Coxeter systems, multiplicity free representations, and twisted Kazhdan-Lusztig Theory

Abstract

This thesis considers three topics related to the representations of Coxeter systems, their Hecke algebras, and related groups. The first topic concerns the construction of generalized involution models, as defined by Bump and Ginzburg. We compute the automorphism groups of all complex reflection groups G(r, p, n) and using this information, we classify precisely which complex reflection groups have generalized involution models. The second topic concerns the set of "unipotent characters" Uch(W) which Lusztig has attached to each finite, irreducible Coxeter system (W, S). We describe a precise sense in which the irreducible multiplicities of a certain W-representation can be used to define a function which serves naturally as a heuristic definition of the Frobenius-Schur indicator on Uch(W). The formula we obtain for this indicator extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which W is a Weyl group. Finally, we study a certain module of the Hecke algebra of a Coxeter system (W, S), spanned by the set of twisted involutions in W. Lusztig has shown that this module has two distinguished bases, and that the transition matrix between these bases defines interesting analogs of the much-studied Kazhdan-Lusztig polynomials of (W, S). We prove several positivity properties related to these polynomials for universal Coxeter systems, using combinatorial techniques, and for finite Coxeter systems, using computational methods.