实约化群和无穷维李代数表示论

In this project, we will study the representation theory of real reductive groups and extended affine Lie algebras. From the point of view of the Langlangs program, real reductive groups are the most intersting groups. For the represenation theory of real reductive groups, we will first study the Schwartz induction of almost linear Nash groups by devolping the theory of infinite dimensional vector bundles on Nash manifolds. Next, we want to prove the uniqueness of Shalika-models by using the generalized functions and the tempered generalized functions. The uniqueness of models is one of the most interesting reseach areas in the representation theory of Lie groups. And the uniqueness of Shalika-models paly an important role in the study of L-functions of genral spin gorups by using the Langlangs lift to the general linear groups. As higher dimensional generalization of affine Kac-Moody algebras, extended affine Lie algebras are studied intensively in the literature in recently years. We will construct the vertex operator representations of the extended affine Lie algebras with nullity 2 by their iterated loop realizations. This will provide the explict constructions of new representations for some extended affine Lie algebras.

本项目主要研究实约化群和扩张仿射李代数的表示理论。从Langlangs纲领的角度来说，实约化群是最有意思的一类李群。关于实约化群表示论，我们将研究的主要内容有：1. 通过发展Nash流形上无穷维向量丛的理论体系来研究几乎线性Nash群的Schwartz诱导理论。Schwartz诱导是实约化群无穷维表示论中亟待完善的一项基础理论。2. 各种李群模型的唯一性问题是近些年来国际上李群表示论中最热门的问题之一。我们将利用广义函数和tempered广义函数来证明Shalika-模型的唯一性。Shalika-模型的唯一性在研究一般spin群的L-函数理论中有着重要的作用。作为仿射Kac-Moody代数的高维推广，扩张仿射李代数是一类已得到广泛重视的无穷维李代数。我们计划利用零度为2 的扩张仿射李代数的迭代loop实现来构造它们的顶点算子表示。这将首次给出几类扩张仿射李代数表示的具体构造。

本项目主要研究了以下问题：.1. 利用仿射 Kac-Moody 代数 loop 模构造的思想，实现了零度为 2 的 toroidal 高维仿射李代数一大类新的不可约可积表示；.2. 将 Drinfeld 的扭量子仿射化推广到一般的单边型量子 Kac-Moody 代数上，并给出这些新的量子代数的顶点表示；.3. 给出了斜环上酉李代数的费米表示并确定了该表示的不可约分解；.4. 证明了扭 Shalika 模型和扭线性周期的唯一性。