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

0.28 MB·English

by  Boris Feigin

OPERS WITH IRREGULAR SINGULARITY AND SPECTRA OF
THE SHIFT OF ARGUMENT SUBALGEBRA
BORIS FEIGIN1,EDWARD FRENKEL2, ANDLEONID RYBNIKOV3
8
0
0 Abstract. Theuniversalenvelopingalgebra of anysimple Liealgebra g con-
2
tains a family of commutative subalgebras, called the quantum shift of argu-

n ment subalgebras [R, FFT]. We prove that generically their action on finite-
a dimensional modules is diagonalizable and their joint spectra are in bijection
J with the set of monodromy-free LG-opers on P1 with regular singularity at

5 onepointandirregularsingularityofordertwoatanotherpoint. Wealsoprove
2 a multi-point generalization of this result, describing the spectra of commut-

ing Hamiltonians in Gaudin models with irregular singulairity. In addition, we
]
A showthatthequantumshiftofargumentsubalgebracorrespondingtoaregular
nilpotent element of g has a cyclic vector in any irreducible finite-dimensional
Q
g-module. As a byproduct, we obtain the structure of a Gorenstein ring on
. any such module. This fact may have geometric significance related to the
h
t intersection cohomology of Schubertvarieties in theaffine Grassmannian.
a
m
[

