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Branching rules of Dolbeault cohomology groups over indefinite Grassmannian manifolds. (English) Zbl 1227.22017

The author studies irreducible unitary representations of the unitary group \(G=U(n,n)\). Algorithms for calculating branching rules of finite-dimensional representations are well known, whereas no such algorithm is known for infinite-dimensional representations when restricted to compact subgroups. In the paper under review a family of singular unitary representations are studied. They are realized in Dolbeault cohomology groups over indefinite Grassmannian manifolds. For these representations the author is able to find a closed formula of irreducible decompositions with respect to reductive symmetric pairs \((A_{2n-1},D_n)\). The associated branching rule is a multiplicity-free sum of infinite-dimensional irreducible representations.

MSC:

22E46 Semisimple Lie groups and their representations
05E15 Combinatorial aspects of groups and algebras (MSC2010)
20G05 Representation theory for linear algebraic groups
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