Modules Of Hecke Algebras Research Articles

Generic Hecke algebra and theta correspondence over finite fields

We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for type I dual pairs over finite fields. For the product of the pair of Hecke algebras under consideration, we show that there is a generic Hecke algebra module whose specializations at prime powers give the Hecke algebra modules and whose specialization at 1 can be explicitly described. As an application, we prove the conservation relation on the first occurrence indices for all irreducible representations. As another application, we recover the results of Aubert-Michel-Rouquier and Pan on the explicit description of theta correspondence between Harish-Chandra series.

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.

Graded Specht Modules as Bernstein–Zelevinsky Derivatives of the RSK Model

AbstractWe clarify the links between the graded Specht construction of modules over cyclotomic Hecke algebras and the Robinson-Schensted-Knuth (RSK) construction for quiver Hecke algebras of type $A$, which was recently imported from the setting of representations of $p$-adic groups. For that goal we develop a theory of crystal derivative operators on quiver Hecke algebra modules that categorifies the Berenstein–Zelevinsky strings framework on quantum groups and generalizes a graded variant of the classical Bernstein–Zelevinsky derivatives from the $p$-adic setting. Graded cyclotomic decomposition numbers are shown to be a special subfamily of the wider concept of RSK decomposition numbers.

The Classification of All Singular Nonsymmetric Macdonald Polynomials

The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form qm=t−n with 2≤n≤N. For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the Hecke algebra module is dn−1,n−1,…,n−1,r for some d≥1. In the present paper, it is shown that there are no other singular polynomials.

Positive Jantzen sum formulas for cyclotomic Hecke algebras

This paper proves a “positive” Jantzen sum formula for the Specht modules of the cyclotomic Hecke algebras of type A and uses it to obtain new bounds on decomposition numbers. Quite remarkably, our results are proved entirely inside the cyclotomic Hecke algebras. Our positive sum formula shows that, in the Grothendieck group, the Jantzen sum formula can be written as an explicit non-negative linear combination of modules [E^{varvec{nu }}_{f,e}], which are the modular reductions of simple modules of related Hecke algebras in characteristic zero. The coefficient of [E^{varvec{nu }}_{f,e}] in the sum formula is determined by the graded decomposition numbers in characteristic zero, which are known, and by the characteristic of the field. As a consequence we give an explicit upper bound for the decomposition numbers in characteristic p>0 in terms of linear combinations of decomposition numbers of a cyclotomic Hecke algebra at ep^rth roots of unity in characteristic zero, for rge 0. Finally, we prove a new and more elegant “classical” Jantzen sum formula for these algebras.

On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

We provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

On simple modules of cyclotomic quiver Hecke algebras of type A

We give a classification of the graded simple modules of cyclotomic quiver Hecke algebras of type A using the diagram calculus of the diagrammatic Cherednik algebra. We also obtain a non-trivial lower bound for the dimension of the simple modules which is independent of the field characteristic.

Affine Hecke Algebras via DAHA

A method is suggested for obtaining the Plancherel measure for Affine Hecke Algebras as a limit of integral-type formulas for inner products in the polynomial and related modules of Double Affine Hecke Algebras. The analytic continuation necessary here is a generalization of "picking up residues" due to Arthur, Heckman, Opdam and others, which can be traced back to Hermann Weyl. Generally, it is a finite sum of integrals over double affine residual subtori; a complete formula is presented for $A_1$ in the spherical case.

Degenerate cyclotomic Hecke algebras and higher level Heisenberg categorification

We associate a monoidal category Hλ to each dominant integral weight λ of slˆp or sl∞. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to λ. We show that, in the sl∞ case, the level d Heisenberg algebra embeds into the Grothendieck ring of Hλ, where d is the level of λ. The categories Hλ can be viewed as a graphical calculus describing induction and restriction functors between categories of modules for degenerate cyclotomic Hecke algebras, together with their natural transformations. As an application of this tool, we prove a new result concerning centralizers for degenerate cyclotomic Hecke algebras.

Hecke modules from metaplectic ice

We present a new framework for a broad class of affine Hecke algebra modules, and show that such modules arise in a number of settings involving representations of p-adic groups and R-matrices for quantum groups. Instances of such modules arise from (possibly non-unique) functionals on p-adic groups and their metaplectic covers, such as the Whittaker functionals. As a byproduct, we obtain new, algebraic proofs of a number of results concerning metaplectic Whittaker functions. These are thus expressed in terms of metaplectic versions of Demazure operators, which are built out of R-matrices of quantum groups depending on the cover degree and associated root system.

