Representation Theory of Reductive Groups

抽象沙希 研究将包括两个部分。 第一部分是关于构造约化李群的小幂幺表示的显式模型，并利用这些模型来研究张量积的分解和这种表示的限制。预计这将为扩展θ-对应提供自然的背景。 建议的第二部分给出了杰克多项式理论的扩展，这些理论本身是GL（n）的球面多项式的推广，其中用任意参数替换根的多重性。我们所描述的多项式是通过改变正根的半和得到的，因此依赖于n个辅助参数。预计这些多项式的研究将导致新的和有趣的结果。 该提案的第一部分将应用于数论，这是一个深刻而美丽的数学领域，经常对各种实际问题产生意想不到的深远影响。例如，在设计网络时，为了便利信息的有效传播，最好具有高度的互联性。事实证明，一些最好的网络是通过使用数论考虑来构建的。 该提案的第二部分与物理学，多元统计学和组合学等不同领域相关。

Abstract Sahi The research will consist of two parts. The first part is concerned with constructing explicit models for small unipotent representations of reductive Lie groups, and with using these models to study the decomposition of tensor products and restrictions of such representations. It is expected that this will provide a natural context for extending the theta-correspondence. The second part of the proposal gives an extension of the theory of Jack polynomials, which are themselves generalizations of spherical polynomials for GL(n) where one replaces the root multiplicity by an arbitrary parameter. The polynomials we describe are obtained by varying the half-sum of positive roots, hence depend on n auxiliary parameters. It is expected that a study of these polynomials will lead to new and interesting results. The first part of the proposal will have applications to Number Theory, a deep and beautiful area of mathematics that has often had unexpected and profound implications for practical problems of various kinds. For example in the design of networks, in order to facilitate efficient dissemination of information, it is desirable to have a high degree of inter-connectivity. It turns out that some of the best networks are constructed by using number-theoretic considerations. The second part of the proposal has relevance to diverse areas such as Physics, Multivariate Statistics, and Combinatorics.