Affine Hecke Algebra Research Articles

On the K-theory stable bases of the Springer resolution

Cohomological and K-theoretic stable bases originated from the study of quantum cohomology and quantum K-theory. Restriction formula for cohomological stable bases played an important role in computing the quantum connection of cotangent bundle of partial flag varieties. In this paper we study the K-theoretic stable bases of cotangent bundles of flag varieties. We describe these bases in terms of the action of the affine Hecke algebra and the twisted group algebra of KostantKumar. Using this algebraic description and the method of root polynomials, we give a restriction formula of the stable bases. We apply it to obtain the restriction formula for partial flag varieties. We also build a relation between the stable basis and the Casselman basis in the principal series representations of the Langlands dual group. As an application, we give a closed formula for the transition matrix between Casselman basis and the characteristic functions.

Morita equivalence for $k$-algebras

We review Morita equivalence for finite type k-algebras A and also a weakening of Morita equivalence which we call stratified equivalence. The spectrum of A is the set of equivalence classes of irreducible A-modules. For any finite type k-algebra A, the spectrum of A is in bijection with the set of primitive ideals of A. The stratified equivalence relation preserves the spectrum of A and also preserves the periodic cyclic homology of A. However, the stratified equivalence relation permits a tearing apart of strata in the primitive ideal space which is not allowed by Morita equivalence. A key example illustrating the distinction between Morita equivalence and stratified equivalence is provided by affine Hecke algebras associated to affine Weyl groups.

Quantum Schur duality of affine type C with three parameters

We establish a three-parameter Schur duality between the affine Hecke algebra of type C and a coideal subalgebra of quantum affine $\mathfrak{sl}_n$. At the equal parameter specializations, we obtain Schur dualities of types BCD.

A Fock space model for decomposition numbers for quantum groups at roots of unity

In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module $\mathcal{F}_{\ell}$ with bar involution by specifying generators and "straightening relations" adapted from those appearing in the Kashiwara-Miwa-Stern formulation of the $q$-Fock space. By relating $\mathcal{F}_{\ell}$ to the corresponding affine Hecke algebra we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan-Lusztig polynomials. This property and the convenient combinatorial labeling of bases of $\mathcal{F}_{\ell}$ by dominant integral weights makes $\mathcal{F}_{\ell}$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.

On quiver W-algebras and defects from gauge origami

In this note, using Nekrasov's gauge origami framework, we study two different versions of the BPS/CFT correspondence – first, the standard AGT duality and, second, the quiver W algebra construction which has been developed recently by Kimura and Pestun. The gauge origami enables us to work with both dualities simultaneously and find exact matchings between the parameters. In our main example of an A-type quiver gauge theory, we show that the corresponding quiver qW-algebra and its representations are closely related to a large-n limit of spherical gln double affine Hecke algebra whose modules are described by instanton partition functions of a defect quiver theory.

Modified affine Hecke algebras and quiver Hecke algebras of type A

We introduce some modified forms for the degenerate and non-degenerate affine Hecke algebras of type A. These are certain subalgebras living inside the inverse limit of cyclotomic Hecke algebras. We construct faithful representations and standard bases for these algebras and give some explicit description of their centers. We show that there are algebra isomorphisms between some generalized Ore localizations of these modified affine Hecke algebras and of the quiver Hecke algebras of type A. As an application, we show that the center conjecture for the cyclotomic quiver Hecke algebra of type A holds if and only if the center conjecture for the cyclotomic Hecke algebra of type A holds.

Towers of Solutions of qKZ Equations and Their Applications to Loop Models

Cherednik’s type A quantum affine Knizhnik–Zamolodchikov (qKZ) equations form a consistent system of linear q-difference equations for V_n-valued meromorphic functions on a complex n-torus, with V_n a module over the mathrm{GL}_n-type extended affine Hecke algebra {mathcal {H}}_n. The family ({mathcal {H}}_n)_{nge 0} of extended affine Hecke algebras forms a tower of algebras, with the associated algebra morphisms {mathcal {H}}_nrightarrow {mathcal {H}}_{n+1}, in the Hecke algebra descending of arc insertion at the affine braid group level. In this paper, we consider qKZ towers (f^{(n)})_{nge 0} of solutions, which consist of twisted-symmetric polynomial solutions f^{(n)} (nge 0) of the qKZ equations that are compatible with the tower structure on ({mathcal {H}}_n)_{nge 0}. The compatibility is encoded by the so-called braid recursion relations: f^{(n+1)}(z_1,ldots ,z_{n},0) is required to coincide up to a quasi-constant factor with the push-forward of f^{(n)}(z_1,ldots ,z_{n}) by an intertwiner mu _{n}{:},V_{n}rightarrow V_{n+1} of {mathcal {H}}_{n}-modules, where V_{n+1} is considered as an {mathcal {H}}_{n}-module through the tower structure on ({mathcal {H}}_n)_{nge 0}. We associate with the dense loop model on the half-infinite cylinder with nonzero loop weights, a qKZ tower (f^{(n)})_{nge 0} of solutions. The solutions f^{(n)} are constructed from specialized dual non-symmetric Macdonald polynomials with specialized parameters using the Cherednik–Matsuo correspondence. In the special case that the extended affine Hecke algebra parameter is a third root of unity, f^{(n)} coincides with the (suitably normalized) ground state of the inhomogeneous dense O(1) loop model on the half-infinite cylinder with circumference n.

