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L. G. Rybnikov Institute for Theoretical and Experimental Physics Moscow, Russia leo.rybnikov@gmail.com

Op ers with irregular singularity and sp ectra of the shift of argument subalgebra
(Joint work with Boris Feigin and Edward Frenkel, math.QA/0712.1183.) Let g be a semisimple complex Lie algebra, and U (g) its universal enveloping algebra. The algebra U (g) bears a natural filtration by the degree with respect to the generators. The associated graded algebra is the symmetric algebra S (g) = C[g ] by the Poincar´ e­Birkhoff­ Witt theorem. The commutator on U (g) defines the Poisson bracket on S (g). Let Z S (g) = S (g)g be the center of S (g) with respect to the Poisson bracket, and let µ g = g be a regular semisimple element. Due to the result of Mischenko and Fomenko (1978) n the algebra Aµ S (g) generated by the elements µ , where Z S (g), is commutative with respect to the Poisson bracket, and has maximal possible transcendence degree equal 1 to 2 (dim g + rk g). If µ is a regular element contained in the Cartan subalgebra h g, then h Aµ , and if µ = f is the principal nilpotent element then the subalgebra Af contains the principal nilpotent subalgebra zg (f ). It was recently shown that the shift of argument subalgebras can be quantized: Fact 1. (R., Feigin ­ Frenkel ­ Toledano Laredo) There exist a family of commutative subalgebras Aµ U (g) (where µ greg ) such that gr Aµ = Aµ . ^ The subalgebra Aµ U (g) is the image of some homomorphism Z (g) - U (g), where ^ ^ Z (g) is the center of the completed enveloping algebra of g at the critical level. The main question we discuss is to describe the spectra of Aµ in finite-dimensional irreducible g-modules. The most interesting case is the "most special" case when µ = f is the principal nilpotent element. The principal gradation on U (g) is defined on the generators as follows. deg
pr

e = -(, ) ,

deg

pr

h = 0 h h.

The generators of Af are homogeneous with respect to this gradation. Note that the Poincar´ e series of Af with respect to the principal gradation is equal to that of the algebra U (n- ). I shall discuss the following main Theorem 1. (Feigin ­ Frenkel ­ R.) For any integral dominant weight the highest vector of V is a cyclic vector for Af acting on V . Thus the space V is natural ly identified with a quotient of Af by a certain ideal I Af . Af /I is a complete intersection. ^ The spectrum of the center at the critical level Z (g) is identified with the space of L G-opers on the punctured formal disc (where L G is the Langlands dual group for G). L G-opers are connections in the principal GL -bundle satisfying a certain transversality condition. Namely, for a curve U = Spec R and some coordinate t on U , the space OpL G (U ) of L G-opers is the quotient of the space of L G-connections of the form d + (p
-1

+ v(t))dt,

v(t) L b(R) maximal unipotent subgroup, GL = GLr , and GL -opers are with the algebra of polynomial Frenkel and Toledano Laredo

by the action of the group L N (R), where L N L G is the and L b L g is the Borel subalgebra. For G = GLr , we have simply differential operators of the degree r. ^ Since Aµ is a quotient of Z (g), the algebra Aµ is identified functions on a certain space of opers. It is shown by Feigin, that Spec Aµ is the set of L G-opers, which

1


1. are defined globally on CP 1 \{0, }, 2. have regular singularity at 0 3. have an irregular singularity of the degree 2 at with the 2-residue µ. Moreover, the image of Aµ in End(V ) factors through the opers, which 1. have the residue at 0; 2. have trivial monodromy representation. For every dominant weight , the no-monodromy condition of corresponding L G-opers is a finite set of polynomial relations in the generators of Af . The number of such relations is equal to the number of positive roots of g. These relations have the degrees ( , + ) with respect to the principal grading. We prove that the no-monodromy conditions generate the ideal I . Note that this agrees with the q -analog of the Weyl dimension formula. Namely, the Poincar´ series of any irreducible finite-dimensional g-module V with respect to the principal e grading is 1 - q ( ,+) . (q ) = 1 - q ( ,) >0 We note that the non-central generators of Af have the degrees ( , ) with respect to the principal grading, and the no-monodromy relations have the degrees ( , + ), and hence Af /I has the same Poincar´ series. e

2