Gelfand-Tsetlin algebras and cohomology rings of Laumon spaces
Boris Feigin, Michael Finkelberg, Igor Frenkel, Leonid Rybnikov
aa r X i v : . [ m a t h . AG ] M a r Gelfand-Tsetlin algebras and cohomologyrings of Laumon spaces
Boris Feigin, Michael Finkelberg, Igor Frenkel and LeonidRybnikov
To the memory of Izrail Moiseevich Gelfand
Abstract.
Laumon moduli spaces are certain smooth closures of themoduli spaces of maps from the projective line to the flag variety of GL n . We calculate the equvariant cohomology rings of the Laumonmoduli spaces in terms of Gelfand-Tsetlin subalgebra of U ( gl n ), andformulate a conjectural answer for the small quantum cohomology ringsin terms of certain commutative shift of argument subalgebras of U ( gl n ).
1. Introduction
The moduli spaces Q d were introduced by G. Laumon in [13] and [14]. Theyare certain compactifications of the moduli spaces of degree d maps from P to the flag variety B n of GL n . The original motivation of G. Laumon was tostudy the geometric Eisenstein series, but later the Laumon moduli spacesproved useful also in the computation of quantum cohomology and K -theoryof B n , see e.g. [10], [2]. The aim of the present note is to calculate the coho-mology rings of the Laumon moduli spaces, and to formulate a conjecturalanswer for the quantum cohomology rings.The main tool is the action of the universal enveloping algebra U ( gl n )by correspondences [7] on the direct sum (over all degrees) of cohomology of Q d . More precisely, we consider the localized equivariant cohomology B := L d H • GL n × C ∗ ( Q d ) ⊗ H • GLn × C ∗ ( pt ) Frac( H • GL n × C ∗ ( pt )) where C ∗ acts as “looprotations” on the source P , while GL n acts naturally on the target B n . Wealso consider a “local version” Q d of the Laumon moduli space, which is acertain closure of the moduli space of based maps of degree d from P to B n .This local version does not carry the action of the whole group GL n × C ∗ ,but only of the Cartan torus e T × C ∗ . Accordingly, we consider the equivariant Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikovcohomology (resp. localized equivariant cohomology) ′ V = L d H • e T × C ∗ ( Q d )(resp. V = L d H • e T × C ∗ ( Q d ) ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt ))).According to [1] (cf. also [19] and our Theorem 2.7), the above action of U ( gl n ) identifies V with the universal Verma module V . Similarly, B carriesthe action of two copies of U ( gl n ) by correspondences, and can be identifiedwith the tensor square B of V (Theorem 5.8). The nonlocalized cohomology ′ V is identified with a certain integral form V of V , a version of the uni-versal dual Verma module (Theorem 3.5). We were unable to describe the nonlocalized equivariant cohomology of F d Q d as a U ( gl n ) -module, but wepropose a conjecture 5.14 in this direction; it is an equivariant generalizationof Conjecture 6.4 of [7]. The description of the cohomology rings is given in representation theoreticterms. Namely, the universal enveloping algebra of gl n contains the Gelfand-Tsetlin subalgebra G (a maximal commutative subalgebra). For a given degree d , the equivariant cohomology ′ V d is identified with the weight subspace V d .The identity element 1 d of the cohomology ring goes to the weight component v d of the Whittaker vector v ∈ V . It turns out that the vector 1 d ∈ ′ V is cyclicfor G ; hence the equivariant cohomology ring H • e T × C ∗ ( Q d ) is identified witha quotient of the Gelfand-Tsetlin subalgebra (Corollary 3.7). Similar resultshold for the localized equivariant cohomology of Q d and Q d (Propositions 2.17and 5.12).The proof uses two ingredients. First, the localized equivariant coho-mology has a natural basis of classes of the torus fixed points. We check thatunder the identification V ≃ V , this basis goes to the Gelfand-Tsetlin basisof V . Also, the cohomology of Q d contain the (K¨unneth components of the)Chern classes of the universal tautological vector bundles on Q d × P . Theoperators of multiplication by these Chern classes are diagonal in the fixedpoint basis, and by comparison to the Gelfand-Tsetlin basis it is possible toidentify these operators with the action of certain generators of G . Finally,since the diagonal class of Q d is decomposable, the above Chern classes gen-erate the cohomology ring of Q d .In a similar vein, in Proposition 6.7 we compute the localized equivariant K -ring of Q d in terms of the “quantum Gelfand-Tsetlin algebra”. The Picard group of the local Laumon space Q d is free of rank n − d are nonzero. It possesses the set of distinguished generators:the classes of determinant line bundles D , . . . , D n − . Let T be a torus withthe cocharacter lattice Pic( Q d ), and let q i , ≤ i ≤ n −
1, be the coordinateson T corresponding to D i . We conjecture a formula for the operator M D i of quantum multiplication by the first Chern class c ( D i ). A priori this operatorlies in End( V d )[[ q , . . . , q n − ]], but according to Conjecture 4.6, it is the Taylorexpansion of a rational End( V d )-valued function on T . Moreover, this functionelfand-Tsetlin algebras and cohomology rings of Laumon spaces 3arises from the action of a universal element QC i ∈ U ( gl n ) (depending on q , . . . , q n − ) on the weight space V d of the universal Verma module. Thecommutant of the collection of all such elements { QC i ( q , . . . , q n − ) } is a shiftof argument subalgebra A q ⊂ U ( gl n ) (a maximal commutative subalgebra,see [21]).We consider the flat End( V d )-valued connection on T : ∇ = P n − i =2 q i ∂∂q i + QC i (the quantum connection ). Conjecture 4.6 (recentlyproved by A. Negut) implies that ∇ is induced by the Casimir connec-tion [4, 6, 24, 16] on the Cartan subalgebra h ⊂ sl n under an embedding T ֒ → h . In particular, ∇ has regular singularities, and its monodromy factorsthrough the action of the pure braid group P B n (fundamental group of thecomplement in h to the root hyperplanes) on the weight space V d by the“quantum Weyl group operators”. This note is a result of discussions with many people. In particular, the “re-stricted” correspondences E d,α ij , E d,α ij , E ∞ d,α ij (see 3.3, 5.13) and the actionof U ( gl n ) ⋉C [ gl n ] on nonequivariant cohomology of F d Q d were introduced byA. Kuznetsov in 1998. The idea to consider the action of correspondences onthe equivariant cohomology of Laumon spaces was proposed by V. Schecht-man in 1997. I. F. would like to thank A. Licata and A. Marian for many usefuldiscussions of representation theory based on Laumon’s spaces. In fact, someof the calculations associated with the double gl n action of Section 5 werecarried out independently by A. Licata and A. Marian, also in a nonequiv-ariant setting. Furthermore, we learnt from A. Marian that the Chern classesof the tautological bundles generate the cohomology ring of Q d . A. I. Molevprovided us with references on Gelfand-Tsetlin bases. Above all, our interestin quantum cohomology of Laumon spaces was inspired by R. Bezrukavnikov,A. Braverman, A. Negut, A. Okounkov and V. Toledano Laredo during thebeautiful special year 2007/2008 at IAS organized by R. Bezrukavnikov. Weare deeply grateful to all of them. Finally, we are obliged to A. Tsymbal-iuk for the careful reading of the first version of our note and spottingseveral mistakes. B. F. was partially supported by the grants RFBR 05-01-01007, RFBR 05-01-01934, and NSh-6358.2006.2. I. F. was supported bythe NSF grant DMS-0457444. M. F. was partially supported by the OswaldVeblen Fund, RFBR grant 09-01-00242, the Science Foundation of the SU-HSE awards No.T3-62.0 and 10-09-0015, and the Ministry of Education andScience of Russian Federation, grant No. 2010-1.3.1-111-017-029. The workof L. R. was partially supported by RFBR grants 07-01-92214-CNRSL-a,05-01-02805-CNRSL-a, 09-01-00242, the Science Foundation of the SU-HSEawards No.T3-62.0 and 10-09-0015, and the Ministry of Education and Sci-ence of Russian Federation, grant No. 2010-1.3.1-111-017-029. He gratefullyacknowledges the support from Deligne 2004 Balzan prize in mathematics. Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikov
2. Local Laumon spaces
