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Let $G$ be a semisimple Lie group. Embed it as a subgroup into a special linear group of suitable rank, $SL(n)$ (real or complex). The question is: is it always possible to find such an embedding, that the Bruhat cells in $G$ are equal to the intersections of $G$ with the Bruhat cells in $SL(n)$? How can such embeddings be characterized, if they exist?

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    $\begingroup$ It's probably better to start with a complex linear algebraic group, where the ideas about Bruhat decomposition are worked out in more detail. Though Bruhat developed his ideas first in the Lie group setting, the work of Chevalley and Borel-Tits on semisimple algebraic groups was more systematic. In any case, things get much more complicated if the underlying field is not algebraically closed. $\endgroup$
    – Jim Humphreys
    Oct 9, 2014 at 22:54
  • $\begingroup$ This is not quite what I need, but if anybody knows the answer in the case of complex linear algebraic groups, I will be happy to hear it! $\endgroup$
    – user59308
    Oct 11, 2014 at 8:07

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