Admissible Sequences for Twisted Involutions in Weyl Groups

Let W be a Weyl group, Sigma a set of simple reflections in W related to a basis Delta for the root system Phi associated with W and theta an involution such that theta(Delta) = Delta. We show that the set of theta-twisted involutions in W, J(theta) = {w is an element of W vertical bar theta(w) = w(-1)} is in one to one correspondence with the set of regular involutions J(Id). The elements of J(theta) are characterized by sequences in Sigma which induce an ordering called the Richardson-Springer Poset. In particular, for Phi irreducible, the ascending Richardson-Springer Poset of J(theta), for nontrivial theta is identical to the descending Richardson-Springer Poset of J(Id).