A note on a symplectic structure on the space of G- monopoles

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1 University of Massachusetts Amherst Amherst Mathematics and Statistics Department Faculty Publication Series Mathematics and Statistics 1999 A note on a symplectic structure on the space of G- monopoles M Finkelberg A Kuznetsov N Markarian I Mirkovic University of Massachusetts - Amherst, mirkovic@math.umass.edu Follow this and additional works at: Part of the Physical Sciences and Mathematics Commons Recommended Citation Finkelberg, M; Kuznetsov, A; Markarian, N; and Mirkovic, I, "A note on a symplectic structure on the space of G-monopoles" (1999). COMMUNICATIONS IN MATHEMATICAL PHYSICS Retrieved from This Article is brought to you for free and open access by the Mathematics and Statistics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in Mathematics and Statistics Department Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contact scholarworks@library.umass.edu.

2 arxiv:math/ v2 [math.ag] 19 Apr 1998 A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES MICHAEL FINKELBERG, ALEXANDER KUZNETSOV, NIKITA MARKARIAN, AND IVAN MIRKOVIĆ 1. Introduction 1.1. Let G be a semisimple complex Lie group with the Cartan datum (I, ) and the root datum (Y, X,...). Let H B = B +,B G be a Cartan subgroup and a pair of opposite Borel subgroups respectively. Let X = G/B be the flag manifold of G. Let C = P 1 be the projective line. Let α = i I a ii N[I] H 2 (X, Z). The moduli space of G-monopoles of topological charge α (see e.g. [4]) is naturally identified with the space M b (X, α) of based maps from (C, ) to (X,B + ) of degree α. The moduli space of G-monopoles carries a natural hyperkähler structure, and hence a holomorphic symplectic structure. We propose a simple explicit formula for the symplectic structure on M b (X, α). It generalizes the well known formula for G = SL 2 [1] Recall that for G = SL 2 we have (X,B + ) = (P 1, ). Recall the natural local coordinates on M b (P 1, a) (see [1]). We fix a coordinate z on C such that z( ) =. Then a based map φ : (C, ) (P 1, ) of degree a is a rational function p(z) q(z) where p(z) is a degree a polynomial with the leading coefficient 1, and q(z) is a degree < a polynomial. Let U be the open subset of based maps such that the roots x 1,..., x a of p(z) are multiplicity free. Let y k be the value of q(z) at x k. Then x 1,..., x a, y 1,..., y a form an étale coordinate system on U. The symplectic form Ω on M b (P 1, a) equals dy k dx k. In other words, the Poisson brackets of these coordinates are as follows: a k=1 y k {x k, x m } = 0 = {y k, y m }; {x k, y m } = δ km y m. For an arbitrary G and i I let X i X be the corresponding codimension 1 B -orbit (Schubert cell), and let X i X i be its closure (Schubert variety). For φ M b (X, α) we define x 1 i,..., x a i i A 1 as the points of intersection of φ(p 1 ) with X i. This way we obtain the projection π α : M b (X, α) A α (the configuration space of I-colored divizors of degree α on A 1 ). Let U M b (X, α) be the open subset of based maps such that φ(p 1 ) X i X i for any i I, and x k i x l j for any i, j I, 1 k a i, 1 l a j. Locally in X the cell X i is the zero divizor of a function ϕ i (globally, ϕ i is a section of the line bundle L ωi corresponding to the fundamental weight X). The rational function Date: March M.F. and A.K. were partially supported by CRDF grant RM