1. Introduction
2
v
Let g be a simple Lie algebra over C. The symmetric algebra S(g) carries a
3
8 naturalPoissonstructure. APoissoncommutativesubalgebra A of S(g) =C[g∗],
µ
1 called the classical shift of argument subalgebra, was defined in [MF] (see also
1
[Ma]). It is generated by the derivatives of all orders in the direction of µ ∈ g∗
.
2 of all elements of the algebra of g-invariants in C[g∗]. Recently, this algebra
1
was quantized in [R, FFT]. More precisely, a commutative subalgebra A of the
7 µ
0 universal enveloping algebra U(g) of g was constructed and it was proved that for
v: all regular µ ∈ g∗ the associated graded of Aµ, with respect to the order filtration
i on U(g) is A . Thealgebra A is called thequantum shift of argument subalgebra
X µ µ
of U(g). This is a free polynomial algebra in 1(dimg+rkg) generators for any
r 2
a regular µ ∈ g∗ (see [R, FFT]).
For any regular µ the algebra A contains the centralizer of µ in g. In par-
µ
ticular, if µ is regular semi-simple, then A contains a Cartan subalgebra of g.
µ
Hence it acts on weight subspaces of g-modules. We note that, as shown in [R], a
certain limit of A in the case when g = sl may be identified with the Gelfand–
µ n
Zetlin algebra. Hence the algebra A may be thought of as a generalization of
µ
the Gelfand–Zetlin algebra to an arbitrary simple Lie algebra. It is an interesting
question to describe the joint (generalized) eigenvalues of A on g-modules.
µ
Date: December2007.
1 Supportedby thegrants RFBR 05-01-01007, RFBR 05-01-02934 and NSh-6358.2006.2.
2 Supportedby DARPAand AFOSRthrough thegrant FA9550-07-1-0543.
3 Supported by the grants RFBR 05 01 00988-a, RFBR 05-01-02805-CNRSL-a, and Deligne
2004 Balzan prize in mathematics. The work was finished during L.R.’s stay at theInstitute for
AdvancedStudysupported bythe NSFgrant DMS-0635607.
1
The first steps towards answering this question were taken in [FFT], where it
was shown that A is isomorphic to the algebra of functions on a certain space of
µ
LG-opers on the projective line P1. More precisely, this is the space of LG-opers
on P1 with regular singularity at the point 0 and irregular singularity of order 2
at the point ∞, with a fixed 2-residue determined by µ (see [FFT] for details).
Here LG istheLanglands dual group of G (LG is taken to beof adjointtype), and
LG-opers are connections on a principal LG-bundle over P1 satisfying a certain
transversality condition, as defined in [BD]. The appearance of the Langlands
dual group is not accidental, but is closely related to the geometric Langlands
correspondence, through a description of the center of the completed enveloping
algebraoftheaffineKac–Moodyalgebra g atthecriticallevelintermsof LG-opers
on the punctured disc [FF1, Fr2, Fr4].
Thus, we obtain that the spectra of A on a g-module M are encoded by
b µ
LG-opers on P1 satisfying the above properties. Furthermore, in [FFT] it was
shownthat if M = V , the irreduciblefinite-dimensional g-modulewith dominant
λ
integral highest weight λ, then these LG-opers satisfy two additional properties:
they have a fixed residue at the point 0 (where the oper has regular singularity),
determined by λ (we denote the space of such opers by Op (P1)λ ), and
LG π(−µ)
they have trivial monodromy. It was conjectured in [FFT] that in fact there is a
bijection between the spectra of A on V and the set of monodromy-free opers
µ λ
from Op (P1)λ .
LG π(−µ)
In this paper we prove this conjecture. Furthermore, we prove the following
statement (see Corollary 4):
Theorem A. For generic regular semi-simple µ ∈ g∗ and any dominant integral
λ, the quantum shift of argument subalgebra A ⊂ U(g) is diagonalizable and has
µ
simple spectrum on the irreducible g-module V . Moreover, its joint eigenvalues
λ
(and hence eigenvectors, up to a scalar) are in one-to-one correspondence with
monodromy-free opers from Op (P1)λ .
LG π(−µ)
In this paper we do not address the question of constructing the eigenvectors of
A . Conjecturally, for generic µ, they may be constructed by the Bethe Ansatz
µ
method described in [FFT], but we do not attempt to prove this conjecture here,
nor do we use Bethe Ansatz in the proof of Theorem A. Instead, we rely on the
isomorphism between the algebra A and the algebra of functions on opers and
µ
the study of opers with irregular singularity and trivial monodromy.
The crucial step in our proof is the analysis of the action of A in the case
µ
when µ = f, a regular nilpotent element of g∗ ≃ g. The algebra A contains the
f
centralizer a of f in g. Elements of a and more general elements of A are
f f f
not diagonalizable operators on V , but nilpotent operators. Hence it is natural
λ
to view them as “creation operators” and ask whether they generate V from
λ
its highest weight vector. Theorem 1 implies that the answer to this question is
affirmative (see Corollary 2 for more details).
Theorem B. The g-module V is cyclic as an A -module. The annihilator of V
λ f λ
in A is generated by the no-monodromy conditions on opers from Op (P1)λ.
f LG 0
This provides a natural structure of a Gorenstein ring on any finite-dimensional
irreducible g-module V . We hope that this fact has a geometric interpretation.
λ
2
Namely, due to the results of [Gi, MV], the space V may be naturally identi-
λ
fied with the global cohomology of the irreducible perverse sheaf on the Schubert
variey Gr in the affine Grassmannian of the Langlands dual group LG. On the
λ
other hand, the universal enveloping algebra U(a ) of the centralizer a of the
f f
principal nilpotent element f is identified with the cohomology ring of the affine
Grassmannian. Henceitacts onthecohomology ofourperversesheaf. Thisaction
is precisely the action of U(a ) on V [Gi]. But U(a ) is a subalgebra of the com-
f λ f
mutative algebra A . This leads us to a natural question: what is the geometric
f
meaning of A ? Perhaps, if one could answer this question, one could derive the
f
cyclicity of V as a A -module by geometric means.
λ f
The shift of argument subalgebra A has a multi-point generalization, denot-
µ
ed by A (z ,...,z ), where z ,...,z are distinct points on P1\∞. This is a
µ 1 N 1 N
commutative subalgebra of U(g)⊗N, which consists of the Hamiltonians of the
Gaudin model with irregular singularity [R, FFT]. We show that this algebra is
isomorphic to the algebra of functions on the space of LG-opers on P1 with reg-
ular singularities at z ,...,z and irregular singularity of order 2 at ∞, with
1 N
the 2-residue determined by µ. We also prove a conjecture of [FFT] that the set
of joint eigenvalues of this algebra on the tensor product V ⊗...⊗V of irre-
λ1 λN
ducible finite-dimensional g-modules is in bijection with the set of opers of this
kind with fixed residues determined by the highest weights λ ,...,λ and the
1 N
no-monodromy condition (see Corollary 6). This is a multi-point generalization of
Theorem A. We also prove a multi-point analogue of Theorem B (see Corollary 5).
We note that for the ordinary Gaudin model (with regular singularity) a de-
scription of the spectrum of the Hamiltonians in terms of an appropriate set of
monodromy-free opers analogous to Theorem A was conjectured in [Fr3], Conjec-
ture 1 (it was proved in [Fr3], Theorem 2.7,(3) that the spectrum does embed into
this set of monodromy-free opers). In the course of writing this paper we learned
that a variant of this conjecture was proved in [MTV] in the case when g = gl ,
M
by a detailed analysis of intersections of Schubert varieties in the Grassmannians.
Theorem A in the case of g = gl should be related to this statement via the
N
duality of [TL].
Finally, we expect that the results of this paper may be generalized to the
affine Kac–Moody algebras. As explained in [FF2], the affine analogue of the shift
of argument subalgebra corresponding to regular semi-simple µ is the algebra of
quantum integrals of motion of the AKNS hierarchy of soliton equations. Such an
algebra may also bedefinedfor aregular nilpotent µ. In this case, its action is not
diagonalizable, but it gives rise to a commutative algebra of creation operators.
