Bethe algebra and the algebra of functions on the space of differential operators of order two with polynomial kernel

Abstract.

We show that the algebra of functions on the scheme of monic linear second-order ordinary differential operators with prescribed n + 1 regular singular points, prescribed exponents $\Lambda^{(1)},\ldots,\Lambda^{(n)},\Lambda^{(\infty)}$ at the singular points, and having the kernel consisting of polynomials only, is isomorphic to the Bethe algebra of the Gaudin model acting on the vector space Sing $L_{\Lambda^{(1)}}\,\otimes\,\cdots\,\otimes L_{\Lambda^{(n)}}[\Lambda^{(\infty)}]$ of singular vectors of weight Λ^(∞) in the tensor product of finite-dimensional polynomial ${\mathfrak{g}}{\mathfrak{l}}_2$-modules with highest weights $\Lambda^{(1)},\ldots,\Lambda^{(n)}$.