Siegel upper half-space
In mathematics, given a positive integer <math>g</math>, the Siegel upper half-space <math>\mathcal H_g</math> of degree <math>g</math> is the set of <math>g \times g</math> symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Template:Harvs. The space <math>\mathcal H_g</math> is the symmetric space associated to the symplectic group <math>\mathrm{Sp}(2g,\mathbb R)</math>. When <math>g=1</math> one recovers the Poincaré upper half-plane.
The space <math>\mathcal H_g</math> is sometimes called the Siegel upper half-plane.<ref>Template:Cite journal</ref>
Definitions
As a complex domain
The space <math>\mathcal H_g</math> is the subset of <math>M_g(\mathbb C)</math> defined by :
- <math>\mathcal H_g = \{ X+iY : X, Y \in M_g(\mathbb R), X^t = X,\, Y^t = Y,\, Y \text{ is definite positive} \}. </math>
It is an open subset in the space of <math>g\times g</math> complex symmetric matrices, hence it is a complex manifold of complex dimension <math>\tfrac{g(g+1)} 2</math>.
This is a special case of a Siegel domain.
As a symmetric space
The symplectic group <math>\mathrm{Sp}(2g, \mathbb R)</math> can be defined as the following matrix group:
- <math>\mathrm{Sp}(2g, \mathbb R) = \left\{ \begin{pmatrix} A & B \\ C & D \end{pmatrix}: A, B, C, D \in M_g(\mathbb R) ,\, AB^t- BA^t = 0, CD^t - DC^t = 0, AD^t - BC^t = 1_g \right\}. </math>
It acts on <math>\mathcal H_g</math> as follows:
- <math>Z\mapsto (AZ+B)(CZ+D)^{-1} \text{ where } Z\in\mathcal{H}_g, \begin{pmatrix}A&B\\ C&D\end{pmatrix}\in \mathrm{Sp}_{2g}(\mathbb{R}).</math>
This action is continuous, faithful and transitive. The stabiliser of the point <math>i1_g \in \mathcal H_g</math> for this action is the unitary subgroup <math>U(g)</math>, which is a maximal compact subgroup of <math>\mathrm{Sp}(2g, \mathbb R)</math>.Template:Sfn Hence <math>\mathcal H_g</math> is diffeomorphic to the symmetric space of <math>\mathrm{Sp}(2g, \mathbb R)</math>.
An invariant Riemannian metric on <math>\mathcal H_g</math> can be given in coordinates as follows:
- <math>d s^2 = \text{tr}(Y^{-1} dZ Y^{-1} d \bar{Z}),\, Z = X+iY.</math>
Relation with moduli spaces of Abelian varieties
Siegel modular group
The Siegel modular group is the arithmetic subgroup <math>\Gamma_g = \mathrm{Sp}(2g, \mathbb Z)</math> of <math>\mathrm{Sp}(2g, \mathbb R)</math>.
Moduli spaces
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} The quotient of <math>\mathcal H_g</math> by <math>\Gamma_g</math> can be interpreted as the moduli space of <math>g</math>-dimensional principally polarised complex Abelian varieties as follows.Template:Sfn If <math>\tau = X+iY \in \mathcal H_g</math> then the positive definite Hermitian form <math>H</math> on <math>\mathbb C^g</math> defined by <math>H(z, w) = w^*Y^{-1} z</math> takes integral values on the lattice <math>\mathbb Z^g + \mathbb Z^g \tau</math>We view elements of <math>\mathbb Z^g</math> as row vectors hence the left-multiplication. Thus the complex torus <math>\mathbb C^g / \mathbb Z^g + \mathbb Z^g \tau</math> is a Abelian variety and <math>H</math> is a polarisation of it. The form <math>H</math> is unimodular which means that the polarisation is principal. This construction can be reversed, hence the quotient space <math>\Gamma_g \backslash \mathcal H_g</math> parametrises principally polarised Abelian varieties.
See also
- Paramodular group, a generalization of the Siegel modular group
- 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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