Taussky-Todd Lecture

In my talk I will discuss a new approach to representation theory of algebraic groups. In the usual approach one starts with an algebraic group $\mathcal{G}$ over some local (or finite) field $F$, considers the group $G = \mathcal{G}(F)$ of its $F$-points as a topological group and studies some category $Rep(G)$ of continuous representations of the group $G$. I will argue that more correct objects to study are some kind of sheaves on the stack $B\mathcal{G}$ corresponding to the group $\mathcal{G}$. I will show that this point of view naturally requires to change the formulation of some basic problems in Representation Theory. In particular this approach might explain the appearance of representations of all pure forms of a group $G$ in Vogan's formulation of Langlands' correspondence.