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Critical points of the product of powers of linear functions and families of bases of singular vectors. (English) Zbl 0842.17044
The quasiclassical asymptotics of Varchenko-Schechtman solutions for the \({\mathfrak {sl}}_2\)-Knizhnik-Zamolodchikov equation are studied. It is shown that asymptotics of basic solutions can be described as certain matrix-valued functions evaluated at critical points of special rational functions. The solutions have to be singular vectors in the tensor product of \({\mathfrak {sl}}_2\)-Verma modules and are the eigenfunctions of a family of commuting operators of Gaudin type. The Shapovalov norm of the asymptotic solutions is calculated. It is shown how to deduce the norm of Bethe eigenvectors in the Gaudin model from this result.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
33D80 Connections of basic hypergeometric functions with quantum groups, Chevalley groups, \(p\)-adic groups, Hecke algebras, and related topics
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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