The universal enveloping algebra of any simple Lie algebra 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 with regular singularity at one point and irregular singularity of order 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 has a cyclic vector in any irreducible finite-dimensional -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.
"Opers with irregular singularity and spectra of the shift of argument subalgebra." Duke Math. J. 155 (2) 337 - 363, 1 November 2010. https://doi.org/10.1215/00127094-2010-057