We recall the setup of [7], [2]. Let C be a smooth projective curve of genuszero. We fix a coordinate z on C , and consider the action of C ∗ on C suchthat v ( z ) = v − z . We have C C ∗ = { , ∞} .We consider an n -dimensional vector space W with a basis w , . . . , w n .This defines a Cartan torus T ⊂ G = GL n ⊂ Aut ( W ). We also considerits 2 n -fold cover, the bigger torus e T , acting on W as follows: for e T ∋ t =( t , . . . , t n ) we have t ( w i ) = t i w i . We denote by B the flag variety of G .Given an ( n − d = ( d , . . . , d n − ), weconsider the Laumon’s quasiflags’ space Q d , see [14], 4.2. It is the modulispace of flags of locally free subsheaves0 ⊂ W ⊂ . . . ⊂ W n − ⊂ W = W ⊗ O C such that rank( W k ) = k , and deg( W k ) = − d k .It is known to be a smooth projective variety of dimension 2 d + . . . +2 d n − + dim B , see [13], 2.10.We consider the following locally closed subvariety Q d ⊂ Q d (quasiflagsbased at ∞ ∈ C ) formed by the flags0 ⊂ W ⊂ . . . ⊂ W n − ⊂ W = W ⊗ O C such that W i ⊂ W is a vector subbundle in a neighbourhood of ∞ ∈ C , andthe fiber of W i at ∞ equals the span h w , . . . , w i i ⊂ W .It is known to be a smooth quasiprojective variety of dimension 2 d + . . . + 2 d n − . The group G × C ∗ acts naturally on Q d , and the group e T × C ∗ acts naturally on Q d . The set of fixed points of e T × C ∗ on Q d is finite; we recall its descriptionfrom [7], 2.11.Let e d be a collection of nonnegative integers ( d ij ) , i ≥ j , such that d i = P ij =1 d ij , and for i ≥ k ≥ j we have d kj ≥ d ij . Abusing notation wedenote by e d the corresponding e T × C ∗ -fixed point in Q d : W = O C ( − d · w , W = O C ( − d · w ⊕ O C ( − d · w ,. . . . . . . . . , W n − = O C ( − d n − , · w ⊕ O C ( − d n − , · w ⊕ . . . ⊕ O C ( − d n − ,n − · w n − . For i ∈ { , . . . , n − } , and d = ( d , . . . , d n − ), we set d + i := ( d , . . . , d i +1 , . . . , d n − ). We have a correspondence E d,i ⊂ Q d × Q d + i formed by the pairs( W • , W ′• ) such that for j = i we have W j = W ′ j , and W ′ i ⊂ W i , see [7], 3.1.In other words, E d,i is the moduli space of flags of locally free sheaves0 ⊂ W ⊂ . . . W i − ⊂ W ′ i ⊂ W i ⊂ W i +1 . . . ⊂ W n − ⊂ W elfand-Tsetlin algebras and cohomology rings of Laumon spaces 5such that rank( W k ) = k , and deg( W k ) = − d k , while rank( W ′ i ) = i , anddeg( W ′ i ) = − d i − E d,i is a smooth projective algebraic variety ofdimension 2 d + . . . + 2 d n − + dim B + 1.We denote by p (resp. q ) the natural projection E d,i → Q d (resp. E d,i → Q d + i ). We also have a map r : E d,i → C , (0 ⊂ W ⊂ . . . W i − ⊂ W ′ i ⊂ W i ⊂ W i +1 . . . ⊂ W n − ⊂ W ) supp( W i / W ′ i ) . The correspondence E d,i comes equipped with a natural line bundle L i whose fiber at a point(0 ⊂ W ⊂ . . . W i − ⊂ W ′ i ⊂ W i ⊂ W i +1 . . . ⊂ W n − ⊂ W )equals Γ( C , W i / W ′ i ).Finally, we have a transposed correspondence T E d,i ⊂ Q d + i × Q d .Restricting to Q d ⊂ Q d we obtain the correspondence E d,i ⊂ Q d × Q d + i together with line bundle L i and the natural maps p : E d,i → Q d , q : E d,i → Q d + i , r : E d,i → C − ∞ . We also have a transposed correspondence T E d,i ⊂ Q d + i × Q d . It is a smooth quasiprojective variety of dimension 2 d + . . . + 2 d n − + 1. We denote by ′ V the direct sum of equivariant (complexified) cohomol-ogy: ′ V = ⊕ d H • e T × C ∗ ( Q d ). It is a module over H • e T × C ∗ ( pt ) = C [ t ⊕ C ] = C [ x , . . . , x n , ~ ]. Here t ⊕ C is the Lie algebra of e T × C ∗ . We define ~ astwice the positive generator of H C ∗ ( pt, Z ). Similarly, we define x i ∈ H e T ( pt, Z )in terms of the corresponding one-parametric subgroup. We define V = ′ V ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt )).We have an evident grading V = ⊕ d V d , V d = H • e T × C ∗ ( Q d ) ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt )). We denote by U the universal enveloping algebra of gl n over the field C ( t ⊕ C ).For 1 ≤ j, k ≤ n we denote by E jk ∈ gl n ⊂ U the usual elementary matrix.The standard Chevalley generators are expressed as follows: e i := E i +1 ,i , f i := E i,i +1 , h i := E i +1 ,i +1 − E ii (note that e i is represented by a lower triangular matrix). Note also that U isgenerated by E ii , ≤ i ≤ n, E i,i +1 , E i +1 ,i , ≤ i ≤ n −
1. We denote by U ≤ the subalgebra of U generated by E ii , ≤ i ≤ n, E i,i +1 , ≤ i ≤ n −
1. It actson the field C ( t ⊕ C ) as follows: E i,i +1 acts trivially for any 1 ≤ i ≤ n −
1, and E ii acts by multiplication by ~ − x i + i −
1. We define the universal Vermamodule V over U as U ⊗ U ≤ C ( t ⊕ C ). The universal Verma module V is anirreducible U -module. Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikov The grading and the correspondences T E d,i , E d,i give rise to the followingoperators on V (note that though p is not proper, p ∗ is well defined on thelocalized equivariant cohomology due to the finiteness of the fixed point setsand smoothness of E d,i ): E ii = ~ − x i + d i − − d i + i − V d → V d ; h i = ~ − ( x i +1 − x i ) + 2 d i − d i − − d i +1 + 1 : V d → V d ; f i = E i,i +1 = p ∗ q ∗ : V d → V d − i ; e i = E i +1 ,i = − q ∗ p ∗ : V d → V d + i . Theorem 2.7.
The operators e i = E i +1 ,i , E ii , f i = E i,i +1 on V defined in 2.6satisfy the relations in U , i.e. they give rise to the action of U on V . There is aunique isomorphism Ψ of U -modules V and V carrying ∈ H e T × C ∗ ( Q ) ⊂ V to the lowest weight vector ∈ C ( t ⊕ C ) ⊂ V . The proof is entirely similar to the proof of Theorem 2.12 of [2]; cf.also [19]. (cid:3)
According to the Localization theorem in equivariant cohomology (seee.g. [3]), restriction to the e T × C ∗ -fixed point set induces an isomorphism H • e T × C ∗ ( Q d ) ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt )) → H • e T × C ∗ ( Q e T × C ∗ d ) ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt ))The fundamental cycles [ e d ] of the e T × C ∗ -fixed points e d (see 2.2) form abasis in ⊕ d H • e T × C ∗ ( Q e T × C ∗ d ) ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt )). The embedding of apoint e d into Q d is a proper morphism, so the direct image in the equivariantcohomology is well defined, and we will denote by [ e d ] ∈ V d the direct imageof the fundamental cycle of the point e d . The set { [ e d ] } forms a basis of V .The matrix coefficients of the operators e i , f i in the basis { [ e d ] } werecomputed in [2]; cf. also [19] 8.2. The result is: Proposition 2.9.
The matrix coefficients of the operators e i , f i in the basis { [ e d ] } are as follows: e i [ e d, e d ′ ] = − ~ − Y j = k ≤ i ( x j − x k +( d i,k − d i,j ) ~ ) − Y k ≤ i − ( x j − x k +( d i − ,k − d i,j ) ~ ) if d ′ i,j = d i,j + 1 for certain j ≤ i ; f i [ e d, e d ′ ] = ~ − Y j = k ≤ i ( x k − x j + ( d i,j − d i,k ) ~ ) − Y k ≤ i +1 ( x k − x j + ( d i,j − d i +1 ,k ) ~ ) if d ′ i,j = d i,j − for certain j ≤ i ;All the other matrix coefficients of e i , f i vanish. The proof is entirely similar to that of Corollary 2.20 of [2].elfand-Tsetlin algebras and cohomology rings of Laumon spaces 7
We will follow the notations of [17] on the Gelfand-Tsetlin bases in rep-resentations of gl n . To a collection e d = ( d ij ) , n − ≥ i ≥ j we asso-ciate a Gelfand-Tsetlin pattern
Λ = Λ( e d ) := ( λ ij ) , n ≥ i ≥ j as follows: λ nj := ~ − x j + j − , n ≥ j ≥ λ ij := ~ − x j + j − − d ij , n − ≥ i ≥ j ≥ V ∋ ξ e d = ξ Λ := ( − ~ ) −| d | Ψ[ e d ]. According to Proposition 2.9,the matrix coefficients of the operators e i , f i in the basis { ξ e d } are as follows: e i, Λ( e d ) , Λ( e d ′ ) = Y j = k ≤ i ( x j − x k + ( d i,k − d i,j ) ~ ) − Y k ≤ i − ( x j − x k + ( d i − ,k − d i,j ) ~ )if d ′ i,j = d i,j + 1 for certain j ≤ i ; f i, Λ( e d ) , Λ( e d ′ ) = − ~ − Y j = k ≤ i ( x k − x j +( d i,j − d i,k ) ~ ) − Y k ≤ i +1 ( x k − x j +( d i,j − d i +1 ,k ) ~ )if d ′ i,j = d i,j − j ≤ i ;All the other matrix coefficients of e i , f i vanish.The above matrix coefficients, under appropriate specialization of x , . . . , x n , coincide with the matrix coefficients of e i , f i in the Gelfand-Tsetlin basis of an irreducible finite dimensional gl n -module, cf. formulas (2.7), (2.6)of Theorem 2.3 of [17]. For this reason we suggest to call the basis { ξ e d } (overall collections e d ) of V the Gelfand-Tsetlin basis . Algebraically, ξ e d = ξ Λ ∈ V can be defined by the formulas (2.9)–(2.11) of [17] (where ξ = ξ = 1 ∈ V ).Up to proportionality, the Gelfand-Tsetlin basis can also be defined as aneigenbasis of the Gelfand-Tsetlin subalgebra of U .For a future reference, let us formulate once again the relation betweenthe fixed point base of V and the Gelfand-Tsetlin base of V : Theorem 2.11.
