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On Lusztig's 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
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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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Stefan Dawydiak
Affiliation: Department of Mathematics, University of Toronto, Toronto, Ontario M5S 2E4 Canada
Email: stefand@math.utoronto.ca

DOI: https://doi.org/10.1090/proc/15259
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