Abstract
We prove measure rigidity for the action of maximal horospherical subgroups onhomogeneous spaces over a field of positive characteristic. In thecase when the lattice is uniform we prove the action of any horosphericalsubgroup is uniquely ergodic.
Highlights
Let K be a global function field and let S be a finite set of places in K
We let P + denote the group generated by all KS-split unipotent subgroups of P = P(KS), see section 2 for more details
Μ is the probability Haar measure on the closed orbit G+Γ/Γ where the closure is with respect to the Hausdorff topology
Summary
Let K be a global function field and let S be a finite set of places in K. There exists some Kparabolic subgroup P of G and some g ∈ G such that g−1U g ⊂ P = P(KS) and μ is the Σ-invariant measure on the closed orbit Σ gΓ where Σ = gP +(◦P ∩ Γ)g−1 and the closure is with respect to the Hausdorff topology Using this theorem and the Linearization techniques, see [DM93], we conclude, as a corollary, equidistribution of orbits of subgroups satisfying the assumptions of Theorem 1.1. The idea of using mixing property to prove measure classification for horospherical flows, used in this paper, are by no means new the results in positive characteristic are new, see [R83]. Another possible approach to this problem would be to use representation theoretic approach of M. What is surprising is that, to the best of our knowledge, no simpler proof is known even for this very special case
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