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

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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.