On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂

Abstract

Let G G be a split connected reductive algebraic group, let H H be the corresponding affine Hecke algebra, and let J J be the corresponding asymptotic Hecke algebra in the sense of Lusztig. When G = S L 2 G=\mathrm {SL}_2 , and the parameter q q is specialized to a prime power, Braverman and Kazhdan showed recently that for generic values of q q , H H has codimension two as a subalgebra of J 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 } \{t_w\} of J J , and further invert the canonical isomorphism between the completions of H H and J J , obtaining explicit formulas for each basis element t w t_w in terms of the basis { T w } \{T_w\} of H H . We conjecture some properties of this expansion for more general groups. We conclude by using our formulas to prove that J J acts on the Schwartz space of the basic affine space of S L 2 \mathrm {SL}_2 , and produce some formulas for this action.