Iwahori Hecke Algebra Research Articles

Genuine pro-$p$ Iwahori–Hecke algebras, Gelfand–Graev representations, and some applications

We study the Iwahori-component of the Gelfand–Graev representation of a central cover of a split linear reductive group and utilize our results for three applications. In fact, it is advantageous to begin at the pro- p level. Thus to begin we study the structure of a genuine pro- p Iwahori–Hecke algebra, establishing a Bernstein presentation. With this structure theory we first describe the pro- p part of the Gelfand–Graev representation and then the more subtle Iwahori part. For the first application we relate the Gelfand–Graev representation to the metaplectic representation of Sahi–Stokman–Venkateswaran, which conceptually realizes the Chinta–Gunnells action from the theory of Weyl group multiple Dirichlet series. For the second we compute the Whittaker dimension of the constituents of regular unramified principal series representations; for the third we do the same for unitary unramified principal series representations.

Stability in the category of smooth mod-p representations of

Abstract Let $p \geq 5$ be a prime number, and let $G = {\mathrm {SL}}_2(\mathbb {Q}_p)$ . Let $\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $\mathcal {R} \subset \Xi \times \Xi $ denote the support of the pro-p Iwahori ${\mathrm {Ext}}$ -algebra of G, viewed as a $(Z,Z)$ -bimodule. We show that the locally ringed space $\Xi /\mathcal {R}$ is a projective algebraic curve over ${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $\Xi /\mathcal {R}$ , we construct a stable localising subcategory $\mathcal {L}_U$ of the category of smooth k-linear representations of G.

Kato’s irreducibility criterion for Kac-Moody groups over local fields

In 2014, Braverman, Kazhdan, Patnaik and Bardy-Panse, Gaussent and Rousseau associated Iwahori-Hecke algebras to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we defined and studied their principal series representations. In 1982, Kato provided an irreducibility criterion for these representations, in the reductive case. We had obtained partially this criterion in the Kac-Moody case. In this paper, we prove this criterion in the Kac-Moody case.

Homological stability for Iwahori-Hecke algebras of type Bn

We show that homological stability holds for the family of Iwahori-Hecke algebras of type Bn, where homology is identified with the relevant Tor group. This family of algebras is related to the Coxeter groups of type Bn, which are groups of signed permutations. The result builds on Hepworth's result for Iwahori-Hecke algebras of type An.

Categorifying Hecke algebras at prime roots of unity, part I

We equip the type A A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p p it becomes a p p -dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p p -dg Grothendieck group. We conjecture that the p p -dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p p -canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree + 2 +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p p -dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A A . We also examine a particular Bott-Samelson bimodule in type A 7 A_7 , which is indecomposable in characteristic 2 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p p -dg setting of being indecomposable.

Skew cellularity of the Hecke algebras of type 𝐺(ℓ,𝑝,𝑛)

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer’s cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a “cellular algebra Clifford theory” for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are graded skew cellular algebras. In the special case when p = 2 p = 2 this implies that the Hecke algebras of type G ( ℓ , 2 , n ) G(\ell ,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck’s result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G ( ℓ , p , n ) G(\ell ,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of “adjustment matrices” in positive characteristic.

The pro-p-Iwahori invariants of the universal quotient representation for GL2(F)

Let [Formula: see text] be a finite extension of [Formula: see text]. Every supersingular mod [Formula: see text] representation of [Formula: see text] is a quotient of a certain universal module, say [Formula: see text] The space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] is instrumental in classifying the supersingular mod [Formula: see text] representations of [Formula: see text]. For [Formula: see text] the space of pro-[Formula: see text]-Iwahori invariants of [Formula: see text] has been determined by Schein and Hendel using the spherical Hecke algebra. An alternative way to study mod [Formula: see text] representations of [Formula: see text] is introduced by Anandavardhanan and Borisagar using the Iwahori–Hecke algebra. In this paper, we give a canonical isomorphism between the universal modules for [Formula: see text] obtained from the spherical approach and the Iwahori–Hecke approach. We also compute an explicit basis of the pro-[Formula: see text]-Iwahori invariant space of [Formula: see text] for [Formula: see text] using the Iwahori–Hecke approach. This method allows us to extend Hendel’s computation slightly in the range of [Formula: see text]-adic digits from [Formula: see text] to [Formula: see text]

On the derived category of the Iwahori–Hecke algebra

We state a conjecture that relates the derived category of smooth representations of a $p$-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of ${\rm GL}_n$ by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and of Emerton and Helm.

Centers of Hecke algebras of complex reflection groups

We provide a dual version of the Geck–Rouquier Theorem (Geck and Rouquier in Finite Reductive Groups (Luminy, 1994), Progr. Math., vol. 141, Birkhäuser Boston, Boston, pp. 251–272, 1997) on the center of an Iwahori–Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank 2, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra H. These models enable concrete calculations inside H. For each of the eight groups, we compute an explicit integral basis of the center of H.

Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings

We obtain a complete characterisation of factorial multiparameter Hecke von Neumann algebras associated with right-angled Coxeter groups. Considering their ℓp-convolution algebra analogues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier for group Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us to draw conclusions on spherical representation theory of groups acting on right-angled buildings, which are in strong contrast to behaviour of spherical representations in the affine case. We also investigate certain graph product representations of right-angled Coxeter groups and note that our von Neumann algebraic structure results show that these are finite factor representations. Further classifying a suitable family of them up to unitary equivalence allows us to reveal high-dimensional Euclidean subspaces of the space of extremal characters of right-angled Coxeter groups.

