A Lie-theoretic construction of some representations of the degenerate affine and double affine Hecke algebras of type 𝐵𝐶_{𝑛}

Abstract

Abstract. Let G = GL(N ), K = GL(p) × GL(q), where p + q = N , and let n be a positive integer. We construct a functor from the category of HarishChandra modules for the pair (G, K) to the category of representations of the degenerate affine Hecke algebra of type B n , and a functor from the category of K-monodromic twisted D-modules on G/K to the category of representations of the degenerate double affine Hecke algebra of type BC n ; the second functor is an extension of the first one.

Representations of Gan–Ginzburg algebras

“…Remark 2.5 It is not hard to check that the algebra H t,k,c ((Z/mZ) n S n , C n ) can be obtained by rational degeneration from the (trigonometric) degenerate double affine Hecke algebra (dDAHA) H H(t, k 1 , k 2 , k 3 ) described in [4], Sect. 3.1.…”

“…3.1. Roughly speaking, the rational degeneration is obtained by setting X i = ehx i andỹ i =h y i , where X i and y i are the generators of H H(t, k 1 , k 2 , k 3 ) in [4], and by taking the limith → 0. The generatorsx i ,ỹ i of the rational degeneration are related to the generators x i , y i in Definition 2.3 by the formulas x i =ỹ i and y i =x i .…”

“…In [4], Etingof, Freund, and Ma generalize the construction of [1] from type A to type BC. Namely, the authors define a functor from the category of twisted D-modules on the symmetric space GL N /GL p × GL q (with p + q = N ) to the category of representation of a corresponding dDAHA of type BC n (see [4], Sect.…”

“…We thus get a functor F n,χ from the category of W α -modules to the category of representations of A n,λ,−1 (Q). When Q is the Jordan quiverà 0 (the quiver with one vertex and one loop), our construction reduces to the one of [1], while when Q is the cyclic quiver of length two (that is the quiver with two vertices and two arrows pointing in opposite directions) it reduces to the rational limit of the functor from [4]. Another interesting aspect of Gan-Ginzburg algebras is their relation with rational generalized double affine Hecke algebras (rational GDAHA).…”

“…In Sect. 3, we construct the functor F n,χ and compare it with the functors from [1] and [4] in the special case of the Cherednik algebra of type A n−1 and BC n , respectively. In Sect.…”