Flag-transitive subgroups of Chevalley groups with Abelian stabilizers (Q1057366)

The groups named in the title are shown to leave a ''polarity'' of the associated Tits building invariant (under an additional technical hypothesis). This is used to prove the following theorem: Let G be a Chevalley group of rank \(\geq 2\) over the reals resp. over an algebraically closed field K. Then the flag-transitive subgroups of G with Abelian stabilizers are exactly the maximal compact subgroups of G resp. the groups \(\bar G(\)k) where k is a subfield of K with \([K:k]=2\), \(\bar G\) is a simple algebraic group defined over K with \(\bar G(\)K)\(=G\) and \(\bar G(\)k) is the group of points of \(\bar G\) over k for some k- compact k-structure on \(\bar G.\)