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

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