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

TL;DR

This paper derives Harish-Chandra's volume formula for flag manifolds using Weyl's law and the Euler-Maclaurin formula, linking geometric volume to algebraic properties of Lie groups and touching on the Atiyah-Singer index theorem.

Contribution

It introduces a novel derivation of Harish-Chandra's volume formula from Weyl's law using the Euler-Maclaurin formula, revealing new connections in Lie theory.

Findings

Derived Harish-Chandra's volume formula using Weyl's law and Euler-Maclaurin

Connected volume calculations to algebraic properties of Lie groups

Highlighted a mystery related to the Atiyah-Singer index theorem

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.