structure via convolution, see [CG10]. The equivariant cohomology H * G (T * B) admits a canonical action by H G * (St), and hence by the affine Hecke algebra H , due to an isomorphism H ≃ H G * (St) of C[ ]-algebras proved by Lusztig [Lus88]. The isomorphism can be explicitly described as follows. ...
In this paper, we introduce quantum Demazure--Lusztig operators acting by ring automorphisms on the equivariant quantum cohomology of the Springer resolution. Our main application is a presentation of the torus-equivariant quantum cohomology in terms of generators and relations. We provide explicit descriptions for the classical types. We also recover Kim's earlier results for the complete flag varieties by taking the Toda limit.
... We analyze the graded Hecke algebra H " H µσ pΨ σ q defined by the relations in (3.2.7) for the case σ " 1. We will consider only the Hecke algebras arising by the constructions of [Lus88;Lus95b]. Let G be a complex connected reductive group, with a fixed Borel subgroup B, and maximal torus A Ă B. The Lie algebras will be denoted by the corresponding Gothic letters. ...
... The construction of standard modules using equivariant homology H M pnq‚ is in §8 in loc. cit., see also [Lus95b,pp. 10.7, 10.12]). ...
The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra-Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig's category of unipotent representations. Our formula generalizes the main result of arXiv:2112.14354, where this formula was obtained in the Iwahori-spherical case. We deduce that for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton, verifying a conjecture of M\oeglin and Waldspurger.
... G,c (X) (1 X , ω X ) which come with the analogous push, pull and convolution operations (see also [Lus88]). ...
We give a K-theoretic realization of all affine Hecke algebras with two unequal parameters including exceptional types. This extends the celebrated work of Kazhdan and Lusztig, who gave a K-theoretic realization of affine Hecke algebras with equal parameters, and complements results of Kato, who extended this construction to the three-parameter affine Hecke algebra of type C. A key idea behind our new construction is to exploit the reducibility of the adjoint representation in small characteristic. We also show that under suitable geometric conditions, our construction leads to a Deligne-Langlands style classification of simple modules. We verify these geometric conditions for thereby obtaining a full geometric classification of the simple modules for the affine Hecke algebra of with two parameters away from roots of unities.
... The Heckman-Opdam central character is defined in Definition 2.7, and is in terms of some combinatorial data of the root system. It is closely related to the notion of distinguished nilpotent orbits in the theory of Lusztig geometric graded Hecke algebra [Lu88,Lu95]. Our first main result: One of our key theoretic input in proving Theorem 1.1 is the construction of the minimally induced module from discrete series in Theorem 9.2. ...
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.
... By [33] G ∨,• u -conjugacy classes of such triples are naturally in bijection with Irr (ވG ∨,• u , T ∨ , triv, log(q F )/2) . Let us write that as ...
... The term "Drinfeld orbifold algebras" alludes to the subject's origins in [Dri86], where Drinfeld introduced a broad class of algebras to serve as noncommutative coordinate rings for singular orbifolds. When the group is a Coxeter group acting by its reflection representation, Drinfeld's algebras are isomorphic (see [RS03]) to the graded Hecke algebras from [Lus88], which arise from a filtration of an affine Hecke algebra when the group is crystallographic (see [Lus89]). The representation theory of these algebras is useful in understanding representations and geometric structure of reductive p-adic groups. ...
Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincar\'{e}-Birkhoff-Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces.
... Many authors have studied representations of graded Hecke algebras, their subalgebras generated by certain idempotents, and connections to the geometry of the corresponding orbifolds V /G. (See for example [13,15,18,19].) ...
We generalize graded Hecke algebras to include a twisting two-cocycle for the associated finite group. We give examples where the parameter spaces of the resulting twisted graded Hecke algebras are larger than that of the graded Hecke algebras. We prove that twisted graded Hecke algebras are particular types of deformations of a crossed product.
... In the classical setting of the Kazhdan-Lusztig geometry for affine Hecke algebras [KL], an algorithm does not exist (at least to our knowledge). In the setting of the geometric graded affine Hecke algebras [Lu2], a complete and difficult algorithm was obtained by Lusztig in [Lu3]. In the particular case of the graded affine Hecke algebra H A of type A (which appears for example for the p-adic group GL(n)), there exist exact functors defined by Arakawa and Suzuki [AS] from the category O for gl(n, C) to the category of finite-dimensional H A -modules, and, using results of Lusztig and Zelevinsky, the multiplicity question for H A -modules can be resolved in terms of the usual Kazhdan-Lusztig polynomials for S n (i.e., Schubert cells for GL(n, C)). ...
