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Uniform distribution of orbits of lattices on spaces of frames. (English) Zbl 1067.37004

Let \(G =\text{SL}(n,\mathbb R)\) and \(V_{n,l}\) be the space of \(l\)-frames in \(\mathbb R^n\), \(1\leq l\leq n\). The group \(G\) acts on this space as follows: \(g(\nu_1,\dots,\nu_l)=(g\nu_1,\dots,g\nu_l)\), \(g\in G\) and \((\nu_1,\dots,\nu_n)\in V_{n,l}\). Let \(\Gamma\) be a lattice in \(G\) such that \(\Gamma\setminus G\) has finite volume. The main result of this paper concerns the distribution of \(\Gamma\)-orbits in \(V_{n,l}\). When \(l = n\), every orbit of \(\Gamma\) is discrete. Let \(l< n\), \(F = \text{SL}(n, Z)\) and \(\nu = (\nu_1,\dots,\nu_n)\) be an \(l\)-frame in \(\mathbb R^n\). S. G. Dani and S. Raghavan [Isr. J. Math. 36, 300–320 (1980; Zbl 0457.28008)] proved that the orbit \(\Gamma.\nu\) is dense in \(V_{n,l}\) if and only if the space spanned by \((\nu_i, i =1,\dots,l)\) contains no nonzero rational vectors. W. A. Veech [Am. J. Math. 99, 827–859 (1977; Zbl 0365.28012)] has shown that if \(\Gamma\) is a cocompact lattice in \(G\), then every orbit in \(V_{n,l}\) \(l < n\) is dense. Here, it is shown that dense \(\Gamma\)-orbits are uniformly distributed with respect to an explicitly described measure on \(V_{n,l}\). Dani and Raghvan also considered orbits of frames under \(\text{Sp}(n,\mathbb Z)\). They proved that for an isotropic frame \(\nu = (\nu_1,\dots,\nu_n)\) in \(\mathbb R^{2n}\), \(\Gamma.\nu\) is dense in the space of isotropic \(n\)-frames if and only if the space spanned by \((\nu_i\), \(i =1,\dots,n)\) contains no nonzero rational vectors. Here, this result is improved by showing that the dense orbits of \(\Gamma\) are uniformly distributed. The method of the proof uses the Iwasawa decomposition for \(\text{Sp}(n,\mathbb R)\) and uniform dist of large unipotent subgroups due to Shah. A proof for the results of Dani, Raghvan and Veech based on topological rigidity of unipotent flows is also given.

MSC:

37A15 General groups of measure-preserving transformations and dynamical systems
22E40 Discrete subgroups of Lie groups
37A17 Homogeneous flows
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
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