The study of the representation theoretical aspect of generalized flag varieties

广义旗品种代表性理论研究

基本信息

批准号：
18540162
负责人：
MATUMOTO Hisayosi
金额：
$ 1.09万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2006
资助国家：
日本
起止时间：
2006 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-18540162/
关键词：
unitay representations semisimple Lie groups generalized Verma modules

项目摘要

1 Homomorphisms between scalar generalized Verma modulesWe had classified the homomorphisms between scalar generalized Verma modules associated to maximal parabolic subalgebras and explained how to use the operators constructed in the maximal case to get some operators in general. We conjectures that all the homomorphisms arise in this way.We call a parabolic subalgebra normal, if each parabolic subalgebra which has a common Levi part is conjugate to each other under some inner automorphism. We had proved that for classical algebras and "almost half" of normal parabolic subalgebra, the above conjecture is affirmative for regular infinitesimal characters.In the first year, the principal investigator gave geometrical proof of the above conjecture and removed the assumption "classical". In the second year, we investigated submodules of scalar generalized Verma modules of maximal Gelfand-Krillov dimensions.2 Irreducibility of the space of continuous Whittaker vectorsThe famous "multiplicity one theorem" tells us that the dimension of the space of continuous Whittaker vectors on an irreducible admissible representation of a quasi-split real linear Lie group is at most one. For non quasi-split groups the multiplicity one theorem fails. As a natural extension of the multiplicity one theorem to non quasi-split case, I propose the following conjecture.The space of continuous Whittaker vectors is irreducible as a module over the finite W-algebra.For example, we have an affirmative answer for the type A groups.

1标量广义的Verma模块之间的同态性质分类已将标量广义的Verma模块之间的同态分类，并解释了如何使用在最大情况下构建的操作员在最大情况下构建的1个运算符，以使某些操作员一般。我们猜想所有同态都以这种方式出现。我们称抛物线亚代词架是正常的，如果每个具有共同levi部分的抛物线亚代甲甲状腺代词是在某些内部自动形态下彼此共轭的。我们已经证明，对于经典的代数和正常抛物线亚代词的“几乎一半”，上述猜想对于常规的无穷小特征是肯定的。第一年，首席研究者对上述猜想进行了几何证明，并删除了假设“古典”。在第二年，我们调查了最大Gelfand-krillov维度的标量广义Verma模块的 - 2连续惠特克载体的空间不可约可但可以说，著名的“多重性One oneRem”告诉我们，在不可偿还的linit linit linit line line line line seal ime ye Quasi-seal a Quasi-Spliet a Quasi-Slace a Quasi-Slace a Quasi-Slace a Quasi-Slace a Quasi-Slace seal in y Quasi-Slace seal in y Quasi-Slace上的尺寸不可约束。对于非准分数组，多重性一个定理失败。作为多个定理的自然扩展到非准切片案例，我提出了以下猜想。连续的惠特克矢量的空间是在有限的w-algebra上不可修复的。例如，我们对A类组有一个肯定的答案。