Let be an irreducible smooth complex representation of a general linear p-adic group and let be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that is a representation of a standard Levi subgroup of a p-adic classical group of higher rank. We show that the reducibility of the representation of the appropriate p-adic classical group obtained by (normalized) parabolic induction from does not depend on , if is "separated" from the supercuspidal support of . (Here, "separated" means that, for each factor of a representation in the supercuspidal support of , the representation parabolically induced from is irreducible.) This was conjectured by E. Lapid and M. Tadi\'c. (In addition, they proved, using results of C. Jantzen, that this induced representation is always reducible if the supercuspidal support is not separated.) More generally, we study, for a given set I of inertial orbits of supercuspidal representations of p-adic general linear groups, the category \CC _{I,\sigma} of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by and I, show that this category is equivalent to a category of finitely generated right modules over a direct sum of tensor products of extended affine Hecke algebras of type A, B and D and establish functoriality properties, relating categories with disjoint I's. In this way, we extend results of C. Jantzen who proved a bijection between irreducible representations corresponding to these categories. The proof of the above reducibility result is then based on Hecke algebra arguments, using Kato's exotic geometry.
... The PBW algebras H Q;Ä include the braided Cherednik algebras of Bazlov and Berenstein [6]. In the special case that q ij D 1 for all i; j , they also include Lusztig's graded Hecke algebras [21,22], the symplectic reflection algebras explored by Etingof and Ginzburg [14], the Drinfeld Hecke algebras of [13], and the noncommutative deformations of Kleinian singularities studied by Crawley-Boevey and Holland [12]. ...
We consider quantum (skew) polynomial rings and observe that their graded automorphisms coincide with those of quantum exterior algebras. This allows us to define a quantum determinant that gives a homomorphism of groups acting on quantum polynomial rings. We use quantum subdeterminants to classify the resulting Drinfeld Hecke algebras for the symmetric group, other infinite families of Coxeter and complex reflection groups, and mystic reflection groups (which satisfy a version of the Shephard–Todd–Chevalley theorem). This direct combinatorial approach replaces the technology of Hochschild cohomology used by Naidu and Witherspoon over fields of characteristic zero and allows us to extend some of their results to fields of arbitrary characteristic and also locate new deformations of skew group algebras.
... Let us now define the pseudofunctors which will induce the equivariant bounded derived category of perverse sheaves we consider in this paper; the core ideas we use go back to the perspectives pioneered and championed by [5], [11], [1]. We first need to define the category which controls the equivariant descent which we use to define our equivariant derived categories. ...
In this paper we extend Beilinson's realization formalism for triangulated categories and filtered triangulated categories to a pseudofunctorial and pseudonatural setting. As a consequence we prove an equivariant version of Beilinson's Theorem: for any algebraic group G over an algebraically closed field K and for any G-variety X, there is an equivalence of categories between the equivariant bounded derived category of l-adic sheaves and the equivariant bounded derived category of l-adic perverse sheaves on X. We also show that the equivariant analogues of the other non-D-module aspects of Beilinson's Theorem hold in the equivariant case.
... These algebras can be used to study a wider range of representations of p-adic groups such as unipotent representations [33]. In terms of geometry, graded Hecke algebras arise as certain Ext-algebras in an equivariant derived category of constructible sheaves [3,34]. Some formality results in this direction can be found in [41]. ...
The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character of an equivalence of triangulated categories between a perfect derived category of dg-modules and the triangulated category generated by the corresponding perverse sheaves. The main step in this construction is a formality result that we prove for a wide range of ‘Springer sheaves’.
... If we instead work with non-affine groups, the characterization of smooth free G-varieties as G-varieties withétale locally trivializable actions G × Γ → Γ fails. 18]). We say that Γ is a smooth free G-variety if Γ is a left G-variety (over K) with a geometric quotient 7 (cf. ...
In this paper we show that if is a category and if is a pseudofunctor such that for each object X of the category F(X) is a tangent category and for each morphism f of the functor F(f) is part of a strong tangent morphism (F(f),\!\quot{\alpha}{f}) and that furthermore the natural transformations \!\quot{\alpha}{f} vary pseudonaturally in , then there is a tangent structure on the pseudolimit which is induced by the tangent structures on the categories F(X) together with how they vary through the functors F(f).
