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Hello

Someone knows something about the complex reductive Lie groups (the complexification of a compact Lie group) which are not defined over the real numbers. I would like to know, if is it possible, an simple example.

I appreciate any help or references.

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The way you seem to be defining "complex reductive group" is not the standard procedure, and almost defines the answer to be negative (once you clarify what you mean by "defined over the real numbers"; e.g., the complexification structure you give makes the answer negative tautologically). But even if you use the "right" definition (in terms of the theory of linear algebraic groups) there are no examples because any connected reductive group over an alg. closed field of char. 0 is defined over $\mathbf{Q}$ (e.g., by inspecting the classification of simply connected cases and their centers). –  grp Oct 8 '12 at 4:28
    
Your question has (at least) 2 meanings, depending whether you mean definED or definABLE over reals. The stabilizer in $GL_2(\mathbf{C})$ of a pair of distinct, non-real, non-conjugate lines in $\mathbf{C}^2$ is not defined over $\mathbf{R}$ (but it's certainly definable over $\mathbf{R}$, being a 2-torus). –  YCor Oct 8 '12 at 9:36

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