Symplectic group (awaiting review)

The symplectic group $\Sp(2g,\R)$ of degree $g$ is a subgroup of $\SL(2g,\R)$, $$\Sp(2g,\R) =\left\{ M=\left(\begin{matrix}A&B\\C&D\end{matrix}\right)\in\GL(2g,\R): \,^tMJM=J\right\},$$ where $J=\left(\begin{matrix}0&I\\-I&0\end{matrix}\right)$ and $^tM$ denotes the transpose of $M$. The defining condition $^tMJM=J$ on elements of $\GL(2g,\R)$ forces them to have determinant 1, although this is not immediate.

Thus the integral symplectic group $\Sp(2g,\Z)$, defined similarly and also called the full modular group, is all of $\Sp(2g, \R)\cap{\rm M}(2g,\Z)$.