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.