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Oct 19, 2016 at 8:19 comment added Venkataramana @John. It does not seem possible for a finite index subgroup $\Gamma $ of $Sp_{2g}(O)$ to have only Abelian stabilizers, since such a $\Gamma $ will always contain a finite index subgroup of $SU(h)(O_K)$ for the $h$ as above.
Oct 19, 2016 at 7:29 vote accept John
Oct 19, 2016 at 7:29 comment added John Thank you for your answer. This is actually very surprising (to me). I have only one question: is it possible at all for $\Gamma$ to act with only abelian stabilizers (if $K$ is real quadratic)?
Oct 19, 2016 at 6:50 history answered Venkataramana CC BY-SA 3.0