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

Let (W,S)(W,S) be a Coxeter system and write PW(q)PW(q) for its Poincaré series. Lusztig has shown that the quotient PW(q2)/PW(q)PW(q2)/PW(q) is equal to a certain power series LW(q)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⊂SJ⊂S. This new power series specializes to LW(q)LW(q) when s=−1s=−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 WJWJ 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)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.