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

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.

15 pages. In v4, updated to match published version

References in corpus (2)

• Hecke algebras with unequal parameters
• Remarks on the asymptotic Hecke algebra

Cited by in corpus (1)

• Kazhdan-Laumon Category O, Braverman-Kazhdan Schwartz space, and the semi-infinite flag variety