Hochschild and cyclic homology of finite type algebras

Abstract

Abstract. We study Hochschild and cyclic homology of finite type algebras using abelian stratifications of their primitive ideal spectrum. Hochschild homology turns out to have a quite complicated behavior, but cyclic homology can be related directly to the singular cohomology of the strata. We also briefly discuss some connections with the representation theory of reductive p-adic groups.

Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

“…(See also [18] for similar results on the groups of real points of algebraic groups defined over R.) These periodic cyclic cohomology groups are seen to be isomorphic to K * (C * r (G))⊗C, by combining results from [1,16] to those of [12]. See also [15].…”

“…Fix now k, and let φ n : F k → C be 1 on the set 15) and vanishes outside this set. (Here q is the number of elements of the residual field of F, and the non-Archimedean norm "| |" is normalized such that its range is {0} ∪ {q…”

Hecke algebras and harmonic analysis

“…In the split case this is a theorem of Baum and Nistor [10]. The proof in [10] is very interesting and is based on Lusztig's asymptotic morphism in combination with techniques from [35]. For unequal label Hecke algebras one may connect this with Lusztig's conjectures from [43], see [4].…”

“…Under certain additional assumptions we in fact have a proof of the statement (in fact, Solleveld has recently announced a general proof (private communication)). This proof is based on the philosophy explained in the important paper [35] and the Langlands classification for general affine Hecke algebras [23].…”

The Hecke algebra of a reductive p-adic group: a geometric conjecture

“…This is a special case of the Feigin-Tsygan theorem; for a proof of this theorem which proceeds by reduction to the case of smooth varieties, see [25]. Let E σ be the maximal compact subgroup of the complex torus D σ , so that E σ is a compact torus.…”