Contravariant forms on Whittaker modules

TL;DR

This paper classifies contravariant forms on nondegenerate Whittaker modules for complex semisimple Lie algebras, revealing their structure, dimension, and induction procedures, with implications for Verma modules and degenerate Whittaker modules.

Contribution

It provides a complete classification of contravariant forms on Whittaker modules and describes their dimensions and induction methods, extending understanding of these modules.

Findings

Contravariant forms form a vector space with dimension equal to the Weyl group cardinality.

A procedure for parabolically inducing contravariant forms is established.

The existence and formula for the dimension of contravariant forms on degenerate Whittaker modules are derived.

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.