Distribution of lattice orbits on homogeneous varieties

Abstract

Given a lattice Γ in a locally compact group G and a closed subgroup H of G, one has a natural action of Γ on the homogeneous space V = H\G. For an increasing family of finite subsets {Γ T : T > 0}, a dense orbit v·Γ, v ∈ V and compactly supported function ϕ on V , we consider the sums S ϕ,v (T ) = γ∈ΓT ϕ(vγ). Understanding the asymptotic behavior of S ϕ,v (T ) is a delicate problem which has only been considered for certain very special choices of H, G and {Γ T }. We develop a general abstract approach to the problem, and apply it to the case when G is a Lie group and either H or G is semisimple. When G is a group of matrices equipped with a norm, we have S ϕ,v (T ) ∼ GT ϕ(vg)dg, where G T = {g ∈ G : g < T } and Γ T = G T ∩Γ. We also show that the asymptotics of S ϕ,v (T ) is governed by V ϕ dν, where ν is an explicit limiting density depending on the choice of v and · .

Lattice actions on the plane revisited

“…Write u = u 1 u 2 ∈ R 2 , and define a norm | · | and a 'product' on R 2 by |v| = max {|v 1 |, |v 2 |} , v u = −u 2 v 1 u 1 v 1 −u 2 v 2 u 1 v 2 , where v = v 1 v 2 . Let dx denote the Lebesgue measure on P. It was shown in the above-mentioned papers (see particularly [GW,§12.4]) that S f,u (T )…”

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

“…The results stated in [25] implies an asymptotic for each intersection of these sets, whence also an asymptotic for their union. 20 Now, let us indicate the proof of the second assertion of (6.12).…”

“…Having established basic definitions concerning tempered and 1 m -tempered representations, we now show that certain naturally occurring examples, viz., representations of the real points of an algebraic group, on the real points of an (algebraic) homogeneous spaces, have such properties. 20 The reference [25] assumes that S connected. It also proves "Theorem 2.7" only in a special case, deferring the general case to another paper.…”

“…Throughout, we fix a Sobolev norm S d , where d is sufficiently large (depending only on G, H). 25 We may suppose (adjusting x 1 or x 2 by an element of S) that x 1 = x 2 exp(r), with r ∈ r, and will establish a lower bound on r .…”

Ergodic theory and the duality principle on homogeneous spaces

“…We also use the same notation for the corresponding set of quadratic forms. It is known that for almost all quadratic forms Q in the space of nondegenerate quadratic forms of given dimension, the setQ (F d [GW,Sec. 1.5.1], and here we show that the asymptotic formula from [GW] holds with an exponentially decaying error term for almost all quadratic forms Q ∈ Q p,q (∆).…”