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University of Sydney Algebra Seminar

Apoorva Khare

Monday 3 April, 12-1pm, Place: Quad S421

Weight-formulas for highest weight modules, and the Weyl-Kac character formula for higher order Vermas

We introduce higher order Verma modules over a Kac-Moody algebra \g (one may assume this to be sl(n) throughout the talk, without sacrificing novelty). Using these, we present positive formulas (without cancellations) for the weights of arbitrary highest weight \g-modules. The key ingredient is that of "higher order holes" in the weights, which we introduce and explain. We also discuss the higher order parabolic BGG category O, and explain how BGG reciprocity does not always hold even in rank 2, yet the Cartan matrix is symmetric. We will end with BGG resolutions and Weyl-Kac type character formulas, for higher order Verma modules in certain cases. (Joint work with G.V.K. Teja.)