Differential operators of gradient type on symmetric spaces and representations of Lie algebras

本项目的目的是利用黎曼对称空间上齐次向量丛上的梯度型不变微分算子，研究不可约Harish-Chandra模嵌入半单李群的各种诱导表示.这样的微分算子的核实现了对偶Harish-Chandra模的最大整体化，嵌入的确定被归结为在这个核空间中确定等变函数。首先，广义Gelfand-Graev表示形成一族由幂零轨道参数化的诱导模。对于Hermitian型单李群的最高权Harish-Chandra模，给出了与全纯幂零轨道相联系的广义Whittaker模型.也就是说，它表明，每个最高的重量模块嵌入，与非零和有限的多重性，到广义Gelfand-Graev表示附加到唯一的开放轨道在其相关的品种。对于酉最高权模，嵌入空间可以完全用梯度型微分算子的主符号来描述。其次，我们考虑了四元数型单李群。利用Schmid梯度型微分算子刻画了Borelde Siebenthal离散级数的第0个n-同调空间，或等价地刻画了Borelde Siebenthal离散级数在主级数中的嵌入.特别地，当群的真实的秩不为1时，n-同调空间恰好有两个指数。第三，对于某些具有不可约相伴簇的Harish-Chandra模，阐明了相伴圈的重数与梯度型微分算子之间的关系。重数可用梯度型微分算子的主符号表示。

The purpose of this project is to study the embeddings of irreducible Harish-Chandra modules into various induced representations of a semisimple Lie group, by using the invariant differential operators of gradient type on certain homogeneous vector bundles over the Riemannian symmetric space. The kernel of such a differential operator realizes the maximal globalization of the dual Harish-Chandra module, and the determination of the embeddings in question is reduced to specifying the equivariant functions in this kernel space.First, the generalized Gelfand-Graev representations form a family of induced modules parametrized by the nilpotent orbits. Concerning the Harish-Chandra modules with highest weights for a simple Lie group of Hermitian type, the generalized Whittaker models associated with the holomorphic nilpotent orbits are specified. Namely, it is shown that each highest weight module embeds, with nonzero and finite multiplicity, into the generalized Gelfand-Graev representation attached to the unique open orbit in its associated variety. As for the unitary highest weight module, the space of the embeddings can be completely described in terms of the principal symbol of the differential operator of gradient type.Second, we consider a simple Lie group of quaternionic type. The 0th n-homology spaces, or equivalently, the embeddings into the principal series, of the Borelde Siebenthal discrete series are described, by using the Schmid differential operator of gradient type. We find in particular that the n-homology space has exactly two exponents if the real rank of the group is not one.Third, the relationship between the multiplicities in the associated cycles and the differential operators of gradient type are clarified for certain Harish-Chandra modules with irreducible associated varieties. The multiplicity can be written down by means of the principal symbol of a gradient type differential operator.