We expect that these operators generate highest weight modules over affine Kac–
Moody algebras; for example, the irreducible integrable representations. In the
latter case, we expect that the generators of the corresponding annihilating ideal
are given by the no-monodromy conditions on the corresponding affine opers, by
analogy with Theorem B in the finite-dimensional case. We plan to discuss this in
more detail in our next paper.
3
The present paper is organized as follows. In section 2 we collect basic facts on
Gaudin models and opers. In section 3 we formulate and discuss the main result
of the paper. The detailed proof is given in the last section 4.
Acknowledgements. We thank A. Glutsyuk, Yu. Ilyashenko, and V. Toledano
Laredo for useful discussions.
2. Preliminaries
2.1. Gaudin algebras. Gaudin model was introduced in [Ga1] as a spin model
relatedtotheLiealgebra sl ,andgeneralizedtothecaseofanarbitrarysemisimple
2
Lie algebra in [Ga2], Section 13.2.2. For any x ∈ g, set
x(i) =1⊗···⊗1⊗x⊗1⊗···⊗1∈ U(g)⊗N
(x atthe ithplace). Let {x }, a = 1,...,dimg,beanorthonormalbasisof g with
a
respect to Killing form, and let z ,...,z be pairwise distinct complex numbers.
1 N
TheHamiltoniansofGaudinmodelarethefollowingmutuallycommutingelements
of U(g)⊗N:
dimg (i) (k)
x x
a a
(1) H = .
i
z −z
i k
k6=i a=1
X X
In [FFR], a large commutative subalgebra A(z ,...,z ) containing the H ’s
1 N i
was constructed with the help of the affine Kac–Moody algebra g, which is the
universalcentral extension of g((t)). Namely, according to[FF1,Fr2], thecomplet-
ed enveloping algebra Uκc(g) of g at the critical level (in the notabtion of [FFT])
contains a large center Z(g). Set g = g[[t]] ⊂ g and g = t−1g[t−1] ⊂ g. The
+ −
natural homomorphismeZ(bg) −→b(Uκc(g)/Uκc(g) · g+)bg+ is surjective [FF1, Fr2].
Every element of the latterbquotienbt has a uniquebrepresbentative in U(g−)b. Thus
we obtain the following natbural emebeddbinge b b
(U (g)/U (g)·g )bg+ ֒→ U(g ). b
κc κc + −
Let z(g) ⊂ U(g ) be the image of this embedding. The commutative subalgebra
A(z ,...,z ) ⊂−U(g)⊗N eis thben ethe bimagbe of z(g) ⊂bU(g ) under the homomor-
1 N −
phism
b b
(2) U(g )−→ U(g)⊗Nb b
−
of evaluation at the points z ,...,z (see [FFR]).
1 N
b
This construction is generalized as follows (see [FFT, R]. For different approach
in the case g = gl see [ChT]). One constructs a family of homomorphisms
N
(3) U(g )−→ U(g)⊗N ⊗S(g)
−
generalizing the evaluation homomorphisms (2) in the sense that composing it
with the natural augmentationbS(g) −→ C we obtain (2). Now, for any collection
z ,...,z , the image of z(g) under this homomorphism is a certain commutative
1 N
subalgebra U(g)⊗N⊗S(g). Evaluatingatanypoint µ ∈ g∗ = SpecS(g), weobtain
a commutative subalgebrabAµ(z1,...,zN) ⊂ U(g)⊗N depending on z1,...,zN
4
and µ ∈ g∗. These subalgebras contain the following “inhomogeneous” Gaudin
hamiltonians:
dimg (i) (k) dimg
x x
H = a a + µ(x )x(i).
i z −z a a
i k
k6=i a=1 a=1
X X X
In particular, A(z ,...,z ) = A (z ,...,z ) corresponding to µ = 0.
1 N 0 1 N
In particular, for N = 1, we obtain a family of commutative subalgebras
A (z ) ⊂ U(g) which does not depend on z . We will set z = 0 in this case
µ 1 1 1
and denote this algebra simply by A . It is proved in [FFT, R] that the associ-
µ
ated graded algebra of A (with respect to the PBW filtration) for regular µ is
µ
the (classical) shift of argument subalgebra A ⊂ S(g). This Poisson commuta-
µ
tive subalgebra was first constructed by Mishchenko and Fomenko in [MF], in the
following way. Let
(4) S(g)g= C[P ,...,P ],
1 ℓ
where ℓ = rank(g) be the center of S(g) with respect to the Poisson bracket,
and the P are chosen so that they are homogeneous with respect to the natural
i
grading on S(g). Let µ ∈ g∗ be a regular semisimple element. Then the subal-
gebra A ⊂ S(g) is generated by the elements ∂nP , where k = 1,...,ℓ, n =
µ µ k
0,...,degP −1, (or, equivalently, generated by central elements of S(g) = C[g∗]
k
shifted by tµ for all t ∈ C). These elements are algebraically independent (for a
unform proof, see [FFT], Theorem 3.11). Hence the subalgebra A ⊂ S(g) is a
µ
free polynomial algebra in 1(dimg+rkg) generators (and therefore has maximal
2
possible transcendence degree). Since grA = A , we obtain the following
µ µ
Lemma 1. The algebra A is a free polynomial algebra in 1(dimg+rkg) gener-
µ 2
ators.
Let
g = n⊕h⊕n
−
beaCartandecomposition of g. Denote by gR bethecompactreal formof g, and
let hR = gR ∩h. Any irreducible finite-dimensional g-module has a gR-invariant
Hermitian form. We shall use the following
Lemma 2. For z ,...,z ∈ R and µ ∈ ih∗ ⊂ h∗, the algebra A (z ,...,z ) ⊂
1 N R µ 1 N
U(g)⊗N acts by Hermitian operators on any irreducible finite-dimensional g-
module.
Proof. This follows from the fact that the center at the critical level and the
homomorphism(3)aredefinedover R, andhence Aµ(z1,...,zN) ⊂ U(gR)⊗N. (cid:3)
Wealsoneedthefollowingfactsonthelimitpointsofthefamily A (z ,...,z ).
µ 1 N
Namely,
Proposition 1. [R] lim A (sz ,...,sz )= lim A (z ,...,z )= A(1)⊗···⊗
s−→∞ µ 1 N s−→∞ sµ 1 N µ
A(N) ⊂ U(g)⊗N for regular semisimple µ ∈g∗.
µ
We have A = A for any t ∈ C×,ξ ∈ g∗. Thus, the subalgebras A ⊂ U(g)
ξ tξ ξ
and A ⊂ U(g) are conjugate for any g ∈ G,t ∈ C∗,ξ ∈ g. Let Π be the set
tAd(g)ξ
5
of simple roots of g. We will choose generators {e } of the lower nilpotent
−α α∈Π
subalgebra n . Let
−
f = e ∈g
−α
α∈Π
X
be a principal nilpotent element. From now on we will identify g and g∗ using a
non-degenerate invariant inner product.
Lemma 3. For any regular µ ∈ g, the closure of the family A ⊂ U(g)
tAd(g)ξ
(g ∈ G,t ∈ C∗,ξ ∈ g) contains the subalgebra A ⊂ U(g).
f
Proof. Let {e, h, f} be a principal sl -triple in g, where h ∈ h. Consider
2
Kostant’s slice of regular elements
g := f +z (e).
can g
For anyregular µ ∈ g, the AdG-orbit of µ intersects g (this isaclassical result
can
due to Kostant [K]). Thus, there exists x ∈ z (e) such that A belongs to the
g f+x
family A . Since x∈ n, we have
tAd(g)ξ
lim t2Adexp(−th)(f +x)= f,
t−→∞
and therefore A is a limit point of the family A . (cid:3)
f tAd(g)ξ
The ”most degenerate” subalgebra A ⊂ U(g) among the those corresponding
µ
toregular µ isthesubalgebra A . Itisafreecommutativealgebrawithgenerators
f
Π(n) such that grΠ(n) = ∂nP ∈ S(g), where i = 1,...,l, n = 0,1,...,d =
i i e i i
degP −1, P are the generators of S(g)g (see formula (4)).
i i
The element h defines the principal gradation deg on U(g) such that
pr
deg e = −deg e = 1, α ∈ Π, deg h = 0, h ∈ h.
pr α pr α pr
The generators of A are homogeneous with respect to this gradation, with
f
deg Π(n) = −n. Thus, the algebra A is graded by the principal gradation:
pr i f
A = A(−n). Note that the Poincar´e series of A with respect to the princi-
f n≥0 e f
pal gradation is equal to that of the algebra U(n ). Thus, it is natural to expect
−
that irLreducible highest weight g-modules are cyclic as A -modules (having the
f
highest weight vector as a cyclic vector).
2.2. Opers on the projective line. Now let us describe the spectra of Gaudin
algebras following [FFT].
Consider the Langlands dual Lie algebra Lg whose Cartan matrix is the trans-
pose of the Cartan matrix of g. By LG we denote the group of inner automor-
phisms of Lg. We fix a Cartan decomposition
Lg = Ln⊕Lh⊕Ln .
−
The Cartan subalgebra Lh is naturally identified with h∗. We denote by L∆,
L∆ , and LΠ the root system of Lg, the set of positive roots, and the set of
+
simple roots, respectively.
Set
p−1 = e−α∨ ∈ Lg.
α∨∈Π∨
X
6
Let
1
ρ = α ∈ h∗ = Lh.
2
αX∈∆+
The operator Adρ defines the principal gradation on Lg, with respect to which
we have a direct sum decomposition Lb = L b . Let p be the unique element
i≥0 i 1
of degree 1 in Ln such that {p ,2ρ,p } is an sl -triple (note that p has the
−1 1 L 2 −1
degree −1). Let
ℓ
V = V
can can,i
i=1
M
be the space of Adp -invariants in Ln, decomposed according to the principal
1
gradation. Here V has degree d , the ith exponent of Lg (and of g). In
can,i i
particular, V is spanned by p . Now choose a linear generator p of V .
can,1 1 j can,j
Consider the Kostant slice in Lg,
ℓ
Lg = p + y p , y ∈ C .
can −1 j j
 
