The Category $\mathcal O$ for Lie Algebras of Vector Fields (I): Tilting Modules and Character Formulas

Abstract

In this article, we exploit the theory of graded module categories with semi-infinite character developed by Soergel (Character formulas for tilting modules over Kac–Moody algebras, Represent. Theor. 2 (1998), 432–448) to study representations of infinite-dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n) (n \geq 2)$, and obtain a description of indecomposable tilting modules. The character formulas for those tilting modules are determined.