Category $\mathcal O$ for the Virasoro algebra: Cohomology and Koszulity

We investigate blocks of the Category $\mathcal O$ for the Virasoro algebra over the complex numbers. We demonstrate that the blocks have Kazhdan-Lusztig theories, and that the truncated blocks give rise to interesting Koszul algebras. The simple modules have BGG resolutions, and from this we compute the extensions between Verma modules and simple modules, and between pairs of simple modules.