Speaker
Description
Affine Hecke algebras play a prominent role in the representation theory of a $p$-adic group $G$, with the principal Bernstein block being equivalent to modules over the Iwahori-Hecke algebra. Braverman-Kazhdan proposed Lusztig's asymptotic Hecke algebra $J$ as means to an algebraic version of tempered representations for the principal block. In particular, elements of $J$ are certain rapidly-decaying functions on $G(F)$.
I will explain positivity properties, as conjectured by Braverman-Kazhdan, for two bases of the best understood part of the ring $J$, in terms of K-theory of the flag variety of $\hat{G}$ and spherical Kazhdan-Lusztig polynomials for $G$, and report on joint work
in progress with Bezrukavnikov to give an asymptotic characterization of these functions in terms of the wonderful compactification of $G$. Time permitting, I will explain partial results on positivity properties for two-sided cells corresponding to Levi subgroups of the general linear group with one block size.