  Building on Seidel-Solomon's fundamental work, we define the notion of a
$\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$
where $\mathfrak{g} \subset SH^1(M)$ is a sub-Lie algebra of the symplectic
cohomology of $M$. When $M$ is a (symplectic) mirror to an (algebraic)
homogeneous space $G/P$, homological mirror symmetry predicts that there is an
embedding of $\mathfrak{g}$ in $SH^1(M)$. This allows us to study a mirror
theory to classical constructions of Borel-Weil and Bott. We give explicit
computations recovering all finite dimensional irreducible representations of
$\mathfrak{sl}_2$ as representations on the Floer cohomology of an
$\mathfrak{sl}_2$-equivariant Lagrangian brane and discuss generalizations to
arbitrary finite-dimensional semisimple Lie algebras.