3 2 M. FINKELBERG, A. KUZNETSOV, N. MARKARIAN, AND I. MIRKOVIĆ ϕ i φ on C is of the form p i(z) q i (z) where p i(z) is a degree a i polynomial with the leading coefficient 1, and q i (z) is a degree < a i polynomial. Let yi k be the value of q i (z) at x k i. Then x k i, yi k, i I, 1 k a i, form an étale coordinate system on U. The Poisson brackets of these coordinates are as follows: y k i yl j {x k i, x l j} = 0 = {yi k, yi}; l {x k i, yj} l = δ ij δ kl yj; l {yi k, yj} l = i j x k i xl j for i j It follows that the symmetric functions of the x-coordinates (well defined on the whole M b (X, α)) are in involution. In other words, the projection π α : M b (X, α) A α is an integrable system on M b (X, α). The fibers of π α : U A α are Lagrangian submanifolds of U. It is known that all the fibers of π α : M b (X, α) A α are equidimensional of the same dimension α (see [3]), hence π α is flat, hence all the fibers are Lagrangian Let P B be a parabolic subgroup. The construction of the Poisson structure on M b (X, α) generalizes verbatim to the space of based maps M = M b (G/P, β). In most cases the corresponding map P : T M T M is not an isomorphism, i.e. M splits into nontrivial symplectic leaves. For certain degrees α N[I] we have the natural embedding Π : M b (X, α) M, and the image is a symplectic leaf of P. Moreover, all the symplectic leaves are of the form gπ(m b (X, α)) for certain α N[I], g P, see the Theorem The above Poisson structure is a baby (rational) version of the Poisson structure on the moduli space of B-bundles over an elliptic curve [2]. We learnt of its definition (as a differential in the hypercohomology spectral sequence, see 2) from B.Feigin. Thus, our modest contribution reduces just to a proof of Jacobi identity. Note that the Poisson structure of [2] arises as a quasiclassical limit of elliptic algebras. On the other hand, M b (X, α) is an open subset in the moduli space M b (X, α) of B-bundles on C trivialized at, such that the induced H-bundle has degree α. One can see easily that M b (X, α) is isomorphic to an affine space A 2 α, and the symplectic structure on M b (X, α) extends to the Poisson structure on M b (X, α). The latter one can be quantized along the lines of [2]. It is clear from the above discussion that the present note owes its existense to the generous explanations of B.Feigin Notations. For a subset J I we denote by P J B the corresponding parabolic subgroup. Thus, P = B. Denote by X J = G/P J the corresponding parabolic flag variety; thus, X = X. We denote by : X X J the natural projection. We denote by x X J the marked point (B + ). Let M = M b (X J, α) denote the space of based algebraic maps φ : (C, ) (X J, x) of degree α H 2 (X J, Z). Let g denote the Lie algebra of G. Let g XJ denote the trivial vector bundle with the fiber g over X J and let p XJ g XJ (resp. r XJ p XJ g XJ ) be its subbundle with the

4 A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES 3 fiber over a point P equal to the corresponding Lie subalgebra p g (resp. its nilpotent radical r p g). In case J = we will also denote r X by n X, and p X by b X. Note that the quotient bundle h X := b X /n X is trivial (abstract Cartan algebra). Recall that the tangent bundle TX J of X J (resp. cotangent bundle T X J ) is canonicaly isomorphic to the bundle g XJ /p XJ (resp. r XJ ) Acknowledgments. This paper has been written during the stay of the second author at the Max-Planck-Institut für Mathematik. He would like to express his sincere gratitude to the Institut for the hospitality and the excellent work conditions. 2. The Poisson structure 2.1. The fibers of the tangent and cotangent bundles of the space M at the point φ are computed as follows: T φ M = H 0 (C, (φ TX J ) O C ( 1)) = H 0 (C, (φ g XJ /p XJ ) O C ( 1)), T φ M = H1 (C, (φ T X J ) O C ( 1)) = H 1 (C, (φ r XJ ) O C ( 1)). The second identification follows from the first by the Serre duality. We have a tautological complex of vector bundles on X J : r XJ g XJ g XJ /p XJ (1) The pull-back via φ of this complex twisted by O C ( 1) gives the following complex of vector bundles on C (φ r XJ ) O C ( 1) (φ g XJ ) O C ( 1) (φ g XJ /p XJ ) O C ( 1) (2) Consider the hypercohomology spectral sequence of the complex (2). Since g XJ is the trivial vector bundle we have H (C, (φ g XJ ) O C ( 1)) = 0, hence the second differential of the spectral sequence induces a map d 2 : H 1 (C, (φ r XJ ) O C ( 1)) H 0 (C, (φ g XJ /p XJ ) O C ( 1)) that is a map P X J φ : Tφ M T φm. This construction easily globalizes to give a morphism P X J : T M T M. Theorem 1. P defines a Poisson structure on M. Here we will reduce the Theorem to the case X J = X. This case will be treated in the next section Let : H 2 (X, Z) H 2 (X J, Z) be the push-forward map. The map induces a map Π : M b (X, α) M b (X J, α).

