Dualities of Gaudin models with irregular singularities for general linear Lie (super)algebras

We prove an equivalence between the actions of the Gaudin algebras with irregular singularities for ${\mathfrak{g}\mathfrak{l}}_d$ and ${\mathfrak{g}\mathfrak{l}}_{p+m\mathrm{\mid }q+n}$ on the Fock space of $d(p+m)$ bosonic and $d(q+n)$ fermionic oscillators. This establishes a duality of $({\mathfrak{g}\mathfrak{l}}_d,{\mathfrak{g}\mathfrak{l}}_{p+m\mathrm{\mid }q+n})$ for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for ${\mathfrak{g}\mathfrak{l}}_{p+m\mathrm{\mid }q+n}$ acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over ${\mathfrak{g}\mathfrak{l}}_{p+m\mathrm{\mid }q+n}$ and that the action is diagonalizable with a simple spectrum under a generic condition. We also study the classical versions of Gaudin algebras with irregular singularities and demonstrate a duality of $({\mathfrak{g}\mathfrak{l}}_d,{\mathfrak{g}\mathfrak{l}}_{p+m\mathrm{\mid }q+n})$ for classical Gaudin models.