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Multiplicity-free representations and visible actions on complex manifolds. (English) Zbl 1085.22010
Multiplicity-free representations appear in various contexts of mathematics, and numerous multiplicity-free theorems have been found over several decades. It seems that no single known technique could be applied to all multiplicity-free representations, in both finite and infinite dimensional cases. This paper presents a simple principle based on complex geometry to explain various kinds of multiplicity-free representations in a general framework. The idea is focused on the non-standard geometric perspectives of multiplicity-free representations in both finite and infinite dimensional cases, and new proofs of classical multiplicity-free theorems in various contexts are obtained along with an investigation of complex geometry in which a totally real submanifold meets generic orbits of a group. Thus also a number of new theorems are proved. The main machinery is an abstract multiplicity-free theorem for the section of equivariant holomorphic vector bundles. The notion of visible action on a complex manifold is introduced and discussed in the broad context of symplectic geometry and Riemannian geometry, such as coisotropic actions and polar actions. Some examples of visible actions are given.Various examples of multiplicity-free representations in both finite and infinite dimensional cases are explained. At last, the results on the multiplicity-free theorem of representations reflected by the geometry of coadjoint orbits are briefly considered.

MSC:
22E46 Semisimple Lie groups and their representations
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20G05 Representation theory for linear algebraic groups
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