Projective cocycles over SL(2,R) actions: measures invariant under the upper triangular group

Abstract

We consider the action of SL(2, R) on a vector bundle H preserving an ergodic probability measure nu on the base X. Under an irreducibility assumption on this action, we prove that if (nu) over cap is any lift of nu to a probability measure on the projectivized bunde P(H) that is invariant under the upper triangular subgroup, then (nu) over cap is supported in the projectivization P(E-1) of the top Lyapunov subspace of the positive diagonal semigroup. We derive two applications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycle depend continuously on affine measures, answering a question in [57]. Second, if P(V) is an irreducible, flat projective bundle over a compact hyperbolic surface Sigma, with hyperbolic foliation F tangent to the flat connection, then the foliated horocycle flow on (TF)-F-1 is uniquely ergodic if the top Lyapunov exponent of the foliated geodesic flow is simple. This generalizes results in [13] to arbitrary dimension.