The Virasoro algebra is an interesting Lie algebra from a variety of perspectives. A common way to study Lie algebras is through their modules (or representations), that is, ways that the Lie algebra acts on vectors spaces.

I will give an introduction to the Virasoro algebra and its representation theory, comparing it to the classical Lie algebra . In particular, I will discuss simple modules L and Verma modules M, two fundamental types of modules, and their connections to blocks. My own research has focused on the Virasoro algebra module . I'll present some of my results on the decomposition of this module by blocks and the potential connections to Hecke algebras and Kazhdan-Lusztig polynomials for the Viraso algebra, concepts which capture the geometric structure of blocks.