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On modular Harish-Chandra series of finite unitary groups


Author: Emily Norton
Journal: Represent. Theory 24 (2020), 483-524
MSC (2010): Primary 20C33; Secondary 17B65, 05E10, 20C20
DOI: https://doi.org/10.1090/ert/549
Published electronically: October 7, 2020
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Abstract: In the modular representation theory of finite unitary groups when the characteristic $ \ell $ of the ground field is a unitary prime, the $ \widehat {\mathfrak{sl}}_e$-crystal on level $ 2$ Fock spaces graphically describes the Harish-Chandra branching of unipotent representations restricted to the tower of unitary groups. However, how to determine the cuspidal support of an arbitrary unipotent representation has remained an open question. We show that for $ \ell $ sufficiently large, the $ \mathfrak{sl}_\infty $-crystal on the same level $ 2$ Fock spaces provides the remaining piece of the puzzle for the full Harish-Chandra branching rule.


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Additional Information

Emily Norton
Affiliation: Department of Mathematics, TU Kaiserslautern, Gottlieb-Daimler-Strasse 48, 67663 Kaiserslautern, Germany
Email: norton@mathematik.uni-kl.de

DOI: https://doi.org/10.1090/ert/549
Received by editor(s): November 4, 2019
Received by editor(s) in revised form: May 29, 2020
Published electronically: October 7, 2020
Additional Notes: During the first two weeks of work on this paper, the author was supported by Max Planck Institute for Mathematics, Bonn. The rest of the time the author was supported at TU Kaiserslautern by the grant SFB-TRR 195
Article copyright: © Copyright 2020 American Mathematical Society