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Isospectrality of Margulis-Smilga spacetimes for irreducible representations of real split semisimple Lie groups

Sourav Ghosh

TL;DR

This work analyzes affine actions of finitely generated groups into semidirect products $\mathsf{G}\ltimes_{\mathtt{R}}\mathsf{V}$, where $\mathsf{G}$ is a real split semisimple group with trivial center and $\mathtt{R}$ is a faithful irreducible representation admitting zero as a weight. By constructing algebraic (polynomial) expressions from Margulis invariants, the authors prove isospectral rigidity: representations with identical Margulis (or Jordan–Margulis) invariants are conjugate under natural automorphisms, provided certain density and rank conditions hold. The novelty lies in making Margulis invariants algebraic when $\mathtt{R}$ preserves a bilinear form, enabling rigorous algebraicity arguments and extending prior results to a broad class of Smilga-type affine actions, including Margulis–Smilga spacetimes. Applications show that Margulis invariant spectra uniquely determine Margulis–Smilga manifolds up to conjugacy in key cases, linking spectral data to geometric properness of actions and broadening the scope beyond previously known examples.

Abstract

In this article, we look at real split semisimple algebraic groups $\mathsf{G}$ with trivial center and faithful irreducible algebraic representations $\mathtt{R}$ of $\mathsf{G}$ on some vector space $\mathsf{V}$ which admit zero as a weight and which are self-contragredient (for example, adjoint representation of $\mathsf{PSL}(n,\mathbb{R})$). We show that, there exist polynomials made out of Margulis invariants of $(g,X)\in\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$ which are also rational expressions in $(g,X)$. Moreover, we show that any Zariski dense finitely generated subgroup of $\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$, for which the linear parts of the non-identity elements are loxodromic, is isospectrally rigid with respect to the Margulis invariants. In particular, we show that Margulis--Smilga spacetimes are isospectrally rigid too.

Isospectrality of Margulis-Smilga spacetimes for irreducible representations of real split semisimple Lie groups

TL;DR

This work analyzes affine actions of finitely generated groups into semidirect products , where is a real split semisimple group with trivial center and is a faithful irreducible representation admitting zero as a weight. By constructing algebraic (polynomial) expressions from Margulis invariants, the authors prove isospectral rigidity: representations with identical Margulis (or Jordan–Margulis) invariants are conjugate under natural automorphisms, provided certain density and rank conditions hold. The novelty lies in making Margulis invariants algebraic when preserves a bilinear form, enabling rigorous algebraicity arguments and extending prior results to a broad class of Smilga-type affine actions, including Margulis–Smilga spacetimes. Applications show that Margulis invariant spectra uniquely determine Margulis–Smilga manifolds up to conjugacy in key cases, linking spectral data to geometric properness of actions and broadening the scope beyond previously known examples.

Abstract

In this article, we look at real split semisimple algebraic groups with trivial center and faithful irreducible algebraic representations of on some vector space which admit zero as a weight and which are self-contragredient (for example, adjoint representation of ). We show that, there exist polynomials made out of Margulis invariants of which are also rational expressions in . Moreover, we show that any Zariski dense finitely generated subgroup of , for which the linear parts of the non-identity elements are loxodromic, is isospectrally rigid with respect to the Margulis invariants. In particular, we show that Margulis--Smilga spacetimes are isospectrally rigid too.

Paper Structure

This paper contains 15 sections, 36 theorems, 62 equations.

Table of Contents

  1. Objects of study
  2. Statements of main results
  3. Comparison with previous papers
  4. Preliminaries
  5. Jordan decomposition
  6. Weights and eigenspaces
  7. Algebraicity
  8. Characteristic polynomial
  9. Unit eigenspace projection
  10. Margulis invariant
  11. Isospectral rigidity
  12. Zero spectrum
  13. General case
  14. Applications
  15. Normal subgroups

Key Result

Theorem 1

Suppose $(\mathsf{G},\mathsf{V},\mathtt{R})$ satisfy Hypothesis hyp.1, with $\mathsf{rank}(\mathsf{G})\geq2$, and $\rho,\varrho:\Gamma\to\mathsf{G}\ltimes_\mathtt{R}\mathsf{V}$ both satisfy Hypothesis hyp.2. Then, if $\rho$ and $\varrho$ are isospectral with respect to the Jordan--Margulis invariant

Theorems & Definitions (78)

  • Theorem 1: see Corollary \ref{['cor.JM']}
  • Theorem 2: see Corollary \ref{['cor.marlin']}
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Theorem 1.1.1: Jordan decomposition
  • Definition 1.1.2
  • Remark 1.1.3
  • Definition 1.1.4
  • Remark 1.1.5
  • ...and 68 more