Bernstein–Zelevinsky Derivatives: a Hecke Algebra Approach

Let $G$ be a general linear group over a $p$-adic field. It is well known that Bernstein components of the category of smooth representations of $G$ are described by Hecke algebras arising from Bushnell-Kutzko types. We describe the Bernstein components of the Gelfand-Graev representation of $G$ by explicit Hecke algebra modules. This result is used to translate the theory of Bernstein-Zelevinsky derivatives in the language of representations of Hecke algebras, where we develop a comprehensive theory.

Cell Decompositions of Quiver Flag Varieties for Nilpotent Representations of the Cyclic Quiver

Generalizing Schubert cells in type A and a cell decomposition of Springer fibres in type A found by L. Fresse we prove that varieties of complete flags in nilpotent representations of a cyclic quiver admit an affine cell decomposition parametrized by multi-tableaux. We show that they carry a torus operation with finitely many fixpoints. As an application of the cell decomposition we obtain a vector space basis of certain modules (for quiver Hecke algebras of nilpotent representations of this quiver), similar modules have been studied by Kato as analogues of standard modules.

Infinitesimal change of stable basis

The purpose of this note is to study the Maulik–Okounkov K-theoretic stable basis for the Hilbert scheme of points on the plane, which depends on a “slope” $$m \in {\mathbb {R}}$$ . When $$m = \frac{a}{b}$$ is rational, we study the change of stable matrix from slope $$m-\varepsilon $$ to $$m+\varepsilon $$ for small $$\varepsilon >0$$ , and conjecture that it is related to the Leclerc–Thibon conjugation in the q-Fock space for $$U_q\widehat{{\mathfrak {gl}}}_b$$ . This is part of a wide framework of connections involving derived categories of quantized Hilbert schemes, modules for rational Cherednik algebras and Hecke algebras at roots of unity.

Isomorphisms between simple modules of degenerate cyclotomic Hecke algebras

We give explicit isomorphisms between simple modules of degenerate cyclotomic Hecke algebras defined via various cellular bases. A special case gives a generalized Mullineux involution in the degenerate case.

Iwahori component of the Gelfand–Graev representation

Let G be a split reductive group over a p-adic field F. Let B be a Borel subgroup and U the maximal unipotent subgroup of B. Let \(\psi \) be a Whittaker character of U. Let I be an Iwahori subgroup of G. We describe the Iwahori–Hecke algebra action on the Gelfand–Graev representation \((\mathrm {ind}_{U}^{G}\psi )^I\) by an explicit Hecke algebra module.

An involution based left ideal in the Hecke algebra

We show that the Hecke algebra module carried by the involutions in a Weyl group (defined by the author and Vogan) can be identified with a left ideal in the Hecke algebra. An analogous result is proved for any Coxeter group.

Duality for Ext-groups and extensions of discrete series for graded Hecke algebras

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 Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between a tempered module and a discrete series vanish.

Connection Problems for Quantum Affine KZ Equations and Integrable Lattice Models

Cherednik attached to an affine Hecke algebra module a compatible system of difference equations, called quantum affine Knizhnik-Zamolodchikov (KZ) equations. In case of a principal series module we construct a basis of power series solutions of the quantum affine KZ equations. Relating the bases for different asymptotic sectors gives rise to a Weyl group cocycle, which we compute explicitly in terms of theta functions. For the spin representation of the affine Hecke algebra of type C the quantum affine KZ equations become the boundary qKZ equations associated to the Heisenberg spin-1/2 XXZ chain. We show that in this special case the results lead to an explicit 4-parameter family of elliptic solutions of the dynamical reflection equation associated to Baxter's 8-vertex face dynamical R-matrix. We use these solutions to define an explicit 9-parameter elliptic family of boundary quantum Knizhnik-Zamolodchikov-Bernard (KZB) equations.

Some remarks on Gill's theorems on Young modules

In a recent paper, [10], Gill, using methods of the modular representation theory of finite groups, describes some results on the tensor product of Young modules for symmetric groups. We here give an alternative approach using the polynomial representation theory of general linear groups and the Schur functor. The main result is a formula for the multiplicities of Young modules in a tensor product in terms of the characters of the simple polynomial modules for general linear groups. Our approach is also valid for Young modules for Hecke algebras of type A at roots of unity and here the formula involves the characters of the simple polynomial modules for quantised general linear groups.

Homological properties of quantised Borel–Schur algebras and resolutions of quantised Weyl modules

We continue the development of the homological theory of quantum general linear groups previously considered by the first author. The development is used to transfer information to the representation theory of quantised Schur algebras. The acyclicity of induction from some rank-one modules for quantised Borel–Schur subalgebras is deduced. This is used to prove the exactness of the complexes recently constructed by Boltje and Maisch, giving resolutions of the co-Specht modules for Hecke algebras.