Semiclassical analysis on principal bundles

TL;DR

This paper develops a semiclassical Borel-Weil calculus for analyzing $G$-equivariant operators on principal bundles, leading to new results in dynamical systems, spectral theory, and quantum ergodicity.

Contribution

It introduces a novel semiclassical framework for $G$-principal bundles, enabling explicit conditions for mixing and spectral properties, with broad applications in dynamics and analysis.

Findings

Conditions for rapid mixing of flows on principal bundles.

Spectral analysis showing hypoellipticity of horizontal Laplacians.

Quantum ergodicity results under ergodicity assumptions.

Abstract

Let $G$ be a compact Lie group. We introduce a semiclassical framework, called Borel-Weil calculus, to investigate $G$-equivariant (pseudo)differential operators acting on $G$-principal bundles over closed manifolds. In this calculus, the semiclassical parameters correspond to the highest roots in the Weyl chamber of the group $G$ that parametrize irreducible representations, and operators are pseudodifferential in the base variable, with values in Toeplitz operators on the flag manifold associated to the group. This monograph unfolds two main applications of our calculus. Firstly, in the realm of dynamical systems, we obtain explicit sufficient conditions for rapid mixing of volume-preserving partially hyperbolic flows obtained as extensions of an Anosov flow to a $G$-principal bundle (for an arbitrary $G$). In particular, when $G = \mathrm{U}(1)$, we prove that the flow on the extension is rapid mixing whenever the Anosov flow is not jointly integrable, and the circle bundle is not torsion. When $G$ is semisimple, we prove that ergodicity of the extension is equivalent to rapid mixing. Secondly, we study the spectral theory of sub-elliptic Laplacians obtained as horizontal Laplacians of a $G$-equivariant connection on a principal bundle. When $G$ is semisimple, we prove that the horizontal Laplacian is globally hypoelliptic as soon as the connection has a dense holonomy group in $G$. Notably, this result encompasses all flat bundles with a dense monodromy group in $G$. We also prove a quantum ergodicity result for flat (and in some situations non-flat) principal bundles under a suitable ergodicity assumption. We believe that this monograph will serve as a cornerstone for future investigations applying the Borel-Weil calculus across different fields.