We develop a (2-)categorical generalization of the theory of group representations and characters. We categorify the concept of the trace of a linear transformation, associating to any endofunctor of any small category a set called its categorical trace. In a linear situation, the categorical trace is a vector space and we associate to any two commuting self-equivalences a number called their joint trace. For a group acting on a linear category V we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors. We call this function the 2-character of V. Such functions of commuting pairs (and more generally, n-tuples) appear in the work of Hopkins, Kuhn and Ravenel [HKR00] on equivariant Morava E-theories. We define the concept of induced categorical representation and show that the corresponding 2-character is given by the same formula as was obtained in [HKR00] for the transfer map in the second equivariant Morava E-theory.

We calculate equivariant elliptic cohomology of the partial flag variety G/H, where H \subseteq G are compact connected Lie groups of equal rank. We identify the RO(G)-graded coefficients Ell_G^* as powers of Looijenga's line bundle and prove that transfer along the map {\pi}: G/H -\rightarrow pt is calculated by the Weyl-Kac character formula. Treating ordinary cohomology, K-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N.Ganter and A.Ram, Elliptic Schubert calculus. In preparation].Comment: 44 page

We formulate the axioms of an orbifold theory with power operations. We define orbifold Tate K-theory, by adjusting Devoto's definition of the equivariant theory, and proceed to construct its power operations. We calculate the resulting symmetric powers, exterior powers and Hecke operators and put our work into context with orbifold loop spaces, level structures on the Tate curve and generalized Moonshine.

We generalize the definition of orbifold elliptic genus, and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H ∞ -map into the Morava-Lubin-Tate theory E h , then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus [DMVV97]. It depends only on h and not on the genus.

Contents 1. Introduction 1.1. Background on generalized Moonshine 1.2. Background on equivariant elliptic cohomology and statement of results 1.3. Acknowledgements 2. Principal bundles over complex elliptic curves 2.1. The moduli space of principal G-bundles 2.2. Line bundles over M G 2.3. Construction and properties of the Freed-Quinn line bundle L α 2.4. Sections of L α 3. Abelian groups and level structures 3.1. Isogenies 3.2. Duals 3.3. Cyclic groups 4. Symmetric groups and coverings 5. Hopkins-Kuhn-Ravenel character theory 5.1. Products 5.2. Change of groups 6. Hecke operators and power operations 6.1. The Hecke correspondence 6.2. Power operations in K-theory 6.3. Symmetric and exterior powers on Ell α G 6.4. Replicability 7. The Witten genus 7.1. Twisted Thom isomorphisms 7.2. A few words about physics 7.3. The Witten genus and replicability? Appendix A. Generalities on principal bundles References