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29 March 2024
 
  » arxiv » math.QA/0510364

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Bispectral and $(glN,glM)$ Dualities
E. Mukhin ; V. Tarasov ; A. Varchenko ;
Date 18 Oct 2005
AbstractLet $V = < p_{ij}(x)e^{la_ix}, i=1,...,n, j=1, ..., N_i >$ be a space of quasi-polynomials of dimension $N=N_1+...+N_n$. 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 $U = < q_{ab}(u)e^{z_au} >$ whose regularized fundamental operator is the differential operator $sum_{i=0}^N u^i A_{N-i}(partial_u)$. The space $U$ is constructed from $V$ 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 $(glN,glM)$ dual Gaudin models as well as between the corresponding Bethe vectors.
Source arXiv, math.QA/0510364
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