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

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.