The spectrum of Anosov representations

Abstract

Given a $P$-Anosov representation into a noncompact semisimple real algebraic group $G$, where $P < G$ is a parabolic subgroup, we construct a natural resonance spectrum for the classical dynamics associated with the representation. This spectrum is a complex analytic hypersurface in $(\mathfrak{a}_P^*)_{\mathbb{C}}$, the complexified dual of the Lie algebra of the split component of the associated Levi group $L < P$. We reinterpret several objects from the theory of Anosov representations within this spectral framework and investigate, in higher rank, questions that are classically related to Pollicott-Ruelle theory in the rank-one setting. In particular, the ``leading resonance''--which is now a hypersurface--is identified with the critical hypersurface of the representation and, as in the rank-one case, the resonant states lead to continuous families of invariant measures. We prove that the multivariate zeta functions and Poincar\'e series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_P^*)_{\mathbb{C}}$. We also establish an asymptotic expansion in inverse powers of time for the correlation function of the diagonal flow under a Diophantine condition on the representation. Most of our results concerning Anosov representations are obtained as a byproduct of a general theory of Axiom A actions of type $(k,1)$, where $k+1:=\dim \mathfrak{a}_P$, that we introduce in the article.