Opers with irregular singularity and spectra of the shift of argument subalgebra

Abstract

The universal enveloping algebra of any simple Lie algebra $\mathfrak{g}$ contains a family of commutative subalgebras, called the quantum shift of argument subalgebras. We prove that generically their action on finite-dimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromy-free $^LG$-opers on ${\mathbb{P}}^1$ with regular singularity at one point and irregular singularity of order $2$ at another point. We also prove a multipoint generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singularity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of $\mathfrak{g}$ has a cyclic vector in any irreducible finite-dimensional $\mathfrak{g}$-module. As a by-product, we obtain the structure of a Gorenstein ring on any such module. This fact may have geometric significance related to the intersection cohomology of Schubert varieties in the affine Grassmannian.