 Siegel upper half-space

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

The Siegel upper half-space, typically denoted as \( \mathcal{H}_g \), is a concept from several complex variables and algebraic geometry. It is a generalization of the upper half-plane concept found in one complex variable and is an important object in the study of several complex variables, algebraic curves, and arithmetic geometry.