Weak cuspidality and Howe correspondence

TL;DR

This paper investigates how the Howe correspondence influences Harish-Chandra series and weak cuspidality for type I dual pairs over finite fields, introducing a bijection with minimal unipotent support.

Contribution

It defines a new bijection derived from the Howe correspondence that preserves minimal unipotent support and explores its interaction with weak cuspidality.

Findings

Established a bijection with minimal unipotent support

Analyzed the effect on Harish-Chandra series

Examined the interaction with weak cuspidality

Abstract

We study the effect of the Howe correspondence on Harish-Chandra series for type I dual pairs over finite fields with odd characteristic. We define a bijection obtained from this correspondence, and enjoying the property of ``having minimal unipotent support''. Finally, we examine the interaction between the Howe correspondence and weak cuspidality.