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

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