A new proof of Harish-Chandra's integral formula

Abstract

We present a new proof of Harish-Chandra's formula $$\Pi(h_1) \Pi(h_2) \int_G e^{\langle \mathrm{Ad}_g h_1, h_2 \rangle} dg = \frac{ [ \! [ \Pi, \Pi ] \!] }{|W|} \sum_{w \in W} \epsilon(w) e^{\langle w(h_1),h_2 \rangle},$$ where $G$ is a compact, connected, semisimple Lie group, $dg$ is normalized Haar measure, $h_1$ and $h_2$ lie in a Cartan subalgebra of the complexified Lie algebra, $\Pi$ is the discriminant, $\langle \cdot, \cdot \rangle$ is the Killing form, $[ \! [ \cdot, \cdot ] \!]$ is an inner product that extends the Killing form to polynomials, $W$ is a Weyl group, and $\epsilon(w)$ is the sign of $w \in W$. The proof in this paper follows from a relationship between heat flow on a semisimple Lie algebra and heat flow on a Cartan subalgebra, extending methods developed by Itzykson and Zuber for the case of an integral over the unitary group $U(N)$. The heat-flow proof allows a systematic approach to studying the asymptotics of orbital integrals over a wide class of groups.