Affine Hecke Algebra Research Articles

On J-folded Alcove Paths and Combinatorial Representations of Affine Hecke Algebras

AbstractWe introduce the combinatorial model of J-folded alcove paths in an affine Weyl group and construct representations of affine Hecke algebras using this model. We study boundedness of these representations, and we state conjectures linking our combinatorial formulae to Kazhdan-Lusztig theory and Opdam’s Plancherel Theorem.

Generalized double affine Hecke algebra for double torus

We propose a generalization of the double affine Hecke algebra of type-C∨C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^\\vee C_1$$\\end{document} at specific parameters by introducing a “Heegaard dual” of the Hecke operators. Shown is a relationship with the skein algebra on double torus. We give automorphisms of the algebra associated with the Dehn twists on the double torus.

A comparison of endomorphism algebras

Let F be a non-archimedean local field and G be a connected reductive group over F. For a Bernstein block in the category of smooth complex representations of G(F), we have two kinds of progenerators: the compactly induced representation indKG(F)(ρ) of a type (K,ρ), and the parabolically induced representation IPG(ΠM) of a progenerator ΠM of a Bernstein block for a Levi subgroup M of G. In this paper, we construct an explicit isomorphism of these two progenerators. Moreover, we compare the description of the endomorphism algebra EndG(F)(indKG(F)(ρ)) for a depth-zero type (K,ρ) in [20] with the description of the endomorphism algebra EndG(F)(IPG(ΠM)) in [33], that are described in terms of affine Hecke algebras.

Higher rank (𝑞,𝑡)-Catalan polynomials, affine Springer fibers, and a finite rational shuffle theorem

We introduce the higher rank ( q , t ) (q,t) -Catalan polynomials and prove they equal truncations of the Hikita polynomial to a finite number of variables. Using affine compositions and a certain standardization map, we define a d i n v \mathtt {dinv} statistic on rank r r semistandard ( m , n ) (m,n) -parking functions and prove c o d i n v \mathtt {codinv} counts the dimension of an affine space in an affine paving of a parabolic affine Springer fiber. Combining these results, we give a finite analogue of the Rational Shuffle Theorem in the context of double affine Hecke algebras. Lastly, we also give a Bizley-type formula for the higher rank Catalan numbers in the non-coprime case.

Erratum to: The cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups

Erratum to: The cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups

Symmetry and Pieri rules for the bisymmetric Macdonald polynomials

Bisymmetric Macdonald polynomials can be obtained through a process of antisymmetrization and t-symmetrization of non-symmetric Macdonald polynomials. Using the double affine Hecke algebra, we show that the evaluation of the bisymmetric Macdonald polynomials satisfies a symmetry property generalizing that satisfied by the usual Macdonald polynomials. We then obtain Pieri rules for the bisymmetric Macdonald polynomials where the sums are over certain vertical strips.

The cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups

The cyclotomic degenerate affine Hecke algebras of arbitrary Weyl groups

Parameters of Hecke algebras for Bernstein components of [formula omitted]-adic groups

Let G be a reductive group over a non-archimedean local field F. Consider an arbitrary Bernstein block Rep(G)s in the category of complex smooth G-representations. In earlier work the author showed that there exists an affine Hecke algebra H(O,G) whose category of right modules is closely related to Rep(G)s. In many cases this is in fact an equivalence of categories, like for Iwahori-spherical representations.In this paper we study the q-parameters of the affine Hecke algebras H(O,G). We compute them in many cases, in particular for principal series representations of quasi-split groups and for classical groups.Lusztig conjectured that the q-parameters are always integral powers of the cardinality of the residue field of F, and that they coincide with the q-parameters coming from some Bernstein block of unipotent representations. We reduce this conjecture to the case of absolutely simple p-adic groups, and we prove it for most of those.

DG-Enhanced Hecke and KLR Algebras

We construct DG-enhanced versions of the degenerate affine Hecke algebra and of the affine Hecke algebra. We extend Brundan-Kleshchev and Rouquier's isomorphism and prove that after completion DG-enhanced versions of affine Hecke algebras (degenerate or nondegenerate) are isomorphic to completed DG-enhanced versions of KLR algebras for suitably defined quivers. As a byproduct, we deduce that these DG-algebras have homologies concentrated in degree zero. These homologies are isomorphic respectively to the degenerate cyclotomic Hecke algebra and the cyclotomic Hecke algebra.

On some double Nahm sums of Zagier

Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no q-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a q-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first q-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011 and whose modularity was proved by Cherednik and Feigin in 2013 via nilpotent double affine Hecke algebras.

Tangles in affine Hecke algebras

Framed oriented [Formula: see text]-tangle diagrams in the annulus, subject to the Homfly skein relations, are used to produce an algebra isomorphic to the affine Hecke algebras [Formula: see text] of type [Formula: see text]. The use of closed curves and braids gives neat pictures for central elements in the algebras.

