= Siegel upper half-space
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{title2=$\mathfrak H_r=\{Z=Z^{\mathsf T}:\operatorname{Im}Z>0\}$}
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Positive complex <Lagrangian subspaces> for the integral symplectic form $\left(\begin{smallmatrix}0&-I\\ I&0\end{smallmatrix}\right)$ are graphs $H_Z=\{(Zu,u):u\in\mathbb C^r\}$. Isotropy is the symmetry of $Z$, and $iQ(v,\bar v)=2u^{\mathsf T}(\operatorname{Im}Z)\bar u$ gives positivity. For $g=\left(\begin{smallmatrix}A&B\\ C&D\end{smallmatrix}\right)$ in the <symplectic group>, the action is $Z\mapsto(AZ+B)(CZ+D)^{-1}$.
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