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

对称空间上梯度型微分算子及李代数的表示

• 批准号: 09440002
• 负责人: YAMASHITA Hiroshi
• 金额: $ 8.9万
• 依托单位: Hokkaido University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440002/
• 关键词: semisimple Lie group; Harish-Chandra module; nilpotent orbit; differential operator of gradient type; multiplicity; generalized Whittaker model; discrete series; highest weight representation; 随伴多様体; 四元数型対称空間; Borel de-Siebenthal離数系列; 一般化されたWhi

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.

在黎曼对称空间上，利用梯度型不变微分算子在齐次向量束上研究不可约Harish-Chandra模嵌入半单李群的各种诱导表示。该微分算子的核实现了对偶Harish-Chandra模的最大全局化，并将所讨论的嵌入的确定简化为在该核空间中指定等变函数。首先，广义Gelfand-Graev表示形成了由幂零轨道参数化的诱导模族。摘要针对一类简单李群的最重权的Harish-Chandra模，给出了与全纯幂零轨道相关的广义Whittaker模型。也就是说，证明了每个最高权模以非零和有限的多重性嵌入到与唯一开轨道相关变化的广义Gelfand-Graev表示中。对于幺正最高权模，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，我们考虑一个四元数型的简单李群。利用梯度型的Schmid微分算子，描述了Borelde Siebenthal离散级数的第0个n-同调空间，或等价地，在主级数中的嵌入。我们特别发现，当群的实秩不是1时，n-同调空间恰好有两个指数。第三，明确了一类具有不可约关联变量的Harish-Chandra模的关联环的多重度与梯度型微分算子之间的关系。多重性可以用梯度型微分算子的主符号表示。

项目成果

Yamashita H.: "Reduced Schur henctions and Littlewood-Richardson coefficients"J.of London Math.Soc.(2). 59・2. 396-406 (1999)

Yamashita H.：“约化 Schur 向量和 Littlewood-Richardson 系数”J.of London Math.Soc.(2) 396-406 (1999)。

H. Yamashita: "Associated variety, Kostant-Sekuguchi correspondence, and locally free U(n)-action on Harish-Chandra modules"J. Math. Soc. Japan. 51-No.1. 129-149 (1999)

H. Yamashita：“关联簇、Kostant-Sekuguchi 对应以及 Harish-Chandra 模块上的局部自由 U(n) 作用”J.

齋藤睦: "Grobner deformations of regular holonomic systems" Proc.Japan Acad.74・7. 111-113 (1998)

Mutsumi Saito：“正则完整系统的 Grobner 变形”Proc.Japan Acad.74・7（1998）。

M. Saito: "Hypergeometric polynomials and integer programming"Compositio Math.. 115-No.2. 185-204 (1999)

M. Saito：“超几何多项式和整数规划”Compositio Math.. 115-No.2。

平井武: "Relations between unitary representations of diffeomorphism groups and those of infinite symmetric group or of related permutation groups" J.Math.Kyoto Univ.37・2. 261-316 (1997)

Takeshi Hirai：“微分同胚群的酉表示与无限对称群或相关排列群的酉表示之间的关系”J.Math.Kyoto Univ.37・2（1997）。