Higher rank quantum-classical correspondence

Abstract

For a compact Riemannian locally symmetric space $\Gamma \setminus G∕K$ of arbitrary rank we determine the location of certain Ruelle–Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle–Taylor resonances and establish a spectral gap which is uniform in $\Gamma$ if $G∕K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e., a one-to-one correspondence between horocyclically invariant Ruelle–Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G∕K$.