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. 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
 * $$d s^2 = \text{tr}(Y^{-1} dZ Y^{-1} d \bar{Z}).$$

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