Algebro-Geometric Derived Categories and Applications

I will discuss some applications of ideas from derived algebraic geometry (DAG) to representation theory in joint work with David Nadler. First I'll review the theory of Drinfeld centers of tensor categories and its generalization to derived algebraic geometry (joint also with John Francis), followed by it application to a Langlands duality statement for Lusztig's character sheaves. Second I will discuss how DAG sees the local Langlands correspondence for real groups as a base change from the complex case.