Traces on Iwahori-Hecke algebras and counting rational points

0.1. Let W be a Weyl group with length function w 7→ |w| and let H be the Iwahori-Hecke algebra over Q(q) (q is an indeterminate) attached to W . Recall that H is the Q(q)-vector space with basis {Tw;w ∈ W} with multiplication given by TwTw′ = Tww′ if w,w ′ ∈ W satisfy |ww| = |w|+ |w| and (Tt − q)(Tt + 1) = 0 if t ∈ W, |t| = 1. For w ∈ W let τ(w,q) ∈ Z[q] be the trace of the linear map H −→ H given by h 7→ Twh. The “trace polynomials” τ(w,q) appear in relation with counting Fq-rational points in certain algebraic varieties; three apparitions are in [L78],[L21],[L85], see 0.2(a), 0.3(a), 0.4(a); the fourth one is new, see 0.5. (We denote by Fq the subfield with q elements of an algebraic closure k of Fp; q is a power of a prime number p.)