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 gl_d and gl_p+m|q+n on the Fock space of d(p+m) bosonic and d(q+n) fermionic oscillators. This establishes a duality of (gl_d,gl_p+m|q+n) for Gaudin models. As an application, we show that the Gaudin algebra with irregular singularities for gl_p+m|q+n acts cyclically on each weight space of a certain class of infinite-dimensional modules over a direct sum of Takiff superalgebras over gl_p+m|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 (gl_d,gl_p+m|q+n) for classical Gaudin models.