5 4 M. FINKELBERG, A. KUZNETSOV, N. MARKARIAN, AND I. MIRKOVIĆ Proposition 1. The map Π respects P, that is the following square is commutative T M b (X, α) Π Π T M b (X J, α) P X T M b (X, α) Π P X J Π T M b (X J, α) Proof. We have the following commutative square on X n X g X g X /b X r XJ g X g X /b X r XJ g XJ (g XJ /p XJ ) Consider its pull-back via φ M b (X, α) twisted by O C ( 1). Let d 2 denote the second differential of the hypercohomology spectral sequence of the middle row. Then we evidently have P X Π = d 2, Π d 2 = P X J and the Proposition follows. Now, assume that we have proved that P X defines a Poisson structure. For any β H 2 (X J, Z) we can choose α H 2 (X, Z) such that α = β and the map Π is open. Then the algebra of functions on M b (X J, β) is embedded into the algebra of functions on M b (X, α) and the Proposition 1 shows that the former bracket is induced by the latter one. Hence it is also a Poisson bracket. 3. The case of X In this section we will denote M b (X, α) simply by M Since h is a trivial vector bundle on X the exact sequences induce the isomorphisms 0 n X b X h X 0, 0 h X g X /n X g X /b X 0 T φ M = H 0 (C, (φ g X /b X ) O C ( 1)) = H 0 (C, (φ g X /n X ) O C ( 1)), Tφ M = H1 (C, (φ n X ) O C ( 1)) = H 1 (C, (φ b X ) O C ( 1)) Applying the construction of 2.1 to the following tautological complex of vector bundles on X b X g X h X g X /n X (3) and taking into account the above isomorphisms we get a map P X φ : T φ M T φm.

6 A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES 5 Lemma 1. We have P X φ = P X φ. Proof. The same reasons as in the proof of the Proposition 1 work if we consider the following commutative diagram n X g X g X /b X n X g X g X /n X b X g X h X g X /n X It will be convenient for us to use the complex (3) for the definition of the map P X φ instead of (1) Here we will describe the Plücker embedding of the space M. Let X = I be the set of fundamental weights: i, ω j = δ ij. We denote by (, ) the scalar product on X such that (i, j ) = i j for the simple roots i, j. For a dominant weight λ X we denote by V λ the irreducible G-module with highest weight λ. Recall that X is canonically embedded into the product of projective spaces X ω P(V ω ) This induces the embedding M ω M b (P(V ω ), α, ω ) Note that the marked point of the space P(V ω ) is just the highest weight vector v ω with respect to the Borel subgroup B. A degree d based map φ ω : (C, ) (P(V ω ), v ω ) can be represented by a V ω -valued degree d polynomial in z, taking the value v ω at infinity. Let us denote the affine space of such polynomials by R d (V ω ). The Plücker embedding of the space M is the embedding into the product of affine spaces M ω R α,ω (V ω ). A map φ M will be represented by a collection of polynomials (φ ω R α,ω (V ω )) ω.

7 6 M. FINKELBERG, A. KUZNETSOV, N. MARKARIAN, AND I. MIRKOVIĆ 3.3. The coordinates. The dual representation Vω decomposes into the sum of weight subspaces Vω = λ XV λ ω. We choose a weight base (fω λ) of V ω ω, such that fω (v ω) = 1. Suppose i, ω = 1. Then dim V ω ω = dim V ω+i ω = 1, and dim V ω+i +j ω = 0 if i j = 0, and dimv ω+i +j ω = 1 if i j 0. Hence, in the latter case, the vectors fω ω, f i ω ω and f i +j ω ω are defined uniquely up to multiplication by a constant. Let E i, F i, H i be the standard generators of g. Then we will take f i ω ω := E i fω ω, +j ω fi ω := E j E i fω ω. We consider the polynomials φ λ ω := fλ ω (φ ω): the λ weight components of φ ω. In particular, φ ω s the degree α, ω unitary polynomial and φ i ω s the degree < α, ω polynomial. Let x 1 ω,..., x ω α,ω be the roots of φ ω ω and yω, 1...,y ω α,ω be the values of φ i ω ω at the points x 1 ω,...,x α,ω ω respectively. Consider the open subset U M formed by all the maps φ such that all x k ω are distinct and all yk ω are non-zero. On this open set we have α,ω φ ω ω (z) = k=1 The collection of 2 α functions (z x k ω), φ i ω ω (z) = α,ω k=1 y k ω φ ω ω (z) (φ ω ω ) (x k ω )(z xk ω ). (x k ω, yk ω ), (ω, 1 k α, ω ) (4) is an étale coordinate system in U. One can either check this straightforwardly, or just note that the matrix of P X in these coordinates has a maximal rank, see the Remark 2 below. So let us compute the map P X in these coordinates The action of g on V nduces an embedding g X /n X V ω L ω ω of vector bundles over X and the dual surjection Vω L ω b X, ω where L ω stands for the line bundle, corresponding to the weight ω. Hence we have the following complex Vω L ω g X h X V ω L ω (5) ω ω Remark 1. The differentials in the above complex in the fiber over a point B X are computed as follows: ( ) ( ) ϕ Vω ϕ(ξ k v )ξ k ω(h i )ϕ(v )h i g h, ξ h g h ξv ω(h)v V ω.

