The spectrum of Anosov representations

TL;DR

This paper introduces a spectral framework for Anosov representations, constructing a resonance spectrum that generalizes classical dynamical concepts to higher rank groups, with applications to zeta functions, Poincaré series, and flow mixing estimates.

Contribution

It develops a new resonance spectrum for Anosov representations, extending Ruelle-Pollicott theory to higher rank and linking spectral data with geometric and dynamical invariants.

Findings

Constructed a complex resonance spectrum as a hypersurface in the dual of the split component.

Proved meromorphic extension of zeta functions and Poincaré series for Anosov representations.

Established sharp mixing estimates under Diophantine conditions.

Abstract

Given a $\vartheta$-Anosov representation into a real reductive group $G$, we construct a natural resonance spectrum associated with the representation. This spectrum is a complex analytic variety of codimension $1$ in $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$, the complexified dual of the split component of the associated Levi group $L_\vartheta < G$. 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 Ruelle-Pollicott 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. As a corollary of our work, we prove that the zeta functions and Poincar\'e series associated with Anosov representations admit a meromorphic extension to $(\mathfrak{a}_\vartheta^*)_{\mathbb{C}}$. We also establish sharp mixing estimates for the refraction 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 free Abelian cocycles over hyperbolic flows. This article is intended as a foundational work toward more advanced results such as higher-rank quantum/classical correspondence, the detection of topological invariants of representations via the value at zero of Poincar\'e series or the order of vanishing of zeta functions, sharp counting results for the Lyapunov spectrum, etc.