Measure Rigidity for forms of SL(2,R)^k

We outline a recently finished joint work with Elon Lindenstrauss concerning measure rigidity for diagonal actions on irreducible quotients of SL(2,R)^k. Allowing general diagonal subgroups a new argument was required. We explain this new argument relying on quantitative recurrence properties along stable subgroups, and utilising the arithmetic nature of the space.