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We study the rational Cherednik algebra attached to the complex reflection group $G(r,1,2)$. Each irreducible representation $S^\lambda$ of $G(r,1,2)$ corresponds to a standard module $\Delta(\lambda)$ for the rational Cherednik algebra. We give necessary and sufficient conditions for the existence of morphism between two of these modules and explicit formulas for them when they exist.

We give a non-negative combinatorial formula, in terms of Littlewood-Richardson numbers, for the homology of the unitary representations of the cyclotomic rational Cherednik algebra, and as a consequence, for the graded Betti numbers for the ideals of a class of subspace arrangements arising from the reflection arrangements of complex reflection groups., 41 pages

We prove an asymptotic version of a conjecture by Varagnolo and Vasserot on an equivalence between the category \({\mathcal {O}}\) for a cyclotomic rational Cherednik algebra and a suitable truncation of an affine parabolic category \({\mathcal {O}}\) that, in particular, implies Rouquier’s conjecture on the decomposition numbers in the former. Our proof uses two ingredients: an extension of Rouquier’s deformation approach as well as categorical actions on highest weight categories and related combinatorics.

We consider the construction of refined Chern–Simons torus knot invariants by M. Aganagic and S. Shakirov from the DAHA viewpoint of I. Cherednik. We give a proof of Cherednik's conjecture on the stabilization of superpolynomials, and then use the results of O. Schiffmann and E. Vasserot to relate knot invariants to the Hilbert scheme of points on C 2 . Then we use the methods of the second author to compute these invariants explicitly in the uncolored case. We also propose a conjecture relating these constructions to the rational Cherednik algebra, as in the work of the first author, A. Oblomkov, J. Rasmussen and V. Shende. Among the combinatorial consequences of this work is a statement of the m n shuffle conjecture.

We study holonomic D -modules on SL n ( C ) × C n , called mirabolic modules , analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of Verdier's specialization functor for representations of Cherednik algebras which agrees, on category O , with the restriction functor of Bezrukavnikov and Etingof. In type A , we also consider a Verdier specialization functor on mirabolic D -modules. We show that Hamiltonian reduction intertwines specialization functors on mirabolic D -modules with the corresponding functors on representations of the Cherednik algebra. This allows us to apply known Hodge-theoretic purity results for nearby cycles in the setting considered by Bezrukavnikov and Etingof.

We introduce parabolic degenerations of rational Cherednik algebras of complex reflection groups, and use them to give necessary conditions for finite-dimensionality of an irreducible lowest weight module for the rational Cherednik algebra of a complex reflection group, and for the existence of a non-zero map between two standard modules. The latter condition reproduces and enhances, in the case of the symmetric group, the combinatorics of cores and dominance order, and in general shows that the c-ordering on category O may be replaced by a much coarser ordering. The former gives a new proof of the classification of finite dimensional irreducible modules for the Cherednik algebra of the symmetric group., 35 pages

We study equivalences for category Op of the rational Cherednik algebras Hp of type G`(n) = (μ`) n o Sn: a highest weight equivalence between Op and Oσ(p) for σ ∈ S` and an action of S` on a non-empty Zariski open set of parameters p; a derived equivalence between Op and Op′ whenever p and p′ have integral difference; a highest weight equivalence between Op and a parabolic category O for the general linear group, under a non-rationality assumption on the parameter p. As a consequence, we confirm special cases of conjectures of Etingof and of Rouquier.

The Calogero-Moser families are partitions of the irreducible characters of a complex reflection group derived from the block structure of the corresponding restricted rational Cherednik algebra. It was conjectured by Martino in 2009 that the generic Calogero-Moser families coincide with the generic Rouquier families, which are derived from the corresponding Hecke algebra. This conjecture is already proven for the whole infinite series G(m,p,n) and for the exceptional group G4. A combination of theoretical facts with explicit computations enables us to determine the generic Calogero-Moser families for the nine exceptional groups G4, G5, G6, G8, G10, G23=H3, G24, G25, and G26. We show that the conjecture holds for all these groups - except surprisingly for the group G25, thus being the first and only-known counter-example so far., Comment: Accepted for publication in Algebras and Representation Theory. In V2: Rewritten introduction and some minor corrections (thanks to the reviewer!). 32 pages. Comments welcome

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

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 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.