The isomorphism
Ψ : V ∼ −→ V of Theorem 2.7 takes [ e d ] to ( − ~ ) | d | ξ e d where | d | = d + . . . + d n − . Remark 2.12.
One can prove that the isomorphism Ψ : V ∼ −→ V of Theo-rem 2.7 takes [ e d ] to ξ e d up to proportionality without explicitly computingthe matrix coefficients. In effect, the Gelfand-Tsetlin basis is uniquely (up toproportionality) characterized by the property that the matrix coefficients of e k , f k with respect to ξ Λ , ξ Λ ′ vanish if λ ij = λ ′ ij for some i > k . Now it isimmediate to see that the matrix coefficients of e k , f k with respect to [ e d ] , [ e d ′ ]vanish if d ij = d ′ ij for some i > k . We consider the line bundle D k on Q d whose fiber at the point ( W • ) equalsdet R Γ( C , W k ). Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikov Lemma 2.14. D k is a e T × C ∗ -equivariant line bundle, and the character of e T × C ∗ acting in the fiber of D k at a point e d = ( d ij ) equals P j ≤ k (1 − d kj ) x j + d kj ( d kj − ~ .Proof. Straightforward. (cid:3)
Let
Cas k = k P i,j =1 E ij E ji be the quadratic Casimir element of U ( gl k )naturally embedded into U ( gl n ) ⊂ U . The operator Cas k is diagonal in theGelfand-Tsetlin basis, and the eigenvalue of Cas k on the basis vector ξ e d = ξ Λ is P j ≤ k λ kj ( λ kj + k − j + 1). We define the following element of U : ] Cas k := Cas k + (2 − k ) k X j =1 E jj − k X j =1 ~ − x j ( ~ − x j −
1) + k ( k − k − . The eigenvalue of this element on the basis vector ξ e d is P j ≤ k − d kj ) x j ~ − + d kj ( d kj − Corollary 2.15. a) The operator of multiplication by the first Chern class c ( D k ) in V is diagonal in the basis { [ e d ] } , and the eigenvalue correspondingto e d = ( d ij ) equals P j ≤ k (1 − d kj ) x j + d kj ( d kj − ~ .b) The set { c ( D k ) : k ≥ , d k = 0 = d k − } forms a basis in the nonequivariant cohomology H ( Q d ) .c) The isomorphism Ψ : V ∼ −→ V carries the operator of multiplicationby c ( D k ) to the operator ~ ] Cas k .Proof. a) follows from Lemma 2.14.b) It follows e.g. from [7] that dim H ( Q d ) = ♯ { k ≥ , d k = 0 = d k − } . Now it is easy to see from Lemma 2.14 that the classes { [ D k ] : k ≥ , d k = 0 = d k − } in Pic( Q d ) are linearly independent, and hence the classes { c ( D k ) : k ≥ , d k = 0 = d k − } are linearly independent in H ( Q d ).c) Straightforward from a) and formula for eigenvalue of ] Cas k on ξ Λ . (cid:3) It is known that a completion b V of the universal Verma module V containsa unique Whittaker vector v = P d v d such that v = 1 (the lowest weightvector), and f i v = ~ − v for any 1 ≤ i ≤ n −
1. Let us denote by 1 d ∈ H e T × C ∗ ( Q d ) ⊂ V d the unit element of the cohomology ring. Then Ψ(1 d ) = v d .The proof is entirely similar to the proof of Proposition 2.31 of [2], and goesback to [1].Recall that the Gelfand-Tsetlin subalgebra G ⊂ End( V ) is generatedby the Harish-Chandra centers of the universal enveloping algebras gl , gl , . . . , gl n (embedded into gl n as the upper left blocks) over the field C ( t ⊕ C ). We denote by I d ⊂ G the annihilator ideal of the vector v d ∈ V ,elfand-Tsetlin algebras and cohomology rings of Laumon spaces 9and we denote by G d the quotient algebra of G by I d . The action of G on v d gives rise to an embedding G d ֒ → V d . Proposition 2.17. a) G d ∼ −→ V d .b) The composite morphism Ψ : H • e T × C ∗ ( Q d ) ⊗ C [ t ⊕ C ] C ( t ⊕ C ) = V d ∼ −→ V d ∼ −→ G d is an algebra isomorphism.c) The algebra H • e T × C ∗ ( Q d ) ⊗ C [ t ⊕ C ] C ( t ⊕ C ) is generated by { c ( D k ) : k ≥ , d k = 0 = d k − } .Proof. c) The algebra H • e T × C ∗ ( Q d ) ⊗ C [ t ⊕ C ] C ( t ⊕ C ) consists of operators onthe space V d which are diagonal in the basis of fixed points [ e d ]. On theother hand, the operators Cas k ∈ G , k ≥
2, are diagonal in the Gelfand-Tsetlin basis ξ e d and have different joint eigenvalues on different ξ e d . Hencethe images of Cas k in End( V d ) generate the algebra of operators which arediagonal in the Gelfand-Tsetlin basis, and in particular, the images of Cas k , k ≥
2, in G d generate G d . By Theorem 2.11, the isomorphism Ψ : V d → V d carries [ e d ] to ( − ~ ) | d | ξ e d . By Corollary 2.15, c ( D k ) is Ψ − ( ~ Cas k ) up toan additive constant. Hence the elements c ( D k ) = Ψ − ( ~ Cas k ) + const ∈ H • e T × C ∗ ( Q d ) ⊗ C [ t ⊕ C ] C ( t ⊕ C ) generate the algebra H • e T × C ∗ ( Q d ) ⊗ C [ t ⊕ C ] C ( t ⊕ C ).a-b) Since Ψ(1 d ) = v d , the (surjective) homomorphismΨ − : C [ Cas , . . . , Cas n − ] → H • e T × C ∗ ( Q d ) ⊗ C [ t ⊕ C ] C ( t ⊕ C ) factorsthrough G d . Hence (a) and (b). (cid:3)
3. Integral forms
We denote by U ⊂ U the C [ t ⊕ C ]-subalgebra generated by the set { E ij := ~ E ij , ≤ i < j ≤ n ; E ij , ≤ j < i ≤ n ; E ′ ii := E ii − ~ − x i , i = 1 , . . . , n } .We denote by U ≤ the subalgebra of U generated by { E ′ ii , ≤ i ≤ n ; E ij , ≤ i < j ≤ n } . It acts on the ring C [ t ⊕ C ] as follows: E ij acts trivially for any i < j , and E ′ ii acts by multiplication by i −
1. We define the integral formof the universal Verma module V ⊂ V over U as V := U ⊗ U ≤ C [ t ⊕ C ].We define the integral form of the universal dual Verma module V ∗ ⊂ V as V ∗ := { u ∈ V : ( u, u ′ ) ∈ C [ t ⊕ C ] for any u ′ ∈ V } (where ( u, u ′ ) stands forthe Shapovalov form). Clearly, V ∗ is a U -module.Note that the Whittaker vector v ∈ b V lies inside the completion of V ∗ ,and is uniquely characterized by the properties a) f i v = v where f i := E i,i +1 ;b) the highest weight component of v equals 1 ∈ C [ t ⊕ C ].Finally, we denote by G ⊂ G the integral form of the Gelfand-Tsetlinsubalgebra, generated by the centers of the algebras U , U , . . . , U n = U con-structed from the Lie algebras gl , gl , . . . , gl n (embedded into gl n as theupper left blocks) the same way as U is constructed from gl n . Recall that theHarish-Chandra isomorphism identifies the center of U ( gl k ) with the ring ofsymmetric polynomials in k variables. Namely, to any symmetric polynomial0 Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikov P one assigns a central element HC ( P ), whose PBW degree equals deg P ,acting on the Verma module with the highest weight λ = ( λ , . . . , λ k ) asthe scalar operator with the eigenvalue P ( λ , . . . , λ i − i + 1 , . . . , λ k − k + 1).Clearly, the central element HC ( P ) := ~ deg P HC ( P ) lies in U ( gl k ). More-over, the difference HC ( P ) − P ( x , . . . , x k ) is divisible by ~ in U ( gl k ), hence ~ − ( HC ( P ) − P ( x , . . . , x k )) also lies in the center of U ( gl k ).We denote by I d ⊂ G the annihilator ideal of the vector v d ∈ V ∗ , andwe denote by G d the quotient algebra of G by I . The action of G on v d