# Reference request: a verification of a nonstandard subgroup being a Tits subgroup.

I have a particular infinite-index subgroup $H$ of the genus 2 symplectic group $Sp(2, \mathbb{R})$. This subgroup is self-normalizing (ie. $gHg^{-1}=H$ only if $g\in H$). I am looking to determine whether or not this subgroup is a Tits subgroup of $Sp_2$. That is, I want to know whether or not the pair $(G,H)$ extends to a Tits system, ie. whether $H \backslash G/H$ admits a cellular decomposition $\coprod_{w\in W} HwH$ where $W$ is some set (which can be endowed with group structure following Bourbaki Ch IV.2 Ex.3).

Besides the instances where $H$ is obviously a Borel or parabolic subgroup, can anyone refer me to an instance in the literature where a (possibly ad-hoc) verification of a nonstandard subgroup being a Tits subgroup is performed?

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Is the subgroup $H$ supposed to be closed (in the Zariski or real topology)? Otherwise it's hard to get any help from the known structure of the ambient group as the group of real points of an algebraic group or as a Lie group. – Jim Humphreys May 6 '12 at 13:59
The subgroup $H$ is closed, being an extension of the standard embedding of $SL_2 \times SL_2$ into $Sp(2, \mathbb{R})$by an order two cyclic group. Explicitly $H$ is the stabilizer in $Sp(2, \mathbb{R})$ of the standard $\omega$-orthogonal splitting $\mathbb{R}(e_1, f_1) \oplus \mathbb{R}(e_2, f_2)$ of the 4-dimensional linear symplectic space $(\mathbb{R}^4, \omega)$. I am presently trying to ''organize'' the action of $H$ on the building of $Sp(2, \mathbb{R})$, ie. the flag complex of totally isotropic subspaces. – J. Martel May 6 '12 at 22:23