8 A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES 7 Here v is a highest weight vector of V ω with respect to B ; (ξ k ), (ξ k ) are dual (with respect to the standard scalar product) bases of g; and (h i ), (h i ) are dual bases of h In order to compute the brackets of the coordinates (4) at a point φ M we need to take the pull-back of the complex (5) via φ, twist it by O C ( 1) and compute the second differential of the hypercohomology spectral sequence. The following Lemma describes this differential in general situation. Lemma 2. Consider a complex K = (F space and Consider f A O C f Hom(F, A O C ) = A H 0 (C, F ), g Hom(A O C, G) = A H 0 (C, G). g G) on C, where A is a vector D = tr(f g) H 0 (C, F ) H 0 (C, G) = H 0 (C C, F G), where tr : A A C is the trace homomorphism. Then 1) The restriction of D to the diagonal C C vanishes, hence D = D for some D H 0 (C C, (F G)( )) = H 0 (C, F ( 1)) H 0 (C, G( 1)) = = H 1 (C, F( 1)) H 0 (C, G( 1)). 2) The second differential d 2 : H 1 (C, F( 1)) H 0 (C, G( 1)) of the hypercohomology spectral sequence of K O C ( 1) is induced by the section D. Proof. The first statement is evident. To prove the second statement consider the following commutative diagram on C C F( 1) O C D f( 1) 1 (1 g) A O C C ( 1, 0) (OC ( 1) G) 1 g O C ( 2) G( 1) O C ( 1) G (OC ( 1) G) Both rows are complexes with acyclic middle term, hence the second differentials of the hypercohomology spectral sequences commute with the maps induced on cohomology by the vertical arrows: H 1 (C C, F( 1) O C ) D H 1 (C C, O C ( 2) G( 1)) d 2 H 0 (C C, (O C ( 1) G) ) d 2 H 0 (C C, (O C ( 1) G) ) Now it remains to note that H 1 (C C, F( 1) O C ) = H 1 (C, F( 1)), H 1 (C C, O C ( 2) G( 1)) = H 0 (C, G( 1)), H 0 (C C, (O C ( 1) G) ) = H 0 (C, G( 1)),

9 8 M. FINKELBERG, A. KUZNETSOV, N. MARKARIAN, AND I. MIRKOVIĆ and that the map H 0 (C, G( 1)) H 0 (C, G( 1)) induced by the map d 2 in the second row of the above diagram is identity Consider the pullback of (5) via φ M, and twist it by O C ( 1). We want to apply Lemma 2 to compute the (, ω j )-component of the second differential of the hypercohomology spectral sequence. In notations of the Lemma we have D ωi,ω j (z, w) = ξ k φ ωi (z) ξ k φ ωj (w) (h i )φ ωi (z) ω j (h i )φ ωj (w) = = ξ k φ ωi (z) ξ k φ ωj (w) (, ω j )φ ωi (z) φ ωj (w) V ωi V ωj (z, w). Lemma 3. The operator ξ k ξ k (, ω j ) acts as a scalar multiplication on every irreducible summand V λ V ωi V ωj. On V ωi +ω j it acts as a multiplication by 0. If = ω j, then on V 2ωi i V V ωi it acts as a multiplication by ( 2). If i j, i j 0 then on V ωi +ω j i j V V ωj it acts as a multiplication by (i j 2). Proof. It is easy to check that ξ k ξ k commutes with the natural action of g on V ωi V ωj. The first part of the Lemma follows. The rest of the Lemma can be checked by the straightforward computation of the action of ξ k ξ k on the highest vectors of the corresponding subrepresentations If we want to compute the brackets of the coordinates (4) we are interested in the components of D ωi,ω j (z, w) in the weights + ω j, + ω j i, + ω j j, + ω j i j. (6) The following Lemma describes the corresponding weight components of the tensor product V ωi V ωj. Lemma 4. The embedding V ωi +ω j V ωi V ωj induces an isomorphism in the weights (6) with the following two exceptions: (1) (V ωi V ωi ) 2 i = V 2 i 2 V 2 i 2 i ; the G-projection to the second summand is given by the formula a(v ωi F i v ωi ) + b(f i v ωi v ωi ) a b 2 (v F i v ωi F i v ωi v ωi ). (2) (V ωi V ωj ) +ω j i j = V +ω j i j +ω j V +ω j i j +ω j i j if i j and i j 0; the G-projection to the second summand is given by the formula a(v ωi F i F j v ωj ) + b(f i v ωi F j v ωj ) + c(f j F i v ωi v ωj ) Proof. Straightforward. b a c i j 2 (v F i F j v ωj + (i j)f i v ωi F j v ωj + F j F i v ωi v ωj ).

