Appears in collection : 2025 - T1 - WS1 - Intertwining operators and geometry

With the aim of uniform treatment of multiplicity-free representations of Lie groups, T. Kobayashi introduced the notion of visible action for holomorphic actions of Lie groups on complex manifolds. His propagation theorem of the multiplicity-freeness property produces various kinds of multiplicity-free theorems for unitary representations realized in the space of holomorphic sections of an equivariant holomorphic vector bundle whose base space admits a visible action of a Lie group. Kobayashi has indicated two directions of generalizations of his multiplicity-free theorem. One is a generalization to infinite dimensional manifolds and has been done by Miglioli and Neeb. The other is a generalization to cohomology spaces, which is the main concern of this talk.

I would like to talk about a cohomology version of Kobayashi's theorem and its application to multiplicity-free restrictions of Zuckerman derived functor modules to reductive subgroups.