Category O

Summary

Category $\mathcal{O}$ is a category of modules of a complex semisimple Lie algebra that contains the finite-dimensional modules and the Verma modules. It was introduced by Bernstein, Gelfand and Gelfand. It is an abelian category in which every object has finite length. So it is large enough to be flexible and small enough to be studied in detail. For integral weights, the multiplicities of simple modules in Verma modules are given by the Kazhdan–Lusztig conjecture.