# On Measures Invariant Under Diagonalizable Groups on Quotients of Semi-simple Groups

# On Measures Invariant Under Diagonalizable Groups on Quotients of Semi-simple Groups

Actions of diagonalizable algebraic groups (which are referred to as tori in the theory of algebraic groups, though in cases of interest to us are not compact) on arithmetic quotient spaces play an important role in many number theoretic and other applications. I will present a joint result with Einsiedler (extending earlier work with A. Katok) where we classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple $S$-algebraic groups invariant under a maximal split diagonalizable group (torus) in at least one simple factor, and show that the algebraic support of such a measure splits into the product of four homogeneous spaces: an algebraic 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.