A generating function for the trace of the Iwahori-Hecke algebra

Abstract

The Iwahori-Hecke algebra has a ``natural'' trace $\tau$. This trace is the evaluation at the identity element in the usual interpretation of the Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic semi-simple group. The Iwahori-Hecke algebra contains an important commutative sub-algebra $\mathbf{C}[{\theta }_x]$, that was described and studied by Bernstein, Zelevinski and Lusztig. In this note we compute the generating function for the value of $\tau$ on the basis ${\theta }_x$.