Siegel upper half-space

In mathematics, given a positive integer ${\displaystyle g}$, the Siegel upper half-space ${\displaystyle {\mathcal {H}}_{g}}$ of degree ${\displaystyle g}$ is the set of ${\displaystyle g\times g}$ symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939). The space ${\displaystyle {\mathcal {H}}_{g}}$ is the symmetric space associated to the symplectic group ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$. When ${\displaystyle g=1}$ one recovers the Poincaré upper half-plane.

The space ${\displaystyle {\mathcal {H}}_{g}}$ is sometimes called the Siegel upper half-plane.

The space ${\displaystyle {\mathcal {H}}_{g}}$ is the subset of ${\displaystyle M_{g}(\mathbb {C} )}$ defined by :

It is an open subset in the space of ${\displaystyle g\times g}$ complex symmetric matrices, hence it is a complex manifold of complex dimension ${\displaystyle {\tfrac {g(g+1)}{2}}}$.

This is a special case of a Siegel domain.

The symplectic group ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$ can be defined as the following matrix group:

It acts on ${\displaystyle {\mathcal {H}}_{g}}$ as follows:

This action is continuous, faithful and transitive. The stabiliser of the point ${\displaystyle i1_{g}\in {\mathcal {H}}_{g}}$ for this action is the unitary subgroup ${\displaystyle U(g)}$, which is a maximal compact subgroup of ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$. Hence ${\displaystyle {\mathcal {H}}_{g}}$ is diffeomorphic to the symmetric space of ${\displaystyle \mathrm {Sp} (2g,\mathbb {R} )}$.