How to compute the Frobenius–Schur indicator of a unipotent character of a finite Coxeter system

Abstract

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 Broué, Lusztig, and Malle in the remaining cases. This paper concerns a certain W -representation ̺ W in the vector space generated by the involutions of W . Our main result is to show that the irreducible multiplicities of ̺ W are given by the Fourier transform of a unique function ǫ : 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 ǫ 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 ̺ W derived by Kottwitz when (W, S) is classical, and prove that ̺ W defines a Gelfand model if and only if (W, S) has type A n , H 3 , or I 2 (m) with m odd. We show finally that a conjecture of Kottwitz connecting the decomposition of ̺ 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.

Positivity conjectures for Kazhdan-Lusztig theory on twisted involutions: the universal case

“…We also mention that when W is finite, the irreducible decomposition of M q 2 has a surprising interpretation in terms of the "Fourier transform" of a set of "unipotent characters" attached to (W, S). This phenomenon, which is studied in various cases in the articles [3,7,15,20,23], gives one more indication that M q 2 deserves consideration not only in the crystallographic case.…”

Frobenius–Schur indicators of unipotent characters and the twisted involution module

“…By an argument similar to that in the proof of Proposition 2.3, we obtain: Let us write ρ for the character of the representation R, and similarlyρ for that of R. We are interested in the decomposition ofρ as a class function on the coset W.σ. In the case that σ = id, this decomposition was obtained by Kottwitz [6], Casselman [1] for Weyl groups and by Marberg [17] for the non-crystallographic Coxeter groups. Here, we discuss the case when σ = id.…”

“…Finally, the construction ofρ also works for the case where W is any dihedral group and σ = id. Thus, inspired by Marberg [17], our computations also allow us to formally define Frobenius-Schur indicators for "unipotent characters" of twisted dihedral groups; see Theorem 5.4.…”

“…3. Finally, as in [17], we now formally define Frobenius-Schur indicators for the combinatorially introduced unipotent characters of dihedral groups with non-trivial automorphism to obtain an analogue of Theorem 5.1 in this case and show a unicity statement. For this recall that there is a way to attach a set Uch( 2 I 2 (m)) of combinatorial objects, called "unipotent characters", to the dihedral groups I 2 (m) (m ≥ 3) with non-trivial Coxeter automorphism of order 2 (see [12]), such that each χ ∈ Uch( 2 I 2 (m)) has a degree χ(1) ∈ C[q], and a Frobenius eigenvalue Fr(χ) (a root of unity).…”

“…For this recall that there is a way to attach a set Uch( 2 I 2 (m)) of combinatorial objects, called "unipotent characters", to the dihedral groups I 2 (m) (m ≥ 3) with non-trivial Coxeter automorphism of order 2 (see [12]), such that each χ ∈ Uch( 2 I 2 (m)) has a degree χ(1) ∈ C[q], and a Frobenius eigenvalue Fr(χ) (a root of unity). Following [17] in the untwisted case, we propose to introduce Frobenius-Schur indicators ν(χ) satisfying the following properties:…”