Siegel upper half-space

Template:Short description 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 Template:Harvs. It is the symmetric space associated to the symplectic group Template:Math.

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 Template:Math. 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 Template:Math = Template:Math, the Siegel upper half-space has only one metric up to scaling whose isometry group is Template:Math. 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 Template:Math are proportional to

$d{s}^2=\text{tr}(Y^{-1}dZ{Y}^{-1}d\overline{Z}).$

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 $J\in \mathrm{H}\mathrm{o}\mathrm{m}(V)$ such that $J^2=-1$ and $\omega (Jv,v)>0$ for all vectors $v\ne 0$.

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

${\mathscr{H}}_g=\mathrm{S}\mathrm{p}(2g,ℝ)/\mathrm{U}(n),$

where we used that $\mathrm{U}(n)=\mathrm{S}\mathrm{p}(2g,ℝ)\cap \mathrm{G}\mathrm{L}(g,ℂ)$ is the maximal torus. Since the isometry group of a symmetric space $G/K$ is $G$, we recover that the isometry group of ${\mathscr{H}}_g$ is $\mathrm{S}\mathrm{p}(2g,ℝ)$. An isometry acts via a generalized Möbius transformation

$Z\mapsto (AZ+B)(CZ+D{)}^{-1}\text{ where }Z\in {\mathscr{H}}_g,\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\in {\mathrm{S}\mathrm{p}}_{2g}(ℝ).$

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

See also

• Moduli of abelian varieties
• Paramodular group, a generalization of the Siegel modular group
• Siegel domain, a generalization of the Siegel upper half space
• 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

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