Methods of supergeometry in representation theory of supergroups

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The project addresses structure problems in representation theory of supergroups and suggests geometric methods to approach them. Two fundamental questions of representation theory of any algebraic structure are the description of irreducible representations and the description of indecomposable representations. The second question is trivial in semi-simple case; however, since finite-dimensional representations of a simple algebraic supergroup are not completely reducible, both questions turns out to be very difficult: naive constructions give representations which are not irreducible, and irreducibles allow nontrivial extensions. (The situation is similar to one for modular representations.) In the last 25 years works of many mathematicians brought a lot of progress. Now, the characters of irreducible representations are known for general linear and orthosymplectic supergroups. The proposal suggests to adopt the geometric ideas of localization and support variety for supergroups to address the question of extensions. In particular, the proposal constructs a graph which conjecturally encodes a lot of information about extensions and characters. This graph appears naturally in calculation of cohomology groups of invertible sheaves on flag supervarieties and in attempt to generalize a Borel--Weil--Bott theory for supervarieties. A complementary geometric approach is via a functor associating to each module over a Lie superalgebra a quasicoherent sheaf on the cone of selfcommuting odd elements. The support of this sheaf has two counterparts in other branches of representation theory: the associated varieties of Harish-Chandra modules and the rank varieties in modular case. The conjecture is that the complexity of a simple module grows with the dimension of the support of the corresponding sheaf, in particular, one can prove the Kac-Wakimoto conjecture on superdimension this way. The proposal also suggests first steps for "odd" geometric quantization of the self-commuting cone. Recently, supersymmetric spaces became very popular in relation to sigma models; several physical papers develop particular examples of harmonic analysis on such spaces. The proposal contains a conjecture about the structure of modules of regular functions on supersymmetric spaces. The final part of the proposal concerns infinite-dimensional Lie superalgebras; character formulae for affine superalgebras were conjectured by Kac and Wakimoto. The aim here is to single out the cases when the conjectures hold, and to prove them in these cases.In modern particle physics, the principle of supersymmetry plays a very important role already for decades. A relatively new development is that, during the last decade, physicists realized that using supergroup symmetries makes feasible solutions of certain problems in the theory of condensed matter as well, e.g., in (super)conductivity. The methods of supersymmetry factor the questions interesting for physicists through the theory of representations of supergroups and superalgebras. The proposal puts forward several geometric methods, which, when developed, would answer a lot of currently pending concrete questions needed by physicists.