Structural and geometric study of representations with applications to categorification

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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.