Fe b 20 06 REPRESENTATION AND CHARACTER THEORY IN 2 - CATEGORIES

defined on n-tuples of commuting elements of G and invariant under simultaneous conjugation. Thus, in [HKR00] the study of nth Morava E-theory at a prime p leads to χ(g1, ..., gn) defined on gi of p-power order. On the other hand, the construction of equivariant elliptic cohomology in [Dev96] (which corresponds to n = 2) uses functions χ(g1, g2) as above but without restrictions on the order. The motivation here comes from equivariant string theory. Earlier 2-class functions came up in Grojnowki’s work on equivariant elliptic cohomology (unpublished). What is lacking in the above approaches is an analog of the notion of representation which would produce the generalized characters by means of some kind of trace construction. In this paper we supply such a notion for n = 2. It turns out that the right object to consider is an action of G on a category instead of a vector space or, more generally, on an object of a 2-category. Our approach fits naturally with and was partially motivated by the recent works [TS04], [BDR04] aiming at geometric definitions of elliptic cohomology. All of these works involve 2-categorical constructions. For a subgroup H of a finite group G Hopkins, Kuhn and Ravenel introduced in [HKR00] a natural induction map from n-class functions