Morita's Duality for Split Reductive Groups

TL;DR

This paper extends Morita's duality theory to split reductive groups over p-adic fields, constructing and analyzing key representations and their dualities in this broader setting.

Contribution

It generalizes Morita's duality framework from symplectic groups to all split reductive groups over p-adic fields, introducing new constructions of holomorphic and principal series representations.

Findings

Established duality between sub-representations of holomorphic and principal series.

Constructed explicit models of these representations for split reductive groups.

Extended Morita's theory to a broader class of algebraic groups.

Abstract

In this paper, we extend the work in "Morita's Theory for the Symplectic Groups" to split reductive groups. We construct and study the holomorphic discrete series representation and the principal series representation of a split reductive group $\mathrm {G}$ over a $p$-adic field $F$ as well as a duality between certain sub-representations of these two representations.