Spectral Theory of Horizontal Laplacians

In this talk I will consider a principal G-bundle over a closed Riemannian manifold (M, g), and equip P with a G-equivariant connection from which one defines the horizontal Laplacian \Delta_H; \Delta_H is a second order sub-elliptic operator on P. I will discuss several spectral theoretic properties of  \Delta_H: 1) (global) hypoellipticity under a density assumption of the holonomy group of P; 2) lower bounds on the first eigenvalue of \Delta_H on Fourier modes; and 3) a quantum ergodicity result for \Delta_H on Fourier modes. The main idea is to introduce a semiclassical (microlocal) framework that we call the Borel-Weil calculus which allows to study G-equivariant operators on principal bundles. The semiclassical parameter in the calculus is the highest weight in the Weyl chamber of the group G, and the operators are pseudodifferential in the base variable, with values in Toeplitz operators on the flag manifold. This is joint work with Thibault Lefeuvre.