Simplicity of spectra for Bethe subalgebras in $Y(\mathfrak{gl}_2)$

Abstract: We consider Bethe subalgebras B(C) in the Yangian $Y(\mathfrak{gl}_2)$ with $C$ regular $2\times 2$ matrix. We study the action of Bethe subalgebras of $Y(\mathfrak{gl}_2)$ on finite-dimensional representations of $Y(\mathfrak{gl}_2)$. We prove that $B(C)$ with real diagonal $C$ has simple spectrum on any irreducible $Y(\mathfrak{gl}_2)$-module corresponding to a disjoint union of real strings. We extend this result to limits of Bethe algebras. Our main tool is the computation of Shapovalov-type determinant for the nilpotent degeneration of $B(C)$.