On measures invariant under tori on quotients of semisimple groups


We classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple $S$-algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces: a torus, a homogeneous space on which the measure is (up to finite index) the Haar measure, a product of homogeneous spaces on each of which the action degenerates to a rank one action, and a homogeneous space in which every element of the action acts with zero entropy.


Manfred Einsiedler

ETH Zürich, Zürich, Switzerland

Elon Lindenstrauss

Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel