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

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• 作者：Eric Marberg ^{emarberg@math.mit.edu}
• 关键词：Unipotent character ; Coxeter system ; Frobenius&ndash ; Schur indicator ; Fourier transform ; Finite reductive group ; Fusion datum ; Left cell
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：20 June, 2013
• 年：2013
• 卷：240
• 期：Complete
• 页码：484-519
• 全文大小：421 K

文摘

For each finite, irreducible Coxeter system , Lusztig has associated a set of ¡°unipotent characters¡± . There is also a notion of a ¡°Fourier transform¡± on the space of functions , due to Lusztig for Weyl groups and to Brou¨¦, Lusztig, and Malle in the remaining cases. This paper concerns a certain -representation in the vector space generated by the involutions of . Our main result is to show that the irreducible multiplicities of are given by the Fourier transform of a unique function , which for various reasons serves naturally as a heuristic definition of the Frobenius-Schur indicator on . The formula we obtain for extends prior work of Casselman, Kottwitz, Lusztig, and Vogan addressing the case in which is a Weyl group. We include in addition a succinct description of the irreducible decomposition of derived by Kottwitz when is classical, and prove that defines a Gelfand model if and only if has type , , or with odd. We show finally that a conjecture of Kottwitz connecting the decomposition of to the left cells of 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 and its attached Fourier transform.