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Closures of locally divergent orbits of maximal tori and values of homogeneous forms

Part of: Geometry of numbers Ergodic theory Diophantine approximation, transcendental number theory Lie groups Noncompact transformation groups

Published online by Cambridge University Press:  05 November 2020

GEORGE TOMANOV
GEORGE TOMANOV*
Affiliation:
Institut Camille Jordan, Université Claude Bernard – Lyon I, Bâtiment de Mathématiques, 43 Bld. du 11 Novembre 1918, 69622Villeurbanne Cedex, France (e-mail: [email protected])
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Abstract

Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $ \overline{T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e. $\overline{T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

Keywords

locally divergent orbitshomogeneous orbit closurevalues of forms at integer points

MSC classification

Primary: 22F30: Homogeneous spaces 22E40: Discrete subgroups of Lie groups
Secondary: 11J68: Approximation to algebraic numbers 11H46: Products of linear forms 37A17: Homogeneous flows
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 10 , October 2021 , pp. 3142 - 3177
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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