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

A. N. Varchenko   ; A. M. Slinkin   ; D. Thompson

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 4, pp. 653-684

Résumé

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. Bibliography: 25 titles.

Détail

Keywords: commuting differential operators, eigenfunctions, Weyl group invariance, Bethe Ansatz, Wronskian equation
Mots-clés : quasi-polynomials.

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