We use this observation to show that the forgetful 2-functor creates and preserves pseudolimits indexed by 1-categories. As an application, this allows us to describe how equivariant descent interacts with the tangent structures on the category of smooth (real) manifolds and on various categories of (algebraic) varieties over a field.
... These algebras can be used to study a wider range of representations of p-adic groups such as unipotent representations [Lus95a]. In terms of geometry, graded Hecke algebras arise as certain Ext-algebras in an equivariant derived category of constructible sheaves [Lus95b,AMS18]. Some formality results in this direction can be found in [Sol22]. There also is a coherent categorical Deligne-Langlands correspondence [BZCHN20] which works without fixing a central character but replaces constructible sheaves with a certain category of coherent sheaves. ...
The Deligne-Langlands correspondence parametrizes irreducible representations of the affine Hecke algebra by certain perverse sheaves. We show that this can be lifted to an equivalence of triangulated categories. More precisely, we construct for each central character of an equivalence of triangulated categories between a perfect derived category of dg-modules and the triangulated category generated by the corresponding perverse sheaves. The main step in this construction will be a formality result that prove for a wide range of `Springer sheaves'.
... In [Lu95b], Lusztig established an equivalence between unipotent representations of a reductive group over a non-Archimedian local field and the module theory for certain affine Hecke algebras with unequal parameters. Lusztig's fruitful strategy [Lus88,Lus89,Lus95a] for classifying the simple modules of these affine Hecke algebras was to reduce them to certain twisted graded Hecke algebras, obtained by taking m-adic completions and associated gradeds for maximal ideals m of the center. In turn, these graded Hecke algebras have natural geometric realizations as the Borel-Moore homology on certain "fixed-point" versions of the Steinberg stack, i.e. the self-Exts of "fixed-point" versions of the equivariant Springer sheaf. ...
In this paper we establish an equivalence between the category of -equivariant coherent sheaves on the formal loop space of a smooth stack X and the category of coherent filtered -modules on X, generalizing results from the case where X is a smooth variety. This equivalence interpolates between categories of coherent sheaves on stacks appearing as categorical traces in algebro-geometric settings, and categories of equivariant constructible sheaves in topological settings. Unlike in the case of varieties, the subcategory of coherent -modules on a smooth stack X is larger than the subcategory of compact objects; we identify the corresponding Koszul dual subcategory as the category of -equivariant ind-continuous coherent sheaves on .
... By[25, Theorem 6.5], the subalgebra k[z 1 , . . . , zn] Sn ⊂ R(n) is nothing else but the center of R(n). ...
Let be a pair of symplectically dual conical symplectic singularities. Assume that Y has a symplectic resolution . The equivariant Hikita conjecture (that we call Hikita-Nakajima conjecture) claims that there should be an isomorphism of (graded) algebras , here is the torus acting on Y preserving the Poisson structure, is the universal deformation of , is a generic one-dimensional torus acting on and is the algebra of schematic -fixed points of . We prove the Hikita-Nakajima conjecture for Gieseker variety (ADHM space). We produce the isomorphism explicitly on generators and relations. It turns out that if we realize as a Coulomb branch, then Chern classes of the tautological bundle on correspond to Chern classes of the tautological bundle on the corresponding variety of triples. We also describe the Hikita-Nakajima isomorphism above using realization of as a Nakajima quiver variety and as the spectrum of the center of rational Cherednik algebra corresponding to and identify all the algebras that appear with the center of degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo and Vasserot). Finally, we formulate as a conjecture that when X is a Nakajima quiver variety then the Hikita-Nakajima isomorphism should identify Chern classes of tautological bundles on X with Chern classes of tautological bundles on the corresponding variety of triples and describe possible approaches towards the proof.
... • When the parameters k are of "geometric type", one can use families of representations from [Lus1,Lus2,AMS1]. • For most real-valued k one can use the technique with parameter deformations from [Sol6,proof of Lemma 6.4], to reduce to the previous case. ...
