Geometric construction of Heisenberg–Weil representations for finite unitary groups and Howe correspondences

被引：0
作者：
Naoki Imai
Takahiro Tsushima
机构：
[1] The University of Tokyo,Graduate School of Mathematical Sciences
[2] Chiba University,Department of Mathematics and Informatics, Faculty of Science
来源：
European Journal of Mathematics | 2023年 / 9卷
关键词：
Weil representation; Howe correspondence; Lusztig induction; Finite unitary group; 20C33; 11F27;
D O I：
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摘要：
We give a geometric construction of the Heisenberg–Weil representation of a finite unitary group by the middle étale cohomology of an algebraic variety over a finite field, whose rational points give a unitary Heisenberg group. Using also a Frobenius action, we give a geometric realization of the Howe correspondence for (Sp2n,O2-)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\textrm{Sp}_{2n},\textrm{O}_2^-)$$\end{document} over any finite field including characteristic two. As an application, we show that unipotency is preserved under the Howe correspondence.
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