Session S10 - Categorification, Higher Representation Theory, and Homological Knot Invariants

Friday, July 16, 19:20 ~ 19:55 UTC-3

Costandard Whittaker modules and contravariant pairings

Anna Romanov

University of Sydney, Australia   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

In 1997, Milicic—Soergel introduced a category N of modules over a semisimple Lie algebra which includes both category O and all Whittaker modules. In many ways, the structure of this category is similar to category O: objects are finite-length, simple objects arise as unique irreducible quotients of parabolically-induced standard modules, and composition multiplicities are given by Kazhdan—Lusztig polynomials. However, in other ways, the category is surprising: the objects are not weight modules, and standard modules do not admit unique contravariant forms. In particular, the lack of weight-space decompositions means that the duality in category O cannot be naively extended to category N. In ongoing work with Brown, we classify contravariant pairings between standard Whittaker modules and Verma modules, which leads to a natural algebraic definition of costandard objects in category N. We show that these costandard objects align with costandard (twisted) Harish-Chandra sheaves under Beilinson—Bernstein localization, and that with this set of costandard objects, category N has the structure of a highest weight category.

Joint work with Adam Brown (Institute of Science and Technology, Austria).

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