Characters, coadjoint orbits and Duistermaat-Heckman integrals

The asymptotics of characters $\chi_{k\lambda}(\exp(h/k))$ irreducible representations a compact Lie group $G$ for large values the scaling factor $k$ are given by Duistermaat-Heckman (DH) integrals over coadjoint orbits $G$. This phenomenon generalises to central extensions loop groups $\widehat{LG}$ and diffeomorphisms circle $\widehat{\rm Diff}(S^1)$. We show that integrable modules affine Kac-Moody algebras Virasoro algebra factorize into divergent contribution standard form convergent which can be interpreted as formal DH orbital integral. For some modules, our results match recently computed Stanford Witten. In this case, $k$-scaling has same origin one gives rise classical conformal blocks. Furthermore, we consider reduced spaces suggest new invariant replaces symplectic volume in infinite dimensional situation. also other (in particular, corresponding minimal models) obtain DH-type expressions do not correspond any orbits. study functions $V(x)$ algebra. they related Hankel transform spectral densities $\rho(E)$ studied Saad, Shenker Stanford.