Higher rank quantum-classical correspondence

For a compact Riemannian locally symmetric space $\mathrm{\Gamma }\mathrm{\backslash }G\mathrm{/}K$ 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 $\mathrm{\Gamma }$ if $G\mathrm{/}K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G\mathrm{/}K$.