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Isotropy representation of flag manifolds

Alekseevsky, D. V. — 1998

Proceedings of the 17th Winter School "Geometry and Physics"

A flag manifold of a compact semisimple Lie group G is defined as a quotient M = G / K where K is the centralizer of a one-parameter subgroup exp ( t x ) of G . Then M can be identified with the adjoint orbit of x in the Lie algebra 𝒢 of G . Two flag manifolds M = G / K and M ' = G / K ' are equivalent if there exists an automorphism φ : G G such that φ ( K ) = K ' (equivalent manifolds need not be G -diffeomorphic since φ is not assumed to be inner). In this article, explicit formulas for decompositions of the isotropy representation for all flag manifolds...

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