Dimensions of Affine Deligne–Lusztig Varieties: A New Approach via Labeled Folded Alcove Walks and Root Operators

Let G be a reductive group over the field F=k((t)), where k is an algebraic closure of a finite field, and let W be the (extended) affine Weyl group of G. The associated affine Deligne-Lusztig varieties $X_x(b)$, which are indexed by elements b in G(F) and x in W, were introduced by Rapoport. Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension. We use techniques inspired by geometric group theory and representation theory to address these questions in the case that b is a pure translation, and so prove much of a sharpened version of Conjecture 9.5.1 of Gortz, Haines, Kottwitz, and Reuman. Our approach is constructive and type-free, sheds new light on the reasons for existing results in the case that b is basic, and reveals new patterns. Since we work only in the standard apartment of the building for G(F), our results also hold in the p-adic context, where we formulate a definition of the dimension of a p-adic Deligne-Lusztig set. We present two immediate consequences of our main results, to class polynomials of affine Hecke algebras and to affine reflection length.

Arakawa-Suzuki functors for Whittaker modules

In this paper we construct a family of exact functors from the category of Whittaker modules of the simple complex Lie algebra of type An to the category of finite-dimensional modules of the graded affine Hecke algebra of type Aℓ. Using results of Backelin [2] and of Arakawa-Suzuki [1], we prove that these functors map standard modules to standard modules (or zero) and simple modules to simple modules (or zero). Moreover, we show that each simple module of the graded affine Hecke algebra appears as the image of a simple Whittaker module. Since the Whittaker category contains the BGG category O as a full subcategory, our results generalize results of Arakawa-Suzuki [1], which in turn generalize Schur-Weyl duality between finite-dimensional representations of SLn(C) and representations of the symmetric group Sn.

Double Affine Hecke Algebra of Rank 1 and Orthogonal Polynomials on the Unit Circle

An infinite-dimensional representation of the double affine Hecke algebra of rank 1 and type $$(C_1^{\vee },C_1)$$ in which all generators are tridiagonal is presented. This representation naturally leads to two systems of polynomials that are orthogonal on the unit circle. These polynomials can be considered as circle analogs of the Askey–Wilson polynomials. The corresponding polynomials orthogonal on an interval are constructed and discussed.

Modular Representation Theory of Affine and Cyclotomic Yokonuma-Hecke Algebras

We explore the modular representation theory of affine and cyclotomic Yokonuma-Hecke algebras. We provide an equivalence between the category of finite dimensional representations of the affine (resp. cyclotomic) Yokonuma-Hecke algebra and that of an algebra which is a direct sum of tensor products of affine Hecke algebras of type A (resp. Ariki-Koike algebras). As one of the applications, the irreducible representations of affine and cyclotomic Yokonuma-Hecke algebras are classified over an algebraically closed field of characteristic p. Secondly, the modular branching rules for these algebras are obtained; moreover, the resulting modular branching graphs for cyclotomic Yokonuma-Hecke algebras are identified with crystal graphs of irreducible integrable representations of affine Lie algebras of type A.

DAHA and skein algebra of surfaces: double-torus knots

We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combining these two types of DAHA, we construct the DAHA representation for the skein algebra on a genus-two surface, and we propose a DAHA polynomial for a double-torus knot, which is a simple closed curve on a genus two Heegaard surface in S3. Discussed is a relationship between the DAHA polynomial and the colored Jones polynomial.

($${{\mathbf {t}}},{{\mathbf {q}}}$$)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators

We introduce the natural (t, q)-deformation of the Q-system algebra in type A. The q-Whittaker limit $$t\rightarrow \infty $$ gives the quantum Q-system algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [DFK17]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [HKO+99]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [FL99]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [DFK14]). We show that the (q, t)-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type $${\mathfrak {gl}}_N$$ . Moreover, we describe the kernel of the surjective homomorphism from the quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [Mik07]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [SV11]) to this new algebra. It is generated by (q, t)-determinants, new objects which are a deformation of the quantum determinant associated with the quantum Q-system. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the $$SL_2({\mathbb {Z}})$$ -action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [BGLX16]) in the limit $$N\rightarrow \infty $$ . Thus, the (q, t)-deformation of the Q-system cluster algebra leads directly to Macdonald theory.

On the reducibility of induced representations for classical p-adic groups and related affine Hecke algebras

Let π be an irreducible smooth complex representation of a general linear p-adic group and let σ be an irreducible complex supercuspidal representation of a classical p-adic group of a given type, so that π ⨁ σ 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 π ⨁ σ does not depend on σ, if σ is “separated” from the supercuspidal support of π. (Here, “separated” means that, for each factor ρ of a representation in the supercuspidal support of π, the representation parabolically induced from ρ ⨁ σ is irreducible.) This was conjectured by E. Lapid and M. Tadic. (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 CI,σ of smooth complex finitely generated representations of classical p-adic groups of fixed type, but arbitrary rank, and supercuspidal support given by σ and I, and 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 AB 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.

