The classical and quantum particle on a flag manifold

TL;DR

This paper explores geodesics and the spectrum of the Laplace-Beltrami operator on flag manifolds, providing explicit solutions and finite-dimensional approximations that facilitate analysis of these geometric and spectral properties.

Contribution

It introduces a family of invariant metrics on flag manifolds that simplify the analysis of geodesics and spectra, and constructs finite-dimensional approximations for spectral calculations.

Findings

Explicit solutions for geodesics on flag manifolds.

Finite-dimensional approximations for spectral analysis.

Invariant metrics that simplify complex geometric problems.

Abstract

In the present paper we consider two related problems, i.e. the description of geodesics and the calculation of the spectrum of the Laplace-Beltrami operator on a flag manifold. We show that there exists a family of invariant metrics such that both problems can be solved simply and explicitly. In order to determine the spectrum of the Laplace-Beltrami operator, we construct natural, finite-dimensional approximations (of spin chain type) to the Hilbert space of functions on a flag manifold.