Fully commutative elements in the Weyl and affine Weyl groups

Let W be a Weyl or an affine Weyl group and let Wc be the set of fully commutative elements in W. We associate each w Wc to a digraph G(w). By using G(w), we give a graph-theoretic description for Lusztigʹs a-function on Wc and describe explicitly all the distinguished involutions of Wc. The results verify two conjectures in our case: one was proposed by myself in [Adv. Sci. China Math. 3 (1990) 79–98, Conjecture 8.10] and the other by Lusztig in [T. Asai et al., Open problems in algebraic groups, in: R. Hotta (Ed.), Problems from the conference on Algebraic Groups and Representations, Katata, August 29–September 3, 1983].