Bispectral and $(\text{glN},\text{glM})$ Dualities

Let $V=<p_{ij}(x)e^{{\text{la}}_{i}x},i=1,\mathrm{.}\mathrm{.}\mathrm{.},n,j=1,\mathrm{.}\mathrm{.}\mathrm{.},N_i>$ be a space of quasi-polynomials of dimension $N=N_1+\mathrm{.}\mathrm{.}\mathrm{.}+N_n$. Define the regularized fundamental operator of $V$ as the polynomial differential operator $D = \sum_{i=0}^N A_{N-i}(x)\p^i$annihilating$V$ and such that its leading coefficient $A_0$ is a polynomial of the minimal possible degree. We construct a space of quasi-polynomials $U=<q_{ab}(u)e^{z_{a}u}>$ whose regularized fundamental operator is the differential operator $\sum_{i=0}^N u^i A_{N-i}(\partial_u)$\mathrm{.}Thespace$U$isconstructedfrom$V$ by a suitable integral transform. Our integral transform corresponds to the bispectral involution on the space of rational solutions (vanishing at infinity) to the KP hierarchy, see \cite{W}. As a corollary of the properties of the integral transform we obtain a correspondence between critical points of the two master functions associated with the $(\text{glN},\text{glM})$ dual Gaudin models as well as between the corresponding Bethe vectors.