givesrise to an embedding G d ֒ → V ∗ d . Lemma 3.2. G d ∼ −→ V ∗ d .Proof. By graded Nakayama lemma, it suffices to prove the surjectivity of G d / ( x , . . . , x n , ~ = 0) → V ∗ d / ( x , . . . , x n , ~ = 0). We denote by g > ⊂ gl n the Lie subalgebra spanned by the set { E ij , ≤ j < i ≤ n } . We denoteby g ≥ ⊂ gl n (resp. g ≤ ⊂ gl n , g > ⊂ gl n , g < ⊂ gl n ) the Lie subalgebraspanned by the set { E ij , ≤ j ≤ i ≤ n } (resp. { E ij , ≤ i ≤ j ≤ n } , { E ij , ≤ j < i ≤ n } , { E ij , ≤ i < j ≤ n } ). The Killing form identifiesthe vector space g > with the dual of g < , and gives rise to an isomorphismSym( g < ) ≃ C [ g > ]. The universal enveloping algebra of g ≥ over C [ t ] liesinside U and is denoted by U ≥ . Evidently, U ≥ ≃ U ( g ≥ ) ⊗ C [ t ]. We have U / ( x , . . . , x n , ~ = 0) ≃ C [ g > ] ⋊ U ( g ≥ ). Here the semidirect product is takenwith respect to the adjoint action of g ≥ on g > (and the induced action onthe algebra of functions).Let V denote the space of distributions on g > supported at the ori-gin, that is cohomology with support of the structure sheaf H n ( n − { } ( g > , O ).The algebra C [ g > ] ⋊ U ( g ≥ ) acts on V naturally. As a C [ g > ]-module, V is cofree, and its completion is naturally isomorphic to C [ g > ] ∗ . Clearly, V ∗ d / ( x , . . . , x n , ~ = 0) as a module over U / ( x , . . . , x n , ~ = 0) ≃ C [ g > ] ⋊ U ( g ≥ ) is isomorphic to V . The value of the Whittaker vector v | ~ =0 in thecompletion of V is the functional χ : C [ g > ] → C which sends P ∈ C [ g > ]to P ( f ) ∈ C , where f = n − P i =1 f i is the principal nilpotent element. The adjoint G ≥ -orbit of f is dense in g > , hence the submodule generated by the Whit-taker vector is dense in the completion of V . This means that each weightspace of V is generated by the component of the Whittaker vector.Consider the Whittaker module W over C [ g > ] ⋊ U ( g ≥ ), that is, in-duced from the character χ : C [ g > ] → C . The module W is free with respectto U ( g ≥ ), and hence has natural filtration coming from the PBW filtrationon U ( g ≥ ). The associated graded gr W is naturally a C [ g ≥ ] ⊗ S ( g > ) = C [ g ]-module. It is easy to see that gr W = C [ f + g ≥ ]. The restriction of theGelfand-Tsetlin subalgebra in C [ g ] to the affine subspace f + g ≥ is known tobe an isomorphism onto C [ f + g ≥ ] (see [12], [23]). Thus the module W is gen-erated by the Whittaker vector as a G d / ( x , . . . , x n , ~ = 0)-module. Hencethe G d / ( x , . . . , x n , ~ = 0)-submodule generated by the Whittaker vector v is dense in the completion of V . This means that each weight space of V elfand-Tsetlin algebras and cohomology rings of Laumon spaces 11is generated by the component of the Whittaker vector with respect to theaction of the Gelfand-Tsetlin subalgebra. (cid:3) We consider a correspondence E d,i ⊂ Q d × Q d + i defined as r − { } . In otherwords, E d,i is a closed subvariety of E d,i where we impose a condition that thequotient flag is supported at { } ∈ C . It is a smooth quasiprojective varietyof dimension 2 d + . . . +2 d n − . We denote by p : E d,i → Q d , q : E d,i → Q d + i the natural projections. Note that both p and q are proper.More generally, for 1 ≤ i < j ≤ n we denote by d ± α ij the sequence( d , . . . , d i − , d i ± , . . . , d j − ± , d j , . . . , d n − ). We have a correspondence ◦ E d,α ij ⊂ Q d × Q d + α ij formed by the pairs ( W • , W ′• ) such that a) W ′ k ⊂ W k for any 1 ≤ k ≤ n −
1; b) The quotient W • / W ′• is supported at { } ∈ C ; c) For i ≤ k < j the natural map W k / W ′ k → W k +1 / W ′ k +1 is an isomorphism (of one-dimensional vector spaces). We define a correspondence E d,α ij ⊂ Q d × Q d + α ij as the closure of ◦ E d,α ij . According to Lemma 5.2.1 of [7], E d,α ij is irreducibleof dimension 2 d + . . . + 2 d n − + j − i −
1. We denote by p ij : E d,α ij → Q d , q ij : E d,α ij → Q d + α ij the natural projections. Note that both p ij and q ij are proper.Also, we consider the correspondences E d,α ij ⊂ Q d × Q d + α ij definedexactly as E d,α ij ⊂ Q d × Q d + α ij in 3.3 but with condition b) removed (i.e. weallow the quotient flag to be supported at an arbitrary point of C − ∞ ). Inparticular, E d,α i,i +1 = E d,i . We denote by p ij : E d,α ij → Q d , q ij : E d,α ij → Q d + α ij the natural projections. Note that q ij is proper, while p ij is not. Recall that ′ V = ⊕ d ′ V d := ⊕ d H • e T × C ∗ ( Q d ). The grading and the correspon-dences E d,α ij give rise to the following operators on ′ V : E ii = x i + ( d i − − d i + i − ~ : ′ V d → ′ V d ; h i = ( x i +1 − x i ) + (2 d i − d i − − d i +1 + 1) ~ : ′ V d → ′ V d ; f i = E i,i +1 = p ∗ q ∗ : ′ V d → ′ V d − i ; e i = E i +1 ,i = − q ∗ p ∗ : ′ V d → ′ V d + i ; E ij = p ij ∗ q ∗ ij : ′ V d → ′ V d − α ij (1 ≤ i < j ≤ n ); E ji = ( − j − i q ij ∗ p ∗ ij : ′ V d → ′ V d + α ij (1 ≤ i < j ≤ n ); E ji = ( − j − i q ij ∗ p ∗ ij : ′ V d → ′ V d + α ij (1 ≤ i < j ≤ n ). Theorem 3.5. a) The operators { E ij , ≤ i ≤ j ≤ n ; E ij , ≤ j < i ≤ n } on ′ V defined in 3.4 satisfy the relations in U , i.e. they give rise to the action of U on ′ V .b) There is a unique isomorphism Φ of U -modules ′ V and V ∗ carrying ∈ H e T × C ∗ ( Q ) ⊂ ′ V to the lowest weight vector ∈ C [ t ⊕ C ] ⊂ V ∗ .Proof. a) We define the operators E ij = p ij ∗ q ∗ ij : V d → V d − α ij (1 ≤ i < j ≤ n ) . (1)2 Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid RybnikovIt is clear that E d,α ij ≃ E d,α ij × ( C − ∞ ). It follows that for any 1 ≤ i, j ≤ n we have E ij = ~ E ij . Furthermore, the operators E i,i ± are exactlythose defined in 2.6, and they satisfy the relations of U (and generate it)by Theorem 2.7. Finally, according to Proposition 5.6 of [7], the elements E ij ∈ U , ≤ i = j ≤ n act in V by the same named operators of (1) and 3.4.This proves a).b) We have ′ V ⊂ V ∼ −→ V ⊃ V ∗ , so we have to check that Ψ( ′ V ) = V ∗ ,and then Φ = Ψ | ′ V . Recall that Ψ( v d ) = 1 ∈ H e T × C ∗ ( Q d ). By the virtue ofLemma 3.2 it suffices to prove that H • e T × C ∗ ( Q d ) is generated by the actionof the integral form G of the Gelfand-Tsetlin subalgebra on the vector 1 ∈ H e T × C ∗ ( Q d ).For any 1 ≤ i ≤ n − W i the tautological i -dimensionalvector bundle on Q d × C . By the K¨unneth formula we have H • e T × C ∗ ( Q d × C ) = H • e T × C ∗ ( Q d ) ⊗ ⊕ H • e T × C ∗ ( Q d ) ⊗ τ where τ ∈ H C ∗ ( C ) is the first Chernclass of O (1). Under this decomposition, for the Chern class c j ( W i ) we have c j ( W i ) =: c ( j ) j ( W i ) ⊗ c ( j − j ( W i ) ⊗ τ where c ( j ) j ( W i ) ∈ H j e T × C ∗ ( Q d ), and c ( j − j ( W i ) ∈ H j − e T × C ∗ ( Q d ).The following Lemma goes back to [5] : Lemma 3.6.
