Siegel upper half-space

In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939).

The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case g=1. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group Sp(2g, C). Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group SL(2, C) = Sp(2, C), the Siegel upper half-space has only one metric up to scaling whose isometry group is Sp(2g, C). Writing a generic matrix Z in the Siegel upper half-space in terms of its real and imaginary parts as Z = X + iY, all metrics with isometry group Sp(2g, C) are proportional to

${\displaystyle ds^{2}={\text{tr}}(Y^{-1}dZY^{-1}d{\bar {Z}}).}$