Important results in the theory of representation of Lie groups and Lie algebras have been proved initially using geometry. The geometric interpretation is given via the category of perverse sheaves on a flag variety. Algebraic proofs were available later with the developing of the theory of Soergel Bimodules, and more generally, with the theory of Diagrammatic Categories of Elias-Williamson.

The geometric context is very rich in tools and intuition, and one hopes to have something similar in the context of Diagrammatic Categories. In a recent paper with P. Achar and S. Riche [https://arxiv.org/abs/1802.07651], we explain how to reintroduce perverse sheaves in the diagrammatic context.

In this seminar, we review the definition of Diagrammatic Categories, we define perverse sheaves in this context and we shall prove that they satisfy lots of the properties of their counterpart in geometry., which are properties of the category of representations $\mathcal{O}$ of a semisimple Lie algebra.