A New Proof of Harish-Chandra’s Integral Formula

Abstract

We present a new proof of Harish-Chandra’s formula (Harish-Chandra in Am J Math 79:87–120, 1957)

$$\Pi(h_1) \Pi(h_2) \int_G e^{\langle {\mathrm{Ad}}_g h_1,h_2 \rangle} dg = \frac{ [ [ \Pi, \Pi ] ] }{|W|} \sum_{w \in W}\epsilon(w) e^{\langle w(h_1),h_2 \rangle},$$

where G is a compact, connected, semisimple Lie group, dg is normalized Haar measure, h₁ and h₂ 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 (J Math Phys 21:411–421, 1980) 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.