Bispectral and (\mathfrak{gl}_{N},\mathfrak{gl}_{M}) dualities

Let be a space of quasipolynomials of dimension N=N ₁+⋅⋅⋅+N _n . We define the regularized fundamental operator of V as the polynomial differential operator D=∑ _i=0 ^N A _N−i (x)∂ _x ⁱ annihilating V and such that its leading coefficient A ₀ is a polynomial of the minimal possible degree. We apply a suitable integral transformation to V to construct a space of quasipolynomials whose regularized fundamental operator is the differential operator ∑ _i=0 ^N u ⁱ A _N−i (∂ _u ). Our integral transformation corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) of the KP hierarchy. As a corollary of the properties of the integral transformation, we obtain a correspondence between critical points of the two master functions associated with the -dual Gaudin models and also between the corresponding Bethe vectors. The research of E. M. was supported in part by the NSF (Grant No. DMS-0140460). The research of A. V. was supported in part by the NSF (Grant No. DMS-0244579).