nLab Siegel upper half-space




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.


The Siegel upper half space of degree gg (or genus gg) is the set of g×gg\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 gg may be obtained as the quotient g=Sp 2g()/U(g)\mathcal{H}_{g}=\Sp_{2g}(\mathbb{R})/\U(g) of the symplectic group Sp 2g()\Sp_{2g}(\mathbb{R}) by its maximal compact subgroup (and the stabilizer of i𝟙 gi\cdot\mathbb{1}_{g}), the unitary group U(g)\U(g), embedded into Sp 2g()\Sp_{2g}(\mathbb{R}) as follows:

U(g)={(A B B A)Sp 2g():AA t+BB t=𝟙 g}.\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.


Last revised on November 23, 2022 at 04:58:26. See the history of this page for a list of all contributions to it.