Abstract
We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space S=Gamma backslash G/K is compact. More precisely, given a sequence of homogeneous probability measures on S, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including S itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when G={mathbf{SL}}_3({{mathbb {R}}}) and Gamma ={mathbf{SL}}_3({{mathbb {Z}}}).
Highlights
1.1 Background and motivationsThe study of sequences of measures invariant under unipotent flows has been a central theme in homogeneous dynamics, and the deep theorems obtained have had several important arithmetic applications
Such that the projection π1 on to the first factor is given by restricting to the corresponding subtorus. It follows that the restriction of (·, ·) to X ∗(AI )Q is a non-degenerate scalar product that is invariant with respect to the action of NMI (AI )(Q)
Μ is the homogeneous probability measure on \G associated with a connected algebraic subgroup H of G of type H and an element g ∈ G, and, Hn is contained in H for all n large enough
Summary
The study of sequences of measures invariant under unipotent flows has been a central theme in homogeneous dynamics, and the deep theorems obtained have had several important arithmetic applications Prototypical in this respect is Margulis’s proof of the Oppenheim Conjecture concerning the values of irrational indefinite quadratic forms at integral vectors [12]. In [10], building on the earlier work of Dani–Margulis [8], Eskin–Mozes–Shah proved a nondivergence criterion for sequences of homogeneous measures and, motivated by a counting problem for lattice points on homogeneous varieties, applied this to show in [9] that, when, for all n ∈ N, Hn = H, for a fixed reductive subgroup H of G not contained in a proper Q-parabolic subgroup of G, any weak limit μ of μn = μH,gn is homogeneous. This paper deals principally with the convergence of measures on general locally symmetric spaces and their Satake compactifications, but we hope to discuss the Baily–Borel compactification, and possible applications, in a future work
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