  This paper studies the self-Floer theory of a monotone Lagrangian submanifold
$L$ of a symplectic manifold $X$ in the presence of various kinds of symmetry.
First we suppose $L$ is $K$-homogeneous and compute the image of low
codimension $K$-invariant subvarieties of $X$ under the length-zero closed-open
string map. Next we consider the group $\mathrm{Symp}(X, L)$ of
symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the
Oh spectral sequence to coefficients of arbitrary characteristic, incorporating
its action on the classes of holomorphic discs. This imposes constraints on the
differentials which force them to vanish in certain situations. These
techniques are combined to study a family of homogeneous Lagrangians in
products of projective spaces, which exhibit some unusual properties.