Let A be a \C-algebra with an action of a finite group G, let be a 2-cocycle on G and consider the twisted crossed product A \rtimes \C [G,\natural]. We determine the Hochschild homology of A \rtimes \C [G,\natural] for two classes of algebras A: - rings of regular functions on nonsingular affine varieties, - graded Hecke algebras. The results are achieved via algebraic families of (virtual) representations and include a description of the Hochschild homology as module over the centre of A \rtimes \C [G,\natural]. This paper prepares for a computation of the Hochschild homology of the Hecke algebra of a reductive p-adic group.
... Degree-zero deformations of skew group algebras are called Drinfeld graded Hecke algebras in recognition of their origins in [4] (see also [15]). These include the important special cases when G acts on a symplectic vector space, and more particularly when G is a complex reflection group acting by the sum of a reflection representation and its dual (a doubled representation). ...
The wavefront set is a fundamental invariant of an admissible representation arising from the Harish-Chandra–Howe local character expansion. In this paper, we give a precise formula for the wavefront set of an irreducible representation of real infinitesimal character in Lusztig’s category of unipotent representations in terms of the Deligne–Langlands–Lusztig correspondence. Our formula generalises the main result of [D. Ciubotaru, L. Mason-Brown and E. Okada, Wavefront sets of unipotent representations of reductive 𝑝-adic groups I, preprint (2021), https://arxiv.org/abs/2112.14354 ], where this formula was obtained in the Iwahori-spherical case. We deduce that, for any irreducible unipotent representation with real infinitesimal character, the algebraic wavefront set is a singleton. In the process, we establish new properties of the generalised Springer correspondence in relation to Lusztig’s families of unipotent representations of finite reductive groups.
Let be an affine Nakajima quiver variety, and let be the corresponding BFN Coulomb branch. Assume that can be resolved by the (smooth) Nakajima quiver variety . The Hikita-Nakajima conjecture claims that there should be an isomorphism of (graded) algebras , where is a torus acting on preserving the Poisson structure, is the (Poisson) deformation of over , is a generic one-dimensional torus acting on , and is the algebra of schematic -fixed points of . We prove the Hikita-Nakajima conjecture for Gieseker variety (ADHM space). We produce the isomorphism explicitly on generators. We also describe the Hikita-Nakajima isomorphism above using the realization of as the spectrum of the center of the rational Cherednik algebra corresponding to and identify all the algebras that appear in the isomorphism with the center of the degenerate cyclotomic Hecke algebra (generalizing some results of Shan, Varagnolo, and Vasserot).
This paper addresses the question: given a scalar group, can we determine all the additions that transform this scalar group into a (near-)field? A key approach to addressing this problem involves transporting (near-)field structures via multiplicative automorphisms. We compute the set of continuous multiplicative automorphisms of the real and complex fields and analyze their structures. Additionally, we characterize the endo-bijections on the scalar group that define these additions.
Graded Hecke algebras can be constructed geometrically, with constructible sheaves and equivariant cohomology. The input consists of a complex reductive group G (possibly disconnected) and a cuspidal local system on a nilpotent orbit for a Levi subgroup of G. We prove that every such “geometric” graded Hecke algebra is naturally isomorphic to the endomorphism algebra of a certain -equivariant semisimple complex of sheaves on the nilpotent cone in the Lie algebra of G.
From there we provide an algebraic description of the -equivariant bounded derived category of constructible sheaves on . Namely, it is equivalent with the bounded derived category of finitely generated differential graded modules of a suitable direct sum of graded Hecke algebras. This can be regarded as a categorification of graded Hecke algebras.
This paper prepares for a study of representations of reductive p-adic groups with a fixed infinitesimal central character. In sequel papers [34, 35], that will lead to proofs of the generalized injectivity conjecture and of the Kazhdan–Lusztig conjecture for p-adic groups.
Motivated by relating the representation theory of the split real and p-adic forms of a connected reductive algebraic group G, we describe a subset of orbits on the complex flag variety for a certain symmetric subgroup. (Here r is the semisimple rank of G.) This set of orbits has the property that, while the closure of individual orbits are generally singular, they are always smooth along other orbits in the set. This, in turn, implies consequences for the representation theory of the split real group.
The main purpose of this paper is to identify the tempered modules for the affine Hecke algebra of type with arbitrary, non-root of unity, unequal parameters, in the exotic Deligne-Langlands correspondence in the sense of Kato. Our classification has several applications to the Weyl group module structure of the tempered Hecke algebra modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sign representation of the Weyl group. This last combinatorial classification was expected from the work of Heckman-Opdam and Slooten.
