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). It is the symmetric space associated to the symplectic group Sp(2g, R).

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, R). 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, R) = Sp(2, R), the Siegel upper half-space has only one metric up to scaling whose isometry group is Sp(2g, R). 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, R) are proportional to

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

The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure ${\displaystyle \omega }$, on the underlying ${\displaystyle 2n}$ dimensional real vector space ${\displaystyle V}$, that is, the set of ${\displaystyle J\in Hom(V)}$ such that ${\displaystyle J^{2}=-1}$ and ${\displaystyle \omega (Jv,v)>0}$ for all vectors ${\displaystyle v\neq 0}$.^[1]