We generalize Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the graded double affine Hecke algebra (DAHA) on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the degenerate graded DAHA on the cohomology of parabolic Hitchin fibers, providing the first step towards a global Springer theory.

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 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 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.

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 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

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.

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 .

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”.

We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

We prove a transcendence result of Hecke operator $T({\mathfrak{l}})$ over the ring generated by diamond operators over $\mathbb {Q}$ in a non-CM component of the cyclotomic ordinary big $p$ -adic Hecke algebra, and we discuss its conjectural implication of a modulo- $p$ version to the vanishing of the cyclotomic $\mu$ -invariant.

In the first part of this paper we study normal forms of elements of the imprimitive complex reflection group G(e,1,n). This allows to prove a conjecture of Broue on basis elements and the canonical symmetrizing form of the associated cyclotomic Hecke algebra. Secondly we introduce a root system for G(e,1,n) and study the associated length function. This has many properties in common with the usual length function for finite Weyl groups.

Let G be a finite group. To a set of subgroups of order two we associate a mod 2 Hecke algebra and construct a homomorphism, ψ, from its units to the class-group of Z[G]. We show that this homomorphism takes values in the subgroup, D(Z[G]). Alternative constructions of Chinburg invariants arising fromthe Galois module structure of higher-dimensional algebraic K-groups of rings of algebraic integers often differ by elements in the image of ψ. As an application we show that two such constructions coincide.

This paper gives a review of Cherednik’s results on the representation-theoretic approach to Macdonald polynomials and related special functions. Macdonald polynomials are a remarkable 2-parameter family of polynomials which can be associated to every root system. As special cases, they include the Schur functions, the q q -Jacobi polynomials, and certain spherical functions on real and p p -adic symmetric spaces. They have a number of elegant combinatorial properties, which, however, are extremely difficult to prove. In this paper we show that a natural setup for studying these polynomials is provided by the representation theory of Hecke algebras and show how this can be used to prove some of the combinatorial identities for Macdonald polynomials.

We compute Plancherel measures and formal degrees for unipotent representations of p -adic groups, using Hecke algebra isomorphisms. The results verify special cases of the conjectural uniformity of formal degrees in an L -packet.

We prove that the Hecke algebra H ( W , W J ) \mathcal {H}(W,{W_J}) , where W is a Weyl group of spherical type and W J {W_J} is a standard parabolic subgroup of W of corank ≥ 2 \geq 2 , is noncommutative.

Following the work of Kashiwara-Rouquier and Gan-Ginzburg, we define a family of exact functors from category $\mathcal O$ for the rational Cherednik algebra in type $A$ to representations of certain "coloured braid groups" and calculate the dimensions of the representations thus obtained from standard modules. To show that our constructions also make sense in a more general context, we also briefly study the case of the rational Cherednik algebra corresponding to complex reflection group $\mathbb Z/l\mathbb Z$., Comment: Revised to improve exposition, giving more details on the construction of the microlocal local systems and providing background information on twisted D-modules in an appendix

We prove that Macdonald polynomials are characters of irreducible Cherednik algebra modules., Comment: 6 pages; v2 logical content essentially the same, but main theorem is now explicitly an equivalence between the n! theorem and the character formula for irreducible H-modules

The purpose of this paper is to calculate the decomposition numbers for Hecke algebras of typeF4 (andC3) with unequal parameters. The problem is reduced to specifying decomposition maps of the generic Hecke algebras. The concept of Kazhdan-Lusztig polynomials and left cells serves to determine their irreducible representations. We also prove a result about the minimal ring over which the irreducible representations of the generic algebras can be realized.

The Dedekind eta-function is defined for any τ in the upper half-plane bywhere x = exp(2πiτ) and x1/24 = exp(2πiτ/24). By an eta-product we shall mean a functionwhere N ≥ 1 and eachrδ ∈ ℤ. In addition, we shall always assume that is an integer. Using the Legendre-Jacobi symbol (—), we define a Dirichlet character ∈ bywhen a is odd. If p is a prime for which ∈(p) ≠ 0and if F is a function with a Fourier seriesthen we define a Hecke operator Tp bywhereand