On Lusztigโs asymptotic Hecke algebra for $\mathrm {SL}_2$
Author:
Stefan Dawydiak
Journal:
Proc. Amer. Math. Soc. 149 (2021), 71-88
MSC (2020):
Primary 20C08; Secondary 22E50
DOI:
https://doi.org/10.1090/proc/15259
Published electronically:
October 20, 2020
MathSciNet review:
4172587
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Abstract | References | Similar Articles | Additional Information
Abstract: Let $G$ be a split connected reductive algebraic group, let $H$ be the corresponding affine Hecke algebra, and let $J$ be the corresponding asymptotic Hecke algebra in the sense of Lusztig. When $G=\mathrm {SL}_2$, and the parameter $q$ is specialized to a prime power, Braverman and Kazhdan showed recently that for generic values of $q$, $H$ has codimension two as a subalgebra 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 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.
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Additional Information
Stefan Dawydiak
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4 Canada
Email:
stefand@math.utoronto.ca
Received by editor(s):
September 4, 2019
Received by editor(s) in revised form:
April 29, 2020, and May 1, 2020
Published electronically:
October 20, 2020
Communicated by:
Alexander Braverman
Article copyright:
© Copyright 2020
American Mathematical Society