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  "body": "\n## Description\n\nLet $g \\ge 1$ be an integer.\nThe *Siegel upper half-space* of degree $g$ is\n$$\\mathbb{H}_g = \\set{Z \\in M_{g \\times g}(\\C) : Z^T = Z,\\ \\operatorname{Im} Z \\text{ is positive definite}}.$$\nIt is an open subset of the space of complex symmetric $g \\times g$ matrices, so it is a complex manifold of dimension $g(g+1)/2$.\nFor $g = 1$ it is the [[upper half-plane|X6XGGH5FPW]].\n\nThe real [[symplectic group|X12FS2JAP0]] $Sp(2g, \\R)$, the group of real $2g \\times 2g$ matrices $M = \\begin{pmatrix} A \u0026 B \\\\ C \u0026 D \\end{pmatrix}$ with $M^T J M = J$ for $J = \\begin{pmatrix} 0 \u0026 I_g \\\\ -I_g \u0026 0 \\end{pmatrix}$, acts on $\\mathbb{H}_g$ by\n$$M \\cdot Z = (AZ + B)(CZ + D)^{-1}.$$\nFor $Z = X + iY \\in \\mathbb{H}_g$, the matrix $\\begin{pmatrix} Y^{1/2} \u0026 X Y^{-1/2} \\\\ 0 \u0026 Y^{-1/2} \\end{pmatrix}$ is symplectic and sends $iI_g$ to $Z$, so the action is transitive.\nThe stabilizer of $iI_g$ is the subgroup with $A = D$ and $C = -B$, which is isomorphic to the unitary group $U(g)$ via $A + iB$.\nHence $\\mathbb{H}_g \\cong Sp(2g, \\R)/U(g)$.\n\n## Remarks\n\nThe Siegel upper half-space is a [[Hermitian symmetric space|NT4797Z9H8]] of noncompact type, the type $III_g$ in Cartan's list.\nThe Cayley transform $Z \\mapsto (Z - iI)(Z + iI)^{-1}$ maps it biholomorphically onto the bounded domain $\\set{W : W^T = W,\\ I - W^* W \u003e 0}$.\nThe quotient $Sp(2g, \\Z) \\backslash \\mathbb{H}_g$ is the moduli space of principally polarized abelian varieties of dimension $g$: the point $Z$ corresponds to the complex torus $\\C^g / (\\Z^g + Z \\Z^g)$.\nSiegel modular forms are functions on $\\mathbb{H}_g$ with a transformation law under $Sp(2g, \\Z)$.\n\n## Explanations\n\n### L1 human\n\nThe upper half-plane is the set of complex numbers with positive imaginary part.\nThe Siegel upper half-space replaces the number by a symmetric matrix and \"positive\" by \"positive definite\".\nJust as the upper half-plane parametrizes elliptic curves, the Siegel space parametrizes higher-dimensional complex tori with a polarization, through their period matrices.\n",
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