Homology of graded Hecke algebras

Abstract

Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W . We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H-modules with real central character forms a Q-basis of the representation ring of W .Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety.

Graded Hecke Algebras for Disconnected Reductive Groups

“…We obtain a bijection between G • -conjugacy classes of triples (y, σ 0 , ρ • ) and W • L -association classes of pairs (σ 0 , π) with σ 0 ∈ t and (π, V π ) ∈ Irr((W • L ) σ 0 ). It is well-known, see for example [Sol2,Theorem 1.1], that the latter set in a bijection with Irr(W • L ×S(t * )) via…”

Topological K-theory of affine Hecke algebras

“…We consider the Grothendieck group R Z (H) of finite length modules of a graded Hecke algebra H with parameters k. We show that it is the direct sum of the subgroup spanned by modules induced from proper parabolic subalgebras and an elliptic part R Z (H). We prove that R Z (H) is isomorphic to the elliptic part of the representation ring of the Weyl group associated to H. By Paragraph 1.1, R Z (H) is free abelian and does not depend on the parameters k. The main ingredients are the author's work [Sol3] on the periodic cyclic homology of graded Hecke algebras, and the study of discrete series representations by Ciubotaru,Opdam and Trapa [CiOp2,COT].…”

“…We can also regard it as the composition of representations with the algebra homomorphism (18) for ǫ = 0, then its image consists of O(t) ⋊ W -representations on which O(t) acts via 0 ∈ t. Let Irr 0 (H) be the set of irreducible tempered H(R, k)-modules with central character in a/W . It is known from [Sol3,Theorem 6.5] that, for real-valued k, r induces a bijection (20)…”

Parabolically Induced Representations of Graded Hecke Algebras

“…Since H is a deformation of S(t * ) W, there should be a relation between the spectra of these two algebras. An appropriate theory to make this precise is periodic cyclic homology H P * , since there is an isomorphism H P * (H) ∼ = H P * (S(t * ) W) [26]. As the periodic cyclic homology of an algebra can be regarded as a kind of cohomology of the spectrum of that algebra, we would like to understand the geometry of Irr(H) better.…”

“…Theorem 4 and part of the above are worked out in the sequel to this paper [26]. Basically the author divided the material between these two papers such that all the required representation theory is in this one, while the homological algebra and cohomological computations are in [26].…”