Is the group of symplectomorphisms of a symplectic manifold the symplectic group?

Question

The group of symplectomorphisms of a symplectic manifold $M$ is a subgroup of the group of diffeomorphisms $GL(n)$, actually it is a subgroup of $SL(n)$. My question is whether this group of symplectomorphisms is actually $Sp(n) \subset SL(n) \subset GL(n)$. On the other hand I read that the group of symplectomorphisms is larger than the isometry group of $M$ which is contradictory.

Answer 1

The group of linear symplectomorphisms from an $n$-dimensional symplectic vector space to itself is a subgroup of $GL(n)$, namely $Sp(n)$, but the group of symplectomorphisms from an $n$-dimensional symplectic manifold to itself, $\operatorname{Sympl}(M, \omega)$ is not contained in $GL(n)$. In fact, $\operatorname{Sympl}(M, \omega)$ is an infinite-dimensional Lie group.

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I just realised that you said that $GL(n)$ is the group of diffeomorphisms. That is not what $GL(n)$ denotes, it denotes the general linear group.