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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.
MSC:
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
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