# 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.

### About Pure mathematics seminars

We present regular seminars on a range of pure mathematics interests. Students, staff and visitors to UQ are welcome to attend, and to suggest speakers and topics.

Seminars are usually held on Tuesdays from 3pm to 3.50pm.

Talks comprise 45 minutes of speaking time plus five minutes for questions and discussion.

#### Information for speakers

Researchers in all pure mathematics fields attend our seminars, so please aim your presentation at a general mathematical audience.

#### Contact us

To volunteer to talk or to suggest a speaker, email Ole Warnaar or Travis Scrimshaw.