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

Johannes Droschl

TL;DR

This work analyzes the $\ell$-modular theta correspondence in type II for $GL_n$ and $GL_m$ over a non-archimedean field, establishing that multiplicities in $\mathrm{Hom}(S(\mathrm{Mat}_{n,m}), \pi\otimes\pi')$ are governed by a skeleton map $sk$ to $\mathrm{Par}^\ell\times\mathrm{Par}^\ell$ derived from modular symmetric-group data. It blends symmetric-group representation theory, Bernstein–Zelevinsky’s geometric lemma, and Fourier-analytic techniques on matrix spaces to reduce the problem to modular branching via partial permutations and multisegment combinatorics, yielding an explicit multiplicity formula $\dim \mathrm{Hom} = d_{\mathrm{sk}(\pi,\pi')}$. Under the rank constraint $n,m \le d\ell$ and especially when $d|n,m$ with $\ell d > \max(n,m)$, the paper provides computable Pieri-type algorithms and sharp bounds, recovering the characteristic-zero behavior in the appropriate limit. Overall, the results advance modular Howe duality by translating multiplicities into a structured, recursive combinatorial framework tied to residual and segment data, with potential applications to congruences and $\gamma$-factors in automorphic contexts.

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.

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

TL;DR

This work analyzes the -modular theta correspondence in type II for and over a non-archimedean field, establishing that multiplicities in are governed by a skeleton map to derived from modular symmetric-group data. It blends symmetric-group representation theory, Bernstein–Zelevinsky’s geometric lemma, and Fourier-analytic techniques on matrix spaces to reduce the problem to modular branching via partial permutations and multisegment combinatorics, yielding an explicit multiplicity formula . Under the rank constraint and especially when with , the paper provides computable Pieri-type algorithms and sharp bounds, recovering the characteristic-zero behavior in the appropriate limit. Overall, the results advance modular Howe duality by translating multiplicities into a structured, recursive combinatorial framework tied to residual and segment data, with potential applications to congruences and -factors in automorphic contexts.

Abstract

In this paper we compute the multiplicities appearing in the -modular theta correspondence in type II over a non-archimedean field , where is a prime not dividing the residue cardinality of . 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 -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 is the order of the residue cardinality of in , and the rank of the involved general linear groups is bounded above by , the behavior of the theta correspondence can be predicted via explicit algorithms coming from Pieri's Formula.
Paper Structure (14 sections, 33 theorems, 140 equations)

This paper contains 14 sections, 33 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. The setup
  3. Partitions
  4. Partial permutations and the symmetric group
  5. Representation theory of $G_n$
  6. Geometric Lemma of Bernstein and Zelevinsky
  7. Cuspidal representations
  8. Multisegments
  9. Intertwining operators
  10. Representations and their Endomorphisms
  11. Theta correspondence in type II
  12. Geometry of $\mathrm{Mat}_{n,m}$
  13. Fourier-theoretic aspects
  14. Combining the approaches

Key Result

Theorem 1

Assume that $\mathrm{char}(K)=0$, $n\le m$, and let $\pi$ be an irreducible and smooth $K$-representation of $G_n$. Then there exists an, up to isomorphism unique, irreducible, smooth $K$-representation $\pi'$ of $G_m$ with $\mathrm{Hom}(S_K(\mathrm{Mat}_{n,m}),\pi\otimes\pi')\neq 0$. In this case and if $n=m$, we have $\pi'\cong \pi^\lor$.

Theorems & Definitions (53)

  • Theorem 1: Min08
  • Theorem 2
  • Theorem 3
  • Lemma 2.1
  • Theorem 2.4: Pieri's Formula
  • Theorem 2.5
  • Corollary 2.1
  • Lemma 2.2: BerZel77 Lemma 2.11
  • Lemma 2.3
  • Theorem 2.6: MinSec14, Zel80
  • ...and 43 more