Contravariant forms on Whittaker modules

Abstract

Let g \mathfrak {g} be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker g \mathfrak {g} -modules Y ( χ , η ) Y(\chi , \eta ) introduced by Kostant. We prove that the set of all contravariant forms on Y ( χ , η ) Y(\chi , \eta ) forms a vector space whose dimension is given by the cardinality of the Weyl group of g \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 ( χ , η ) M(\chi , \eta ) introduced by McDowell.

Errata to “contravariant forms on Whittaker modules”

“…As stated, Lemma 3.10 of [BR21], and the following equations (3.7) and (3.8), are false. The remaining results of [BR21] can be recovered with the following corrections.…”

“…(2) Replace the proof of part (a) of [BR21, Lemma 3.12] with the following proof, which no longer refers to Lemma 3.10 or equation (3.8) (with all of the notation described in [BR21]):…”