TY - JOUR
T1 - Opers with irregular singularity and spectra of the shift of argument subalgebra
AU - Feigin, Boris
AU - Frenkel, Edward
AU - Rybnikov, Leonid
PY - 2010/11
Y1 - 2010/11
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=78049314589&partnerID=8YFLogxK
U2 - 10.1215/00127094-2010-057
DO - 10.1215/00127094-2010-057
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AN - SCOPUS:78049314589
SN - 0012-7094
VL - 155
SP - 337
EP - 363
JO - Duke Mathematical Journal
JF - Duke Mathematical Journal
IS - 2
ER -