Let G be a complex reductive algebraic group. In arxiv:2108.03453 Ivan Losev, Lucas mason-Brown and the third-named author suggested a symplectic duality between nilpotent Slodowy slices in and affinizations of certain G-equivariant covers of special nilpotent orbits. In this paper, we study the various versions of Hikita conjecture for this pair. We show that the original statement of the conjecture does not hold for the pairs in question and propose a refined version. We discuss the general approach towards the proof of the refined Hikita conjecture and prove this refined version for the parabolic Slodowy varieties, which includes many of the cases considered in arxiv:2108.03453 and more. Applied to the setting of arxiv:2108.03453, the refined Hikita conjecture explains the importance of special unipotent ideals from the symplectic duality point of view. We also discuss applications of our results. In the appendices, we discuss some classical questions in Lie theory that relate the refined version and the original version. We also explain how one can use our results to simplify some proofs of known results in the literature. As a combinatorial application of our results we observe an interesting relation between the geometry of Springer fibers and left Kazhdan-Lusztig cells in the corresponding Weyl group.
We prove a microlocal characterisation of character sheaves on a reductive Lie algebra over an algebraically closed field of sufficiently large positive characteristic: a perverse irreducible G-equivariant sheaf is a character sheaf if and only if it has nilpotent singular support and is quasi-admissible. We also present geometric proofs, in positive characteristic, of the equivalence between being admissible and being a character sheaf, and various characterisations of cuspidal sheaves, following the work of Mirković.
Mirkovi\'c introduced the notion of character sheaves on a Lie algebra. Due to their simple geometric characterization, character sheaves on Lie algebras can be thought of as a simplified model for Lusztig's theory of character sheaves on algebraic groups. We extend the theory to the case of modular coefficients. Along the way, we will reprove some of Mirkovi\'c's results and provide connections with the modular generalized Springer correspondence of Achar, Juteau, Riche, and Williamson.
Let C be a unipotent class of , an irreducible G-equivariant local system on C. The generalized Springer representation appears in the top cohomology of some variety. Let be the representation obtained by summing over all cohomology groups of this variety. It is well known that appears in with multiplicity 1 and that its Springer support C is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of . Suppose C is parametrized by an orthogonal partition with only odd parts. We prove that (resp. ) has a unique multiplicity 1 “maximal” subrepresentation (resp. “minimal” subrepresentation ), where is the sign representation. These are analogues of results for by Waldspurger.
If π is a representation of a p-adic group G(F), and φ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of φ that will detect when π is generic? In this paper we show that if G is classical or if we assume the Kazhdan-Lusztig hypothesis for G, then the answer is yes, and the property is that the orbit of φ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter φ for a p-adic group G(F), the ABV-packet Π ABV φ (G(F)) contains a generic representation if and only if the local adjoint L-function L(s, φ, Ad) is regular at s = 1, and show that this condition is equivalent to the "open parameter" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on G. We show that, in this case, the ABV-packet for φ coincides with its L-packet. Finally, we prove Vogan's conjecture on A-packets for tempered parameters.
We show that certain semi-simple extensions of degenerate affine Hecke algebras of arbitrary Weyl groups admit analogous Khovanov–Lauda–Rouquier generators after some localization. As an application, we obtain the Brundan–Kleshchev–Rouquier-like isomorphisms between direct sums of blocks of cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups and the cyclotomic quotients of some graded subalgebras of the corresponding semi-simple extensions.
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.
We develop a “Soergel theory” for Bruhat-constructible perverse sheaves on the flag variety G∕B of a complex reductive group G, with coefficients in an arbitrary field . Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a “multiplicative” coinvariant algebra and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of “Soergel modules” over this algebra. We also obtain a description of the derived category of unipotently T monodromic sheaves on G∕U (where U, T ⊂ B are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in , where is the -torus dual to T.
In this article we propose a geometric description of Arthur packets for p p -adic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p p -adic groups and propose a geometric description of the transfer coefficients that appear in Arthur’s main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan’s work, including the prediction that Arthur packets are ABV-packets for p p -adic groups. We gather evidence for these conjectures by verifying them in numerous examples.
