Morita's Theory for the Symplectic Groups

TL;DR

This paper constructs and analyzes holomorphic and principal series representations of symplectic groups over p-adic fields, extending Morita's duality concept from SL(2, F) to higher-dimensional symplectic groups.

Contribution

It generalizes Morita and Murase's constructions of these representations and introduces an algebraic duality for symplectic groups, expanding the understanding of their representation theory.

Findings

Constructed holomorphic discrete series representations.

Established a duality between sub-representations.

Extended Morita's duality from SL(2, F) to symplectic groups.

Abstract

We construct and study the holomorphic discrete series representation and the principal series representation of the symplectic group $\mathrm{Sp}(2n,F)$ over a $p$-adic field $F$ as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for $\mathrm{SL}(2, F)$ defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.