Duke Mathematical Journal

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

Boris Feigin, Edward Frenkel, and Leonid Rybnikov

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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.

Article information

Duke Math. J., Volume 155, Number 2 (2010), 337-363.

First available in Project Euclid: 27 October 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E46: Semisimple Lie groups and their representations
Secondary: 82B23: Exactly solvable models; Bethe ansatz 34M40: Stokes phenomena and connection problems (linear and nonlinear)


Feigin, Boris; Frenkel, Edward; Rybnikov, Leonid. Opers with irregular singularity and spectra of the shift of argument subalgebra. Duke Math. J. 155 (2010), no. 2, 337--363. doi:10.1215/00127094-2010-057. https://projecteuclid.org/euclid.dmj/1288185458

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