The equivariant cohomology ring H • e T × C ∗ ( Q d ) is generated by theclasses c ( j ) j ( W i ) , c ( j − j ( W i ) , ≤ j ≤ i ≤ n − (over the algebra C [ t ⊕ C ] ).Proof. By the graded Nakayama lemma, it suffices to prove thatthe nonequivariant cohomology ring H • ( Q d ) is generated bythe K¨unneth components of the (nonequivariant) Chern classes c ( j ) j ( W i ) , c ( j − j ( W i ) , ≤ j ≤ i ≤ n −
1. The locally closed embedding Q d ֒ → Q d induces a surjection on the cohomology rings (see e.g. thecomputation of cohomology of Q d in [7]), so it suffices to prove that thecohomology ring of the compact smooth variety H • ( Q d ) is generated bythe K¨unneth components of the Chern classes of the tautological bundles.But this follows from Theorem 2.1 of [5], since the fundamental class ofthe diagonal in Q d × Q d can be expressed via the Chern classes of thetautological vector bundles (cf. [20], section 5). (cid:3) Returning to the proof of the theorem, if suffices to check that theoperators of multiplication by c ( j ) j ( W i ) , c ( j − j ( W i ) , ≤ j ≤ i ≤ n −
1, inthe equivariant cohomology ring H • e T × C ∗ ( Q d ) = ′ V d lie in the integral form G of the Gelfand-Tsetlin subalgebra. To this end we compute these operatorsexplicitly in the fixed point basis { [ e d ] } (alias Gelfand-Tsetlin basis { ξ e d } ) of V d = V d .The set of eigenvalues of t ⊕ C in the fiber of W i at a point ( e d, ∞ )(resp. ( e d, {− x , . . . , − x i } (resp. {− x + d i, ~ , . . . , − x i + d i,i ~ } ). We have learnt of it from A. Marian. elfand-Tsetlin algebras and cohomology rings of Laumon spaces 13For 1 ≤ j ≤ i , let e ∞ ji (resp. e ji ( e d )) stand for the sum of products ofthe j -tuples of distinct elements of the set {− x , . . . , − x i } (resp. of theset {− x + d i, ~ , . . . , − x i + d i,i ~ } ). Then the operator of multiplication bythe Chern class c j ( W i ) is diagonal in the basis { [ e d, ∞ ] , [ e d, } with eigenval-ues { e ∞ ji , e ji ( e d ) } . It follows that the operator of multiplication by c ( j ) j ( W i )(resp. by c ( j − j ( W i ) , ≤ j ≤ i ≤ n −
1) is diagonal in the basis { [ e d ] } with eigenvalues { ( e ∞ ji + e ji ( e d )) } (resp. { ~ − ( e ∞ ji − e ji ( e d )) } ). Note that e ∞ ji ∈ C [ t ⊕ C ], and e ji ( e d )) is precisely the eigenvalue of the central element HC ( e ji ) ∈ U ( gl i ) corresponding to the j -th elementary symmetric function e j via the Harish-Chandra isomorphism, on the Verma module with high-est weight { λ i ~ , . . . , λ ii ~ } . Hence the operator of multiplication by c ( j ) j ( W i )(with eigenvalues { ( e ∞ ji + e ji ( e d )) } ) lies in the integral form G of the Gelfand-Tsetlin subalgebra. Moreover, e ∞ ji − HC ( e ji ) is divisible by ~ in U ( gl i ). Hencethe operator of multiplication by c ( j − j ( W i ) lies in G as well. (cid:3) Corollary 3.7.
The composition of isomorphisms G d ∼ −→ V ∗ d Φ − ←− H • e T × C ∗ ( Q d ) is an isomorphism of algebras. (cid:3)
4. Speculation on equivariant quantum cohomology of Q d According to Theorem 3 of [10], the variety Q d is Calabi-Yau. For the reader’sconvenience we recall a proof. First note that if d = ( d , . . . , d n − ), and d k = 0, then Q d = Q d ′ × Q d ′′ where d ′ = ( d , . . . , d k − ), and Q d ′ is thecorresponding Laumon moduli space for GL k , while d ′′ = ( d k +1 , . . . , d n − ),and Q d ′′ is the corresponding Laumon moduli space for GL n − k . Hence wemay assume that all the integers d , . . . , d n − are strictly positive.For 1 ≤ i ≤ n −
1, we consider the locally closed subvariety of Q d formedby all the quasiflags which have a defect of degree exactly i . We denote by D i ⊂ Q d the closure of this subvariety. It is a divisor. We denote by [ O ( D i )]the class of the corresponding line bundle in Pic( Q d ). Lemma 4.2.
Assume all the integers d , . . . , d n − are strictly positive. Then [ O ( D )] = [ D ] , . . . , [ O ( D i )] = [ D i +1 ] − [ D i ] , . . . , [ O ( D n − )] = − [ D n − ] .Proof. Recall the morphism π d : Q d → Z d to Drinfeld’s Zastava space (asmall resolution of singularities). For 1 ≤ k ≤ n −
2, choose a point s ∈ Z d with defect of degree exactly k + ( k + 1). We denote by P k the preimage π − d ( s ) ⊂ Q d . By a GL -calculation, P k is a projective line. The fundamentalclasses [ P ] , . . . , [ P n − ] form a basis of H ( Q d , C ). It is easy to see that therestriction D i | P k is trivial if i = k + 1, while D k +1 | P k ≃ O (1) (this is again a GL -calculation). Furthermore, it is easy to see that the restriction O ( D i ) | P k i = k, k + 1, while O ( D k ) | P k ≃ O (1), and O ( D k +1 ) | P k ≃ O ( − GL -calculation). The lemma is proved. (cid:3) Let ◦ Q d ⊂ Q d denote the open subspace formed by all the quasiflagswithout defect. Recall the symplectic form Ω on ◦ Q d constructed in [9]. Notethat the complement Q d \ ◦ Q d equals the union of divisors S ≤ i ≤ n − D i .Thus the top power ω := Ω top is a meromorphic volume form on Q d withpoles at S ≤ i ≤ n − D i . The formula for Ω given in Remark 3 of loc. cit. showsthat the order of the pole of ω at D i is 2 for any 1 ≤ i ≤ n −
1. Hence thecanonical class of Q d in Pic( Q d ) equals 2 P ≤ i ≤ n − [ O ( D i )]. By Lemma 4.2,the class 2 P ≤ i ≤ n − [ O ( D i )] = 0. We have proved Corollary 4.3. (Givental, Lee [10] ) The canonical class of Q d is trivial. To each regular element µ of the Cartan subalgebra h of a semisimple Liealgebra g one can assign a space Q µ of commuting quadratic elements of theuniversal enveloping algebra U ( g ). Namely, Q µ = { X α ∈ ∆ + h h, α ih µ, α i e α e − α , | h ∈ h } , where ∆ + is the set of positive roots, and e α , e − α are nonzero elements ofthe root spaces such that ( e α , e − α ) = 1. Note that the space Q µ does notchange under dilations of µ , hence we have a family of spaces of commutingquadratic operators, parametrized by the regular part of P ( h ).These quadratic elements appear as the quasiclassical limit of the Casimir flat connection on the trivial bundle on the regular part of theCartan subalgebra with the fiber V d , (cf. [4, 6, 24, 16]). This connection isgiven by the formula ∇ = d + κ X α ∈ ∆ + e α e − α dαα , where κ is a parameter. Since every element of U ( g ) of the form e α e − α commutes with the Cartan subalgebra h , this connection remains flat afteradding any closed U ( h )-valued 1-form.The centralizer in U ( g ) of this space of commuting quadratic operatorsis the so-called shift of argument subalgebra A µ ⊂ U ( g ), which is a free com-mutative subalgebra with (dim g + rk g ) generators. For g = sl n the familyof commutative subalgebras A µ ⊂ U ( g ) is an ( n − Consider the shift of argument subalgebra for g = gl n , µ = n − P i =1 q i +1 q i +2 . . . q n ω i with ω i being the fundamental weights of gl n . Sincethe shift of argument algebra does not change under dilations of µ , we canelfand-Tsetlin algebras and cohomology rings of Laumon spaces 15assume that q n = 1. Taking h k = k − P i =1 q i q i +1 . . . q n ω i , we find that the space Q µ is generated by the elements X α ∈ ∆ + h h k , α ih µ, α i e α e − α = X i Ψ : V d → V d carries the operator M D k of quantum multiplication by c ( D k ) to the operator ~ QC k . Corollary 4.7. The localized equivariant quantum cohomology ring of Q d isisomorphic to the quotient of the shift of argument subalgebra A µ by theannihilator of v d . Let A µ denote the integral form A µ ∩ U . Conjecture 4.8. The equivariant quantum cohomology ring of Q d is isomor-phic to the quotient of A µ by the annihilator of v d . Remark 4.9. It is natural to expect 4.8 since the analogue of