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

We consider the space $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$ 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 $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$, constructed by Rubtsov, Silantyev, and Talalaev in 2009. We describe the relations between the action of $\mathcal B$ on $\operatorname{Fun}_{\mathfrak{sl}_2}V[0]$ and spaces of pairs of quasi-polynomials.
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