Variations of the Poincar\'e series for affine Weyl groups and q-analogues of Chebyshev polynomials

Let $(W,S)$ be a Coxeter system and write $P_W(q)$ for its Poincar\'e series. Lusztig has shown that the quotient $P_W(q^2)/P_W(q)$ is equal to a certain power series $L_{W}(q)$, 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 $L^J_W(s,q) \in \mathbb{Z}[[s,q]]$ depending on a subset $J\subset S$. This new power series specializes to $L_W(q)$ 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 $W_J$ in $W$. When $ W$ is an affine Weyl group, we consider the renormalized power series $T_{ W}(s,q) = L^J_W(s,q) / L_W(q)$ 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 $T_W(s,q)$ 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.