Variations of the Poincaré series for affine Weyl groups and q-analogues of Chebyshev polynomials

Abstract

Let (W,S) be a Coxeter system and write PW(q) for its Poincaré series. Lusztig has shown that the quotient PW(q2)/PW(q) is equal to a certain power series LW(q), defined by specializing one variable in the generating function recording the lengths and absolute lengths of the involutions in W. The simplest inductive method of proving this result for finite Coxeter groups suggests a natural bivariate generalization LWJ(s,q)∈Z[[s,q]] depending on a subset J⊂S. This new power series specializes to LW(q) when s=−1 and is given explicitly by a sum of rational functions over the involutions which are minimal length representatives of the double cosets of the parabolic subgroup WJ in W. When W is an affine Weyl group, we consider the renormalized power series TW(s,q)=LWJ(s,q)/LW(q) with J given by the generating set of the corresponding finite Weyl group. We show that when W is an affine Weyl group of type A, the power series TW(s,q) is actually a polynomial in s and q with nonnegative coefficients, which turns out to be a q-analogue recently studied by Cigler of the Chebyshev polynomials of the first kind, arising in a completely different context.