Geometry and Harmonic Analysis on Nilpotent Orbits

(1)用不变理论描述了幂零轨道的theta提升，并证明了一些好的提升的闭包是正规簇，且代数群在它们上的作用是无重的。(2)对于不定酉群U(p，p)，我们对球面幂零轨道进行了完全分类，并描述了其上函数环的结构。(3)将theta提升的概念推广到一般轨道而不是幂零轨道，从而为轨道对应的完全理解研究提供了一个新的思路。这包括了Sekiguchi，Djokovic，Zhao所研究的双线性形式的么模同余类的例子。最后，我们总结了每个研究者的研究。Sekiguchi从不变理论的角度研究了双线性形式的么模同余类。Ohta阐明了复约化代数群的轨道与其实形式之间的对应。更一般地，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的伴随圈、它们的迷向表示与广义Whittaker向量之间的关系。

(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.