Hecke algebras and involutions in Weyl groups

(Py,w;i ∈ N, u is an indeterminate) were defined and computed in terms of an algorithm for any y ≤ w in W . These polynomials are of interest for the representation theory of complex reductive groups, see [6]. Let I = {w ∈ W ;w2 = 1} be the set of involutions in W . In this paper we introduce some new polynomials P σ y,w = ∑ i≥0 P σ y,w;iu i (P σ y,w;i ∈ Z) for any pair y ≤ w of elements of I. These new polynomials are of interest in the theory of unitary representations of complex reductive groups, see [1]; they are again computable in terms of an algorithm, see 4.5. For y ≤ w in I and i ∈ N there is the following relation between Py,w;i and P σ y,w;i: there exist ai, bi ∈ N such that Py,w;i = ai + bi, P σ y,w;i = ai − bi. Let A = Z[u, u−1] and let H be the free A-module with basis (Tw)w∈W with the unique A-algebra structure with unit T1 such that (i) TwTw′ = Tww′ if l(ww ) = l(w) + l(w) (l : W → N is the standard length function) and (Ts + 1)(Ts − u) = 0 for all s ∈ S. Let H be the A-algebra with the same underlying A-module as H but with multiplication defined by the rules (i) and