 Xj=1 
By [K], the adjoint orbit of any regular element in the Lie algebra Lg con-
tains a unique element which belongs to Lg . Thus, we have an isomorphism
can
Lg −→˜Lg/LG = h/W = g/G.
can
In [FF1], the center Z(g) of the completed enveloping algebra U (g) at the
κc
critical level is identified with the algebra of polynomial functions on the space
OpLG(D×) of LG-opers onbthe disc D = SpecC((t)). e b
The notion of opers was introduced in [BD]. We refer the reader to [FFT] for
details. Here we will only say that for U = SpecR and some coordinate t on U,
the space Op (U) of LG-opers is the quotient of the space of LG-connections of
LG
the form
d+(p +v(t))dt, v(t) ∈ Lb(R)
−1
by the action of the group LN(R).
Following [FFT], we denote by Op (P1) the space of LG-opers on
LG (zi);π(−µ)
P1\{z ,...z ,∞} with regular singularities at the points z ,i = 1,...,N, and
1 N i
with irredular singularity of order 2 at the point ∞ with the 2-residue π(−µ) ∈
Lg/LG = g∗/G, where π :g∗ −→ g∗/G is the porjection. Each oper from this space
may be uniquely represented in the following form:
ℓ N ℓ dj
d+ p + µ p + u(i)(t−z )−n−1p dt,
 −1 j j j,n i j
j=1 i=1 j=1n=0
X XXX
 
