Let g be a Kac-Moody Lie algebra. We discuss a family of universal highest weight g-modules, termed “higher-order Verma modules”. With G.V.K. Teja, we show the weights of every highest weight module equal those of a higher-order Verma module. In certain cases, these modules also admit a BGG-type resolution and thus a “higher order” Weyl-Kac character formula.

Restricting to type A, we also explore the log-concavity of the weight-multiplicities of these modules. For usual Verma modules and finite-dimensional simples, this was carried out by Huh-Matherne-Meszaros-St.Dizier in 2022. With J. Matherne and A. St.Dizier, we show this extends to every parabolic Verma module, but not more generally.

Place: Room 204, RIMS