An algebra group over a field k is a group of the form G(k) = 1+A(k) where A(k) is a finite-dimensional associative k-algebra. The (standard) supercharacter theory for G(k) is an extension of the usual character theory, and is defined in terms of the action of G(k) on A(k) given by left multiplication and its dual action on the group of linear characters of A(k). The construction is independent of the field k, and suggests that there may exist a uniform construction in terms of certain geometric objects defined over an algebraic closure K of k. In this talk, we define the notion of a supercharacter sheaf of the algebra group G(K) = 1+A(K), and show how the supercharacters of the finite group G(k') can be obtained for every finite extension k' of k by a unified process from the supercharacter sheaves of G(K). Our definition is based on the character sheaves on A(K) as defined by M. Boyarchenko and V. Drinfeld and on the natural action of G(K) on these, and corresponds to a certain notion of induction of (rank one) character sheaves. (Part of this talk is based on work in collaboration with João Dias.)
