The heat trace expansion on compact Lie groups and the volume formula
Let $G$ be a compact connected Lie group with a bi-invariant metric. Let $T$ be a maximal torus of $G$. Using the representation theory of compact Lie groups and Weyl's law for the heat trace, we verify Harish-Chandra's formula for $\vol(G)/\vol(T)$. The main calculation tools used are the Euler-Maclaurin formula and the Weyl integration formula.