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Wavefront sets and descent method for finite unitary groups

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Let G be a connected reductive algebraic group defined over a finite field \(\mathbb {F}_q\), and let \(\mathfrak {g}\) be the Lie algebra of G. In the 1980s, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group \(G^F\) in the case where q is a power of a good prime for \(G^F\), and defined (Kawanaka) wavefront sets of the irreducible representations \(\pi \) of \(G^F\) by GGGRs. In Lusztig [(Adv Math 94(2):139–179, 1992), Theorem 11.2], Lusztig showed that if a nilpotent element \(X\in \mathfrak {g}^F\) is “large” for an irreducible representation \(\pi \), then the representation \(\pi \) appears with “small” multiplicity in the GGGR associated to X. In this paper, we prove that for unitary groups, if X is the wavefront of \(\pi \), the multiplicity equals one, which generalizes the multiplicity one result of usual Gelfand-Graev representations. Moreover, we give an algorithm to decompose GGGRs for \(\textrm{U}_n(\mathbb {F}_q)\) and calculate the \(\textrm{U}_4(\mathbb {F}_q)\) case by this algorithm.

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Acknowledgements

We acknowledge generous support provided by National Natural Science Foundation of PR China (Nos. 12071326 and 12201444), China Postdoctoral Science Foundation (Nos. 2021TQ0233, 2021M702399), and Jiangsu Funding Program for Excellent Postdoctoral Talent (No. 2022ZB283029).

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  1. School of Mathematical Science, Soochow University, Suzhou, 310027, Jiangsu, People’s Republic of China

    Zhifeng Peng & Zhicheng Wang

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Peng, Z., Wang, Z. Wavefront sets and descent method for finite unitary groups. Math. Z. 305, 65 (2023). https://doi.org/10.1007/s00209-023-03391-7

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