Quadratic and cubic Gaudin Hamiltonians and super Knizhnik-Zamolodchikov equations for general linear Lie superalgebras

We show that under a generic condition, the quadratic Gaudin Hamiltonians associated to gl(p+m|q+n) are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight gl(p+m|q+n)-modules. Moreover, every joint eigenbasis of the Hamiltonians can be obtained from some joint eigenbasis of the quadratic Gaudin Hamiltonians for the general linear Lie algebra gl(r+k) on the corresponding singular weight space in the tensor product of some finite-dimensional irreducible gl(r+k)-modules for r and k sufficiently large. After specializing to p = q = 0, we show that similar results hold as well for the cubic Gaudin Hamiltonians associated to gl(m|n). We also relate the set of singular solutions of the (super) Knizhnik-Zamolodchikov equations for gl(p+m|q+n) to the set of singular solutions of the Knizhnik-Zamolodchikov equations for gl(r+k) for r and k sufficiently large.