  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.
