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.

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