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We introduce families \( \mathcal{B}_n^S\left( {{z_1},\ldots,{z_n}} \right) \) and \( \mathcal{B}_{{n,\hbar}}^S\left( {{z_1},\ldots,{z_n}} \right) \) of maximal commutative subalgebras, called Bethe subalgebras, of the group algebra \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \) of the symmetric group. Bethe subalgebras are deformations of the Gelfand−Zetlin subalgebra of \( \mathbb{C}\left[ {\mathfrak{S}n} \right] \). We describe various properties of Bethe subalgebras.

Let W be a complex reflection group and H_c(W) the Rational Cherednik algebra for $W$ depending on a parameter c. One can consider the category O for H_c(W). We prove a conjecture of Rouquier that the categories O for H_c(W) and H_{c'}(W) are derived equivalent provided the parameters c,c' have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analog of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse., Comment: 32 pages, preliminary version; v2 33 pages some proofs rewritten; v3 32 pages, proofs of the main results substantially simplified, other minor changes; v4 some proofs were modified, other minor changes

We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at t=0. Their representation theory is connected to the geometry of the Calogero-Moser space, and there is a lot of evidence that they contain certain information about Hecke algebras even though the precise connection is so far unclear. We outline the basic theory along with some open problems and conjectures, and give explicit results in the cyclic and dihedral cases.

Given a complex reflection group W we compute the support of the spherical irreducible module of the rational Cherednik algebra of W in terms of the simultaneous eigenfunction of the Dunkl operators and Schur elements for finite Hecke algebras., Comment: 21 pages

We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using $R$-matrices for $U_q(\mathfrak{sl}_N)$. Our construction is motivated by an analogous construction of Silvia Montarani in the rational case. Using the Drinfeld-Kohno theorem for Knizhnik-Zamolodchikov differential equations, we prove that the explicit representations we produce correspond to Montarani's representations under a monodromy functor introduced by Etingof, Gan, and Oblomkov.

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals that are invariant under the action of the corresponding rational Cherednik algebra hence form submodules in the polynomial module. We show that a quantum integrable system can be defined for every such ideal for a real reflection group W. This leads to known and new integrable systems of Calogero–Moser type which we explicitly specify. In the case of classical Coxeter groups, we also obtain generalized Calogero–Moser systems with added quadratic potential.

The theory of congruences for the partition function p ( n ) depends heavily on the properties of half-integral weight Hecke operators. The subject has been complicated by the absence of closed formulas for the Hecke images P ( z ) | T ( l 2 ) , where P ( z ) is the relevant modular generating function. We obtain such formulas using Eulerʼs Pentagonal Number Theorem and the denominator formula for the Monster Lie algebra. As a corollary, we obtain congruences for certain powers of Ramanujanʼs Delta-function.

Let W a be an irreducible affine Weyl group with W 0 the associated Weyl group. The present paper is to study the second lowest two-sided cell Ω qr of W a . Let n qr be the number of left cells of W a in Ω qr . We conjecture that the equality n qr = 1 2 | W 0 | should always hold. When W a is either A ˜ n − 1 , n ⩾ 2 , or of rank ⩽4, this equality can be verified by the existing data (see 0.3). Then the main result of the paper is to prove the inequality n qr ⩽ 1 2 | W 0 | in all cases.

This paper concerns the combinatorics of the orbit Hecke algebra associated with the orbit of a two sided Weyl group action on the Renner monoid of a finite monoid of Lie type, $M$. It is shown by Putcha in \cite{Putcha97} that the Kazhdan-Lusztig involution (\cite{KL79}) can be extended to the orbit Hecke algebra which enables one to define the $R$-polynomials of the intervals contained in a given orbit. Using the $R$-polynomials, we calculate the M\"obius function of the Bruhat-Chevalley ordering on the orbits. Furthermore, we provide a necessary condition for an interval contained in a given orbit to be isomorphic to an interval in some Weyl group., Comment: 12 pages

We apply the Dunkl-Opdam operators and generalized Jack polynomials to study category O for the rational Cherednik algebra of type G(r,1,n). We determine the set of aspherical values, and answer a question of Iain Gordon on the ordering of category O., 20 pages; v2 updated based on reviewer feedback, now with somewhat expanded exposition and proofs

We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q., Latex file, 24 pages, minor corrections. To appear in Journal of Combinatorial Theory A

We introduce the generic central character of an irreducible discrete series representation of an affine Hecke algebra. Using this invariant we give a new classification of the irreducible discrete series characters for all abstract affine Hecke algebras (except for the types E) with arbitrary positive parameters and we prove an explicit product formula for their formal degrees (in all cases)., 68 pages, 5 figures. In the second version an appendix was added

In recent joint work with Wang, we have constructed graded Specht modules for cyclotomic Hecke algebras. In this article, we prove a graded version of the Lascoux-Leclerc-Thibon conjecture, describing the decomposition numbers of graded Specht modules over a field of characteristic zero., Comment: 57 pages; final version

We derive a parameterization of simple modules for the cyclotomic Hecke algebras of type G ( r , p , n ) with p > 1 and n ⩾ 3 over fields of any characteristic coprime to p. We give explicit formulas for the number of simple modules over these cyclotomic Hecke algebras.

