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 .[2] Hence is diffeomorphic to the symmetric space of .
An invariant Riemannian metric on can be given in coordinates as follows:
The quotient of by can be interpreted as the moduli space of -dimensional principally polarised complex Abelian varieties as follows.[3] 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.
Siegel modular form, a type of automorphic form defined on the Siegel upper half-space
Siegel modular variety, a moduli space constructed as a quotient of the Siegel upper half-space
References
^Friedland, Shmuel; Freitas, Pedro J. (2004). "Revisiting the Siegel upper half plane. I". Linear Algebra Appl. 376: 19–44. doi:10.1016/S0024-3795(03)00662-1.