where
ℓ
p + µ p
−1 j j
j=1
X
is the unique element of the LG-orbit of µ ∈ g contained in Lg . Thus,
can
Op (P1) is an affine space of dimension 1(dimg+rkg)N.
LG (zi);π(−µ) 2
7
2.3. Gaudin algebra and opers. By Theorem 5.7,(4) of [FFT], the algebra
A (z ,...,z ) is isomorphic to a quotient of the algebra of polynomial functions
µ 1 N
onthespace Op (P1) ,whichisapolynomialalgebrain 1(dimg+rkg)N
LG (zi);π(−µ) 2
generators. Ontheotherhand,Proposition1andLemma1implythatthealgebra
A (z ,...,z ) is a free polynomial algebra in the same number of generators
µ 1 N
(see also [R], Theorem 2 and Corollary 4). Indeed, due to the same reason as
in Lemma 3, it suffices to prove this for the principal nilpotent µ = f. The
principal gradation on U(g)⊗N determines a filtration on A (z ,...,z ) such
f 1 N
that grA (z ,...,z ) = A⊗N (since lim A (z ,...,z ) = A(1) ⊗···⊗A(N) ⊂
f 1 N f s−→∞ sµ 1 N µ µ
U(g)⊗N, by Proposition 1). Thus trdegA (z ,...,z ) = trdegA⊗N = 1(dimg+
f 1 N f 2
rkg)N.
Thus, we obtain the following assertion which was conjectured in [FFT], Con-
jecture 3.
Proposition 2. There is an isomorphism
A (z ,...,z ) ≃ C[Op (P1) ].
µ 1 N LG (zi);π(−µ)
In particular, the algebra A is identified with the algebra of polynomial func-
µ
tions on the space Op (P1) of LG-opers on P1 with regular singularity at
LG π(−µ)
the point 0 and with singularity of order 2 at ∞, with 2-residue π(−µ), where
π(−µ) is the image of −µ∈ Lg in Lg/LG = g/G (see [FFT], Theorem 5.8). This
space has the following realization (see [FFT], Section 5.4).
On the punctured disc D× at ∞ (with the coordinate s = t−1) each element
∞
of Op (P1) may be uniquely represented by a connection of the form
LG π(−µ)
ℓ dj
d− p − (s−2djµ +s−2dj−1u (s))p ds, u (s)= u sn.
−1 j j j j j,n
 
