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.