Asymptotic Hecke algebras and involutions

Abstract

In [11], a Hecke algebra module structure on a vector space spanned by the involutions in a Weyl group was defined and studied. In this paper this study is continued by relating it to the asymptotic Hecke algebra introduced in [6]. In particular we define a module over the asymptotic Hecke algebra which is spanned by the involutions in the Weyl group. We present a conjecture relating this module to equivariant vector bundles with respect to a group action on a finite set. This gives an explanation (not a proof) of a result of Kottwitz [3] in the case of classical Weyl groups, see 2.5. We also present a conjecture which realizes the module in [11] terms of an ideal in the Hecke algebra generated by a single element, see 3.4.