Geometry and Harmonic Analysis on Nilpotent Orbits

(1)用不变理论描述了幂零轨道的提升，证明了一些好的提升的闭包是正态变化，代数群对它们的作用是无多重性的。我们还得到了用积分表示的幂零轨道的度数公式。建立了相干轮系起升的一般理论，特别证明了相干轮系支撑的多重性是守恒的。(2)对于不定酉群U(p,p)，我们完全分类了球面幂零轨道，同时描述了其上函数环的结构。(3)将提升的概念推广到除幂零轨道以外的一般轨道，为全面理解轨道对应关系的研究提供了启示。这包括由Sekiguchi， Djokovic， Zhao研究的双线性形式的单模同余类的一个例子。最后，对各研究者的研究进行了总结。Sekiguchi从不变理论的角度研究了双线性形式的单模同余类。Ohta澄清了复约代数群的轨道与其实形式之间的对应关系。更一般地说，他将他的研究扩展到对称对的轨道。Yamashita研究了半单李群的酉表示、各向同性表示和广义Whittaker向量的关联环之间的关系。

(1)We describe the theta lifting of nilpotent orbits in the terms of invariant theory, and prove that (the closures of) some good liftings are normal variety and the action of the algebraic groups on them are multiplicity free. We also obtain a degree formula of the nilpotent orbits expressed by integrals.A general theory of the lifting of coherent sheaves are constructed, and in particular we prove the preservation of the multiplicities of the support of coherent sheaves.(2)For the indefinite unitary group U(p,p), we completely classify the spherical nilpotent orbits and at the same time we describe the structure of the function rings on them.(3) We generalize the notion of theta lifting to general orbits other than nilpotent ones, and shed a light on the research of the complete understanding of the orbit correspondence. This includes an example of the unimodular congruence classes of the bilinear forms studied by Sekiguchi, Djokovic, Zhao.At last, we summarize the research of each investigator.Sekiguchi has studied the unimodular congruence classes of the bilinear forms from the view point of the invariant theory.Ohta has clarified the correspondence between the orbits of complex reductive algebraic groups and its real forms.More generally, he extends his research to the orbits of symmetric pairs.Yamashita has investigated the relations between associated cycles of the unitary representations of semisimple Lie groups, their isotropy representations and generalized Whittaker vectors.