Summary: Building on P. Seidel and J. P. Solomon’s fundamental work [Geom. Funct. Anal. 22, No. 2, 443–477 (2012; Zbl 1250.53078)], 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.