Affine Hecke Algebra Research Articles

ADMISSIBLE SUBSETS AND LITTELMANN PATHS IN AFFINE KAZHDAN–LUSZTIG THEORY

The center of an extended affine Hecke algebra is known to be isomorphic to the ring of symmetric functions associated to the underlying finite Weyl group W0. The set of Weyl characters sλ forms a basis of the center and Lusztig showed in [11] that these characters act as translations on the Kazhdan–Lusztig basis element $$ {C}_{w_0} $$ where w0 is the longest element of W0, that is, we have $$ {C}_{w_0}{\mathrm{s}}_{\uplambda}={C}_{w_0{t}_{\uplambda}} $$ . As a consequence, the coefficients that appear when decomposing $$ {C}_{w_0{t}_{\uplambda}}{\mathrm{s}}_{\tau } $$ in the Kazhdan–Lusztig basis are tensor multiplicities of the Lie algebra with Weyl group W0. The aim of this paper is to explain how admissible subsets and Littelmann paths, which are models to compute such multiplicities, naturally appear when working out this decomposition.

Canonical Left Cells and the Lowest Two-sided Cell in an Affine Weyl Group

We give some discussions to the relations between canonical left cells and the lowest two-sided cell of an affine Weyl group. In particular, we use the relations to construct irreducible modules attached to the lowest two-sided cell and some one dimensional representations of an affine Hecke algebra.

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at $\sigma = \bar\sigma^{-1} = q$. A key step in the proof may be of independent interest: we show that the sum (over $n$) of the Hochschild homologies of the positive affine Hecke algebras $\mathrm{AH}_n^+$ is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the $q$-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.

Quantum character varieties and braided module categories

We compute quantum character varieties of arbitrary closed surfaces with boundaries and marked points. These are categorical invariants int _S{mathcal {A}} of a surface S, determined by the choice of a braided tensor category {mathcal {A}}, and computed via factorization homology. We identify the algebraic data governing marked points and boundary components with the notion of a braided module category for {mathcal {A}}, and we describe braided module categories with a generator in terms of certain explicit algebra homomorphisms called quantum moment maps. We then show that the quantum character variety of a decorated surface is obtained from that of the corresponding punctured surface as a quantum Hamiltonian reduction. Characters of braided {mathcal {A}}-modules are objects of the torus category int _{T^2}{mathcal {A}}. We initiate a theory of character sheaves for quantum groups by identifying the torus integral of {mathcal {A}}={text {Rep}}_{q}G with the category {mathcal {D}}_q(G/G)-mod of equivariant quantum {mathcal {D}}-modules. When G=GL_n, we relate the mirabolic version of this category to the representations of the spherical double affine Hecke algebra {mathbb {SH}}_{q,t}.

ℤ/𝕞ℤ-graded Lie algebras and perverse sheaves, III: Graded double affine Hecke algebra

In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra g \mathfrak {g} together with a Z / m Z \mathbb {Z}/m\mathbb {Z} -grading ⨁ i ∈ Z / m Z g i \bigoplus _{i\in \mathbb {Z}/m\mathbb {Z}}\mathfrak {g}_{i} and a block of D G 0 _ ( g i ) \mathcal {D}_{G_{\underline 0}}(\mathfrak {g}_{i}) as introduced in [J. Represent. Theory 21 (2017), pp. 277-321], we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot [Duke Math J. 126 (2005), pp. 251-323] and Oblomkov-Yun [Adv. Math 292 (2016), pp. 601-706] which correspond to the case of the principal block.

Topological K-theory of affine Hecke algebras

Let H(R,q) be an affine Hecke algebra with a positive parameter function q. We are interested in the topological K-theory of H(R,q), that is, the K-theory of its C*-completion C*_r (R,q). We will prove that $K_* (C*_r (R,q))$ does not depend on the parameter q. For this we use representation theoretic methods, in particular elliptic representations of Weyl groups and Hecke algebras. Thus, for the computation of these K-groups it suffices to work out the case q=1. These algebras are considerably simpler than for q not 1, just crossed products of commutative algebras with finite Weyl groups. We explicitly determine $K_* (C*_r (R,q))$ for all classical root data R, and for some others as well. This will be useful to analyse the K-theory of the reduced C*-algebra of any classical p-adic group. For the computations in the case q=1 we study the more general situation of a finite group \Gamma acting on a smooth manifold M. We develop a method to calculate the K-theory of the crossed product $C(M) \rtimes \Gamma$. In contrast to the equivariant Chern character of Baum and Connes, our method can also detect torsion elements in these K-groups.

