Methods of supergeometry in representation theory of supergroups

该项目解决了超群表示论中的结构问题，并提出了几何方法来处理它们。 任何代数结构的表示论的两个基本问题是不可约表示的描述和不可分解表示的描述。 第二个问题在半简单的情况下是平凡的;然而，由于简单代数超群的有限维表示不是完全可约的，这两个问题都变得非常困难：朴素构造给出的表示不是不可约的，而不可约允许非平凡的扩展。 (The这种情况类似于模块化表示的情况）。在过去的25年中，许多数学家的工作带来了很大的进展。现在，一般线性和正交辛超群的不可约表示的特征标是已知的。建议采用局部化的几何思想和支持超群的多样性来解决扩张问题。特别是，该提案构建了一个图，它对有关扩展名和字符的大量信息进行了编码。这个图自然出现在计算上同调群的可逆层旗超品种，并试图推广一个博雷尔-韦尔-博特理论的超品种。 一个互补的几何方法是通过一个函子关联到李超代数上的每一个模上的一个自对易奇元锥上的拟相干层。 这个层的支撑在表示论的其他分支中有两个对应的分支：Harish-Chandra模的相关簇和模情况下的秩簇。 猜想是简单模的复杂性随着相应层的支集维数的增加而增加，特别地，可以用这种方法证明关于超维的Kac-Wakimoto猜想。该提案还建议了自对易锥的“奇”几何量子化的第一步。 最近，超对称空间在sigma模型中变得非常流行;几篇物理论文在这样的空间上发展了调和分析的特殊例子。 该建议包含了一个猜想的结构，模块的正规功能的超对称空间。该建议的最后一部分涉及无限维李超代数;仿射超代数的特征公式由Kac和Wakimoto提出。 这里的目的是找出猜想成立的情况，并在这些情况下证明它们。在现代粒子物理学中，超对称性原理几十年来一直发挥着非常重要的作用。 一个相对较新的发展是，在过去十年中，物理学家意识到使用超群对称性也可以解决凝聚态理论中的某些问题，例如，（超）导电性。超对称的方法通过超群和超代数的表示理论来解决物理学家感兴趣的问题。该提案提出了几种几何方法，这些方法一旦发展起来，将回答物理学家所需要的许多目前悬而未决的具体问题。

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.