Formal Affine Demazure and Hecke Algebras of Kac-Moody Root Systems

We define the formal affine Demazure algebra and a formal affine Hecke algebra associated to a Kac-Moody root system. We prove structure theorems for these algebras, and use them to extend several results and constructions (presentation in terms of generators and relations, coproduct and product structures, filtration by codimension of Bott-Samelson classes, root polynomials and multiplication formulas) that were previously known for finite root systems.

Barbasch-Sahi algebras and Dirac cohomology

We define a class of algebras which are distinguished by a PBW property and an orthogonality condition, and which we call Hopf-Hecke algebras, since they generalize the Drinfeld Hecke algebras defined by Drinfeld. In the course of studying the orthogonality condition and in analogy to the orthogonal group we show the existence of a pin cover for cocommutative Hopf algebras over $\mathbb{C}$ with an orthogonal module or, more generally, pointed cocommutative Hopf algebras over a field of characteristic $0$ with an orthogonal module. Following the suggestion of Dan Barbasch and Siddhartha Sahi, we define a Dirac operator and Dirac cohomology for modules of Hopf-Hecke algebras, generalizing those concepts for connected semisimple Lie groups, graded affine Hecke algebras and symplectic reflection algebras. Using the pin cover, we prove a general theorem for a class of Hopf-Hecke algebras which we call Barbasch-Sahi algebras, which relates the central character of an irreducible module with non-vanishing Dirac cohomology to the central characters occurring in its Dirac cohomology, generalizing a result called Vogan's conjecture for connected semisimple Lie groups, analogous results for graded affine Hecke algebras and for symplectic reflection algebras and Drinfeld Hecke algebras.

The Rectangular Representation of the Double Affine Hecke Algebra via Elliptic Schur–Weyl Duality

Abstract Given a module $M$ for the algebra ${\mathcal{D}}_{\mathtt{q}}(G)$ of quantum differential operators on $G$, and a positive integer $n$, we may equip the space $F_n^G(M)$ of invariant tensors in $V^{\otimes n}\otimes M$, with an action of the double affine Hecke algebra of type $A_{n-1}$. Here $G= SL_N$ or $GL_N$, and $V$ is the $N$-dimensional defining representation of $G$. In this paper, we take $M$ to be the basic ${\mathcal{D}}_{\mathtt{q}}(G)$-module, that is, the quantized coordinate algebra $M= {\mathcal{O}}_{\mathtt{q}}(G)$. We describe a weight basis for $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ combinatorially in terms of walks in the type $A$ weight lattice, and standard periodic tableaux, and subsequently identify $F_n^G({\mathcal{O}}_{\mathtt{q}}(G))$ with the irreducible “rectangular representation” of height $N$ of the double affine Hecke algebra.

Cocenter of p-adic groups, II: Induction map

In this paper, we study some relation between the cocenter H¯(G) of the Hecke algebra H(G) of a connected reductive group G over a nonarchimedean local field and the cocenter H¯(M) of its Levi subgroups M.Given any Newton component of H¯(G), we construct the induction map i¯ from the corresponding Newton component of H¯(M) to it. We show that this map is an isomorphism. This leads to the Bernstein–Lusztig type presentation of the cocenter H¯(G), which generalizes the work [11] on the affine Hecke algebras. We also show that the map i¯ we constructed is adjoint to the Jacquet functor.

The Harish-Chandra isomorphism for quantum $GL_2$

We construct an explicit Harish-Chandra isomorphism, from the quantum Hamiltonian reduction of the algebra \mathcal D_q(GL_2) of quantum differential operators on GL_2 , to the spherical double affine Hecke algebra associated to GL_2 . The isomorphism holds for all deformation parameters q \in \mathbb C^{\times} and t \neq ± i , such that q is not a non-trivial root of unity.

Dualities in the q‐Askey Scheme and Degenerate DAHA

AbstractThe Askey–Wilson polynomials are a four‐parameter family of orthogonal symmetric Laurent polynomials that are eigenfunctions of a second‐order q‐difference operator L, and of a second‐order difference operator in the variable n with eigenvalue . Then, L and multiplication by generate the Askey–Wilson (Zhedanov) algebra. A nice property of the Askey–Wilson polynomials is that the variables z and n occur in the explicit expression in a similar and to some extent exchangeable way. This property is called duality. It returns in the nonsymmetric case and in the underlying algebraic structures: the Askey–Wilson algebra and the double affine Hecke algebra (DAHA). In this paper, we follow the degeneration of the Askey–Wilson polynomials until two arrows down and in four different situations: for the orthogonal polynomials themselves, for the degenerate Askey–Wilson algebras, for the nonsymmetric polynomials, and for the (degenerate) DAHA and its representations.