Siegel upper half-space (awaiting review)

The Siegel upper half-space of degree $g$ is denoted by $\mathcal{H}_g$ and defined as \[ \mathcal{H}_g= \left\{ \tau \in \text{Mat}(g\times g): \tau=\tau^t, \, \text{Im}(\tau)>0 \right\}. \]
It is the set of $g\times g$ complex symmetric matrices which have positive definite imaginary part. It is acted on by the real symplectic group via \[ \left( \begin{matrix} \Sp(2g,\R)\times \mathcal{H}_g & \to & \mathcal{H}_g\\ (M=\left(\begin{matrix}a & b\\ c & d\end{matrix}\right), \tau) &\mapsto & (a\tau+b)(c\tau+d)^{-1} \end{matrix} \right).\] The action of any discrete subgroup of $\Sp(2g,\R)$ on $\mathcal{H}_g$ is properly discontinuous. Note that the integral symplectic group $\Gamma_g$ does not act freely on $\mathcal{H}_g$.