Equivariant K-theory of the semi-infinite flag manifold as a nil-DAHA module

The equivariant K-theory of the semi-infinite flag manifold, as developed recently by Kato, Naito, and Sagaki, carries commuting actions of the nil-double affine Hecke algebra (nil-DAHA) and a q-Heisenberg algebra. The action of the latter generates a free submodule of rank |W|, where W is the (finite) Weyl group. We show that this submodule is stable under the nil-DAHA, which enables one to express the nil-DAHA action in terms of |W|times |W| matrices over the q-Heisenberg algebra. Our main result gives an explicit algebraic construction of these matrices as a limit from the (non-nil) DAHA in simply-laced type. This construction reveals that multiplication by equivariant scalars, when expressed in terms of the Heisenberg algebra, is given by the nonsymmetric q-Toda system introduced by Cherednik and the author.

Schur-Weyl duality for quantum toroidal superalgebras

We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [26] to the super case.

Dimensions of modular irreducible representations of semisimple Lie algebras

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished p p -characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category O \mathcal {O} . For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent p p -character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.

Type A DAHA and doubly periodic tableaux

Analogously to the construction of Suzuki and Vazirani, we construct representations of the GLm-type Double Affine Hecke Algebra at roots of unity. These representations are graded and the weight spaces for the X-variables are parametrized by the combinatorial objects we call doubly periodic tableaux. We show that our representations exhaust all graded X-semisimple representations, and the direct sum of all our representations is faithful. Analogously to the construction of Jordan and Vazirani of rectangular DAHA representations, we show that our representations can be interpreted in terms of ribbon fusion categories associated to Uq(glN) at roots of unity. Combining the ribbon structure with faithfulness we deduce a conjecture of Morton and Samuelson about realization of DAHA as a skein algebra of the torus with base string modulo certain local relations.

Affine Hecke algebras and quantum symmetric pairs

We introduce an affine Schur algebra via the affine Hecke algebra associated to Weyl group of affine type C. We establish multiplication formulas on the affine Hecke algebra and affine Schur algebra. Then we construct monomial bases and canonical bases for the affine Schur algebra. The multiplication formula allows us to establish a stabilization property of the family of affine Schur algebras that leads to the modified version of an algebra K n c {\mathbf K}^{\mathfrak c}_n . We show that K n c {\mathbf K}^{\mathfrak c}_n is a coideal subalgebra of quantum affine algebra U ( g l ^ n ) {\mathbf {U}}(\widehat {\mathfrak {gl}}_n) , and ( U ( g l ^ n ) , K n c ) \big ({\mathbf {U}}(\widehat { \mathfrak {gl}}_n), {\mathbf K}^{\mathfrak c}_n) forms a quantum symmetric pair. The modified coideal subalgebra is shown to admit monomial and stably canonical bases. We also formulate several variants of the affine Schur algebra and the (modified) coideal subalgebra above, as well as their monomial and canonical bases. This work provides a new and algebraic approach which complements and sheds new light on our previous geometric approach on the subject. In the appendix by four of the authors, new length formulas for the Weyl groups of affine classical types are obtained in a symmetrized fashion.

ON THE STRUCTURE OF THE AFFINE ASYMPTOTIC HECKE ALGEBRAS

According to a conjecture of Lusztig, the asymptotic affine Hecke algebra should admit a description in terms of the Grothedieck group of sheaves on the square of a finite set equivariant under the action of the centralizer of a nilpotent element in the reductive group. A weaker form of this statement, allowing for possible central extensions of stabilizers of that action, has been proved by the first named author with Ostrik. In the present paper, we describe an example showing that nontrivial central extensions do arise, thus the above weaker statement is optimal. We also show that Lusztig's homomorphism from the affine Hecke algebra to the asymptotic affine Hecke algebra induces an isomorphism on cocenters and discuss the relation of the above central extensions to the structure of the cocenter.

On the basic representation of the double affine Hecke algebra at critical level

We construct the basic representation of the double affine Hecke algebra at critical level [Formula: see text] associated to an irreducible reduced affine root system [Formula: see text] with a reduced gradient root system. For [Formula: see text] of untwisted type such a representation was studied by Oblomkov [A. Oblomkov, Double affine Hecke algebras and Calogero–Moser spaces, Represent. Theory 8(2004) 243–266] and further detailed by Gehles [K. E. Gehles, Properties of Cherednik algebras and graded Hecke algebras, PhD thesis, University of Glasgow (2006)] in the presence of minuscule weights.

Standard multipartitions and a combinatorial affine Schur-Weyl duality

We introduce the notion of standard (Kleshchev) multipartitions and establish a one-to-one correspondence between standard multipartitions and irreducible representations with integral weights for the affine Hecke algebra of type A with a parameter q∈C× which is not a root of unity. We then extend the correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of integral type. By the affine Schur–Weyl duality, we further extend this to a correspondence between standard multipartitions and Drinfeld multipolynomials of integral type whose associated irreducible polynomial representations completely determine all irreducible polynomial representations for the quantum loop algebra Uq(glˆn). We will see, in particular, the notion of standard multipartitions gives rise to a combinatorial description of the affine Schur–Weyl duality in terms of a column-reading vs. row-reading of residues of a multipartition.

Specializing Koornwinder polynomials to Macdonald polynomials of type B, C, D and BC

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type \((C_n^\vee ,C_n)\) in the sense of Macdonald (Affine Hecke algebras and orthogonal polynomials, Cambridge tracts in mathematics, Cambridge Univ Press, 2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type B, C and D via Ram–Yip type formulas of non-symmetric Koornwinder and Macdonald polynomials.