On orbits of unipotent flows on homogeneous spaces

Let G be a connected Lie group and let F be a lattice in G (not necessarily co-compact). We show that if («,) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of («,) on G/Y is finite. For 'arithmetic lattices' this was proved in [2]. The present generalization is obtained by studying the 'frequency of visiting compact subsets' for unbounded orbits of such flows in the special case where G is a connected semi-simple Lie group of R-rank 1 and Y is any (not necessarily arithmetic) lattice in G. 0. Introduction Let G be a connected Lie group and Y be a lattice in G; that is, G/Y admits a finite G-invariant (Borel) measure. Let («,),eH be a one-parameter subgroup consisting of unipotent elements (that is, each Ad u, is a unipotent linear transformation of the Lie algebra of G). The action of such a one-parameter group on G/Y (on the left) is dubbed a unipotent flow. In the particular case when G = SL (n, R) and Y = SL («, Z) it was proved by Margulis [6] that for any xeG/Y the positive semi-orbit {u,x\ t >0} of a unipotent flow does not tend to infinity; that is, there exists a compact set K, depending on x, such that the set is unbounded. In [2] the present author strengthened this assertion by proving that for a suitable compact set K the set EK(x) is of positive density; that is, lim inf ^/(£*(*) n [0 , r ] )>0 , (0.1) where / is the usual Haar measure on R. Further as an application of this it was deduced that every locally finite ergodic invariant measure of a unipotent flow, and more generally of the action of any subgroup consisting only of unipotent elements, is finite. Using certain standard facts about arithmetic subgroups the above results can easily be generalized to the situation where G is any connected Lie group and Y is an arithmetic lattice in G (cf. [2] for details). In [3], where the above finiteness assertion was used crucially in the classification of invariant measures of maximal use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700002248 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 07 Sep 2017 at 06:22:54, subject to the Cambridge Core terms of