Siegel upper half-space

{{Short description|Set of complex matrices with positive definite imaginary part}}

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 {{harvs|txt|authorlink=Carl Ludwig Siegel|last=Siegel|year=1939}}. It is the symmetric space associated to the symplectic group {{math|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 {{math|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 {{math|SL(2, R)}} = {{math|Sp(2, R)}}, the Siegel upper half-space has only one metric up to scaling whose isometry group is {{math|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 {{math|Sp(2g, R)}} are proportional to

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

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

As a symmetric space of non-compact type, the Siegel upper half space $\backslash mathcal\{H\}\_g$ is the quotient

:$\backslash mathcal\{H\}\_g\; =\; \backslash mathrm\{Sp\}(2g,\backslash mathbb\{R\})/\backslash mathrm\{U\}(n),$

where we used that $\backslash mathrm\{U\}(n)=\backslash mathrm\{Sp\}(2g,\backslash mathbb\{R\})\backslash cap\; \backslash mathrm\{GL\}(g,\backslash mathbb\{C\})$ is the maximal torus.

Since the isometry group of a symmetric space $G/K$ is $G$, we recover that the isometry group of $\backslash mathcal\{H\}\_g$ is $\backslash mathrm\{Sp\}(2g,\backslash mathbb\{R\})$. An isometry acts via a generalized Möbius transformation

:

The quotient space $\backslash mathcal\{H\}\_g/\backslash mathrm\{Sp\}(2g,\backslash mathbb\{Z\})$ is the moduli space of principally polarized abelian varieties of dimension $g$.