We define the elliptic Hecke algebras for arbitrary marked elliptic root systems in terms of the corresponding elliptic Dynkin diagrams and make a ‘dictionary’ between the elliptic Hecke algebras and the double affine Hecke algebras.

For a Hecke pair (G, H) and a finite-dimensional representation σ of H on Vσ with finite range, we consider a generalized Hecke algebra $\mathcal{H}_{\sigma}(G, H)$, which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ of Hσ. There is a (non-full) projection $p_{\sigma} \in C_c(G_{\sigma}, \mathcal{B}(V_{\sigma}))$ such that $\mathcal{H}_{\sigma}(G, H)$ is isomorphic to $p_{\sigma} C_c(G_{\sigma}, \mathcal{B}(V_{\sigma}))p_{\sigma}$. We study the structure and properties of C*-completions of the generalized Hecke algebra arising from this corner realisation, and via Morita–Fell–Rieffel equivalence, we identify, in some cases explicitly, the resulting proper ideals of $C^*(G_{\sigma}, \mathcal{B}(V_{\sigma}))$. By letting σ vary, we can compare these ideals. The main focus is on the case with dim σ = 1 and applications include ax + b-groups and the Heisenberg group.

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., Comment: 31 pages, LaTeX

The quantum affine algebra has two realizations, the usual Drinfeld–Jimbo definition and a new Drinfeld realization given by Drinfeld. In this article, we use the adjoint action to prove that these two realizations are isomorphic for the twisted quantum affine algebra.

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group W = G ( m , p , n ) and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category O for the rational Cherednik algebra of W .

We study geometric representations of G L ( n , R ) for a ring R . The structure of the associated Hecke algebras is analyzed and shown to be cellular. Multiplicities of the irreducible constituents of these representations are linked to the embedding problem of pairs of R -modules x ⊂ y .

Inspired by a profound observation on the Racah–Wigner coefficients of U q ( sl 2 ) , the Askey–Wilson algebras were introduced in the early 1990s. A universal analog △ q of the Askey–Wilson algebras was recently studied. For q not a root of unity, it is known that Z ( △ q ) is isomorphic to the polynomial ring of four variables. A presentation for Z ( △ q ) at q a root of unity is displayed in this paper. As an application, a presentation for the center of the double affine Hecke algebra of type ( C 1 ∨ , C 1 ) at roots of unity is obtained.

We prove the modular branching rule of the cyclotomic Hecke algebras, which has remained open., Comment: 9 pages, last section removed, (v3,v4) minor changes in the introduction, (v5,v6) more detailed citations added, typos corrected

In this report we review some facts about representation theory of Hecke algebras. For Hecke algebras we adapt the approach of A. Okounkov and A. Vershik [Selecta Math., New Ser., 2 (1996) 581], which was developed for the representation theory of symmetric groups. We justify explicit construction of idempotents for Hecke algebras in terms of Jucys-Murphy elements. Ocneanu's traces for these idempotents (which can be interpreted as q-dimensions of corresponding irreducible representations of quantum linear groups) are presented.

On donne une condition necessaire et suffisante pour l'existence de modules de dimension finie sur l'algebre de Cherednik rationnelle associee a un systeme de racines.

We introduce a generalisation of the KP hierarchy, closely related to the cyclic quiver and the Cherednik algebra $H_k(\mathbb Z_m)$. This hierarchy depends on $m$ parameters (one of which can be eliminated), with the usual KP hierarchy corresponding to the $m=1$ case. Generalising the result of G. Wilson, we show that our hierarchy admits solutions parameterised by suitable quiver varieties. The pole dynamics for these solutions is shown to be governed by the classical Calogero-Moser system for the wreath-product $\mathbb Z_m\wr S_n$ and its new spin version. These results are further extended to the case of the multi-component hierarchy., Comment: 40 pages

We study blocks of category O for the Cherednik algebra having the property that every irreducible module in the block admits a BGG resolution, and as a consequence prove a character formula conjectured by Oblomkov-Yun., Comment: 34 pages, color figures

We prove a strong induction theorem for graded Hecke algebras and we classify the tempered and square integrable representations of such algebras using methods of equivariant homology.

In this article, Gouvea's conjecture on controlling the conductor is proven in a special case.

In this short note, we find bases of the centers of generic Hecke algebras associated with certain finite Coxeter groups. Our bases are described using the notion of cell datum of Graham and Lehrer, and the notion of norm.

