Why is the automorphism group of a sympelctic symmetric space a Lie group?


A symplectic symmetric space is a triple $(M, \omega, s)$, where $(M, \omega)$ is a symplectic manifold and $ s \; \colon M \times M \to M $, $(x, y) \mapsto s_x(y)$, is such that $s_x$ is an involutive symplectic diffeomorphism with an isolated fixed point at $x$ and $s_xs_ys_x = s_{s_x(y)} \; \forall \; x, y \in M\; $ $\big($this can be read as $s_x s = ss_x \; \forall x \in M$$\big)$.

An interesting fact about a symplectic symmetric space is the one that once $x$ is an isolated fixed point of $s_x$ and $s_x^2 = Id$, $ds_{x (x)} = -Id_{T_xM}$. Furthermore, a symplectic symmetric space admits a unique affine connection $\nabla$ on it such that $s_x$ is an affinity for every $x \in M$ and such that $\nabla \omega = 0$. This connection has no torsion and is thus a symplectic connection.

The automorphism group $Aut(M, \omega, s)$ of $(M, \omega, s)$ is the group of symplectic automorphisms $\varphi$ of $(M, \omega)$ that satisfy $\varphi \circ s_x = s_{\varphi(x)} \circ \varphi \; \forall \; x \in M$ $\big($ or, in other words, $\varphi \circ s = s\circ \varphi$ $\big)$. It's simple to check that this group is the intersection of the symplectic automorphism group of $(M, \omega)$ and the affine group of $(M, \nabla)$.

My question then is: why is $Aut(M, \omega, s)$ a Lie group?


The affine group of $(M,\nabla)$ is a Lie group $G$ by Kobayashi's theorem that shows that the automorphism group of any affine connection is a Lie group (see Kobayashi and Nomizu's Foundations of Differential Geometry). The dimension of $G$ is at most $n+n^2$ (where $n=\dim M$).

The subgroup $H$ of $G$ consisting of those elements of $G$ that preserve the symplectic structure on $M$ is clearly a closed subgroup, and since every closed subgroup of a Lie group is a Lie group (see, for example, Helgason's proof in his Differential Geometry, Lie Groups, and Symmetric Spaces), it follows that $H$ is a Lie group as well.

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