We define Lie subalgebras of the group algebra of a finite pseudo-reflection group that are involved in the definition of the Cherednik KZ-systems, and determine their structure. We provide applications for computing the Zariski closure of the image of generalized (pure) braid group B inside the representations of the corresponding Hecke algebras. We also get unitarizability results for the representations of B originating from Hecke algebras for suitable parameters, and relate our Lie algebras with the topological closure of B in these compact cases.
In this article we classify all infinitesimal Hecke algebras of so_N. We establish isomorphism of their universal versions and the W-algebras of so_{N+2m+1} with a 1-block nilpotent element of the Jordan type (1,...,1,2m+1). This should be considered as a continuation of the recent joint paper with I. Losev, where the analogous results were obtained for the cases of gl_n and sp_{2n}.
We refine the infinitesimal Hecke algebra associated to a 2-reflection group into a $\Z/2\Z$-graded Lie algebra, as a first step towards a global understanding of a natural $\mathbbm{N}$-graded object. We provide an interpretation of this Lie algebra in terms of the image of the braid group into the Hecke algebra, in connection with unitary structures. This connection is particularly strong when $W$ is a Coxeter group, and when in addition we can use generalizations of Drinfeld's rational even associators. Finally, in the Coxeter ADE case, we provide a generating set of the even part of this L…
For W a finite (2-)reflection group and B its (generalized) braid group, we determine the Zariski closure of the image of B inside the corresponding Iwahori-Hecke algebra. The Lie algebra of this closure is reductive and generated in the group algebra of W by the reflections of W. We determine its decomposition in simple factors. In case W is a Coxeter group, we prove that the representations involved are unitarizable when the parameters of the representations have modulus 1 and are close to 1. We consequently determine the topological closure in this case.
In this paper we consider an infinitesimal Hecke algebra of $sl_2$ in positive characteristic. We show that it is a finitely generated module over its center, and the smooth and the Azumaya loci of its center coincide.
In this paper, we compute the center of the infinitesimal Hecke algebras Hz associated to sl_2 ; then using nontriviality of the center, we study representations of these algebras in the framework of the BGG category O. We also discuss central elements in infinitesimal Hecke algebras over gl(n) and sp(2n) for all n. We end by proving an analogue of the theorem of Duflo for Hz.
We relate completions of infinitesimal Hecke algebras of sl2 to noncommutative deformations of Kleinian singularities of type D of Crawley-Boevey and Holland. As a consequence, we show an analogue of the inequality of Bernstein and simplicity of generic maximal primitive quotients of these algebras. We also establish Skryabin type equivalence for these algebras.
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.
Hopf-Hecke algebras and Barbasch-Sahi algebras were defined by the first named author (2016) in order to provide a general framework for the study of Dirac cohomology. The aim of this paper is to explore new examples of these definitions and to contribute to their classification. Hopf-Hecke algebras are distinguished by an orthogonality condition and a PBW property. The PBW property for algebras such as the ones considered here has been of great interest in the literature and we extend this discussion by further results on the classification of such deformations and by a class of hitherto unex…
The notion of trigonometric spin double affine Hecke algebras (tsDaHa) and trigonometric double affine Hecke-Clifford algebras (tDaHCa) associated to classical Weyl groups are introduced. The PBW basis property is established. An algebra isomorphism relating tDaHCa to tsDaHa is obtained.
We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with real infinitesimal character) into the corresponding p-adic sphe…
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 $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \ci…
Let $\CC^0_{\g}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'(\g)$ and let $R^{A_\infty}\gmod$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty}$. In this paper, we investigate the relationship between the categories $\CC^0_{A_{N-1}^{(1)}}$ and $\CC^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur-Weyl duality functors $\F^{(t)}$ from $R^{A_\infty}\gmod$ to $\CC^0_{A_{N-1}^{(t)}}$ $(t=1,2)$.
We show that Lusztig's homomorphism from an affine Hecke algebra to the direct summand of its asymptotic Hecke algebra corresponding to the lowest two-sided cell is related to the homomorphism constructed by Chriss and Ginzburg using equivariant K-theory by a matrix over the representation ring of the associated algebraic group.
We prove a weak version of Lusztig's conjecture on explicit description of the asymptotic Hecke algebras (both finite and affine), and explain its relation to Lusztig's classification of character sheaves.