10 A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES Hence (see Lemma 3, Lemma 4) when λ is one of the weights (6) the λ-component D ω λ i,ω j (z, w) of the polynomial Dωi,ω j (z, w) = D,ω j (z, w) is zero with the following two z w exceptions D 2 i, = φ (z)φ i (w) φ i z w (z)φ (w) (F i v ωi v ωi v ωi F i v ωi ) (7) D +ω j i j i i φω,ω j = (z)φ ω j j ω j ω j (w) φ i j (z)φ ω j ω j (w) z w (v ωi F i F j v ωj + (i j)f i v ωi F j v ωj + F j F i v ωi v ωj ) (8) (w) φ (z)φ ω j i j Note that the scalar multiplicators of the Lemma 3 canceled with the denominators of the Lemma Now we can compute the brackets. Proposition 2. We have {x k,x l ω j } = 0; {x k,y l ω j } = δ kl δ ij y l ω j ; {y k,x l ω j } = δ kl δ ij y k ; (9) Proof. Note that if p V ωi (z) then dy k (p) = y {yω k i,yω l ω k j } = i j i yω l j, if i j; x k x l ω j {y k,y l } = 0. f i, p(x k ), dx k (p) = where, stands for the natural pairing. Note also that by definition and f i φ f, (x k ) = 0, φ i (x k ) = yω k i p(x k ) (φ ) (x k ), F i v ωi = E i f, F i v ωi = f, E i F i v ωi = 1. Now the Proposition follows from the Lemma 2 and from the formulas of 3.8.,

11 10 M. FINKELBERG, A. KUZNETSOV, N. MARKARIAN, AND I. MIRKOVIĆ Remark 2. The matrix of the bivector field P X in the coordinates (x k ω, yω) k looks as follows 0 diag(yω k) diag(yω) k Since on the open set U this matrix is evidently nondegenerate it follows that the functions (x k ω, y k ω) indeed form an étale coordinate system Now we can prove Theorem 1. Proof of the Theorem 1. The reduction to the case J = has been done in 2.2. The latter case is straightforward by the virtue of Proposition 2. Corollary 1. The map P X provides the space M b (X, α) with a holomorphic symplectic structure. Proof. Since P X gives a Poisson structure it suffices to check that P X is nondegenrate at any point. To this end recall that the hypercohomology spectral sequence of a complex K converges to H (X, K ). Since the only nontrivial cohomology of the complex (1) is h X in degree zero, the complex (2) is quasiisomorphic to (φ h) O C ( 1) in degree zero, hence the hypercohomology sequence of the complex (2) converges to zero, hence Pφ X is an isomorphism. Remark 3. One can easily write down the corresponding symplectic form in the coordinates (4): dy k ω dx k ω + 1 i j dxk dx l ω j yω k 2 x k x l ω j i j 4. Symplectic leaves 4.1. We fix β N[I J] Z[I J] = H 2 (X J, Z), and consider the Poisson structure on M = M b (X J, β). In this section we will describe the symplectic leaves of this structure. Consider α N[I] Z[I] = H 2 (X, Z) such that α = β (see 2.2). Note that is nothing but the natural projection from N[I] to N[I J]. Thus α α N[J]. We will call an element γ N[J] antidominant if γ, j 0 for any j J. We will call α N[I] a special lift of β N[I J] if α = β, and α β is antidominant. Lemma 5. If α is a special lift of β, then the natural projection Π : M b (X J, β) (see 2.2) is an immersion. M b (X, α) Proof. Let φ M b (X, α). Then T φ M b (X, α) = H 0 (C, (φ g X /b X ) O C ( 1)), and T Πφ M b (X J, β) = H 0 (C, ((Πφ) g XJ /p XJ ) O C ( 1)). Hence the kernel of the natural map Π : T φ M b (X, α) T Πφ M b (X J, β) equals H 0 (C, φ ( p XJ /b X ) O C ( 1)). Now p XJ /b X has a natural filtration with the successive quotients of the form L θ where θ is a positive root of the root subsystem spanned by J I. Since α is a

