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

Seunghun Hong

doi:10.1016/j.difgeo.2014.11.006

paper

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

arXiv:1111.2749 · doi:10.1016/j.difgeo.2014.11.006

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.