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Asymptotic properties of infinitesimal characters and applications

Andrés Sambarino

TL;DR

This work develops a comprehensive convex-variation framework for infinitesimal deformations of representations into real semisimple groups, tying together Jordan-variation cones, affine actions via Margulis invariants, and Ledrappier thermodynamic potentials. By proving the cone of Jordan variations and its normalized variants have non-empty interiors under Zariski-density and full loxodromic variation, the authors establish robust links between representation-theoretic data (via the Kostant–Lyapunov–Jordan projection and Kostant lines) and dynamical constructs (pressure forms, entropy, and dynamical intersections) on higher-rank Teichmüller spaces, including Hitchin components. A key achievement is identifying an explicit φ in 𝔞^* that yields pressure forms compatible with Goldman’s symplectic form at Fuchsian points, along with a detailed degeneracy analysis governed by root data and Diophantine equations; the results also show Hessian positivity for Hausdorff-dimension functionals in higher-rank settings. The combination of affine-geometric, cohomological, and thermodynamic methods yields a unified framework linking infinitesimal character variation, properness criteria for affine actions, and geometric-analytic invariants, with applications to pressure forms, level-set deformations, and the geometry of higher Teichmüller spaces.

Abstract

Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group $Γ$ with values in a semi-simple real-algebraic group $\mathsf G$. We prove the \emph{cone of Jordan variations} has non-empty interior and, when $\mathsf G$ is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichmüller spaces. We identify an explicit functional $\varphi\in\mathfrak a^*$ whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation.

Asymptotic properties of infinitesimal characters and applications

TL;DR

This work develops a comprehensive convex-variation framework for infinitesimal deformations of representations into real semisimple groups, tying together Jordan-variation cones, affine actions via Margulis invariants, and Ledrappier thermodynamic potentials. By proving the cone of Jordan variations and its normalized variants have non-empty interiors under Zariski-density and full loxodromic variation, the authors establish robust links between representation-theoretic data (via the Kostant–Lyapunov–Jordan projection and Kostant lines) and dynamical constructs (pressure forms, entropy, and dynamical intersections) on higher-rank Teichmüller spaces, including Hitchin components. A key achievement is identifying an explicit φ in 𝔞^* that yields pressure forms compatible with Goldman’s symplectic form at Fuchsian points, along with a detailed degeneracy analysis governed by root data and Diophantine equations; the results also show Hessian positivity for Hausdorff-dimension functionals in higher-rank settings. The combination of affine-geometric, cohomological, and thermodynamic methods yields a unified framework linking infinitesimal character variation, properness criteria for affine actions, and geometric-analytic invariants, with applications to pressure forms, level-set deformations, and the geometry of higher Teichmüller spaces.

Abstract

Inspired by Benoist, we study objects linked to integrable tangent vectors on the character variety of a semi-group with values in a semi-simple real-algebraic group . We prove the \emph{cone of Jordan variations} has non-empty interior and, when is split, establish non-empty interior of the set of \emph{length-normalized variations}. We apply these techniques to pressure forms on Anosov representations and higher-rank Teichmüller spaces. We identify an explicit functional whose pressure form is compatible with Goldman's symplectic form at Fuchsian points in the Hitchin component. Finally, we show the degeneration of the Hausdorff dimension of \emph{higher}-quasi-circles is governed by a Diophantine equation.
Paper Structure (72 sections, 113 theorems, 351 equations, 6 figures, 3 tables)

This paper contains 72 sections, 113 theorems, 351 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The cone of Jordan variations
  3. Base point Livšic-independence
  4. Pressure forms for higher-rank Teichmüller spaces
  5. Hausdorff dimension of higher-quasi-circles
  6. Preliminaries
  7. Notations from Lie theory
  8. Some $\sl_2$'s of ${\mathfrak g}$
  9. Cartan decomposition
  10. Jordan decomposition
  11. Flag manifolds of $\sf G$
  12. The center of the Levi group $\sf P_{\vartheta}\cap\check{\sf P}_{\vartheta}$
  13. Representations of $\sf G$
  14. Gromov product, additive cross ratio and representations
  15. Proximality
  16. ...and 57 more sections

Key Result

Theorem 1

Let $v\in\sf T_\rho{\mathfrak X(\Upgamma,\sf G)}$ have Zariski-dense base point and full loxodromic variation. Then, $\mathscr{V\!\!J}_{v}$ is convex and has non-empty interior. Moreover, full variation elements of $\rho(\Upgamma)$ are Zariski-dense in $\sf G$ and their Jordan projections intersect

Figures (6)

  • Figure 1: Dynamics of a $(\upphi,X_0)$-compatible element of $\sf G\ltimes V$.
  • Figure 2: Definition of the Affine Ratio
  • Figure 3: Schematic situation in Theorem \ref{['SM']}, the ideally neutral spaces $(FQ)^0$ and $(QF)^0$ are very close (but do not coincide with) $F^+\cap Q^-$ and $Q^+\cap F^-$ respectively.
  • Figure 4: Hasse diagram for the $7$-dimensional irreducible representation of ${\mathsf{G_2}}$, which is the fundamental representation of the short root, together with the corresponding set of weights (in black).
  • Figure 5: The irreducible representation $\mathfrak{so}(3,4)\to\mathfrak{so}(4,4)$.
  • ...and 1 more figures

Theorems & Definitions (216)

  • Theorem 1
  • Corollary : Corollary \ref{['levelsets']} - Deformations along level sets of ${\mathscr h}_{}^{}$ give non-proper actions
  • Theorem 2: Double-density for roots with multiplicity $1$
  • Corollary : No proper actions above level sets of entropy
  • Corollary : Curves with arbitrary small root-variation
  • Corollary : Corollary \ref{['compa1']}
  • Definition 1.1
  • Corollary : Corollary \ref{['pathmetricHitchin']}
  • Definition 1.2
  • Theorem 3: Pressure degenerations are Lie-theoretic
  • ...and 206 more