Distribution of lattice orbits on homogeneous varieties

By Alexander Gorodnik, Barak Weiss. 2004

64 citations recorded by the source corpus.

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 υ· Γ, υ∈V and compactly supported function φ on V, we consider the sums $$ S\varphi ,\upsilon{\left( T \right)} = {\sum\nolimits_{\gamma \in \Gamma _{T} } {\varphi {\left( {\upsilon\gamma } \right)}} } $$ . Understanding the asymptotic behavior of Sφ,υ(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 $$S_{\varphi,\upsilon}(T)\sim \int_{G_{T}}\varphi(\upsilon g) dg,$$ where GT = {g ∈G: ||g|| < T} and Γ T = GT∩ Γ. We also show that the asymptotics of Sφ, υ(T) is governed by $$\int_{V}\varphi d \nu,$$ where ν is an explicit limiting density depending on the choice of υ and || · ||.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fbf6b2c668095f083f813491b11df1046824410d

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Citations in the corpus

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces (2007, background): “at 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. 20The reference [25] assumes that Sconnected. It also proves \Theorem 2.7" only in a special case, deferring th”
Counting lattice points (2009, background): “We recall that a is independent of the choice of norm, see [36] and [62] for the general case.”
The Ergodic Theory of Lattice Subgroups (2006, methodology): “eover, for every p>r≥ 1, there exists θ>0 such that for f∈ Lp(m G/Γ) and a.e. x∈ G/Γ, µx,t(f)− Z G/Γ fdµG/Γ ≤ C(x,f)e−θp,rt with kC(·,f)kLr(µ G/Γ) ≤ CkfkLp(µ G/Γ). Proof. Part (1) was proved in [GW], Section 12.3. To prove part (2), it suﬃces to observe that the representation of G on L2 0(G/Γ”
Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices (2005, methodology): “Counter-examples with the same type of behaviour were given in [DRS],[EM], [ GW ], but here the variety (G) considered is ane symmetric, whereas theirs was not.”