Denominators in the asymptotic Hecke algebra and parahoric-fixed vectors

Contact: Peter Scholze (scholze@mpim-bonn.mpg.de)

The Tits deformation theorem says that deformations of group algebras of
finite groups are trivial away from finitely-many points. In the 1980s,
Lusztig introduced a based ring J, the asymptotic Hecke algebra, that 
implements this theorem in the case of finite Weyl groups, and serves as
a replacement for it in the case of affine Weyl groups. In the latter
case it means that the J ring implements the independence with respect
to q of the representation theory of the affine Hecke algebra, and that
its elements are infinite linear combinations of elements of the affine
Hecke algebra with Laurent series coefficients. Using work of
Braverman-Kazhdan and the Plancherel formula for p-adic groups, we prove
that the above coefficients are in fact rational functions whose
denominators all divide a fixed polynomial that depends only on the
affine Weyl group. We conjecture, and prove in the case that the p-adic
group is GL_n, that this polynomial is precisely the Poincare polynomial
of the finite Weyl group.

We will explain two applications: to the representation theory of affine
Hecke algebras at roots of unity (in progress), and a criterion for
tempered representations to contain vectors fixed under a given
parahoric subgroup of G.