nLab Siegel upper half-space

Idea

The Siegel upper half space is the generalization of the upper half-plane as one passes from the description of complex elliptic curves to that of Riemann surfaces of higher genus.

Definition

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:

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

Related concepts

• Siegel modular form

• symplectic group

References

• Wikipedia, Siegel upper half-space