The $\ell$-modular local theta correspondence in type II and partial permutations

TL;DR

This paper investigates the modular theta correspondence over non-archimedean fields, revealing that multiplicities are governed by symmetric group actions and can be computed using explicit algorithms based on Pieri's Formula.

Contribution

It provides a detailed analysis of multiplicities in the modular theta correspondence, linking them to symmetric group representations and offering algorithms for their computation.

Findings

Multiplicities are governed by symmetric group actions.

Explicit algorithms for predicting theta correspondence behavior.

Reduction to branching problems in modular symmetric group representations.

Abstract

In this paper we compute the multiplicities appearing in the ${\overline{\mathbb{F}}_\ell}$-modular theta correspondence in type II over a non-archimedean field $\mathrm{F}$, where $\ell$ is a prime not dividing the residue cardinality of $\mathrm{F}$. Unlike for representations with complex coefficients, highly non-trivial multiplicities can emerge. We show that these multiplicities are precisely governed by the action of symmetric groups on the set of partial permutations, and the ${\overline{\mathbb{F}}_\ell}$-representation of symmetric groups these give rise to. The problem is thus reduced to certain branching problems in the modular representation theory of symmetric groups. In particular, if $d$ is the order of the residue cardinality of $\mathrm{F}$ in ${\overline{\mathbb{F}}_\ell}$, and the rank of the involved general linear groups is bounded above by $ d\ell$, the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri's Formula.