A new proof of Harish-Chandra's integral formula

We present a new proof of Harish-Chandra’s formula $$\mathrm{\Pi }(h_1)\mathrm{\Pi }(h_2){\int }_G{e}^{⟨{\mathrm{A}\mathrm{d}}_g{h}_1,h_2⟩}dg=\frac{[\text{\hspace{0.17em}⁣}[\mathrm{\Pi },\mathrm{\Pi }]\text{\hspace{0.17em}⁣}]}{\mathrm{\mid }W\mathrm{\mid }}\sum _{w\in W}ϵ(w)e^{⟨w(h_1),h_2⟩},$$ 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, $\mathrm{\Pi }$ is the discriminant, $⟨\cdot ,\cdot ⟩$ is the Killing form, $[\text{\hspace{0.17em}⁣}[\cdot ,\cdot ]\text{\hspace{0.17em}⁣}]$ is an inner product that extends the Killing form to polynomials, $W$ is a Weyl group, and $ϵ(w)$ 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.