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

TL;DR

This paper establishes a connection between the spectra of quantum shift of argument subalgebras in simple Lie algebras and monodromy-free opers, extending to irregular singularities and Gaudin models, with implications for geometric representation theory.

Contribution

It proves the diagonalizability and spectral bijection of quantum shift of argument subalgebras with monodromy-free opers, including multi-point cases and geometric structures.

Findings

Spectra correspond to monodromy-free opers with irregular singularities.

Quantum shift of argument subalgebras have cyclic vectors in finite-dimensional modules.

Modules have a Gorenstein ring structure related to intersection cohomology.

Abstract

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. 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 opers for the Langlands dual group of G on the projective line with regular singularity at one point and irregular singularity of order two at another point. We also prove a multi-point generalization of this result, describing the spectra of commuting Hamiltonians in Gaudin models with irregular singulairity. In addition, we show that the quantum shift of argument subalgebra corresponding to a regular nilpotent element of g has a cyclic vector in any irreducible finite-dimensional g-module. As a byproduct, 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.