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On the existence of homomorphisms between principal series representations of complex semisimple Lie groups. (English) Zbl 1220.22011
The main result of this paper is a neat combinatorial criterion for the existence of homomorphisms between principal series representations of complex semisimple Lie groups and a similar statement for twisted Verma modules over semisimple complex finite dimensional Lie algebras. The criterion is formulated in terms of the combinatorics of weights and the action of the Weyl group. Along the way the author also obtains several interesting vanishing results for the cohomology of principal series representations and twisted Verma modules.

MSC:
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
22E46 Semisimple Lie groups and their representations
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