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

Boris Feigin*, Edward Frenkel, Leonid Rybnikov

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

41 Scopus citations

Abstract

The universal enveloping algebra of any simple Lie algebra 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 P1 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 g has a cyclic vector in any irreducible finite-dimensional 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.

Original languageEnglish
Pages (from-to)337-363
Number of pages27
JournalDuke Mathematical Journal
Volume155
Issue number2
DOIs
StatePublished - Nov 2010
Externally publishedYes

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