DFG-RSF: Geometry and representation theory at the interface of Lie algebras and quivers

该项目的目的是研究各种物体和结构，这些物体和结构被赋予了大的对称性李群，并与箭袋的几何表示理论有关。中心创新的方法是交错的机器的表征理论和方法和方法的李群理论，以描述和研究这样的对象，这是很难处理的一个理论单独。我们将要研究的对象包括旗簇及其退化、Grassmannian簇、球面簇和环面簇、有限维和无限维李代数的表示及其特征、交换李代数和压缩李代数的循环表示。主要目标是：描述旗形变种的复曲面退化并与牛顿-奥昆科夫理论联系，研究A型退化旗形变种和Grassmannian的几何形状，PBW型过滤和相关的最高权分次空间的描述简单和仿射Kac-Moody李代数的表示，量子PBW过滤理论的发展，有限和仿射退化旗簇的代数几何性质的描述，研究了A型旗簇的泛线性退化的全局几何，描述了Borel子代数和子群对自然表示和簇的作用结构，用Macdonald多项式描述了当前和仿射代数的循环表示的分次特征。

The aim of the project is the study of various objects and structures endowed with large Lie groups of symmetries and linked to the geometric representation theory of quivers. The central innovative approach is to interleave the machinery of quiver representation theory and the methods and approaches of Lie theory in order to describe and study such objects, which are hard to deal with within one of the theories alone. The objects we are going to study include flag varieties and their degenerations, quiver Grassmannians, spherical and toric varieties, representations of finite and infinite-dimensional Lie algebras and their characters, and cyclic representations of abelian and contracted Lie algebras.The main objectives are:Description of toric degenerations of flag varieties and links with Newton-Okounkov theory, Study of the geometry of type A degenerate flag varieties and quiver Grassmannians, Description of PBW-type filtrations and associated graded spaces on highest weightrepresentations of simple and affine Kac-Moody Lie algebras, Development of a theory of quantum PBW filtrations, Description of algebro-geometric properties of finite and affine degenerate flag varieties, Study of the global geometry of the universal linear degeneration of type A flag varieties, Description of the structure of actions of Borel subalgebras and subgroups on naturalrepresentations and varieties, Description of graded characters of cyclic representations of current and affine algebras in terms of Macdonald polynomials.