We investigate the generalized involution models of the projective reflection groups G(r, p, q, n). This family of groups parametrizes all quotients of the complex reflection groups G(r, p, n) by scalar subgroups. Our classification is ultimately incomplete, but we provide several necessary and sufficient conditions for generalized involution models to exist in various cases. In the process we solve several intermediate problems concerning the structure of projective reflection groups. We derive a simple criterion for determining whether two groups G(r, p, q, n) and G(r, p′, q′, n) are isomorphic. We also describe explicitly the form of all automorphisms of G(r, p, q, n), outside a finite list of exceptional cases. Building on prior work, this allows us to prove that G(r, p, 1, n) has a generalized involution model if and only if G(r, p, 1, n) ≌ G(r, 1, p, n). We also classify which groups G(r, p, q, n) have generalized involution models when n = 2, or q is odd, or n is odd.

For each finite, irreducible Coxeter system (W, S), Lusztig has associated a set of "unipotent characters" Uch(W). There is also a notion of a "Fourier transform" on the space of functions Uch(W) -> R, due to Lusztig for Weyl groups and to Broue Lusztig, and Malle in the remaining cases. This paper concerns a certain W-representation rho(W) in the vector space generated by the involutions of W. Our main result is to show that the irreducible multiplicities of rho(W) are given by the Fourier transform of a unique function epsilon : Uch(W) -> {-1, 0, 1}, which for various reasons serves naturally as a heuristic definition of the Frobenius-Schur indicator on Uch(W). The formula we obtain for epsilon extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which W is a Weyl group. We include in addition a succinct description of the irreducible decomposition of rho(W) derived by Kottwitz when (W, S) is classical, and prove that rho(W) defines a Gelfand model if and only if (W, S) has type A(n), H-3, or I-2(m) with in odd. We show finally that a conjecture of Kottwitz connecting the decomposition of rho(W) to the left cells of W holds in all non-crystallographic types, and observe that a weaker form of Kottwitz's conjecture holds in general. In giving these results, we carefully survey the construction and notable properties of the set Uch(W) and its attached Fourier transform. (C) 2013 Elsevier Inc. All rights reserved.

It is well known that the representation theory of the finite group of unipotent upper-triangular matrices U-n over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies Diaconis-Isaacs' concept of superinduction in pattern groups. While Superinduction shares many desirable properties with usual induction, it no longer takes characters to characters. We begin by finding Sufficient conditions guaranteeing that Superinduction is in fact induction. It turns out for two natural embeddings of U-m in U-n superinduction is induction. We conclude with all explicit combinatorial algorithm for computing this induction analogous to the Pieri-formulas for the symmetric group. (c) 2009 Elsevier Inc. All rights reserved.