On the existence of homomorphisms between principal series of complex semisimple Lie groups

TL;DR

This paper investigates the conditions under which nonzero homomorphisms exist between principal series representations and twisted Verma modules of complex semisimple Lie groups, advancing understanding of their structural relationships.

Contribution

It provides a comprehensive criterion for the existence of homomorphisms between principal series and twisted Verma modules in complex semisimple Lie groups.

Findings

Criteria for nonzero homomorphisms between principal series representations.

Conditions for homomorphisms between twisted Verma modules.

Enhanced understanding of module interrelations in Lie theory.

Abstract

We determine when there exists a nonzero homomorphism between principal series representations of a complex semisimple Lie group. We also determines the existence of homomorphisms between twisted Verma modules.