Howe duality over finite fields I: The two stable ranges

TL;DR

This paper initiates the study of type I Howe duality over finite fields, focusing on the stable ranges where the rank of one factor exceeds a certain threshold, and aims to explicitly describe the restriction of oscillator representations.

Contribution

It constructs the Howe duality correspondence explicitly in the stable ranges for finite fields, laying groundwork for further applications and generalizations.

Findings

Constructs the duality correspondence in stable ranges.

Lays foundation for applications like multiplicity calculations and character formulas.

Provides explicit descriptions of oscillator representation restrictions.

Abstract

This is the first in a series of papers on type I Howe duality for finite fields, concerning the restriction of an oscillator representation of the symplectic group to a product of a symplectic and an orthogonal group. The goal of the series is describing this restriction completely explicitly. Applications (described in the third paper of the series) include demonstrating that the tensor pairs previously calculated by S.-Y. Pan as occuring with non-zero multiplicity occur with multiplicity 1, proving the type C case of the Gurevich-Howe rank conjecture, and giving a recursive formula for the characters of cuspidal unipotent representations. In this first paper, we construct the correspondence in the two so called stable ranges, where the rank of one of the factors is large enough with respect to the other.