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 -valued connections, for a simply-laced Lie algebra . We prove that subdominant solutions to the ODE defined in different fundamental representations satisfy a set of quadratic equations called -system. This allows us to show that the generalized spectral determinants satisfy the Bethe Ansatz equations.