Harish-Chandra's volume formula via Weyl's Law and Euler-Maclaurin formula

Abstract

Harish-Chandra's volume formula shows that the volume of a flag manifold $G/T$, where the measure is induced by an invariant inner product on the Lie algebra of $G$, is determined up to a scalar by the algebraic properties of $G$. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing the Euler-Maclaurin formula. This approach leads to a mystery that lies under the Atiyah-Singer index theorem.