Involutions in Weyl groups and nil-Hecke algebras

Abstract

0.1. Let W be a Coxeter group and let S be the set of simple reflections of W ; we assume that S is finite. Let w 7→ |w| be the length function on W . Let H be the Iwahori-Hecke algebra attached to W . Recall that H is the free Z[u]-module with basis {Tw;w ∈ W} (u is an indeterminate) with (associative) multiplication characterized by TsTw = Tsw if s ∈ S, w ∈ W, |sw| = |w| + 1, TsTw = u 2 Tsw + (u 2 − 1)Tw if s ∈ S, w ∈ W, |sw| = |w| − 1. Let w 7→ w be an automorphism with square 1 of W preserving S and let I∗ = {w ∈ W ;w ∗ = w} be the set of “twisted involutions” in W . Let M be the free Z[u]-module with basis {ax; x ∈ I∗}. For any s ∈ S we define a Z[u]-linear map Ts : M −→ M by Tsax = uax + (u+ 1)asx if x ∈ I∗, sx = xs , |sx| = |x|+ 1, Tsax = (u 2 − u− 1)ax + (u 2 − u)asx if x ∈ I∗, sx = xs , |sx| = |x| − 1, Tsax = asxs∗ if x ∈ I∗, sx 6= xs , |sx| = |x|+ 1, Tsax = (u 2 − 1)ax + u 2 asxs∗ if x ∈ I∗, sx 6= xs , |sx| = |x| − 1. It is known that the maps Ts define an H-module structure on M . (See [LV12] for the case where W is a Weyl group or an affine Weyl group and [L12] for the general case; the case where W is a Weyl group and u is specialized to 1 was considered earlier in [K00].) When u is specialized to 0, H becomes the free Zmodule H0 with basis {Tw;w ∈ W} with (associative) multiplication characterized by TsTw = Tsw if s ∈ S, w ∈ W, |sw| = |w|+ 1, TsTw = −Tw if s ∈ S, w ∈ W, |sw| = |w| − 1 (a nil-Hecke algebra). From these formulas we see that there is a well defined monoid structure w,w 7→ w • w on W such that for any w,w in W we have TwTw′ = (−1) ′′Tw•w′ (equality in H0). In this monoid we have (a) (w • w) = w • w