The Siegel upper half space of degree $g$ (or genus $g$) is the set of $g\times g$symmetric matrices whose entries are complex numbers with positive-definite imaginary part.

Relation to the Symplectic Group

The Siegel upper half space of degree $g$ may be obtained as the quotient $\mathcal{H}_{g}=\Sp_{2g}(\mathbb{R})/\U(g)$ of the symplectic group$\Sp_{2g}(\mathbb{R})$ by its maximal compact subgroup (and the stabilizer of $i\cdot\mathbb{1}_{g}$), the unitary group$\U(g)$, embedded into $\Sp_{2g}(\mathbb{R})$ as follows:

$\U(g)=\lbrace \begin{pmatrix} A & B \\ -B & A\end{pmatrix}\in\Sp_{2g}(\mathbb{R}):A\cdot A^{t}+B\cdot B^{t}=\mathbb{1}_{g}\rbrace.$