Finite dimensional representations and subgroup actions on homogeneous spaces

Barak Weiss

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10 Scopus citations

Abstract

Let H be an ℝ-subgroup of a ℚ-algebraic group G. We study the connection between the dynamics of the subgroup action of H on G/G and the representation-theoretic properties of H being observable and epimorphic in G. We show that if H is a ℚ-subgroup then H is observable in G if and only if a certain H orbit is closed in G/G; that if H is epimorphic in G then the action of H on G/G is minimal, and that the converse holds when H is a ℚ-subgroup of G; and that if H is a ℚ-subgroup of G then the closure of the orbit under H of the identity coset image in G/G is the orbit of the same point under the observable envelope of H in G. Thus in subgroup actions on homogeneous spaces, closures of 'rational orbits' (orbits in which everything which can be defined over ℚ, is defined over ℚ) are always submanifolds.

Original languageEnglish
Pages (from-to)189-207
Number of pages19
JournalIsrael Journal of Mathematics
Volume106
DOIs
StatePublished - 1 Jan 1998
Externally publishedYes

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