M ay 2 01 6 Stacks in Representation Theory

In this note I would like to introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group G as a sheaf on some geometric object. This point of view necessarily leads to a conclusion that the standard approach to (continuous) representations of algebraic groups should be modified. Let us start with a local or finite field F and fix an algebraic group G defined over F . In the standard approach we consider the set G = G(F ) of F -points of G as a topological group and study an appropriate category Rep(G) of continuous representations of G. The main goal of this note is to explain that this approach is philosophically inconsistent. In fact I will describe how to extend the category Rep(G) to some larger category M(G, F ) that better corresponds to our intuitive understanding of representations of G. We will see that this category can be naturally described as a product of categories Rep(Gi) over all pure inner forms of the group G. On the level of simple objects this means that Irr(M(G, F )) ≈ ∐ Irr(Gi). This agrees with observation by several mathematicians (e.g. by D. Vogan [Vog]) that when we classify irreducible representations it is better to work with the union of sets Irr(Gi) for several forms of the group G than with one set Irr(G).