Bethe Ansatz and the Spectral Theory of Affine Lie Algebra-Valued Connections I. The simply-laced Case

We study the ODE/IM correspondence for ODE associated to ({widehat{mathfrak{g}}})-valued connections, for a simply-laced Lie algebra ({mathfrak{g}}). We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called ({Psi})-system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.