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Representations of solvable Lie groups and differential operators

可解李群和微分算子的表示

基本信息

批准号：
14540194
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：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

项目摘要

So-called "Polynomial conjecture" of Corwin-Greenleaf is a well known difficult conjecture for monomial representations of a connected and simply connected nilpotent Lie group. It has been a central aim of this research project. As there exists a strong parallelism for inducing and restricting representations, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A.Baklouti, G.Lion, J.Ludwig and B.Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.1.Let χ be a unitary character of an analytic subgroup H of G. We consider the monomial representation τ induced by χ up to G. The algebra of invariant differential operators on the line bundle over G/H associated to these data is algebraic over a system of generators of the set of central elements of Corwin-Greenleaf if and only if τ is of finite multiplicities.2.(Polynomial conjecture of Corwin - Greenleaf) Suppose that the monomial representation τ is of finite multiplicities. Then, the algebra of invariant differential operators on the line bundle over G/H associated to these data is isomorphic to the algebra of H-invariant polynomial functions on a certain affine subspace of the linear dual of the Lie algebra of G.3. The above result 1 and the Frobenius reciprocity in distribution sense obtained in the previous research program have their counterpart for the restrictions. We formulated them for the restriction π|K of an irreducible unitary representation π of G to an analytic subgroup K. Then, we proved them in certain particular cases.

Corwin-Greenleaf的所谓“多项式猜想”是关于连通和单连通幂零李群的单项式表示的一个著名的困难猜想。这是本研究项目的核心目标。由于诱导表示和限制表示之间存在着很强的平行性，本文在著名的轨道方法的框架下研究了幂零李群的这种对偶性。在与A. Baklaut、G.Lion、J.Ludwig和B.Magneron的合作中，我得到了下列主要结果。设G是连通单连通幂零李群，1.设χ是G的解析子群H的酉特征标.我们考虑了由χ到G诱导的单项表示τ. G/H上的线丛上的不变微分算子代数是Corwin-Greenleaf中心元集的生成元系上的代数当且仅当τ是有限乘子.（Corwin-Greenleaf的多项式猜想）假设单项表示τ是有限重数的。然后，与这些数据相关联的G/H上的线丛上的不变微分算子代数同构于G的李代数的线性对偶的某个仿射子空间上的H-不变多项式函数代数。上述结果1和以前的研究项目中得到的分布意义下的Frobenius互易性在限制条件下有对应的结果。我们将它们公式化为限制π| G的一个不可约酉表示π到一个解析子群K。然后，我们在某些特定的情况下证明了它们。