# Hodge Theoretic Properties in Representation Theory

A central problem in representation theory is determining the characters of irreducible representations of modules. A complete answer in the case of highest weight modules of complex semisimple Lee algebras is given by the Kazhdan-Lusztig conjecture (a theorem!), which provides an explicit formula to calculate their characters. The original proof of this theorem relies on deep geometrical tools such as the decomposition theorem, which in turn ultimately relies on the Hodge theory of complex varieties. Soergel bimodules give an alternative, completely algebraic approach to the Kazhdan-Lusztig problem. This algebraic approach mimics the geometric one by proving an analogue of the decomposition theorem, where the key step is to show the existance of an algebraic Hodge theory. It is also interesting to look at what happens in the positive characteristic world. Here there is no Hodge theory, yet investigating when certain hodge theoretic properties still hold could help to understand the Lusztig's conjecture on characters of simple representations of algebraic groups