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

摘要奖：DMS-9972065主要研究员：维拉·塞尔加诺娃该项目致力于研究表示理论和D-模理论之间的关系。第一部分推广了Harish-Chandra模的Bernstein-Beilinson结果。Fernando证明了：如果M是有限维李代数G上的不可约模，则作用在M上的局部有限的元集是李子代数。如果相应的李子群K在标志流形上有有限多个轨道，则Bernstein和Beilins证明了这类模可以作为由标志流形上的K-轨道扩张的D-模。主要研究者希望将子群K和M的代数性质与相应的D-模的几何性质联系起来。作为第一个结果，她得到了包含最大环面的每个K的(G，K)-模的几何结构。第二个要攻击的问题是关于Lie超代数的Bernstein-Beilinson定理的推广。简单的概括只适用于非退化的中心人物，而最有趣的表征理论发生在退化的情况下。这位主要研究者希望用Bernstein-Beilinson定理的超模拟来证明她早先关于Kazhdan-Lusztig多项式的猜想。几何对象和物理理论的对称性形成了Lie群，其局部和无限小的结构由李代数描述：具有乘积运算满足简单规则的向量空间。表示论涉及这些李代数和群在几何、分析和代数中的基本分解和表示。项目标题的D-模块是代数结构，它捕捉了微积分和分析的基本方面。主要研究人员认为，Brylinski，Beilinson，Bernstein，Kashiwara，Kazhdan，Lusztig等人工作中建立的表示理论的几何方面可以应用于纯代数表示理论，特别是在Fernando，Futorny和Mathie近期工作中发展的权模理论。人们希望当用D-模的几何语言来完成他们的证明时，他们的技术代数细节可以变得清晰。

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.