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

TL;DR

This paper demonstrates that for certain algebraic groups and their representations, Margulis invariants can be expressed as polynomials and rational functions, leading to isospectral rigidity results for associated spacetimes.

Contribution

It establishes the polynomial and rational nature of Margulis invariants and proves isospectral rigidity for a class of Margulis--Smilga spacetimes in this setting.

Findings

Existence of polynomial expressions for Margulis invariants.

Zariski dense subgroups are isospectrally rigid with respect to Margulis invariants.

Margulis--Smilga spacetimes are shown to be isospectrally rigid.

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.