Howe duality over finite fields II: Explicit stable computation

TL;DR

This paper explicitly describes eta and zeta correspondences in type I Howe duality over finite fields using Lusztig's parametrization, clarifying stable ranges of irreducible representations.

Contribution

It provides an explicit description of eta and zeta correspondences in finite field Howe duality, linking them to Lusztig's parametrization within stable ranges.

Findings

Explicit formulas for eta and zeta correspondences in finite fields.

Identification of stable eta and zeta correspondences among irreducible representations.

Clarification of the occurrence of representations in Howe duality.

Abstract

In this second paper of a series dedicated to type I Howe duality for finite fields, we explicitly describe the eta and zeta correspondences constructed in the first paper in terms of G. Lusztig's parametrization of the irreducible characters of finite groups of Lie type in the two so-called stable ranges. This identifies the stable eta and zeta correspondences among the pairs of irreducible representations whose occurence with non-zero multiplicity in the type I Howe duality correspondence was proved by S.-Y. Pan.