The spectrum of Anosov representations

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 Poincaré 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 Poincaré series or the order of vanishing of zeta functions, sharp counting results for the Lyapunov spectrum, etc.

Figures (3)

• Figure 1: In blue: $\mathfrak{a}^*$, the real part of $\mathfrak{a}^*_{\mathbb{C}}$. In red: the complex leading resonant hypersurface $\mathbf{C}^{(m)}_{\mathbb{C}}$. The intersection of $\mathbf{C}^{(m)}_{\mathbb{C}}$, the red variety, with $\mathfrak{a}^*$, the blue plane, is $\mathbf{C}^{(m)}$, the critical hypersurface.
• Figure 2: Left picture: in $\mathfrak{a}$, the cone $\mathcal{L}$ (in blue) contains the Lyapunov projections $\lambda(\gamma)$ (black crosses). Right picture: in $\mathfrak{a}^*$, the dual cone $\mathcal{L}^*$ (in red) is asymptotic to the critical hypersurface $\mathbf{C}_{\mathbb{R}}^{(d_s)}$ in bold black.
• Figure 3: Left-hand side: Action of the proximal sequence $(\gamma_\ell)_{\ell \geq 0}$ on $G/P_{-\mathscr{C}}$. It has unique attractive and repulsive fixed points $\gamma^+_{-\mathscr{C}}$ and $\gamma^-_{\mathscr{C}}$, respectively (though other fixed points may exist). The basin of attraction of $\gamma^+_{-\mathscr{C}}$ is the set of flags $gP_{-\mathscr{C}}$ that are transverse to $\gamma_{-\mathscr{C}}^-$ (Proposition \ref{['prop:proximal']}). Right-hand side: Action of the same proximal sequence on $G/P_{\mathscr{C}}$. It has unique attractive and repulsive fixed points $\gamma^+_{\mathscr{C}}$ and $\gamma^-_{-\mathscr{C}}$, respectively (possibly with additional fixed points). The basin of attraction of $\gamma_{\mathscr{C}}^+$ is the set of flags $gP_{\mathscr{C}}$ that are tranverse to $\gamma_{\mathscr{C}}^-$ (Proposition \ref{['prop:proximal']}). When $\mathscr{C}$ is $\iota$-invariant, the map $\mathcal{I}_{\mathscr{C}}$ identifies the two spaces as well as their corresponding attractive and repulsive fixed points.

Theorems & Definitions (299)

• Definition 1.1: $\vartheta$-Anosov representation
• Example 1.2
• Example 1.3
• Theorem 1.4: Existence of a domain of discontinuity
• Definition 1.5
• Theorem 1.6: Existence of a resonance spectrum
• Theorem 1.7: Leading resonance hypersurface
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10: Decay of correlation for the refraction flow