Structural and geometric study of representations with applications to categorification

这个项目是“无限和有限维李代数的结构和表示”项目的继续，首席研究员I.Penkov资助了项目“Darstellungstheorie”的第一阶段。在过去三年中，后一个项目的目标已经完全实现，特别是I.Penkov及其合作者在几个研究方向上取得了实质性进展，见下文第2.1节。该项目的继续增加了I.Penkov和A.Huckleberry的一个联合监督博士项目，该项目涉及AG流形上实型半单李群的开放轨道的几何以及与这些轨道的循环空间相关的表示理论。该项目将由一名博士生在2012至2015年间完成。此外，I.Penkov与Vera Serganova(加州大学伯克利分校)和Gregg Zuckerman(耶鲁大学)的合作应继续下去，并在第一阶段筹资期间取得的成果的基础上再接再厉。具体地说，这涉及到无限维李代数和李超代数的可积表示范畴的Koszul性的研究，基于一个猜想，即Zuckerman函子在某些最低权模范畴和某些广义Harish-Chandra模范畴之间建立了等价关系。此外，I.Penkov、V.Serganova和Igor Frenkel(耶鲁大学)也计划对玻色子-费米子通信进行分类。最后，计划在波鸿与Peter Heinzner小组共同举办一次研讨会，讨论几何表示理论中复数几何和代数几何之间的相互作用。

This project is the continuation of the project “Structure and representations of Infinite- and finite-dimensional Lie algebras" with principal investigator I. Penkov funded in the first phase of the project “Darstellungstheorie". In the last three years the goals of this latter project have been fully achieved, in particular essential progress in several directions of study has been made by I. Penkov and his collaborators, see section 2.1 below. The continuation of the project is enriched by the addition of a joint supervision doctoral project of I. Penkov and A. Huckleberry on the geometry of open orbits of real forms of semisimple Lie groups on ag manifolds and the representation theory related to the cycle spaces of these orbits. This project is to be carried out by a doctoral student in 2012 through 2015. In addition, the collaboration of I. Penkov with Vera Serganova (UC Berkeley) and Gregg Zuckerman (Yale University) should continue and build upon results obtained during the first phase of funding. Concretely, this involves the study of Koszulity of categories of integrable representations of infinite-dimensional Lie algebras and Lie superalgebras, work on a conjecture that the Zuckerman functor establishes an equivalence between certain categories of lowest weight modules and certain categories of generalized Harish-Chandra modules. Furthermore I. Penkov, V. Serganova and Igor Frenkel (Yale University) plan to work also on a categorification of the boson-fermion correspondence. Finally, a joint seminar, with the group of Peter Heinzner in Bochum, on the interaction of complex geometry and algebraic geometry in the context of geometric representation theory is planned.