Norm computation and analytic continuation of vector valued holomorphic discrete series representations

TL;DR

This paper explicitly computes the norm of vector-valued holomorphic discrete series representations with specific K-types and explores their properties like unitarizability and reducibility, providing insights into their structure.

Contribution

It provides explicit norm formulas for certain vector-valued holomorphic discrete series representations and analyzes their highest weight modules' properties.

Findings

Explicit norm formulas for specific holomorphic discrete series

Analysis of unitarizability and reducibility of modules

Insights into composition series of highest weight modules

Abstract

In this paper we compute explicitly the norm of the vector-valued holomorphic discrete series representations, when its $K$-type is "almost multiplicity-free." As an application, we discuss the properties of highest weight modules, such as unitarizability, reducibility and composition series.