Abstract: To a symplectic manifold M one can associate various algebraic structures built from pseudo-holomorphic curves, such as the quantum cohomology ring and the Fukaya category. One way to formulate the phenomenon of mirror symmetry is to say that structures associated to the symplectic geometry of M are typically equivalent to structures associated to the algebraic geometry of a rather different space X. In one direction, this allows us to use tools from algebraic geometry to study symplectic manifolds. In the other direction, the symplectic interpretation may make manifest a structure that is surprising from the algebraic point of view. In this talk, I will describe a line of research that incorporates both directions. On the one hand, we can translate an algebraic group action on X into the notion of an equivariant Lagrangian submanifold in M. This gives us representations of Lie algebras in Floer cohomology groups. On the other hand, the symplectic geometry yields certain distinguished bases of these Floer cohomology groups, related to the "theta functions" of Gross-Hacking-Keel, and we expect to the canonical bases arising in representation theory.