Siegel upper half-space

In mathematics, given a positive integer, the Siegel upper half-space of degree is the set of symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by. The space is the symmetric space associated to the symplectic group. When one recovers the Poincaré upper half-plane.
The space is sometimes called the Siegel upper half-plane.

Definitions

As a complex domain

The space is the subset of defined by :
It is an open subset in the space of complex symmetric matrices, hence it is a complex manifold of complex dimension.
This is a special case of a Siegel domain.

The symplectic group can be defined as the following matrix group:
It acts on as follows:
This action is continuous, faithful and transitive. The stabiliser of the point for this action is the unitary subgroup, which is a maximal compact subgroup of. Hence is diffeomorphic to the symmetric space of.
An invariant Riemannian metric on can be given in coordinates as follows:

Relation with moduli spaces of Abelian varieties

Siegel modular group

The Siegel modular group is the arithmetic subgroup of.

Moduli spaces

The quotient of by can be interpreted as the moduli space of -dimensional principally polarised complex Abelian varieties as follows. If then the positive definite Hermitian form on defined by takes integral values on the lattice We view elements of as row vectors hence the left-multiplication. Thus the complex torus is a Abelian variety and is a polarisation of it. The form is unimodular which means that the polarisation is principal. This construction can be reversed, hence the quotient space parametrises principally polarised Abelian varieties.