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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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References:
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