Infinite dimensional quaternionic representations and nilpotent orbits

点击翻译按钮获取中文摘要

The associated variety of an irreducible Harish-Chandra module gives a fundamental nilpotent invariant for the corresponding irreducible admissible representation of a real reductive group. Moreover, the multiplicity in the Harish-Chandra module of an irreducible component of the associated variety can be regarded as the dimension of a certain finite-dimensional representation, called the isotropy representation.The head investigator, Yamashita, has already shown that, in many cases, the isotropy representation can be described, in principle, by means of the principal symbol of a differential operator of gradient-type whose kernel realizes the dual Harish-Chandra module. In this research project, we have begun a systematic study of the isotropy representations attached to Harish-Chandra modules with irreducible associated varieties, including quaternionic representations, discrete series and unitary highest weight modules.The results are summarized as follows:We developed a general the … More ory for the isotropy representations, starting from the Vogan theory on associated cycles. In particular, a criterion for the irreducibility of an isotropy representation is presented. Also, we looked at when the isotropy representation can be described in terms of a differential operator of gradient-type.As for the discrete series, a nonzero quotient of the isotropy representation has been constructed in a unified manner. It seems that this quotient representation is large enough in the whole isotropy module. We have shown that this is the case if the theta-stable parabolic subgroup canonically determined from the discrete series in question admits a Richardson nilpotent orbit with respect to the complexified symmetric pair.The isotropy representation is explicitly described for every singular unitary highest weight module of Hermitian Lie algebras BI, DI and EVII. This allows us to deduce that the isotropy modules are irreducible for all singular unitary highest weight modules of arbitrary simple Hermitian Lie algebra.Principal contribution by the investigators : Saito developed his research on A-hypergeometric system, which is closely related to a realization of unitary highest weight modules. He has established a formula for the rank of a homogeneous A-hypergeometric system. Wachi constructed an analogue of the Capelli identity for generalized Verma modules of scalar type. Nishiyama and Ohta gave a correspondence of nilpotent orbits associated to a symmetric pair, by menas of the moment map with respect to a reductive dual pair. Less