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Mod $\ell$ Weil representations and Deligne--Lusztig inductions for unitary groups

Naoki Imai, Takahiro Tsushima

TL;DR

The paper develops a modular (mod-$\ell$) realization of the Weil representations for finite unitary groups through étale cohomology of Deligne–Lusztig-type varieties and derives a mod-$\ell$ Howe correspondence for the dual pair $(\mathrm{Sp}_{2n},\mathrm{O}_2^-)$. It proves that the relevant modular cohomology representations are irreducible in almost all cases, with a single exceptional extension when $\ell \mid q+1$, and provides precise control of Frobenius action, invariants, and trace data to separate constituents. The results bridge Deligne–Lusztig induction with modular representation theory, offering a robust framework that aligns with, and extends, the ordinary (characteristic-zero) Howe correspondence. Practically, this advances the understanding of modular representations of reductive groups over finite fields and their interconnections via geometric constructions.

Abstract

We study the mod $\ell$ Weil representation of a finite unitary group and related Deligne--Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod $\ell$ Howe correspondence for $(\mathrm{Sp}_{2n},\mathrm{O}_2^-)$ including the case where $p=2$.

Mod $\ell$ Weil representations and Deligne--Lusztig inductions for unitary groups

TL;DR

The paper develops a modular (mod-) realization of the Weil representations for finite unitary groups through étale cohomology of Deligne–Lusztig-type varieties and derives a mod- Howe correspondence for the dual pair . It proves that the relevant modular cohomology representations are irreducible in almost all cases, with a single exceptional extension when , and provides precise control of Frobenius action, invariants, and trace data to separate constituents. The results bridge Deligne–Lusztig induction with modular representation theory, offering a robust framework that aligns with, and extends, the ordinary (characteristic-zero) Howe correspondence. Practically, this advances the understanding of modular representations of reductive groups over finite fields and their interconnections via geometric constructions.

Abstract

We study the mod Weil representation of a finite unitary group and related Deligne--Lusztig inductions. In particular, we study their decomposition as representations of a symplectic group, and give a construction of a mod Howe correspondence for including the case where .

Paper Structure

This paper contains 18 sections, 37 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weil representation of unitary group
  4. General facts on étale cohomology
  5. Lusztig series and $\ell$-blocks
  6. Cohomology as representation of unitary group
  7. Cohomology as representation of symplectic group
  8. Geometric setting
  9. Invariant part
  10. Trace computations
  11. Representation of $\mathop{\mathrm{Sp}}\nolimits_{2n}(\mathbb{F}_q)$
  12. Weil representation of $\mathop{\mathrm{Sp}}\nolimits_{2n}(\mathbb{F}_q)$
  13. Frobenius action
  14. Non-unipotent representation
  15. Unipotent representation
  16. ...and 3 more sections

Key Result

Theorem 1

Assume that $\ell \neq 2$. The $\mathop{\mathrm{Sp}}\nolimits_{2n}(\mathbb{F}_q)$-representations are irreducible except the case where $\ell \mid q+1$ and $(\xi,\kappa)=(1,+)$, in which case $H_{\mathrm{c}}^{2n-1} (Y'_{2n,\overline{\mathbb{F}}_q},\mathscr{K}_{\xi})^{\kappa}$ is a non-trivial extension of the trivial representation by an irreducible representation. Furthermore, the above represe

Theorems & Definitions (74)

  • Theorem : Theorem \ref{['thm']}
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 64 more