Isometric extensions of Anosov flows via microlocal analysis

We revisit the classical framework developed by Brin, Pesin and others to study ergodicity and mixing properties of isometric extensions of volume-preserving Anosov flows, using the microlocal framework developed in the theory of Pollicott-Ruelle resonances. The approach developed in the present note is reinvested in the companion paper [arXiv:2111.14811] in order to show ergodicity of the frame flow on negatively-curved Riemannian manifolds under nearly $1\mathrm{/}4$-pinched curvature assumption (resp. nearly $1\mathrm{/}2$-pinched) in dimension $4$ and $4\mathrm{\ell }+2,\mathrm{\ell }>0$ (resp. dimension $4\mathrm{\ell },\mathrm{\ell }>0$).