Contravariant forms on Whittaker modules,Proceedings of the American Mathematical Society

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Contravariant forms on Whittaker modules
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2020-10-09 , DOI: 10.1090/proc/15205
Adam Brown , Anna Romanov

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.

中文翻译：

Whittaker模块上的逆变形式

摘要：让一个复杂的半简单李代数。我们在由Kostant引入的非退化Whittaker-模块上给出了协变形式的分类。我们证明了所有对变形式的集形成一个向量空间，其维数由的Weyl群的基数给出。我们还描述了抛物线型诱导变体形式的程序。作为推论，我们推导了Verma模块上Shapovalov形式的存在，并为McDowell引入的退化Whittaker模块上的逆形式空间的空间尺寸提供了一个公式。 $\ mathfrak {g}$ $\ mathfrak {g}$ $Y（\ chi，\ eta）$ $Y（\ chi，\ eta）$ $\ mathfrak {g}$ $M（\ chi，\ eta）$

更新日期：2020-11-12

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