Isotropy representations associated with Harish-Chandra modules and nilpotent orbit theory

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The principal purpose of this project is to develop nilpotent orbit theory for Harish-Chandra modules corresponding to irreducible admissible representations of real semisimple Lie groups, in order to get deep understanding on generalized Whittaker models in relation to various nilpotent invariants of representations. We focused our attention to the isotropy representations which give the multiplicities in the associated cycles of Harish-Chandra modules.(1)The isotropy representations have been described explicitly for all singular unitary highest weight modules of simple Hermitian Lie algebras by using the projection to the PRV-components. As a result we have proved the irreducibility of such isotropy representations. A new proof of the Howe duality correspondence for reductive dual pairs is obtained via isotropy representations. For EVII, we have shown that the isotropy representations for certain unitary highest weight modules of non-scalar type give the Dvorski-Sahi correspondence. … More This allows relating the isotropy representations and harmonic analysis on certain compact symmetric spaces of real rank one.(2)General machinery has been established to describe the isotropy representations of Harish-Chandra modules with irreducible associated varieties, by using the principal symbols of differential operators of gradient-type. Applying this machinery, we have revealed an explicit relationship between the isotropy representations for discrete series and the generic fiber of the moment map for the conormal bundle of closed orbits on the flag variety.(3)To identify the generic fiber of the moment map, we have studied the Richardson orbits associated with symmetric pairs. It has been an open problem whether the parabolic subgroups for the Richardson orbits act transitively on the set of Richardson elements in the symmetric part of its nilradical. In this project, we have got a progress to this problem, by giving nice sufficient conditions for the transitivity, and also a counterexample for the Lie groups of type A Less