Equidistribution of expanding translates of curves in homogeneous spaces with the action of $(\mathrm{SO}(n,1))^k$

TL;DR

This paper proves that under certain geometric conditions, the translates of an analytic curve in a homogeneous space with an $( ext{SO}(n,1))^k$ action become equidistributed as the parameter tends to infinity, using representation theory.

Contribution

It establishes equidistribution results for translates of curves in homogeneous spaces under the action of diagonal subgroups, extending previous understanding in this setting.

Findings

Translates of certain curves become equidistributed in the space.

The result applies to typical diagonal subgroup actions.

The proof utilizes linear representation theory of $ ext{SO}(n,1)$ and related groups.

Abstract

Given a homogeneous space $X = G/\Gamma$ with $G$ containing the group $H = (\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$ satisfies certain geometric condition, then for a typical diagonal subgroup $A =\{a(t): t \in \mathbb{R}\} \subset H$ the translates $\{a(t)\phi(I)x: t >0\}$ of the curve $\phi(I)x$ will tend to be equidistributed in $X$ as $t \rightarrow +\infty$. The proof is based on the study of linear representations of $\mathrm{SO}(n,1)$ and $H$.