Character theory for 2-representations

Group representations very comfortably admit a higher categorical generalization. This includes induction and restriction for 2-representations that very directly parallel classical definitions. An appropriate character theory for 2-representations has been more opaque. In this paper we draw from a range of motivations to provide an intuitive character theory for 2-representations. It is built from the bicategorical trace developed to describe topological fixed point theory. We use the secondary traces of Ben-Zvi and Nadler which connect to Riemann-Roch and Lefschetz-type theorems in the context of differential graded categories and Fourier-Mukai transforms. Additional sources of motivation include Lipmans’ work on Grothendieck duality and, crucially, the Bartlett and Ganter–Kapranov approaches to characters. (The second of these was in turn motivated by Hopkins, Kuhn and Ravenel’s work on equivariant Morava $E$-theories.) Our characters are compatible with restriction and induction and, in cases that are sufficiently finite, admit a secondary character.