j=1 n=0
X X
On thepunctured disc D× at 0, each element of Op (P1) may be rep-
0 LG π(−µ)
resented uniquely by a connection of the form
ℓ dj
d+ p + (µ +u t−n−1)p dt.
−1 j j,n j
 
j=1n=0
XX
The 1-residue at 0 of this oper is equal to 
ℓ
1
p + (u + δ )p ∈ Lg ≃ Lg/LG = h/W =g/G.
−1 j,dj 4 j,1 j can
j=1
X
In particular, for the algebra SpecA , we have the following space of opers:
f
ℓ dj
Op (P1) = d− p − s−2dj−1u (s)p )ds, u (s) = u sn ds
LG 0   −1 j j (cid:12) j j,n  
 Xj=1 (cid:12)(cid:12) Xn=0 
 (cid:12) 
ℓ dj (cid:12)
 
= d+ p + u t−n−1p dt(cid:12)
−1 j,n j
   
 Xj=1Xn=0 
 
(here, as before, we omit z , which is set to 0).
 1 
8
Next, it is proved in [FFT], Theorem, 5.7, that for any collection of g-modules
M with highest weights λ , i= 1,...,N, the natural homomorphism
i i
A (z ,...,z ) −→ End(M ⊗···⊗M )
µ 1 N 1 N
factors through the algebra of functions on the subspace Op (P1)(λi) ⊂
LG (zi);π(−µ)
Op (P1) which consists of the opers with the 1-residue π(−λ −ρ) at
LG (zi);π(−µ) i
z . Moreover, for integral dominant λ the action of A (z ,...,z ) on the ten-
i i µ 1 N
sor product of the finite-dimensional modules V factors through the algebra of
λi
functions on monodromy-free opers from Op (P1)(λi) .
LG (zi);π(−µ)
For every integral dominant weight λ, the set of monodromy-free opers on the
punctured disc D× at z with the regular singularity with the residue −λ−ρ at
z
z is defined by finitely many polynomial relations. Namely, each element of the
space Op(D )λ of opers on D× with regular singularity and residue π(−λ−ρ)
z z
may be uniquely represented as
ℓ dj
d+ p + u t−n−1p dt,
−1 j,n j
 
