## Duke Mathematical Journal

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

#### Abstract

The universal enveloping algebra of any simple Lie algebra $\mathfrak{g}$ 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 $\mathbb{P}^1$ 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 $\mathfrak{g}$ has a cyclic vector in any irreducible finite-dimensional $\mathfrak{g}$-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

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

Dates
First available in Project Euclid: 27 October 2010

https://projecteuclid.org/euclid.dmj/1288185458

Digital Object Identifier
doi:10.1215/00127094-2010-057

Mathematical Reviews number (MathSciNet)
MR2736168

Zentralblatt MATH identifier
1226.22017

#### Citation

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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