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

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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.