The present paper is concerned with the connection between the imprimitive reflection groups G ( m , m , n ) G(m,m,n) , m ∈ N m\in \mathbb {N} , and the affine Weyl group A ~ n − 1 \widetilde {A}_{n-1} . We show that A ~ n − 1 \widetilde {A}_{n-1} is a generic version of the groups G ( m , m , n ) G(m,m,n) , m ∈ N m\in \mathbb {N} . We introduce some new presentations of these groups which are shown to have some group-theoretic advantages. Then we define the Hecke algebras of these groups and of their braid versions, each in two ways according to two presentations. Finally we give a new description for the affine root system Φ ¯ \overline {\Phi } of A ~ n − 1 \widetilde {A}_{n-1} such that the action of A ~ n − 1 \widetilde {A}_{n-1} on Φ ¯ \overline {\Phi } is compatible with that of G ( m , m , n ) G(m,m,n) on its root system in some sense.

In this paper we introduce a basic representation for the confluent Cherednik algebras $\mathcal H_{\rm V}$, $\mathcal H_{\rm III}$, $\mathcal H_{\rm III}^{D_7}$ and $\mathcal H_{\rm III}^{D_8}$ defined in arXiv:1307.6140. To prove faithfulness of this basic representation, we introduce the non-symmetric versions of the continuous dual $q$-Hahn, Al-Salam-Chihara, continuous big $q$-Hermite and continuous $q$-Hermite polynomials.

We construct Gaiotto states with fundamental multiplets in $SU(N)$ gauge theories, in terms of the orthonormal basis of spherical degenerate double affine Hecke algebra (SH in short), the representations of which are equivalent to those of $W_n$ algebra with additional $U(1)$ current. The generalized Whittaker conditions are demonstrated under the action of SH, and further rewritten in terms of $W_n$ algebra. Our approach not only consists with the existing literature but also holds for general $SU(N)$ case., 20 pages, 2 figures; v2: minor modifications, typos corrected

The theory of DAHA-Jones polynomials is extended from torus knots to their arbitrary iterations (for any reduced root systems and weights), which incudes the polynomiality, duality and other properties of the DAHA superpolynomials. Presumably they coincide with the reduced stable Khovanov-Rozansky polynomials in the case of non-negative coefficients. The new theory matches well the classical theory of algebraic knots and (unibranch) plane curve singularities; the Puiseux expansion naturally emerges. The corresponding DAHA superpolynomials are expected to coincide with the reduced ones in the Oblomkov-Shende-Rasmussen Conjecture upon its generalization to arbitrary dominant weights. For instance, the DAHA uncolored superpolynomials at a=0, q=1 are conjectured to provide the Betti numbers of the Jacobian factors of the corresponding singularities., v2: Editing, more material on the Betti numbers of Jacobian factors and non-algebraic knots was added, the relation to Jones polynomials was justified

We give a full classification, in terms of periodic skew diagrams, of irreducible semisimple modules in category O for the degenerate double affine Hecke algebra of type A which can be realized as submodules of Verma modules.

We show that the well known homomorphism from any Artin braid group to the Hecke algebra of the same type is injective for the universal coxeter system and that the Burau representation is faithful for all finite coxeter systems of rank two.

A construction of Bernstein associates to each cocharacter of a split p p -adic group an element in the center of the Iwahori-Hecke algebra, which we refer to as a Bernstein function. A recent conjecture of Kottwitz predicts that Bernstein functions play an important role in the theory of bad reduction of a certain class of Shimura varieties (parahoric type). It is therefore of interest to calculate the Bernstein functions explicitly in as many cases as possible, with a view towards testing Kottwitz’ conjecture. In this paper we prove a characterization of the Bernstein function associated to a minuscule cocharacter (the case of interest for Shimura varieties). This is used to write down the Bernstein functions explicitly for some minuscule cocharacters of G l n Gl_n ; one example can be used to verify Kottwitz’ conjecture for a special class of Shimura varieties (the “Drinfeld case”). In addition, we prove some general facts concerning the support of Bernstein functions, and concerning an important set called the “ μ \mu -admissible” set. These facts are compatible with a conjecture of Kottwitz and Rapoport on the shape of the special fiber of a Shimura variety with parahoric type bad reduction.

In this paper, F F is a non-Archimedean local field and G G is the group of F F -points of a connected reductive algebraic group defined over F F . Also, τ \tau is an irreducible representation of a compact open subgroup J J of G G , the pair ( J , τ ) (J,\tau ) being a type in G G . The pair ( J , τ ) (J,\tau ) is assumed to be a cover of a type ( J L , τ L ) (J_{L},\tau _{L}) in a Levi subgroup L L of G G . We give conditions, generalizing those of earlier work, under which the Hecke algebra H ( G , τ ) \scr H(G,\tau ) is the tensor product of a canonical image of H ( L , τ L ) \scr H(L,\tau _{L}) and a sub-algebra H ( K , τ ) \scr H(K,\tau ) , for a compact open subgroup K K of G G containing J J .

We determine the generic degrees of cyclotomic Hecke algebras attached to exceptional finite complex reflection groups. The results are used to introduce the notion of spetsial reflection group, which in a certain sense is a generalization of the finite Weyl group. In particular, to spetsialWWthere is attached a set of unipotent degrees which in the case of a Weyl group is just the set of degrees of unipotent characters of finite reductive groups with Weyl groupWW, and in general enjoys many of their combinatorial properties.

This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r,1,n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of “Kleshchev multipartitions”.