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The real symplectic group of degree $g$ denoted by $\Sp(2g,\R)$ is defined as $\Sp(2g,\R) =\left\{ M=\left(\begin{matrix} a & b\\ c & d\end{matrix}\right)\in\GL(2g,\R): \, M^t J M=J\right\},$ where $J=\left(\begin{matrix}0 & 1_g\\ -1_g & 0\end{matrix}\right)$ and $1_g$ is the identity matrix of size $g$.

By using the following two properties of the Pfaffian: $\text{Pf}(M^t J M)=\text{Pf}(J)$ and $\text{Pf}(M^t J M)=\det(M)\text{Pf}(J)$, we see that $\Sp(2g,\R)$ is a subgroup of $\SL(2g,\R)$.

The integral symplectic group $\Gamma_g=\Sp(2g,\Z)$, also called the Siegel modular group, is defined as $\Gamma_g=\Sp(2g,\R)\cap{\rm M}(2g,\Z)$. Clearly $\Gamma_1=\SL(2,\Z)$.

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• Last edited by Fabien Cléry on 2021-04-27 09:21:02
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