An introduction to the combinatorics of twisted involutions (1)

Let (W,S) be a Coxeter system and let * be an involution of the corresponding Coxeter diagram, that is, a self-inverse automorphism of W which preserves the set of simple generators S. Many fundamental properties of elements in a Coxeter group, such as the notion of a reduced expression, the length function, and the exchange and deletion principles have interesting, nontrivial analogues for just the subset of *-twisted involutions in W, by which we mean the elements w in W with w^* = w^{-1}. The first part of this mini-course will give an introduction to such combinatorial properties of twisted involutions, as developed by Hultman, Richardson, Springer, and others. As motivation, we will review along the way some of the applications of this theory to the study of symmetric varieties. The remainder of the course will be a survey of recent progress and open questions related to some problems in combinatorics and representation theory in which the twisted involutions of a Coxeter group play a central role. Topics will include among the following, as time allows: Lusztig and Vogan's recent study of the "twisted" analogue of the regular representation of an Iwahori-Hecke algebra; Rains and Vazinari's theory of "quasiparabolic sets," of which conjugacy classes of twisted involutions serve as important motivating examples; connections between certain variants of the Poincaré series defined for twisted involutions and q-analogues of orthogonal polynomials as studied by Cigler and others; and reduced expression counting problems for twisted involutions. The course should be accessible to graduate students with some prior exposure to the study of Coxeter groups, or at least with a little knowledge of finite group theory.