Dynamical ${\mathfrak{sl}}_2$ Bethe algebra and functions on pairs of quasi-polynomials

Abstract

We consider the space ${\text{Fun}}_{{\mathfrak{sl}}_2}V[0]$ of functions on the Cartan subalgebra of ${\mathfrak{sl}}_2$ with values in the zero weight subspace $V[0]$ of a tensor product of irreducible finite-dimensional ${\mathfrak{sl}}_2$-modules. We consider the algebra $\mathcal{B}$ of commuting differential operators on ${\text{Fun}}_{{\mathfrak{sl}}_2}V[0]$, constructed by V.Rubtsov, A.Silantyev, D.Talalaev in 2009. We describe the relations between the action of $\mathcal{B}$ on ${\text{Fun}}_{{\mathfrak{sl}}_2}V[0]$ and spaces of pairs of quasi-polynomials.

Latex, 29 pages