Affine Wreath Product Algebras

Abstract We study the structure and representation theory of affine wreath product algebras and their cyclotomic quotients. These algebras, which appear naturally in Heisenberg categorification, simultaneously unify and generalize many important algebras appearing in the literature. In particular, special cases include degenerate affine Hecke algebras, affine Sergeev algebras (degenerate affine Hecke–Clifford algebras), and wreath Hecke algebras. In some cases, specializing the results of the current paper recovers known results, but with unified and simplified proofs. In other cases, we obtain new results, including proofs of two open conjectures of Kleshchev and Muth.

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.

Multivariate Quadratic Transformations and the Interpolation Kernel

We prove a number of quadratic transformations of elliptic Selberg integrals (conjectured in an earlier paper of the author), as well as studying in depth the "interpolation kernel", an analytic continuation of the author's elliptic interpolation functions which plays a major role in the proof as well as acting as the kernel for a Fourier transform on certain elliptic double affine Hecke algebras (discussed in a later paper). In the process, we give a number of examples of a new approach to proving elliptic hypergeometric integral identities, by reduction to a Zariski dense subset of a formal neighborhood of the trigonometric limit.

Completeness of the Bethe Ansatz for an Open $$\varvec{q}$$ q -Boson System with Integrable Boundary Interactions

We employ a discrete integral-reflection representation of the double affine Hecke algebra of type $C^\vee C$ at the critical level q=1, to endow the open finite $q$-boson system with integrable boundary interactions at the lattice ends. It is shown that the Bethe Ansatz entails a complete basis of eigenfunctions for the commuting quantum integrals in terms of Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials.

On the Eigenfunctions for the Multi-species <i>q</i>-Boson System

In a previous paper a multi-species version of the q-Boson stochastic particle system is introduced and the eigenfunctions of its backward generator are constructed by using a representation of the Hecke algebra. In this article we prove a formula which expresses the eigenfunctions by means of the q-deformed bosonic operators, which are constructed from the L-operator of higher rank found in the recent work by Garbali, de Gier and Wheeler. The L-operator is obtained from the universal R-matrix of the quantum affine algebra of type A_{r}^{(1)} by the use of the q-oscillator representation. Thus our formula may be regarded as a bridge between two approaches to studying integrable stochastic systems by means of the quantum affine algebra and the affine Hecke algebra.

SVV algebras

In 2010 Shan, Varagnolo and Vasserot introduced a family of graded algebras in order to prove a conjecture of Kashiwara and Miemietz which stated that the finite-dimensional representations of affine Hecke algebras of type $D$ categorify a module over a certain quantum group. We study these algebras, and in various cases, show how they relate to Varagnolo-Vasserot algebras and to quiver Hecke algebras which in turn allows us to deduce various homological properties.

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.

Tilting modules of affine quasi-hereditary algebras

We discuss tilting modules of affine quasi-hereditary algebras. We present an existence theorem of indecomposable tilting modules when the algebra has a large center and use it to deduce a criterion for an exact functor between two affine highest weight categories to give an equivalence. As an application, we prove that the Arakawa–Suzuki functor (Arakawa–Suzuki, 1998) gives a fully faithful embedding of a block of the deformed BGG category of glm into the module category of a suitable completion of degenerate affine Hecke algebra of GLn.

Affine cellularity of affine Yokonuma–Hecke algebras

We establish an explicit algebra isomorphism between the affine Yokonuma–Hecke algebra Yˆr,n(q) and a direct sum of matrix algebras with coefficients in tensor products of affine Hecke algebras of type A. As an application of this result, we show that Yˆr,n(q) is affine cellular in the sense of Koenig and Xi, and further prove that it has finite global dimension when the parameter q is not a root of the Poincaré polynomial. As another application, we also recover the modular representation theory of Yˆr,n(q) previously obtained in [7].

Young tableaux and representations of Hecke algebras of type ADE

We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of type ADE. Representations are constructed by explicit formulas, in a seminormal form.

Cocenters and representations of affine Hecke algebras

In this paper, we study the relationship 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.

Iterated Torus Knots and Double Affine Hecke Algebras

We give a topological realization of the (spherical) double affine Hecke algebra $\mathrm{SH}_{q,t}$ of type $A_1$, and we use this to construct a module over $\mathrm{SH}_{q,t}$ for any knot $K \subset S^3$. As an application, we give a purely topological interpretation of Cherednik's 2-variable polynomials $P_n(r,s; q,t)$ of type $A_1$ from [Che13] (where $r,s \in \mathbb{Z}$ are relatively prime), and we give a new proof that these specialize to the colored Jones polynomials of the $r,s$ torus knot. We then generalize Cherednik's construction (for $\mathcal{sl}_2$) to all iterated cables of the unknot and prove the corresponding specialization property. Finally, in the appendix we compare our polynomials associated to iterated torus knots to the ones recently defined in [CD14], in the specialization $t=-q^2$.

The universal DAHA of type [formula omitted] and Leonard pairs of q-Racah type

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let F denote an algebraically closed field, and fix a nonzero q∈F that is not a root of unity. The universal double affine Hecke algebra (DAHA) Hˆq of type (C1∨,C1) is the associative F-algebra defined by generators {ti±1}i=03 and relations (i) titi−1=ti−1ti=1; (ii) ti+ti−1 is central; (iii) t0t1t2t3=q−1. We consider the elements X=t3t0 and Y=t0t1 of Hˆq. Let V denote a finite-dimensional irreducible Hˆq-module on which each of X, Y is diagonalizable and t0 has two distinct eigenvalues. Then V is a direct sum of the two eigenspaces of t0. We show that the pair X+X−1, Y+Y−1 acts on each eigenspace as a Leonard pair, and each of these Leonard pairs falls into a class said to have q-Racah type. Thus from V we obtain a pair of Leonard pairs of q-Racah type. It is known that a Leonard pair of q-Racah type is determined up to isomorphism by a parameter sequence (a,b,c,d) called its Huang data. Given a pair of Leonard pairs of q-Racah type, we find necessary and sufficient conditions on their Huang data for that pair to come from the above construction.

A uniform classification of discrete series representations of affine Hecke algebras

We give a new and independent parameterization of the set of discrete series characters of an affine Hecke algebra $\mathcal{H}_{\mathbf{v}}$, in terms of a canonically defined basis $\mathcal{B}_{gm}$ of a certain lattice of virtual elliptic characters of the underlying (extended) affine Weyl group. This classification applies to all semisimple affine Hecke algebras $\mathcal{H}$, and to all $\mathbf{v}\in\mathcal{Q}$, where $\mathcal{Q}$ denotes the vector group of positive real (possibly unequal) Hecke parameters for $\mathcal{H}$. By analytic Dirac induction we define for each $b\in \mathcal{B}_{gm}$ a continuous (in the sense of [OS2]) family $\mathcal{Q}^{reg}_b:=\mathcal{Q}_b\backslash\mathcal{Q}_b^{sing}\ni\mathbf{v}\to\operatorname{Ind}_{D}(b;\mathbf{v})$, such that $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$ (for some $\epsilon(b;\mathbf{v})\in\{\pm 1\}$) is an irreducible discrete series character of $\mathcal{H}_{\mathbf{v}}$. Here $\mathcal{Q}^{sing}_b\subset\mathcal{Q}$ is a finite union of hyperplanes in $\mathcal{Q}$. In the non-simply laced cases we show that the families of virtual discrete series characters $\operatorname{Ind}_{D}(b;\mathbf{v})$ are piecewise rational in the parameters $\mathbf{v}$. Remarkably, the formal degree of $\operatorname{Ind}_{D}(b;\mathbf{v})$ in such piecewise rational family turns out to be rational. This implies that for each $b\in \mathcal{B}_{gm}$ there exists a universal rational constant $d_b$ determining the formal degree in the family of discrete series characters $\epsilon(b;\mathbf{v})\operatorname{Ind}_{D}(b;\mathbf{v})$. We will compute the canonical constants $d_b$, and the signs $\epsilon(b;\mathbf{v})$. For certain geometric parameters we will provide the comparison with the Kazhdan-Lusztig-Langlands classification.