Lemma 3.2 isvalid for A µ as well (and the proof is the same).The map ( q , . . . , q n − ) µ = n − P i =1 q i +1 q i +2 . . . q n − ω i embeds the torus T = C ∗ ( n − with coordinates q , . . . , q n − into the Cartan subalgebra h = C n − of sl n as an open subset of an affine hyperplane. Restricting the Casimir It was recently proved by A. Negut. T and adding an appropriate Cartan term, we obtain the fol-lowing flat connection on T in the coordinates q i : ∇ = d + κ n − X k =2 QC k dq k q k . On the other hand, the trivial vector bundle with the fiber V d over thespace of quantum parameters T is equipped with the quantum connection d + n − P k =2 M D k dq k q k . Corollary 4.10. The isomorphism Ψ carries the quantum connection to theCasimir connection with κ = ~ . Remark 4.11. According to Vinberg [25], the family of subspaces Q µ form anopen subset in the moduli space of ( n − e α e − α in U ( g ). Thus it is natural to expect that theoperators M D k span the space Q µ for some µ depending on q , . . . , q n − (but unfortunately, we have no idea how to prove that the operators M D k are quadratic expressions in the correspondences). Moreover, since the unitin the cohomology ring remains unit in the quantum cohomology ring, thequantum correction has to annihilate the vector v d . Therefore the operatorΨ( M D k ) is QC k up to a change of parametrization. Finally, the flatness of thequantum connection d + n − P k =2 M D k dq k q k is a very restrictive condition on theparametrization µ ( q , . . . , q n − ) — this leaves no other choice for Ψ( M D k )but to coincide with ~ QC k . 5. Global Laumon spaces Recall the setup of 2.3. We define two versions of the correspondences E d,i ⊂ Q d × Q d + i , namely, E d,i := r − { } , E ∞ d,i := r − {∞} . Their projections to Q d (resp. Q d + i ) will be denoted by p , p ∞ (resp. q , q ∞ ). The projection of E d,i to Q d (resp. Q d + i ) will be denoted by p C (resp. q C ).We denote by ′ A (resp. ′ B ) the direct sum of equivariant (complexified)cohomology: ′ A = ⊕ d H • e T × C ∗ ( Q d ) (resp. ′ B = ⊕ d H • G × C ∗ ( Q d )). We define A = ′ A ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt )), and B = ′ B ⊗ H • G × C ∗ ( pt ) Frac( H • G × C ∗ ( pt )).We have an evident grading A = ⊕ d A d , A d = H • e T × C ∗ ( Q d ) ⊗ H • e T × C ∗ ( pt ) Frac( H • e T × C ∗ ( pt )); similarly, B = ⊕ d B d , B d = H • G × C ∗ ( Q d ) ⊗ H • G × C ∗ ( pt ) Frac( H • G × C ∗ ( pt )).Note that H • G × C ∗ ( pt ) = C [ e , . . . , e n , ~ ] where e i is the i -th elementarysymmetric function of x , . . . , x n .elfand-Tsetlin algebras and cohomology rings of Laumon spaces 17 The set of e T × C ∗ -fixed points in Q d is finite; it is described in [7], 2.11. Recallthat this fixed point set is in bijection with the set of collections { b d } of thefollowing data: a) a permutation σ ∈ S n ; b) a matrix ( d ij ) (resp. d ∞ ij ) , i ≥ j of nonnegative integers such that for i ≥ j ≥ k we have d kj ≥ d ij (resp. d ∞ kj ≥ d ∞ ij ) such that d i = P ij =1 ( d ij + d ∞ ij ). Abusing notation we denote by b d the corresponding e T × C ∗ -fixed point in Q d : W = O C ( − d · − d ∞ · ∞ ) w σ (1) , W = O C ( − d · − d ∞ · ∞ ) w σ (1) ⊕ O C ( − d · − d ∞ · ∞ ) w σ (2) ,. . . . . . . . . , W n − = O C ( − d n − , · − d ∞ n − , · ∞ ) w σ (1) ⊕ O C ( − d n − , · − d ∞ n − , ·∞ ) w σ (2) ⊕⊕ . . . ⊕ O C ( − d n − ,n − · − d ∞ n − ,n − · ∞ ) w σ ( n − . Also, we will write b d = ( σ, e d , e d ∞ ).The localized equivariant cohomology A is equipped with the basis ofdirect images of the fundamental classes of the fixed points; by an abuse ofnotation, this basis will be denoted { [ b d ] } . As in the proof of Theorem 3.5 and Corollary 3.7 we see that the H • e T × C ∗ ( pt )-algebra H • e T × C ∗ ( Q d ) is generated by the K¨unneth components of the Chernclasses c ( j ) j ( W C i ) , c ( j − j ( W C i ) , ≤ j ≤ i ≤ n − 1, of the tautological vectorbundles W C i on Q d × C . We compute the operators of multiplication by c ( j ) j ( W C i ) , c ( j − j ( W C i ) , ≤ j ≤ i ≤ n − 1, in the equivariant cohomologyring H • e T × C ∗ ( Q d ) in the fixed point basis { [ b d ] } .We introduce the following notation. For a function f ( x , . . . , x n , ~ ) ∈ C ( t ⊕ C ) and a permutation σ ∈ S n weset f σ ( x , . . . , x n , ~ ) := f ( x σ (1) , . . . , x σ ( n ) , ~ ). Also, we set¯ f ( x , . . . , x n , ~ ) := f ( x , . . . , x n , − ~ ). For 1 ≤ j ≤ i , let e ji ( b d )(resp. e ∞ ji ( b d )) stand for the sum of products of the j -tuples of distinctelements of the set {− x + d i, ~ , . . . , − x i + d i,i ~ } (resp. of the set {− x + d ∞ i, ~ , . . . , − x i + d ∞ i,i ~ } ). Lemma 5.4. The operator of multiplication by c ( j ) j ( W C i ) (resp. by c ( j − j ( W C i ) , ≤ j ≤ i ≤ n − ) is diagonal in the basis { [ b d ] = [( σ, e d , e d ∞ )] } with eigenvalues { ( e ji ( b d ) + e ∞ ji ( b d )) σ } (resp. { ~ − ( e ∞ ji ( b d ) − e ji ( b d )) σ } ).Proof. The same argument as in the proof of Theorem 3.5. (cid:3) Corollary 5.5. The operator of multiplication by c (1)1 ( W C i ) is diagonal in thebasis { [ b d ] = [( σ, e d , e d ∞ )] } with eigenvalues − ( x + . . . + x i ) σ + d i ~ . We denote by U the universal enveloping algebra of gl n ⊕ gl n over the field C ( t ⊕ C ). For 1 ≤ i, j ≤ n we denote by E (1) ij (resp. E (2) ij ) the element( E ij , ∈ gl n ⊕ gl n ⊂ U (resp. the element (0 , E ij ) ∈ gl n ⊕ gl n ⊂ U ).We denote by U ≤ the subalgebra of U generated by E (1) ii , E (2) ii , ≤ i ≤ n, E (1) i,i +1 , E (2) i,i +1 , ≤ i ≤ n − 1. It acts on the field C ( t ⊕ C ) as follows: E (1) i,i +1 , E (2) i,i +1 act trivially for any 1 ≤ i ≤ n − 1, and E (1) ii (resp. E (2) ii ) acts bymultiplication by ~ − x i + i − − ~ − x i + i − universalVerma module B over U as U ⊗ U ≤ C ( t ⊕ C ). The universal Verma module B is an irreducible U -module.For a permutation σ = ( σ (1) , . . . , σ ( n )) ∈ S n (the Weyl group of G = GL n ) we consider a new action of U ≤ on C ( t ⊕ C ) defined as fol-lows: E (1) i,i +1 , E (2) i,i +1 act trivially for any 1 ≤ i ≤ n − 1, and E (1) ii (resp. E (2) ii )acts by multiplication by ~ − x σ ( i ) + i − − ~ − x σ ( i ) + i − B σ over U as U ⊗ U ≤ C ( t ⊕ C ) (with respect to the new action of U ≤ on C ( f ⊕ C )). Finally, we define A := L σ ∈ S n B σ . The grading and the correspondences E d,i , E d,i , E ∞ d,i give rise to the followingoperators on A, B : f (1) i = E (1) i,i +1 = ~ − p ∗ q ∗ : A d → A d − i and B d → B d − i ; f (2) i = E (2) i,i +1 = − ~ − p ∞∗ q ∞∗ : A d → A d − i and B d → B d − i ; e (1) i = E (1) i +1 ,i = − ~ − q ∗ p ∗ : A d → A d + i and B d → B d + i ; e (2) i = E (2) i +1 ,i = ~ − q ∞∗ p ∞∗ : A d → A d + i and B d → B d + i ; f ∆ i = p C ∗ q C ∗ : A d → A d − i and B d → B d − i ; e ∆ i = − q C ∗ p C ∗ : A d → A d + i and B d → B d + i .Furthermore, we define the action of C [ t ⊕ C ] on B as follows: for 1 ≤ i ≤ n − , x i acts on B d by multiplication by c (1)1 ( W C i − ) − c (1)1 ( W C i ) − ( d i − d i − ) ~ (cf. Corollary 5.5); and x n acts by multiplication by e − x − . . . − x n − (recallthat e is the generator of H G ( pt )). Theorem 5.8. a) The operators e (1) i = E (1) i +1 ,i , e (2) i = E (2) i +1 ,i , f (1) i = E (1) i,i +1 , f (2) i = E (2) i,i +1 , along with the action of C ( t ⊕ C ) on B defined in 5.7satisfy the relations in U , i.e. they give rise to the action of U on B ;b) There is a unique isomorphism Ψ of U -modules B and B carrying ∈ H e T × C ∗ ( Q ) ⊂ B to the lowest weight vector ∈ C ( t ⊕ C ) ⊂ B .Proof. We describe the matrix coefficients of e (1) i = E (1) i +1 ,i , e (2) i = E (2) i +1 ,i , f (1) i = E (1) i,i +1 , f (2) i = E (2) i,i +1 in the fixed point basis { [ b d ] } . Recall thematrix coefficients e i [ e d, e d ′ ] , f i [ e d, e d ′ ] computed in Proposition 2.9. The proof ofthe following easy lemma is omitted.elfand-Tsetlin algebras and cohomology rings of Laumon spaces 19 Lemma 5.9. Let b d = ( σ, e d , e d ∞ ) , and b d ′ = ( σ ′ , e d ′ , e d ∞′ ) . The ma-trix coefficients are as follows: e (1) i [ b d, b d ′ ] = δ σ,σ ′ ( e i [ e d , e d ′ ] ) σ ; e (2) i [ b d, b d ′ ] = δ σ,σ ′ (¯ e i [ e d ∞ , e d ∞′ ] ) σ ; f (1) i [ b d, b d ′ ] = δ σ,σ ′ ( f i [ e d , e d ′ ] ) σ ; f (2) i [ b d, b d ′ ] = δ σ,σ ′ (¯ f i [ e d ∞ , e d ∞′ ] ) σ . (cid:3) It follows that the operators e (1) i = E (1) i +1 ,i , e (2) i = E (2) i +1 ,i , f (1) i = E (1) i,i +1 , f (2) i = E (2) i,i +1 on A defined in 5.7 satisfy the relations in U , i.e.they give rise to the action of U on A . Moreover, it follows that the U -module A is isomorphic to A . In order to describe the image of thebasis { [ b d ] } under this isomorphism, we introduce the following notation.First, we introduce a U -module V σ := U ⊗ U ≤ C ( t ⊕ C ) where U ≤ actson C ( t ⊕ C ) as follows: E i,i +1 acts trivially for any 1 ≤ i ≤ n − 1, and E ii acts by multiplication by ~ − x σ ( i ) + i − 1. Similarly, we introduce a U -module ¯ V σ := U ⊗ U ≤ C ( t ⊕ C ) where U ≤ acts on C ( t ⊕ C ) as follows: E i,i +1 acts trivially for any 1 ≤ i ≤ n − 1, and E ii acts by multiplication by − ~ − x σ ( i ) + i − 1. Note that B σ ≃ V σ ⊗ C ( t ⊕ C ) ¯ V σ .Now to a collection e d = ( d ij ), and a permutation σ ∈ S n , we associatea Gelfand-Tsetlin pattern Λ σ = Λ σ ( e d ) := ( λ σij ) , n ≥ i ≥ j , as follows: λ σnj := ~ − x σ ( j ) + j − , n ≥ j ≥ λ σij := ~ − x σ ( j ) + j − − d ij , n − ≥ i ≥ j ≥ ξ σ e d = ξ Λ σ ∈ V σ by the formulas (2.9)–(2.11) of [17] (where ξ = ξ = 1 ∈ V σ ). Similarly, to a collection e d = ( d ij ), and a permutation σ ∈ S n ,we associate a Gelfand-Tsetlin pattern ¯Λ σ = ¯Λ σ ( e d ) := (¯ λ σij ) , n ≥ i ≥ j , asfollows: ¯ λ σnj := − ~ − x σ ( j ) + j − , n ≥ j ≥ 1; ¯ λ σij := − ~ − x σ ( j ) + j − − d ij , n − ≥ i ≥ j ≥ 1. We define ¯ ξ σ e d = ξ ¯Λ σ ∈ ¯ V σ by the formulas (2.9)–(2.11)of [17] (where ξ = ξ = 1 ∈ ¯ V σ ). Finally, to a collection b d = ( σ, e d , e d ∞ ) weassociate an element ξ b d := ξ σ e d ⊗ ¯ ξ σ e d ∞ ∈ V σ ⊗ C ( t ⊕ C ) ¯ V σ = B σ .Theorem 2.11 and Lemma 5.9 implies that under the above isomorphism A ≃ A a basis element [ b d ] goes to ( − ~ ) | d | ξ b d .We are ready to finish the proof of the theorem. Since the action of e T × C ∗ on Q d extends to the action of G × C ∗ , the equivariant cohomol-ogy H • e T × C ∗ ( Q d ) is equipped with the action of the Weyl group S n , and H • G × C ∗ ( Q d ) = H • e T × C ∗ ( Q d ) S n . It follows B = A S n . Since B is closed withrespect to the action of the operators e (1) i = E (1) i +1 ,i , e (2) i = E (2) i +1 ,i , f (1) i = E (1) i,i +1 , f (2) i = E (2) i,i +1 on B , part a) follows. Note however, that the action of S n on A does not commute with the action of the above operators.To prove part b), we describe the action of S n on A explicitly in thebasis { [ b d ] } . For b d = ( σ, e d , e d ∞ ) , σ ′ ∈ S n , f ∈ C ( t ⊕ C ), we have σ ′ ( f [ b d ]) = f σ ′ [( σ ′ σ, e d , e d ∞ )]. We conclude that for b d = (1 , e d , e d ∞ ) (so that ξ b d ∈ B = B )we have (Ψ ) − ( f ξ b d ) = P σ ∈ S n f σ [( σ, e d , e d ∞ )]. This completes the proof ofthe theorem. (cid:3) Remark 5.10. The operators f ∆ i , e ∆ i of 5.7 were introduced in [7]. It is easyto see that f ∆ i = f (1) i + f (2) i , e ∆ i = e (1) i + e (2) i . A completion of B = V ⊗ C ( t ⊕ C ) V contains the Whittaker vector P d b d = b := v ⊗ v . It follows from the proof of Theorem 5.8 that for the unit elementof the cohomology ring 1 d ∈ H G × C ∗ ( Q d ) we have Ψ (1 d ) = b d . The doubleGelfand-Tsetlin subalgebra G := G ⊗ G acts by endomorphisms of B . Wedenote by I d ⊂ G the annihilator ideal of the vector b d ∈ B , and we denoteby G d the quotient of G by I d . The action of G on b d gives rise to anembedding G d ֒ → B d . The same way as in Proposition 2.17.a) one provesthat this embedding is an isomorphism G d ∼ −→ B d . Proposition 5.12. The composite morphism Ψ : H • G × C ∗ ( Q d ) ⊗ H • G × C ∗ ( pt ) Frac( H • G × C ∗ ( pt )) = B d ∼ −→ B d ∼ −→ G d is an algebra isomorphism.Proof. As in the proof of Theorem 3.5 and Corollary 3.7 we see that the H • G × C ∗ ( pt )-algebra H • G × C ∗ ( Q d ) is generated by the K¨unneth components ofthe Chern classes c ( j ) j ( W C i ) , c ( j − j ( W C i ) , ≤ j ≤ i ≤ n − 1, of the tautologicalvector bundles W C i on Q d × C . In order to prove the proposition, it sufficesto check that the operators of multiplication by c ( j ) j ( W C i ) , c ( j − j ( W C i ) , ≤ j ≤ i ≤ n − 1, in the equivariant cohomology ring H • G × C ∗ ( Q d ) lie in G d . Tothis end we compute these operators explicitly in the basis { (Ψ ) − ξ b d , b d =(1 , e d , e d ∞ ) } of B . Lemma 5.4 implies that the operator of multiplicationby c ( j ) j ( W C i ) (resp. by c ( j − j ( W C i ) , ≤ j ≤ i ≤ n − 1) is diagonal in thebasis { (Ψ ) − ξ b d , b d = (1 , e d , e d ∞ ) } with eigenvalues { e ji ( b d ) + e ∞ ji ( b d ) } (resp. { ~ − ( e ∞ ji ( b d ) − e ji ( b d )) } ). As in the proof of Theorem 3.5 we see that e ji ( b d ) isthe eigenvalue of the element of G d ⊗ ⊂ G d , and e ∞ ji ( b d ) is the eigenvalueof the same element in the other copy of the Gelfand-Tsetlin subalgebra1 ⊗ G d ⊂ G d , hence c ( j ) j ( W C i ) and c ( j − j ( W C i ) lie in G d . (cid:3) Recall the notations of 3.3. We consider the correspondences E d,α ij ⊂ Q d × Q d + α ij (resp. E ∞ d,α ij ⊂ Q d × Q d + α ij ) defined exactly as in loc. cit. (resp.replacing the condition in 3.3.b by the condition that W • / W ′• is supportedat ∞ ∈ C ). We denote by p ij : E d,α ij → Q d , q ij : E d,α ij → Q d + α ij , p ∞ ij : E ∞ d,α ij → Q d , q ∞ ij : E ∞ d,α ij → Q d + α ij the natural proper projections. We alsoconsider the correspondences and projections E C d,α ij ⊂ Q d × Q d + α ij , p C ij , q C ij defined as above but without any restriction on the support of W • / W ′• .We consider the following operators on ′ B : E (1) ij = p ij ∗ q ∗ ij : ′ B d → ′ B d − α ij ; E (2) ij = − p ∞ ij ∗ q ∞∗ ij : ′ B d → ′ B d − α ij ;elfand-Tsetlin algebras and cohomology rings of Laumon spaces 21 E (1) ji = ( − i − j q ij ∗ p ∗ ij : ′ B d → ′ B d + α ij ; E (2) ji = − ( − i − j q ∞ ij ∗ p ∞∗ ij : ′ B d → ′ B d + α ij ; E ∆ ij = p C ij ∗ q C ∗ ij : ′ B d → ′ B d − α ij ; E ∆ ji = ( − i − j q C ij ∗ p C ∗ ij : ′ B d → ′ B d + α ij .We have E ∆ ij = ~ − ( E (1) ij + E (2) ij ) , E ∆ ji = ~ − ( E (1) ji + E (2) ji ).We define U ⊂ U as the C [ t ⊕ C ]-subalgebra generated by y (1) , y (2) , y ∆ , y ∈ gl n , where y (1) (resp. y (2) ) stands for ~ ( y, 0) (resp. ~ (0 , y )), and y ∆ stands for ( y, y ). Then it is easy to see that the aboveoperators give rise to the action of U on ′ B .Also, it is easy to check that U / ( ~ = 0) is isomorphic to the algebra U := ( C [ gl n ] ⋊ U ( gl n )) ⊗ C [ t ] (the semidirect product with respect to theadjoint action). Hence ¯ B := ′ B/ ( ~ = 0) = ⊕ d H • G ( Q d ) inherits an action of U . Conjecture 5.14. The U -module ¯ B is isomorphic to H n ( n − { g ≤ } ( gl n , O ) . Underthis isomorphism, the action of H • G ( pt ) on ¯ B corresponds to the action of C [ gl n ] G on H n ( n − { g ≤ } ( gl n , O ) . Conjecture 6.4 of [7] on the direct sum of nonequivariant cohomology of Q d is an immediate corollary of 5.14. We propose a generalization of Conjecture 6.4 