PBW-filtration of representations, degenerate flag varieties and polytopes

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The project is part of a long term program to better understand the geometric, algebraic and combinatorial aspects of the PBW-filtration of a representation. Let g be a simple Lie algebra and let n- be the nilpotent radical of an opposite Borel subalgebra. The PBW-filtration of U(n-) induces a filtration on a finite dimensional irreducible representation Vλ, denote by Vλa the associated graded space. This construction leads naturally to an algebraic group Ga, which can be viewed as a degenerate version of the original group G with Lie algebra g. The group acts on Vλa, and, guided by the embedding of a flag variety Fλ _ P(Vλ), denote by Fλa the closure of the Ga-orbit through the image of highest weight line in P(Vλa).Geometric, algebraic and combinatorial aspects of this construction have been investigated for G = SLn and Sp2m in a series of papers by E. Feigin, M. Finkelberg, G. Fourier and the PI. The aim of this project is to generalize the results to other types as well as to link the newly developed methods and tools to other known constructions in representation theory like crystal bases and generalized Gelfand-Tsetlin patterns.