Continuous Hecke algebras

Abstract

Table of Contents 1. Introduction 2. Continuous Hecke algebras 3. Continuous symplectic reflection algebras and Cherednik algebras 4. Infinitesimal Hecke algebras 5. Representation theory of continuous Cherednik algebras 6. Case of wreath-productsThe theory of PBW properties of quadratic algebras, to which this paper aims to be a modest contribution, originates from the pioneering work of Drinfeld (see [Dr1]). In particular, as we learned after publication of [EG] (to the embarrassment of two of us!), symplectic reflection algebras, as well as PBW theorems for them, were discovered by Drinfeld in the classical paper [Dr2] 15 years before [EG] (namely, they are a special case of degenerate affine Hecke algebras for a finite group G introduced in [Dr2], Section 4).It is our great pleasure to dedicate this paper to Vladimir Drinfeld on the occasion of his 50-th Birthday. Continuous Hecke algebras2.1 Algebraic distributions. Let X be an affine scheme of finite type over C. We shall denote by O(X) the algebra of regular functions on X. An algebraic distribution on X is an element in the dual space O(X) * of O(X). For c ∈ O(X) * and f ∈ O(X), we will denote the value c(f ) by (c, f ).The space O(X) * is naturally equipped with the weak (inverse limit) topology. Note also that O(X) * is a module over O(X): for any f ∈ O(X) and µ ∈ O(X) * we can define the elementLet Z be a closed subscheme of X, and write I(Z) for its defining ideal in O(X). We say that an algebraic distribution µ on X is supported on the scheme Z if µ annihilates I(Z). Clearly, the space of algebraic distributions on X supported on Z is naturally isomorphic to the space of algebraic distributions on Z.Now assume that Z is reduced. We say that µ ∈ O(X) * is scheme-theoretically (respectively, set-theoretically) supported on the set Z if µ annihilates I(Z) (respectively, some power of I(Z)).Example 2.1. For each point a ∈ X, the delta function δ a ∈ O(X) * is defined by δ a (f ) := f (a) where f ∈ O(X). It is scheme-theoretically supported at the point a, and its derivatives are set-theoretically supported at this point.Let G be a reductive algebraic group. Since O(G) is a coalgebra, its dual space O(G) * is an algebra under convolution. The unit of this algebra is the delta function δ 1 of the identity element 1 ∈ G.Note that a continuous representation of the algebra O(G) * is the same thing as a locally finite G-module (i.e. a G-module which is a direct sum of finite dimensional algebraic representations of G).Suppose that G acts on X. Then G acts also on O(X) and O(X) * . We have O(X) = V M V ⊗V , where V runs over the irreducible representations of G, and M V are multiplicity spaces. Thus, O(X) * = V M * V ⊗V * . In particular, O(G) * = V V ⊗V * as a G × G-module.

Quantized symplectic oscillator algebras of rank one

“…However, it is not true in general when dim V > 2. The formula for obtaining PBW deformations of C[V ] U(sp(V )) is given in [8,Theorem 4.2]. For the rest of this paper, H f will always mean the case dim V = 2.…”

“…In [8,Section 4], Etingof, Gan and Ginzburg introduced the family of infinitesimal Hecke algebras H β associated to sp(V ). The algebras H β are PBW deformations of C[V ] U(sp(V )).…”

Center and Representations of Infinitesimal Hecke Algebras of 𝔰𝔩2

“…(Note that k is algebraically closed, of characteristic zero.) As mentioned in [EGG,§4.1], the infinitesimal Hecke algebra H β only exists when im(κ) ⊂ Ug; thus, f xy · c ∈ Ug for all f xy := (y, (1 − g)x) ∈ O(G) (with x ∈ h * , y ∈ h). By the Nullstellensatz, c ∈ Ug, so h ∈ i v i v * i + Ug now; therefore r β is indeed a lift of r n to H β .…”

“…Next, for H β with β at most linear, we refer to [EGG,Examples 4.6,4.7]; thus, H β is the quotient of the above algebra, by the relations…”

Generalized nil-Coxeter algebras, cocommutative algebras, and the PBW property

“…These algebras serve as "noncommutative" coordinate rings of the orbifolds V /G; see [32] for a related setting. Subsequently, Etingof, Ginzburg, and Gan replaced the group by algebraic distributions of a reductive Lie group G. This led to the study of infinitesimal Hecke algebras in [13] (and several recent papers), where U g acts on Sym(V ), with g = Lie(G). These families of deformed algebras continue to be popular and important objects of study, with connections to representation theory, combinatorics, and mathematical physics.…”