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We consider the action of $SL(2,\mathbb{R})$ on a vector bundle $\mathbf{H}$preserving an ergodic probability measure $\nu$ on the base $X$. Under anirreducibility assumption on this action, we prove that if $\hat\nu$ is anylift of $\nu$ to a probability measure on the projectivized bunde$\mathbb{P}(\mathbf{H})$ that is invariant under the upper triangular subgroup,then $\hat \nu$ is supported in the projectivization $\mathbb{P}(\mathbf{E}_1)$of the top Lyapunov subspace of the positive diagonal semigroup. We derive twoapplications. First, the Lyapunov exponents for the Kontsevich-Zorich cocycledepend continuously on affine measures, answering a question in [MMY]. Second,if $\mathbb{P}(\mathbf{V})$ is an irreducible, flat projective bundle over acompact hyperbolic surface $\Sigma$, with hyperbolic foliation $\mathcal{F}$tangent to the flat connection, then the foliated horocycle flow on$T^1\mathcal{F}$ is uniquely ergodic if the top Lyapunov exponent of thefoliated geodesic flow is simple. This generalizes results in [BG] to arbitrarydimension.

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