On Effective Equidistribution of Expanding Translates of Certain Orbits in the Space of Lattices

The latter, by definition, means that the Lie algebra of H is the span of eigenspaces of Ad(gt), t > 0, with eigenvalues bigger than 1 in absolute value. The space X def = G/Γ can be identified with the space of unimodular lattices in R, on which G acts by left translations. Denote by π the natural projection G → X, g 7→ gΓ, and for any z ∈ X let πz : G → X be defined by πz(g) = gz. Also denote by μ̄ the G-invariant probability measure on X and by μ the Haar measure on G such that π∗μ = μ̄. Fix a Haar measure ν on H . Note that the H-orbit foliation is unstable with respect to the action of gt, t > 0. It is well known that for any Borel probability measure ν on H absolutely continuous with respect to ν and for any z ∈ X , gt-translates of (πz)∗ν ′ become equidistributed, that is, weak-∗ converge to μ̄ as t → ∞. An effective version of this statement was obtained in [KM1, Proposition 2.4.8]. In order to state that result, it will be convenient to introduce the following notation: for f ∈ L(H, ν), a bounded continuous function ψ on X , z ∈ X and g ∈ G define