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 $\mathfrak{g}\mathfrak{l}(p+m\mathrm{\mid }q+n)$ are diagonalizable on any singular weight space in any tensor product of unitarizable highest weight $\mathfrak{g}\mathfrak{l}(p+m\mathrm{\mid }q+n)$-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 $\mathfrak{g}\mathfrak{l}(r+k)$ on the corresponding singular weight space in the tensor product of some finite-dimensional irreducible $\mathfrak{g}\mathfrak{l}(r+k)$-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 $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$. We also relate the set of singular solutions of the (super) Knizhnik-Zamolodchikov equations for $\mathfrak{g}\mathfrak{l}(p+m\mathrm{\mid }q+n)$ to the set of singular solutions of the Knizhnik-Zamolodchikov equations for $\mathfrak{g}\mathfrak{l}(r+k)$ for $r$ and $k$ sufficiently large.