One can define formal Duistermaat--Heckman (DH) type integrals for coadjoint orbits of certain infinite dimensional groups including central extensions of loop groups and of diffeomorphisms of the circle. Recently, such DH integrals were considered by Stanford and Witten.

In this talk, we explain the relation between these formal DH integrals and asymptotics of characters of irreducible representations. The talk is based on a joint work with S. Shatashvili.