We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the cate…
In this paper, we show that the elliptic cocenter of the Hecke algebra of a connected reductive group over a nonarchimedean local field is contained in the rigid cocenter. As applications, we prove the trace Paley-Wiener theorem and the abstract Selberg principle for mod-$l$ representations.
The purpose of this note is to study the Maulik-Okounkov $K-$theoretic stable basis for the Hilbert scheme of points on the plane, which depends on a "slope" $m \in \mathbb{R}$. When $m = \frac ab$ is rational, we study the change of stable matrix from slope $m-\varepsilon$ to $m+\varepsilon$ for small $\varepsilon>0$, and conjecture that it is related to the Leclerc-Thibon conjugation in the $q-$Fock space for $U_q\widehat{\mathfrak{gl}}_b$. This is part of a wide framework of connections involving derived categories of quantized Hilbert schemes, modules for rational Cherednik algebras and He…
The main goal of this paper is to show that a wide variety of infinite-dimensional algebras all share a common structure, including a triangular decomposition and a theory of weights. This structure allows us to define and study the BGG Category O, generalizing previous definitions of it. Having presented our axiomatic framework, we present sufficient conditions that guarantee finite length, enough projectives, and a block decomposition into highest weight categories. The framework is strictly more general than the usual theory of O; this is needed to accommodate (quantized or higher rank) inf…
We study the $p$-adic loop group Iwahori-Hecke algebra $\mathcal{H}(G^+,I)$ constructed by Braverman, Kazhdan, and Patnaik and give positive answers to two of their conjectures. First, we algebraically develop the "double coset basis" of $\mathcal{H}(G^+,I)$ given by indicator functions of double cosets. We prove a generalization of the Iwahori-Matsumoto formula, and as a consequence, we prove that the structure coefficients of the double coset basis are polynomials in the order of the residue field. The basis is naturally indexed by a semi-group $\mathcal{W}_\mathcal{T}$ on which Braverman, K…
We use category theory to propose a unified approach to the Schur-Weyl dualities involving the general linear Lie algebras, their polynomial extensions and associated quantum deformations. We define multiplicative sequences of algebras exemplified by the sequence of group algebras of the symmetric groups and use them to introduce associated monoidal categories. Universal properties of these categories lead to uniform constructions of the Drinfeld functor connecting representation theories of the degenerate affine Hecke algebras and the Yangians and of its q-analogue. Moreover, we construct act…
We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $Γ$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $Γ$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$. We partition $X$ according to the path-length distance to both $x$ and $C$. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a $\mathbb{C}$-vector space ${\bf W}$. The universal d…
We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type A III/IV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the …
We introduce loop spaces (in the sense of derived algebraic geometry) into the representation theory of reductive groups. In particular, we apply the theory developed in our previous paper arXiv:1002.3636 to flag varieties, and obtain new insights into fundamental categories in representation theory. First, we show that one can recover finite Hecke categories (realized by D-modules on flag varieties) from affine Hecke categories (realized by coherent sheaves on Steinberg varieties) via S^1-equivariant localization. Similarly, one can recover D-modules on the nilpotent cone from coherent sheave…
We propose to use interpolation categories to study PBW deformations, and demonstrate this idea for the orthosymplectic supergroups. Employing a combinatorial calculus based on pseudographs and partitions which we derive from a suitable Jacobi identity, we classify PBW deformations in (quotients of) Deligne's interpolation categories for the orthosymplectic groups. As special cases, our classification recovers families of infinitesimal Hecke algebras found by Etingof-Gan-Ginzburg (2005) for the symplectic groups and by Tsymbaliuk (2015) for the orthogonal groups together with their respective …
We introduce a general axiomatic framework for algebras with triangular decomposition, which allows for a systematic study of the Bernstein-Gelfand-Gelfand Category $\mathcal{O}$. The framework is stated via three relatively simple axioms; algebras satisfying them are termed "regular triangular algebras (RTAs)". These encompass a large class of algebras in the literature, including (a) generalized Weyl algebras, (b) symmetrizable Kac-Moody Lie algebras $\mathfrak{g}$, (c) quantum groups $U_q(\mathfrak{g})$ over "lattices with possible torsion", (d) infinitesimal Hecke algebras, (e) higher rank…
