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.

Category $\mathcal{O}$ for the Lie algebra of vector fields on the line

“…Remark 3.4. In [7], the authors defined another category O ′ for W. Any module M in O ′ is locally finite over Cd −1 ⊕Cd 0 rather than W + . Similar as ∆(λ), define the W-module:…”

Whittaker category for the Lie algebra of polynomial vector fields

“…They proved that every such nontrivial module M is a simple subquotient of some tensor module T (P, V ). In [11], tilting modules and their character formulas for category O of W n were described. However non-weight modules and weight modules that are not Harish-Chandra modules for W n are not well developed.…”

Irreducible Jet modules for the vector field Lie algebra on $\mathbb{S}^1\times \mathbb{C}$

“…Billig and Futorny classified all irreducible Harish-Chandra W n -module, see [2]. Weight modules for the Lie algebra of vector fields on C n were studied in [11,13,12,24,25,26,29]. Recently, there is a systematic study on representations of the Lie algebra V X for arbitrary smooth affine varieties X, see [3,4,5,6,8].…”