Monomial representation of solvable Lie groups

可解李群的单项式表示

• 批准号: 11640189
• 负责人: FUJIWARA Hidenori
• 金额: $ 1.34万
• 依托单位: Kinki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640189/
• 关键词: nilpotent Lie group; solvable Lie group; unitary representation; irreducible decomposition; orbit method; multiplicity; invariant differential operator; monomial representation; 既約公解

It is well known that there exists a strong parallelism for inducing and restricting representations. In this research, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A. Baklouti, G. Lion and B. Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.1. (Commutativity conjecture of Duflo, Corwin Greenleaf) Let χ be a unitary character of an analytic subgroup H of G. Then, the monomial representation τ induced by χ up to G is of finite multiplicities if and only if the algebra of invariant differential operators on the line bundle over G/H associated to these data is commutative.2. (Frobenius reciprocity) Let π be an irreducible unitary representation of G. The multiplicity of π in the canonical central decomposition of τ is equal to the dimension of the space of (H, χ ) semi-invariant generalized vectors.3. The above result 1 has its counterpart for the restrictions. Namely, let's restrict an irreducible unitary representation of G to an analytic subgroup K. Then, this restriction is of finite multiplicities if and only if the associated algebra of K invariant differential operators is commutative.

众所周知，存在着诱导和限制表示形式的强大并行性。在这项研究中，我在著名的轨道方法的框架中研究了Nilpotent Lie群体的这种双重性。与A. Baklouti，G。Lion和B. Magneron合作，我获得了以下主要结果。令G为连接的，简单地连接的nilpotent Lie Group.1。 （Duflo的可交换性猜想，Corwin Greenleaf）让χ是G的分析亚组H的单一特征。然后，只有当G/H与G/H相关的G/H bundectiant bundectiant tic os t imply of t to Indecy.2时，由χ诱导的单个表示τ是有限的多重性。 （frobenius互惠）让π是G。τ在τ的规范中心分解中π的多样性等于（h，χ）半不变的概括矢量的尺寸。3。上述结果1具有限制的对应物。也就是说，让我们限制G对分析子组K的不可约合的统一表示。然后，当且仅当K相关的k不变差分运算符的相关代数是有限的时限制时的限制。

项目成果

H.Fujiwara,G.Lion,B.Magneron,S.Mehdi: "Un critere de commutativite pour l'algebre des operateurs differentiels invariants sur un espace homogene nilpotent"C.R.Acad.Sci.Paris.Ser.I,Math..

H.Fujiwara、G.Lion、B.Magneron、S.Mehdi：“Un critere de commutativite pour lalgebre des operators Differentiels invariants sur un espace homogene nilpot”C.R.Acad.Sci.Paris.Ser.I，Math..

H. Fujiwara, G. Lion and S. Mehdi: "On the commutativity of the algebra of the algebra of invariant differential operators on certain nilpotent homogeneous spaces"Trans. Amer. Math. Soc.. 353. 4203-4217 (2001)

H. Fujiwara、G. Lion 和 S. Mehdi：“论某些幂零齐次空间上不变微分算子的代数的交换律”Trans。

H.Fujiwara, G.Lion, S.Mehdi: "On the commutativity of the algebra of invariant differential operators on certain nilpotent homogeneous spaces"Trans. Amer. Math. Sic.. 353. 4203-4217 (2001)

H.Fujiwara、G.Lion、S.Mehdi：“论某些幂零齐次空间上不变微分算子代数的交换律”Trans。

H. Fujiwara, G. Lion et B. Magneron: "Algebre de fonctions associees aux representations monomiales des groupes de Lie nilpotents"Prepublication de l'Universite Paris. 13. 2002-02 (2002)

H. Fujiwara、G. Lion 和 B. Magneron：“Algebre de fonctions associees auxrepresentations monomiales des groupes de Lie nilpotons”巴黎大学预出版。

H.Fujiwara, G.Lion, B.Magneren, S.Meholi: "UN critere de commutativite pour l'algebra des operateurs differentiels inveriants sur un espace homogene nilpotent"C. R, Acad. Paris, Ser. I, Math.. 332. 597-600 (2001)

H.Fujiwara、G.Lion、B.Magneren、S.Meholi：“UN critere de commutativite pour lalgebra des operaurs Differentiels inveriant sur un espace homogene nipot”C.