Frobenius-Schur indicators of unipotent characters and the twisted involution module. (Q2836481)

Let \(W\) be a finite Weyl group with \(\sigma\) a graph automorphism of \(W\). \textit{G. Lusztig} and \textit{D. A. Vogan}, jun., [Bull. Inst. Math., Acad. Sin. (N.S.) 7, No. 3, 323-354 (2012; Zbl 1288.20006)], established a connection between the involution module for \(W\) (in the sense of \textit{R. E. Kottwitz}, [see Represent. Theory 4, 1-15 (2000; Zbl 1045.22500)]) and the Frobenius-Schur indicators of the unipotent characters of a corresponding finite group of Lie type in the case where \(\sigma\) is the identity map of \(W\).NEWLINENEWLINE In the paper under review, the authors extend the result to the situation where \(\sigma\) is a non-trivial graph automorphism: they establish a relation between the \(\sigma\)-twisted involution module for \(W\) and the Frobenius-Schur indicators of the unipotent characters of a corresponding twisted finite group of Lie type. The authors also extend a result of \textit{E. Marberg} [Adv. Math. 240, 484-519 (2013; Zbl 1283.20046)] by formally defining Frobenius-Schur indicators for ``unipotent characters'' of twisted dihedral groups.