Abstract: "The Homological Mirror Symmetry conjecture predicts a relationship between the Fukaya category of a symplectic manifold and the category of coherent sheaves on a smooth algebraic variety. A recent result by Gasparim-Grama-San Martin uses Lie theory to construct symplectic Lefschetz fibrations on adjoint orbits of semi-simple Lie algebras. We calculate the Fukaya category of an adjoint orbit of sl(2,C) and show it cannot be equivalent to the derived category of a projective variety, although we can identify it with a triangulated subcategory of the derived category of the second Hirzebruch surface."