  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.
