Opers with irregular singularity and spectra of the shift of argument subalgebra
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 finitedimensional modules is diagonalizable and their joint spectra are in bijection with the set of monodromyfree 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 multipoint 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 finitedimensional gmodule. 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2007
 DOI:
 10.48550/arXiv.0712.1183
 arXiv:
 arXiv:0712.1183
 Bibcode:
 2007arXiv0712.1183F
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematical Physics;
 Mathematics  Algebraic Geometry
 EPrint:
 19 pages