  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.
