In the early 1980s Lusztig defined a ring J that serves as a limit of an
affine Hecke algebra as the parameter v tends to 0, and showed that J
can be realized as a subalgebra of a completion of the affine Hecke
algebra, with elements of J being 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 of G. We will explain a possible
application to the representation theory of affine Hecke algebras at
root of unity.
The talk will be in BA 1200 and also will be available online. Email Joel for a zoom link.