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The Siegel upper half-space of degree gg is denoted by Hg\mathcal{H}_g and defined as Hg={τMat(g×g):τ=τt,Im(τ)>0}. \mathcal{H}_g= \left\{ \tau \in \text{Mat}(g\times g): \tau=\tau^t, \, \text{Im}(\tau)>0 \right\}.
It is the set of g×gg\times g complex symmetric matrices which have positive definite imaginary part. It is acted on by the real symplectic group via (Sp(2g,R)×HgHg(M=(abcd),τ)(aτ+b)(cτ+d)1). \left( \begin{matrix} \Sp(2g,\R)\times \mathcal{H}_g & \to & \mathcal{H}_g\\ (M=\left(\begin{matrix}a & b\\ c & d\end{matrix}\right), \tau) &\mapsto & (a\tau+b)(c\tau+d)^{-1} \end{matrix} \right). The action of any discrete subgroup of Sp(2g,R)\Sp(2g,\R) on Hg\mathcal{H}_g is properly discontinuous. Note that the integral symplectic group Γg\Gamma_g does not act freely on Hg\mathcal{H}_g.

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  • Review status: beta
  • Last edited by Fabien Cléry on 2023-11-17 19:03:31
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