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

TL;DR

This paper investigates the mod $\, ext{ell}$ Weil representations of finite unitary groups, their decomposition, and constructs a mod $\, ext{ell}$ Howe correspondence for specific symplectic and orthogonal groups, including characteristic 2.

Contribution

It introduces a new construction of the mod $\, ext{ell}$ Howe correspondence for $( ext{Sp}_{2n}, ext{O}_2^-)$, expanding understanding of representation theory in characteristic 2.

Findings

Decomposition of mod $\, ext{ell}$ Weil representations as symplectic group representations

Construction of a mod $\, ext{ell}$ Howe correspondence for $( ext{Sp}_{2n}, ext{O}_2^-)$

Extension of results to cases where $p=2$

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 $p=2$.