# The Category $\mathcal{O}$ for Lie algebras of vector fields (I): Tilting modules and character formulas

## TL;DR

This paper studies representations of infinite-dimensional Lie algebras of vector fields, focusing on tilting modules and their character formulas using graded module categories and semi-infinite character theory.

## Contribution

It introduces a new approach to describe indecomposable tilting modules and derive their character formulas for Lie algebras of vector fields.

## Key findings

- Classification of indecomposable tilting modules
- Explicit character formulas derived
- Application of semi-infinite character theory

## Abstract

In this article, we exploit the theory of graded module category with semi-infinite character developed by Soergel in \cite{Soe} to study representations of the infinite dimensional Lie algebras of vector fields $W(n), S(n)$ and $H(n)$ $(n\geq 2)$, and obtain the description of indecomposable tilting modules. The character formulas for those tilting modules are determined.