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.
Categories: math.RT