1 November 2010 Opers with irregular singularity and spectra of the shift of argument subalgebra
Boris Feigin, Edward Frenkel, Leonid Rybnikov
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Duke Math. J. 155(2): 337-363 (1 November 2010). DOI: 10.1215/00127094-2010-057


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.


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Boris Feigin. Edward Frenkel. Leonid Rybnikov. "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


Published: 1 November 2010
First available in Project Euclid: 27 October 2010

zbMATH: 1226.22017
MathSciNet: MR2736168
Digital Object Identifier: 10.1215/00127094-2010-057

Primary: 22E46
Secondary: 34M40 , 82B23

Rights: Copyright © 2010 Duke University Press


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Vol.155 • No. 2 • 1 November 2010
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