We consider Arthur's conjectures for split reductive p-adic groups from the point of view of the wavefront set, a fundamental invariant arising from the Harish-Chandra-Howe local character expansion of an admissible representation. We prove a precise formula for the wavefront set of an irreducible Iwahori-spherical representation with `real infinitesimal character' and determine a lower bound for this invariant in terms of the Kazhdan-Lusztig parameters. We define certain unipotent Arthur packets consisting of representations with minimal allowable wavefront sets, and we prove that their Iwahori-spherical constituents are the Aubert-Zelevinsky duals of tempered representations.
The characters of the (total) Springer representations are identified with the Green functions by Kazhdan (1977), and the latter are identified with Hall-Littlewood’s Q-functions by Green (1955). In this paper, we present a purely algebraic proof that the (total) Springer representations of GL(n) are Ext-orthogonal to each other, and show that it is compatible with the natural categorification of the ring of symmetric functions.
We investigate deformations of skew group algebras arising from the action of the symmetric group on polynomial rings over fields of arbitrary characteristic. Over the real or complex numbers, Lusztig’s graded affine Hecke algebra and analogs are all isomorphic to Drinfeld Hecke algebras, which include the symplectic reflection algebras and rational Cherednik algebras. Over fields of prime characteristic, new deformations arise that capture both a disruption of the group action and also a disruption of the commutativity relations defining the polynomial ring. We classify deformations for the symmetric group acting via its natural (reducible) reflection representation.
We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type A n − 1 A_{n-1} . This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of n n -step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.
We compare the K -theory stable bases of the Springer resolution associated to different affine Weyl alcoves. We prove that (up to relabelling) the change of alcoves operators are given by the Demazure–Lusztig operators in the affine Hecke algebra. We then show that these bases are categorified by the Verma modules of the Lie algebra, under the localization of Lie algebras in positive characteristic of Bezrukavnikov, Mirković, and Rumynin. As an application, we prove that the wall-crossing matrices of the K -theory stable bases coincide with the monodromy matrices of the quantum cohomology of the Springer resolution.
We state a conjecture about the Weyl group action coming from Geometric Satake on zero-weight spaces in terms of equivariant multiplicities of Mirković-Vilonen cycles. We prove it for small coweights in type A. In this case, using work of Braverman, Gaitsgory and Vybornov, we show that the Mirković-Vilonen basis agrees with the Springer basis. We rephrase this in terms of equivariant multiplicities using work of Joseph and Hotta. We also have analogous results for Ginzburg's Lagrangian construction of sln representations.
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.
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.
Continuous cohomology, discrete subgroups and representations CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, III 47 of reductive groups
- A Borel
- N Wallach
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups and representations CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, III 47 of reductive groups, Ann.Math.Stud., vol. 94, Princeton Univ. Press, 1980.
Fourier transforms on a semisimple Lie algebra over Fq, in " Algebraic Groups--Utrecht
- G Lusztzo
G. LuszTzo, Fourier transforms on a semisimple Lie algebra over Fq, in " Algebraic Groups--Utrecht 1986 ", Lecture Note~ in Mathematics, 1271, Springer, 1987, 177-188.
~')) come from vector bundle maps E~. By 8.5, ~l(y) acts on H~,. This corresponds to an action of g-I(y) on V. (It is induced by the adjoint action of M(y) on its Lie algebra
- George Similarly
- H~
- = E
GEORGE LUSZTIG Similarly, E; = (I, | H.~"(~,, s F, = E, |162 E;. By 8.3, the operators A(w), A(~) (resp. A'(w), A'(~)) on n.m~"(~,,.~) (resp. H.~*'(.~',,.~')) come from vector bundle maps A(w), A(~):E-+ E (resp. A'(w), A'(~):E'-~ E'), inducing the identity on the base V. Hence these operators act on each fibre E,, E~. By 8.5, ~l(y) acts on H~,,. This corresponds to an action of g-I(y) on V. (It is induced by the adjoint action of M(y) on its Lie algebra.) Moreover, E, E' are natu-rally ~I(y)-equivariant vector bundles over V and the operators A(w), A(~), A'(w), A'(~) on them are ~/(y)-invariant.
- Ann Reductive Groups
reductive groups, Ann.Math.Stud., vol. 94, Princeton Univ. Press, 1980.
- G Lusztig Character
- V Sheaves
G. Lusztig Character Sheaves,V, Adv. Math. 61 (1986), 103-155.
Representations of finite Chevalley groups,C.B.M.S. Regional Conference series in Math
- G Lusztig