Siegel Upper Half Plane

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 . It is the symmetric space associated to the symplectic group .

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 . 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 = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . 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 are proportional to
:$d s^2 = \text(Y^ dZ Y^ d \bar).$

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

As a symmetric space of non-compact type, the Siegel upper half space $\mathcal_g$ is the quotient
:$\mathcal_g = \mathrm(2g,\mathbb)/\mathrm(n),$
where we used that $\mathrm(n)=\mathrm(2g,\mathbb)\cap \mathrm(g,\mathbb)$ is the maximal torus.
Since the isometry group of a symmetric space $G/K$ is $G$, we recover that the isometry group of $\mathcal_g$ is $\mathrm(2g,\mathbb)$. An isometry acts via a generalized Möbius transformation
:$Z\mapsto (AZ+B)(CZ+D)^ \text Z\in\mathcal_g, \left(\beginA&B\\ C&D\end\right)\in \mathrm_(\mathbb).$
The quotient space $\mathcal_g/\mathrm(2g,\mathbb)$ 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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Complex analysis
Automorphic forms
Differential geometry
1939 introductions

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