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 .
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 . 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 =, the Siegel upper half-space has only one metric up to scaling whose isometry group is . 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 are proportional to
ds2=tr(Y-1dZY-1d\bar{Z}).
The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure
\omega
2n
V
J\inHom(V)
J2=-1
\omega(Jv,v)>0
v\ne0