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”是一个众所周知的困难概念，对于连接且简单地连接的nilpotent Lie组的单体表示。这一直是该研究项目的核心目的。由于存在着诱导和限制表示的强大并行性，因此我在著名的轨道方法的框架中研究了这种二元性。与A.Baklouti，G.Lion，J.LudWig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。令χ是G的分析亚组H的单位特征。我们认为χ直到G诱导的单一表示τ。与这些数据相关的G/H上不变差分算子的代数是代数是在Corwin-Greef的中央元素集合中，如果乘积（如果乘以Corwin-Greeaf的中心元素），则乘积（如果乘积，则次数）。 Corwin -Greenleaf）假设单态表示τ是有限的。然后，与这些数据相关的g/h上的不变差分运算符的代数是与h-crinvariant多项式函数的代数同构，该代数在G.3的Lie代数的线性双重二元组上的某些仿射子空间上。上面的结果1和在上一个研究计划中获得的分配意义上的Frobenius互惠具有其限制对应物。我们为它们的限制π| k制定了g对分析亚组K的不可约合单位表示π。然后，我们在某些特定情况下提供了它们。