Remarks on the asymptotic Hecke algebra

Abstract

Let $G$ be a split reductive $p$-adic group. Let $\mathcal{H}(G)$ be its Hecke algebra and let $\mathcal{C}(G)\supset \mathcal{H}(G)$ be the Harish-Chandra Schwartz algebra. The purpose of this note is to give a spectral interpretation of Lusztig's asymptotic Hecke algebra $J$ (which contains the Iwahori part of $\mathcal{H}(G)$ as a subalgebra), which shows that $J$ is a subalgebra of $\mathcal{C}(G)$. This spectral description also allows to define a version of $J$ beyond the Iwahori component - i.e. we define certain subalgebra $\mathcal{J}(G)$ of $\mathcal{C}(G)$ which contains $\mathcal{H}(G)$. We explain a relation between $\mathcal{J}(G)$ and the Schwartz space of the basic affine space studied by us about 20 years ago.

10 pages; an appendix with an explicit SL(2)-calculation has been added

Cited by in corpus (1)

• On Lusztig's asymptotic Hecke algebra for ${\mathrm{SL}}_2$