= Solution
For $p\in\operatorname{Fix}(\sigma)$, differentiation gives $(d\sigma_p)^2=I$ and $(d\sigma_p)^*\omega_p=-\omega_p$. Thus $d\sigma_p$ is an <anti-symplectic involution> of the <symplectic vector space> $T_pM$. By part (a), its $+1$ <eigenspace> is a <Lagrangian subspace>.
In the local <manifold chart> supplied in the question, $\sigma$ is linear, so its <fixed-point set> is the intersection of the chart with this $+1$ <eigenspace>. Therefore the <fixed-point set> is an <embedded submanifold> and
$$
T_p\operatorname{Fix}(\sigma)=\ker(d\sigma_p-I).
$$
Every such <tangent space> has half the ambient dimension and the <symplectic form> vanishes on it. Hence \b[every nonempty <fixed-point set> is a <Lagrangian submanifold>]. The <fixed-point set> itself need not be <connected>.
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