Centers of centralizers of nilpotent elements in Lie superalgebras sl(m|n) or osp(m|2n)

Let G¯ be the simple algebraic supergroup SL(m|n) or OSp(m|2n) over C. Let g=Lie(G¯)=g0¯⊕g1¯ and let G=G¯(C) where C is considered as a superalgebra concentrated in even degree. Suppose e∈g0¯ is nilpotent. We describe the centralizer ge of e in g and its centre z(ge). In particular, we give bases for ge, z(ge) and (z(ge))Ge. We also determine the labelled Dynkin diagram Δ with respect to e and subsequently describe the relation between (z(ge))Ge and Δ.