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

TL;DR

This paper explicitly describes the basis functions of Lusztig's asymptotic Hecke algebra for SL_2, in terms of the Iwahori-Hecke algebra basis, and explores their properties and actions on Schwartz spaces.

Contribution

It provides explicit formulas connecting the bases of the asymptotic Hecke algebra and the Iwahori-Hecke algebra for SL_2, and investigates their properties and actions.

Findings

Explicit formulas for basis elements $t_w$ in terms of $T_w$

Proof that $J$ acts on the Schwartz space of SL_2's basic affine space

Conjectures on properties of these expansions for general groups

Abstract

Let $H$ be the Iwahori-Hecke algebra and let $J$ be Lusztig's asymptotic Hecke algebra, both specialized to type $\tilde{A}_1$. For $\mathrm{SL}_2$, when the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that a completion of $H$ has codimension two as a subalgebra of a completion of $J$, and described a basis for the quotient in spectral terms. In this note we write these functions explicitly in terms of the basis $\{t_w\}$ of $J$, and further invert the canonical isomorphism between the completions of $H$ and $J$, obtaining explicit formulas for the each basis element $t_w$ in terms of the basis $\{T_w\}$ of $H$. We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that $J$ acts on the Schwartz space of the basic affine space of $\mathrm{SL}_2$, and produce some formulas for this action.