Remarks on the asymptotic Hecke algebra

TL;DR

This paper provides a spectral interpretation of Lusztig's asymptotic Hecke algebra for split reductive p-adic groups, showing it as a subalgebra of the Harish-Chandra Schwartz algebra and extending its definition beyond the Iwahori part.

Contribution

It introduces a spectral description of the asymptotic Hecke algebra and defines a new subalgebra within the Schwartz algebra, linking it to the basic affine space.

Findings

J is a subalgebra of the Schwartz algebra C(G)

A new subalgebra J(G) of C(G) is defined, extending J beyond the Iwahori component

The relation between J(G) and the Schwartz space of the basic affine space is established

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.