Highest weight structures on Whittaker categories

Speaker: Anna Romanov
Affilliation: University of Sydney

Abstract

In 1997, Milicic—Soergel introduced a category N whose simple objects are precisely the irreducible Whittaker modules for a semisimple Lie algebra. Category N can be seen as a generalisation of category O, and it shares many of category O’s nice properties: objects are finite-length, simple objects arise as unique irreducible quotients of parabolically-induced standard objects, and composition series multiplicities are given by Kazhdan—Lusztig polynomials. Category O is the archetypal example of a highest weight category, so it is natural to ask if category N also has a highest weight structure. The main obstacle to answering this question is the lack of duality in category N – objects in N are not weight modules, so duality in category O (which is defined explicitly in terms of weight space decompositions) does not extend to category N in any obvious way. Because of this, there is no existing definition of a costandard object in N which generalises a dual Verma module. In this talk I’ll discuss ongoing work with Adam Brown (IST Austria), where we propose a definition of costandard Whittaker modules using contravariant pairings between standard Whittaker modules and Verma modules. Using our costandard modules, we can endow blocks of N with highest weight structures.