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Stable conjugacy of connected subgroups of real algebraic groups. (English) Zbl 0836.20064
Given a field \(k\) and a connected \(k\)-group \(G\), connected \(k\)-subgroups \(H\), \(H' \subset G\) are called stably conjugate over \(k\) if there exists \(x \in G\) such that \(H' = xHx^{-1}\) and the isomorphism \(H \to H'\), \(h \mapsto xhx^{-1}\), is defined over \(k\). One may ask whether stably conjugate subgroups are \(k\)-conjugate. In general, the answer is known to be negative. The author describes some cases when the answer is positive. Namely, this is the case when \(k\) is a field of characteristic 0 and \(H\), \(H'\) are solvable \(k\)-split subgroups. In the real case, using some results of G. Harder [Math. Z. 92, 396-415 (1966; Zbl 0152.01001)] and D. Shelstad [Compos. Math. 39, 11-45 (1979; Zbl 0431.22011)], one can obtain a slightly more general result replacing the hypothesis on \(H\) and \(H'\) to be \(k\)-split by the following one: a maximal \(\mathbb{R}\)- torus of \(H\) is either a maximal torus of \(G\) or an \(\mathbb{R}\)-split subtorus of \(G\).
As a corollary, one obtains two more cases when the answer to the above question is positive: (1) the unipotent radical of \(H\) is the unipotent radical of a minimal parabolic \(\mathbb{R}\)-subgroup of \(G\); (2) \(G\) is a reductive \(\mathbb{R}\)-group with anisotropic semisimple part and both \(H\) and \(H'\) are \(\mathbb{R}\)-subgroups of maximal rank. The proofs are based on using Galois cohomology.
20G15 Linear algebraic groups over arbitrary fields
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
11E72 Galois cohomology of linear algebraic groups
Full Text: DOI
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