Iwahori–Hecke algebra and unramified local L-functions

In this paper, we compute the Hecke action of a certain test function on the space of an unramified principal series of a connected reductive group over a non-archimedean local field. The key of our computation is the theory of Iwahori–Hecke algebra. As an application, we obtain a new expression of the local L-functions of unramified representations.

A triangular system for local character expansions of Iwahori-spherical representations of general linear groups

For Iwahori-spherical representations of non-Archimedean general linear groups, Chan–Savin recently expressed the Whittaker functor as a restriction to an isotypic component of a finite Iwahori–Hecke algebra module. We generalize this method to describe principal degenerate Whittaker functors. Concurrently, we view Murnaghan’s formula for the Harish-Chandra–Howe character as a Grothendieck group expansion of the same module.

Cells in modified ıquantum groups of type AIII and related Schur algebras

We provide a combinatorial characterization of two-sided cells in modified ıquantum groups of type AIII. Our approach is to lift a corresponding description of two-sided cells in ȷ-Schur algebras associated to Iwahori–Hecke algebras of type B. We further give a combinatorial description of two-sided cells in other two kinds of Schur-type algebras, namely ı-Schur algebras and ı˜-Schur algebras.

Involutions in Weyl groups and nil-Hecke algebras

In a previous article we have defined an action of the Iwahori-Hecke algebra of a Coxeter group W on a free module with basis indexed by the involutions in W. In this paper we show that the specialization of this action at the parameter 0 has a simple description.

Stable Centres of Iwahori-Hecke Algebras of Type A

A celebrated result of Farahat and Higman constructs an algebra FH which “interpolates” the centres \(Z(\mathbb {Z}S_{n})\) of group algebras of the symmetric groups Sn. We extend these results from symmetric group algebras to type A Iwahori-Hecke algebras, Hn(q). In particular, we explain how to construct an algebra FHq “interpolating” the centres Z(Hn(q)). We prove that FHq is isomorphic to \(\mathcal {R}[q,q^{-1}] \otimes _{\mathbb {Z}} {\Lambda }\) (where \(\mathcal {R}\) is the ring of integer-valued polynomials, and Λ is the ring of symmetric functions). The isomorphism can be described as “evaluation at Jucys-Murphy elements”, leading to a proof of a conjecture of Francis and Wang. This yields character formulae for the Geck-Rouquier basis of Z(Hn(q)) when acting on Specht modules.

Homological stability for Iwahori–Hecke algebras

We show that the Iwahori–Hecke algebras H n $\mathcal {H}_n$ of type A n − 1 $A_{n-1}$ satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaoka's homological stability result for the symmetric groups in the case that the defining parameter is equal to 1. We believe that this paper, and our joint work with Boyd on Temperley–Lieb algebras, are the first time that the techniques of homological stability have been applied to algebras that are not group algebras.

The Green polynomials via vertex operators

An iterative formula for the Green polynomial is given using the vertex operator realization of the Hall-Littlewood function. Based on this, (1) a general combinatorial formula of the Green polynomial is given; (2) several compact formulas are given for Green's polynomials associated with upper partitions of length $\leq 3$ and the diagonal lengths $\leq 3$; (3) a Murnaghan-Nakayama type formula for the Green polynomial is obtained; and (4) an iterative formula is derived for the bitrace of the finite general linear group $G$ and the Iwahori-Hecke algebra of type $A$ on the permutation module of $G$ by its Borel subgroup.

Gelfand W-graphs for classical Weyl groups

A Gelfand model for an algebra is a module given by a direct sum of irreducible submodules, with every isomorphism class of irreducible modules represented exactly once. We introduce the notion of a perfect model for a finite Coxeter group, which is a certain set of discrete data (involving Rains and Vazirani's concept of a perfect involution) that parametrizes a Gelfand model for the associated Iwahori-Hecke algebra. We describe perfect models for all classical Weyl groups, excluding type D in even rank. The representations attached to these models simultaneously generalize constructions of Adin, Postnikov, and Roichman (from type A to other classical types) and of Araujo and Bratten (from group algebras to Iwahori-Hecke algebras). We show that each Gelfand model derived from a perfect model has a canonical basis that gives rise to a pair of related W-graphs, which we call Gelfand W-graphs. For types BC and D, we prove that these W-graphs are dual to each other, a phenomenon which does not occur in type A.

An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings

In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bounds for EKR-sets of flags. In this framework, we can reprove and generalize previous upper bounds for EKR-problems in projective and polar spaces. The bounds are obtained by the application of the Delsarte-Hoffman coclique bound to the opposition graph. The computation of its eigenvalues is due to earlier work by Andries Brouwer and an explicit algorithm is worked out. For the classical geometries, the execution of this algorithm boils down to elementary combinatorics. Connections to building theory, Iwahori-Hecke algebras, classical groups and diagram geometries are briefly discussed. Several open problems are posed throughout and at the end.

Principal series representations of Iwahori–Hecke algebras for Kac–Moody groups over local fields

Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. We introduce principal series representations for these algebras. We study these representations and partially generalize Kato and Matsumoto irreducibility criteria.