Siegel upper half-space

Description

Let $g \ge 1$ be an integer. The Siegel upper half-space of degree $g$ is $$\mathbb{H}_g = \set{Z \in M_{g \times g}(\C) : Z^T = Z,\ \operatorname{Im} Z \text{ is positive definite}}.$$ It is an open subset of the space of complex symmetric $g \times g$ matrices, so it is a complex manifold of dimension $g(g+1)/2$. For $g = 1$ it is the upper half-plane.

The real symplectic group $Sp(2g, \R)$, the group of real $2g \times 2g$ matrices $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ with $M^T J M = J$ for $J = \begin{pmatrix} 0 & I_g \\ -I_g & 0 \end{pmatrix}$, acts on $\mathbb{H}_g$ by $$M \cdot Z = (AZ + B)(CZ + D)^{-1}.$$ For $Z = X + iY \in \mathbb{H}_g$, the matrix $\begin{pmatrix} Y^{1/2} & X Y^{-1/2} \\ 0 & Y^{-1/2} \end{pmatrix}$ is symplectic and sends $iI_g$ to $Z$, so the action is transitive. The stabilizer of $iI_g$ is the subgroup with $A = D$ and $C = -B$, which is isomorphic to the unitary group $U(g)$ via $A + iB$. Hence $\mathbb{H}_g \cong Sp(2g, \R)/U(g)$.