j=1n=−∞
X X
 
with
ℓ
p + u ∈ Ad(LG)(−λ−ρ).
−1 j,dj
j=1
X
One can bring this connection to the form
d+ thα∨,λie−α∨ +v(t) dt,
α∨∈Π∨ !
X
where v(t) ∈ t−1Ln + Lb[[t]]. This oper is monodromy-free if and only if
v(t) ∈ Lb[[t]]. Thus,theset ofmonodromy-freeopersis definedby dimLn polyno-
mial relations P (u ) enumerated by positive roots α ∈ ∆ . These polynomial
α j,n +
relations have the degrees (α∨,λ+ρ) with respect to the Z-grading defined by
the formula degu = −n+j (see [FG], Section 2.9, for details).
j,n
Introduce a Z-grading on the algebra
A (z ,...,z )(λi) = C Op (P1)(λi) = C[u(i)]
µ 1 N LG (zi);π(−µ) j,n j=1,...,ℓ; n=0,...,dj; i=1,...,N
h i
(i)
by the formula degu = −n+j. By comparing the degrees of the generators
j,n
of the polynomial algebras Aλ and C[Op (P1)λ], we obtain the following (note
f LG 0
that we abbreviate the notation Op (P1)λ to Op (P1)λ when µ =f):
LG (0);π(−µ) LG 0
Lemma 4. For Aλ = C[Op (P1)λ] this grading coincides with the principal
f LG 0
grading on Aλ.
f
Foranycollectionofintegraldominantweights λ ,...,λ attachedtothepoints
1 N
z ,...,z , we denote by P(z1,...,zN);µ;(λi) the polynomial in u(i) expressing the
1 N zk;α j,n
”no-monodromy” condition at z corresponding to the root α ∈ ∆ .
k +
9
Lemma 5. P(z1,...,zN);µ;(λi) is an inhomogeneous polynomial of highest degree
zk;α
degP(z1,...,zN);µ;(λi) = (α∨,λ + ρ). The leading term of P(z1,...,zN);µ;(λi) is equal
zk;α zk;α
to P(zk);f;(λk).
zk;α
Proof. Written in terms of a local coordinate t at z , an oper from the set
k
Op (P1)(λi) has the form
LG (zi);π(−µ)
ℓ dj
d+ p + v t−n−1p dt, v =u(k)+lower terms.
 −1 j,n j j,n j,n
j=1n=−∞
X X
 
(k)
More precisely, v = u for n ≥ 0, and for n < 0, the polynomial v is a
j,n j,n j,n
(i)
linear combination of scalars and u , m ≥ 0. Therefore, for n < 0, we have
j,m
degv < −n+j. Hence the assertion. (cid:3)
j,n
According to Lemma 5, in order to show that the no-monodromy conditions
define a finite set of opers, it suffices to prove this for the space Op (P1)λ. This
LG 0
will be done in the next section.
3. Main results
3.1. Formulation of the Main Theorem. In section 4, we shall prove the fol-
lowing result.
Theorem 1. The set of monodromy-free opers from Op (P1)λ is 0-dimensional
LG 0
(0);f;(λ)
(equivalently, the trivial monodromy conditions P (u ) form a regular se-
0;α j,n
quence).
3.2. Corollaries. First, let us discuss some corollaries of this Theorem.
Corollary 1. The set of monodromy-free opers from Op (P1)(λi) is 0-
LG (zi);π(−µ)
dimensional.
Proof. This follows directly from Lemma 5. Indeed, the set of monodromy-
free opers from Op (P1)(λi) is the set of common zeros of the polyno-
LG (zi);π(−µ)
mials P(z1,...,zN);µ;(λi). By Lemma 5, the leading terms of these polynomials
zk;α
are P(zk);f;(λk). By Theorem 1, the set of common zeros of P(zk);f;(λk) is 0-
zk;α zk;α
dimensional. Hence he set of common zeros of the polynomials P(z1,...,zN);µ;(λi) is
zk;α
also 0-dimensional. (cid:3)
Corollary 2. The g-module V is cyclic as an A -module. The annihilator of
λ f
V in A is the ideal I ⊂ A = C[Op (P1)λ] generated by the no-monodromy
λ f λ f LG 0
conditions on opers from Op (P1)λ.
LG 0
Proof. Note first that this assertion agrees with the q-analog of the Weyl dimen-
sion formula. Namely, the Poincar´e series of any irreducible finite-dimensional
g-module V with respect to the principal grading is
λ
1−q(α∨,λ+ρ)
χ (q)= .
λ 1−q(α∨,ρ)
α>0
Y
10