Kakutani equivalence of unipotent flows

Abstract

We study Kakutani equivalence in the class of unipotent flows acting on finite-volume quotients of semisimple Lie groups. For every such flow, we compute the Kakutani invariant of M. Ratner, the value being explicitly given by the Jordan block structure of the unipotent element generating the flow. This, in particular, answers a question of M. Ratner. Moreover, it follows that the only loosely Kronecker unipotent flows are given by $\left(\begin{array}{cc}1& t\\ 0& 1\end{array}\right)\times \mathrm{id}$ acting on $(\mathrm{SL}(2,\mathbb{R})\times G^{\prime })∕{\mathrm{\Gamma }}^{\prime }$, where ${\mathrm{\Gamma }}^{\prime }$ is an irreducible lattice in $\mathrm{SL}(2,\mathbb{R})\times G^{\prime }$ (with the possibility that $G^{\prime }=\{e\}$).