Pollicott-Ruelle Resonances on Flag Manifolds

TL;DR

This paper investigates the resonance spectrum of multiflows on flag manifolds induced by Lie group actions, establishing its discreteness, existence of resonant states, and explicit spectra in special cases.

Contribution

It introduces a definition of joint resonance for these flows and provides a detailed spectral characterization in specific flag manifold cases.

Findings

Resonance spectrum is discrete and well-defined.

Existence of resonant states for the multiflow.

Explicit spectral descriptions for projective spaces and full flag manifolds.

Abstract

We study the resonance spectrum of the multiflow induced on a flag manifold by the action, through multiplication by the exponential map, of the Cartan subalgebra of the underlying Lie group. We give a definition of joint resonance for the flow, then prove its discreteness and existence of resonant states. We conclude by explicit characterization of the spectrum in the special cases of Projective spaces and manifolds of full flags.