of [7] in a different direction.Let d = ( d , . . . , d n ) be an n -tuple of integers (not necessarily positive). Let Q d be the moduli stack of flags of locally free sheaves W ⊂ . . . ⊂ W n − ⊂ W n on C such that rk( W i ) = i, deg( W i ) = d i (see [14]). We have a representableprojective morphism π : Q d → Bun , W • W n , where Bun stands for themoduli stack of GL n -bundles on C . The fiber of π over the trivial GL n -bundle(an open point of Bun) is Q d . The correspondences E d,α ij , E ∞ d,α ij , E C d,α ij ofsubsection 5.13 make perfect sense for the stacks Q d in place of Q d . As in loc.cit. , they give rise to the operators E (1) ij , E (2) ij , E ∆ ij , etc. on the constructiblecomplex B := L d ∈ Z n − π ∗ C d on Bun (where C d stands for the constant sheafon Q d ). This constructible complex is the geometric Eisenstein series of [14].The above operators give rise to the action of ¯ U := C [ gl n ] ⋊ U ( gl n ) on B : thisfollows from the results of 5.13 by the argument of [2], 3.8–3.11. In particular,¯ U acts on the stalks of B , and we propose a conjecture describing the resulting¯ U -modules.Recall that the isomorphism classes of GL n -bundles on C areparametrized by the set X + of dominant weights of GL n . For η ∈ X + wedenote by B η the corresponding stalk of B . Also, we will denote by O ( η )the corresponding line bundle on the flag variety B n of GL n . We will keepthe same notation for the lift of O ( η ) to the cotangent bundle T ∗ B n . Wewill denote by L η the direct image of O ( η ) under the Springer resolution2 Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikovmorphism T ∗ B n → N to the nilpotent cone N ⊂ gl n . The cohomology of thecoherent sheaf L η carries a natural action of ¯ U . Conjecture 5.16. The ¯ U -module B η is isomorphic to H n ( n − g < ( N , L η ) (cf. Con-jecture 7.8 of [8] ). 6. Equivariant K -ring of Q d and quantum Gelfand-Tsetlinalgebra We preserve the setup of [2] with the following slight changes of notation.Now e T stands for a 2 n -cover of a Cartan torus of GL n as opposed to SL n in loc. cit. Now U ′ stands for the quantum universal enveloping algebra of gl n over the field C ( e T × C ∗ ), as opposed to the quantum universal envelopingalgebra of sl n in 2.26 of loc. cit. For the quantum universal enveloping algebra of gl n we follow the no-tations of section 2 of [18]. Namely, U ′ has generators t ij , ¯ t ij , ≤ i, j ≤ n ,subject to relations (2.4) of loc. cit. The standard Chevalley generators areexpressed via t ij , ¯ t ij as follows: K i = t i +1 ,i +1 ¯ t ii , E i = ( v − v − ) − ¯ t ii t i +1 ,i , F i = − ( v − v − ) − ¯ t i,i +1 t ii (note that this presentation differs from the one in (2.6) of loc. cit. byan application of Chevalley involution). Note also that U ′ is generated by t ii , ¯ t ii , ≤ i ≤ n ; t i +1 ,i , ¯ t i,i +1 , ≤ i ≤ n − 1. We denote by U ′≤ the sub-algebra of U ′ generated by t ii , ¯ t ii , ¯ t i,i +1 . It acts on the field C ( e T × C ∗ ) asfollows: ¯ t i,i +1 acts trivially for any 1 ≤ i ≤ n − 1, and ¯ t ii = t − ii acts bymultiplication by t − i v − i . We define the universal Verma module M over U ′ as M := U ′ ⊗ U ′≤ C ( e T × C ∗ ).Recall that M = ⊕ d M d , M d = K e T × C ∗ ( Q d ) ⊗ C [ e T × C ∗ ] C ( e T × C ∗ )(cf. [2], 2.7).We define the following operators on M (well defined since the corre-spondences E d,i are smooth, and the e T × C ∗ -fixed point sets are finite): t ii = t i v d i − − d i + i − : M d → M d ; ¯ t ii = t − ii ;¯ t i,i +1 = ( v − − v ) t ii +1 t − i − i v (2 i +1) d i − ( i +1) d i − − id i +1 − i +1 p ∗ q ∗ : M d → M d − i ; t i +1 ,i = ( v − − v ) t − i − i +1 t ii v id i − +( i +1) d i +1 − (2 i +1) d i − q ∗ ( L i ⊗ p ∗ ) : M d → M d + i . According to Theorem 2.12 of [2], these operators satisfy the relations in U ′ , i.e. they give rise to the action of U ′ on M . Moreover, there is a uniqueisomorphism ψ : M → M carrying [ O Q ] ∈ M to the lowest weight vector1 ∈ C ( e T × C ∗ ) ⊂ M .elfand-Tsetlin algebras and cohomology rings of Laumon spaces 23 The construction of Gelfand-Tsetlin basis for the representations of quan-tum gl n goes back to M. Jimbo [11]. We will follow the approach of [18].Given e d and the corresponding Gelfand-Tsetlin pattern Λ = Λ( e d ) (see sub-section 2.10), we define ξ e d = ξ Λ ∈ M by the formula (5.12) of [18]. Accordingto Proposition 5.1 of loc. cit. , the set { ξ e d } (over all collections e d ) forms abasis of M .Recall the basis { [ e d ] } of M introduced in 2.16 of [2]: the direct imagesof the structure sheaves of the torus-fixed points. The following theorem isproved absolutely similarly to Theorem 2.11, using Proposition 5.1 of [18]. Theorem 6.3. The isomorphism ψ : M → M of subsection 6.1 takes [ e d ] to c e d ξ e d where c e d = ( v − −| d | v | d | + P i id i − d i − P i i +12 d i − P i,j d i,j Y i t i ( d i − d i − ) i Y j t P k ≥ j d k,j j . (cid:3) Let Cas vk be the quantum Casimir element of the completion of the quan-tum universal enveloping algebra U v ( gl k ). The quantum Casimir element isdefined in section 6.1. of [15] in terms of the universal R -matrix lying inthe completion of U v ( gl k ) ⊗ U v ( gl k ). According to [15], Proposition 6.1.7,the eigenvalue of Cas vk on the Verma module over U v ( gl k ) with the highestweight λ k is v − ( λ k ,λ k +2 ρ k ) . This means that the operator Cas vk is diagonalin the Gelfand-Tsetlin basis, and the eigenvalue of Cas vk on the basis vector ξ e d = ξ Λ is v − P j ≤ k λ kj ( λ kj + k − j +1) (with t i v j − − d ki = v λ ki ). Consider thefollowing “corrected” Casimir operators ] Cas vk := Cas vk · k Y j =1 t k − jj v k P j =1 ( λ nj − j )( λ nj − j +1) − k ( k − k − Lemma 2.14 admits the following Corollary 6.5. a) The operator of tensor multiplication by the class [ D k ] in M is diagonal in the basis { [ e d ] } , and the eigenvalue corresponding to e d = ( d ij ) equals Q j ≤ k t − d kj j v d kj ( d kj − .b) The isomorphism ψ : M → M carries the operator of tensor multi-plication by [ D k ] to the operator ] Cas vk − .Proof. a) is immediate.b) straightforward from a) and the formula for eigenvalues of Cas vk . (cid:3) K -rings Recall the Whittaker vector k = P d k d of [2] 2.30. According to Proposi-tion 2.31 of loc. cit. , ψ [ O d ] = k d .Consider the “quantum Gelfand-Tsetlin algebra” G ⊂ End( M ) gener-ated by all the ] Cas vk − over the field C ( e T × C ∗ ). We denote by I d ⊂ G the annihilator ideal of the vector k d ∈ V , and we denote by G d the quo-tient algebra of G by I d . 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Boris Feigin, Michael Finkelberg, Igor Frenkel and Leonid Rybnikov Address :B.F.: Landau Institute for Theoretical Physics, Kosygina st 2, Moscow 117940,RussiaM.F.: IMU, IITP, and State University Higher School of Economics,Department of Mathematics,20 Myasnitskaya st, Moscow 101000, RussiaI.F.: Yale University, Department of Mathematics, PO Box 208283, New Haven,CT 06520, USAL.R.: Institute for the Information Transmission Problems andState University Higher School of Economics,Department of Mathematics,20 Myasnitskaya st, Moscow 101000, Russiae-mail::B.F.: Landau Institute for Theoretical Physics, Kosygina st 2, Moscow 117940,RussiaM.F.: IMU, IITP, and State University Higher School of Economics,Department of Mathematics,20 Myasnitskaya st, Moscow 101000, RussiaI.F.: Yale University, Department of Mathematics, PO Box 208283, New Haven,CT 06520, USAL.R.: Institute for the Information Transmission Problems andState University Higher School of Economics,Department of Mathematics,20 Myasnitskaya st, Moscow 101000, Russiae-mail: