A functional model for the tensor product of level \(1\) highest and level \(-1\) lowest modules for the quantum affine algebra \(U_q(\widetilde{\mathfrak{sl}}_2)\) (Q1888295)

The authors study the tensor product of a fundamental integrable highest weight \(U_q(\hat{\mathfrak{sl}}_2)\)-module with a fundamental integrable lowest weight \(U_q(\hat{\mathfrak{sl}}_2)\)-module. This tensor product has a filtration by submodules, generated by the tensor product of a highest weight vector and an extremal vector (which is an element, obtained from the lowest weight vector by the braid group action). It is shown that the subquotients of this filtration are isomorphic to level \(0\) extremal weight modules. This implies a functional realization for the completion of the original tensor product by the above filtration. Further, each \(\mathfrak{sl}_2\)-weight space of the original tensor product is shown to have a nice realization via certain symmetric (Laurent) polynomials. Specializing the parameters \(q\) to \(\sqrt{-1}\) the authors prove a conjecture from [\textit{M. Jimbo, T. Miwa, E. Mukhin} and \textit{Y. Takeyama}, ``Form factors and action of \(U_{\sqrt{-1}}(\widetilde{\mathfrak{sl}}_2)\) on \(\infty\)-cycles'', Commun. Math. Phys. 245, No. 3, 551--576 (2004; Zbl 1106.81044)].