arXiv: Representation Theory | 2019
Contravariant forms on Whittaker modules
Abstract
Let $\\mathfrak{g}$ be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker $\\mathfrak{g}$-modules $Y(\\chi, \\eta)$ introduced by Kostant. We prove that the set of all contravariant forms on $Y(\\chi, \\eta)$ forms a vector space whose dimension is given by the cardinality of the Weyl group of $\\mathfrak{g}$. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules $M(\\chi, \\eta)$ introduced by McDowell.