  We study highest weight representations of shifted Yangians over an
algebraically closed field of characteristic 0. In particular, we classify the
finite dimensional irreducible representations and explain how to compute their
Gelfand-Tsetlin characters in terms of known characters of standard modules and
certain Kazhdan-Lusztig polynomials. Our approach exploits the relationship
between shifted Yangians and the finite W-algebras associated to nilpotent
orbits in general linear Lie algebras.
