Uniformly distributed orbits of certain flows on homogeneous spaces

Let G be a connected Lie group, F be a lattice in G and U = {ut},~R be a unipotent one-parameter subgroup of G, viz. Adu is a unipotent linear transformation for all u ~ U. Consider the flow induced by the action of U (on the left) on G/F. Such a flow is referred as a unipotent flow on the homogeneous space G/F. The study of orbits of unipotent flows has been the subject of several papers. For a nilpotent group G, a result of Green [13] implies that if U has one dense orbit in G/F then every orbit of U is uniformly distributed with respect to a G-invariant measure on G/F. In the case when G = SL(2, R), it was proved by Hedlund that every orbit of the unipotent (horocycle) flow is either dense or periodic; periodic orbits exist only when G/F is non-compact. For a co-compact lattice, this result was strengthened by F/irstenberg [1 lJ proving that every orbit is uniformly distributed with respect to a G-invariant measure. For non-uniform lattices in SL(2, R), using a classification of invariant measures obtained by Dani in [21, Dani and Smillie [31 proved that every non-periodic orbit is uniformly distributed. There are also various results obtained on orbit closures and invariant measures etc. of larger subgroups consisting of unipotent elements, especially the horospherical subgroups. Recently, there was a spurt in the area initiated by Margulis' proof (cf. [151, see also [7]) of Oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows. The reader is referred to the survey articles by Dani [41 and Margulis [14J for various related developments. We now note some conjectures expected to hold for orbits of a unipotent flow, namely the U-action on G/F as above (though we restrict to U being a oneparameter subgroup, the first two conjectures are expected to hold for any subgroup generated by unipotent elements contained in it). A conjecture due to Raghunathan on orbit closures states the following: