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

TL;DR

This paper establishes an isomorphism between the algebra of functions on a scheme of specific second-order differential operators with polynomial solutions and the Bethe algebra acting on singular vectors in tensor products of $gl_2$-modules, linking differential operators with algebraic structures.

Contribution

It demonstrates a novel isomorphism between the algebra of functions on differential operators with polynomial solutions and the Bethe algebra in representation theory.

Findings

Proves the isomorphism between the two algebras.

Connects differential operators with algebraic Bethe structures.

Provides a new perspective on the algebraic structure of differential operators.

Abstract

We show that the following two algebras are isomorphic. The first is the algebra $A_P$ of functions on the scheme of monic linear second-order differential operators on $\C$ with prescribed regular singular points at $z_1,..., z_n, \infty$, prescribed exponents $\La^{(1)}, ..., \La^{(n)}, \La^{(\infty)}$ at the singular points, and having the kernel consisting of polynomials only. The second is the Bethe algebra of commuting linear operators, acting on the vector space $\Sing L_{\La^{(1)}} \otimes ... \otimes L_{\La^{(n)}}[\La^{(\infty)}]$ of singular vectors of weight $\La^{(\infty)}$ in the tensor product of finite dimensional polynomial $gl_2$-modules with highest weights $\La^{(1)},..., \La^{(n)}$.