12 A NOTE ON THE SYMPLECTIC STRUCTURE ON THE SPACE OF G-MONOPOLES 11 special lift, deg φ L θ 0. We conclude that H 0 (C, (φ L θ ) O C ( 1)) = 0, and thus H 0 (C, φ ( p XJ /b X ) O C ( 1)) = 0. The Lemma is proved. Remark 4. For fixed β N[I J] the set of its special lifts is evidently finite. It is nonempty (see e.g. the proof of the Theorem 2) It follows from the Proposition 1 that Π(M b (X, α)) is a symplectic leaf of the Poisson structure P on M, if α is a special lift of β. The group P J acts naturally on M; it preserves P since the complex (2) is P J - equivariant. It follows that for g P J the subvariety gπ(m b (X, α)) M is also a symplectic leaf. Certainly, gπ(m b (X, α)) = Π(M b (X, α)) whenever g B. Theorem 2. Any symplectic leaf of P is of the form gπ(m b (X, α)) where α is a special lift of β, and g P J. Proof. We only need to check that for any ψ M there exists a special lift α, a point φ M b (X, α), and g P J such that ψ = gπφ. In other words, it suffices to find a special lift α, and a point φ M(X, α) (unbased maps!) such that φ( ) P J x (the smallest P J -orbit in X), and ψ = Πφ. Recall that given a reductive group G with a Cartan subgroup H and a set of simple roots X(H), the isomorphism classes of G-torsors over C are numbered by the set X+(H) of the dominant coweights of G : η X+(H) iff η, i 0 for any i. For example, if H = G = H, then X+(H) = Y. If φ M(X, α), we may view φ as a reduction of the trivial G-torsor to B. Let φ H be the corresponding induced H-torsor. Then its isomorphism class equals α. Let us view ψ as a reduction of the trivial G-torsor to P J. Let L J be the Levi quotient of P J, and let ψ L J be the corresponding induced L J -torsor. Let ϕ be the Harder-Narasimhan flag of ψ L J. We may view it as a reduction of ψ to a parabolic subgroup P K, K J. By definition, the isomorphism class η of ϕ L K (as an element of Y ) has the following properties: a) η = β; b) η η, j > 0 for j J K; c) η η, k = 0 for k K. In particular, if L K stands for the quotient of L K by its center, then the induced torsor ϕ L K is trivial. Choosing its trivial reduction to the positive Borel subgroup of L K we obtain a reduction φ of ψ to B. Thus φ is a map from C to X of degree α = η. We see that α is a special lift of β, and φ M b (X, α) has the desired properties. References [1] M.F.Atiyah, N.Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Princeton University Press (1988). [2] B.Feigin, A.Odesskii, Vector bundles on Elliptic Curves and Sklyanin Algebras, to appear in Proceedings of IUM. [3] M.Finkelberg, I.Mirković, Semiinfinite Flags. I. Case of global curve P 1, Preprint alg-geom/

13 12 M. FINKELBERG, A. KUZNETSOV, N. MARKARIAN, AND I. MIRKOVIĆ [4] S.Jarvis, Euclidean monopoles and rational maps, to appear in Proc. Lond. Math. Soc. Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow Russia address: Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow Russia address: Independent Moscow University, 11 Bolshoj Vlasjevskij pereulok, Moscow Russia address: Dept. of Mathematics and Statistics, University of Massachusetts at Amherst, Amherst MA , USA address: