D-Modules Associated with Representation of Reductive Lie Algebras and Superalgebras

AbstractAward：DMS-9972065主要研究者：Vera Serganova该项目致力于研究表示理论与D-模理论之间的关系。 第一部分是关于Harish-Chandra模的Bernstein-Beilinson结果的推广。 Fernando证明了，如果M是有限维李代数G上的不可约模，则局部作用在M上的元素集是李子代数。 如果相应的李群K在旗流形上有有限多个轨道，那么伯恩斯坦和Beilinson证明了这样的模可以作为由旗流形上的K-轨道扩展的D-模得到。 主要的证明者希望将子群K和M的代数性质与相应D-模的几何性质联系起来。 作为第一个结果，她给出了（G，K）-模的几何构造，其中每个K都包含极大环面. 第二个问题是李超代数的伯恩斯坦-贝林森定理的推广。 一个简单的推广只适用于非退化的中心特征标，而最有趣的表示理论发生在退化的情况下。 主要研究者希望使用Bernstein-Beilinson定理的超类比来证明她早先关于Kazhdan-Lusztig多项式的猜想。几何对象和物理理论的对称性形成李群，李群的局部和无穷小结构由李代数描述：具有满足简单规则的乘积运算的向量空间。 表示论关注这些李代数和群在几何、分析和代数中的基本分解和表现。 项目名称的D-模块是代数结构，它捕获了微积分和分析的基本方面。 主要作者认为，在Brylinski，Beilinson，伯恩斯坦，Kashiwara，Kazhdan，Lusztig和其他人的作品中发现的表示论的几何方面可以应用于纯代数表示论，特别是在Fernando，Futorny和Mathieu最近的工作中发展的权模理论中。

AbstractAward: DMS-9972065Principal Investigator: Vera SerganovaThe project is dedicated to the study of relation betweenrepresentation theory and the theory of D-modules. The firstpart concerns a generalization of Bernstein-Beilinson result forHarish-Chandra modules. It was shown by Fernando that if M is anirreducible module over finite-dimensional Lie algebra G then theset of elements which act locally finitely on M is a Liesubalgebra. If the corresponding Lie subgroup K has finitelymany orbits on the flag manifold then Bernstein and Beilinsonshowed that such modules can be obtained as D-modules extendedfrom K-orbits on the flag manifold. The principal investigatorhopes to relate a subgroup K and algebraic properties of M withgeometric properties of the corresponding D-modules. As a firstresult she has a geometric construction of (G,K)-module for everyK containing the maximal torus. The second problem to beattacked is a generalization of Bernstein- Beilinson theorem forLie superalgebras. A straightforward generalization works onlyfor non-degenerate central character, while the most interestingrepresentation theory happens in degenerate case. The principalinvestigator hopes to use a superanalogue of Bernstein-Beilinsontheorem to prove her earlier conjecture for Kazhdan-Lusztigpolynomials in the super case.The symmetries of geometric objects and physical theories formLie groups, whose local and infinitesimal structure are describedby Lie algebras: vector spaces with a product operationsatisfying simple rules. Representation theory concerns basicdecompositions and manifestations of these Lie algebras andgroups in geometry, analysis, and algebra. The D-modules of theproject title are algebraic structures which capture essentialaspects of calculus and analysis. The principal investigatorbelieves that geometric aspects of representation theory foundedin works of Brylinski, Beilinson, Bernstein, Kashiwara, Kazhdan,Lusztig and others can be applied to pure algebraicrepresentation theory, especially in the theory of weight modulesdeveloped by Fernando, Futorny and in recent work of Mathieu.One hopes that technical algebraic details of their proofs canbecome clear when done in geometric language of D-modules.