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 $({\mathfrak{a}}_P^{\ast }{)}_{\mathbb{C}}$, 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 Poincaré series associated with Anosov representations admit a meromorphic extension to $({\mathfrak{a}}_P^{\ast }{)}_{\mathbb{C}}$. 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.

v1. This is an expanded and self-contained version of our work on Anosov representations. It contains several additional results that were omitted from the second version. v2. Substantial revision. Several proofs were streamlined using results from the existing literature, and the introduction was completely rewritten