In , the Siegel upper half-space of degree g (or genus g) (also called the ) is the set of g × g over the whose is . It was introduced by. It is the associated to the Sp(2g, R) . The Siegel upper half-space has properties as a that generalize the properties of the , 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 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 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 such that J^2 = -1 and for all vectors v \ne 0.