Gaiotto-Witten superpotential and Whittaker D-modules on monopoles
aa r X i v : . [ m a t h . AG ] J un Dedicated to the memory of Andrei Zelevinsky
GAIOTTO-WITTEN SUPERPOTENTIAL AND WHITTAKERD-MODULES ON MONOPOLES
ALEXANDER BRAVERMAN, GALYNA DOBROVOLSKA, AND MICHAEL FINKELBERG
Abstract.
Let G be an almost simple simply connected group over C . For a positiveelement α of the coroot lattice of G let ◦ Z α denote the space of maps from P to the flagvariety B of G sending ∞ ∈ P to a fixed point in B of degree α . This space is known to beisomorphic to the space of framed G -monopoles on R with maximal symmetry breakingat infinity of charge α .In [6] a system of (´etale, rational) coordinates on ◦ Z α is introduced. In this note wecompute various known structures on ◦ Z α in terms of the above coordinates. As a byproductwe give a natural interpretation of the Gaiotto-Witten superpotential studied in [9] andrelate it to the theory of Whittaker D-modules discussed in [8]. Introduction
Zastava spaces.
Let G be an almost simple simply connected algebraic group over C . We denote by B the flag variety of G . Let us also fix a pair of opposite Borel subgroups B , B − whose intersection is a maximal torus T (thus we have B = G/B = G/B − ).Let Λ denote the cocharacter lattice of T ; since G is assumed to be simply connected,this is also the coroot lattice of G . We denote by Λ + ⊂ Λ the sub-semigroup spanned bypositive coroots. We say that α ≥ β (for α, β ∈ Λ) if α − β ∈ Λ + .It is well-known that H ( B , Z ) = Λ and that an element α ∈ H ( B , Z ) is representableby an algebraic curve if and only if α ∈ Λ + . Let ◦ Z α denote the space of maps P → B ofdegree α sending ∞ ∈ P to B ∈ B . It is known [6] that this is a smooth symplectic affinealgebraic variety, which can be identified with the space of framed G -monopoles on R withmaximal symmetry breaking at infinity of charge α [10], [11].The scheme ◦ Z α is endowed with a number of remarkable structures (listed below). Onthe other hand in [6] the authors introduce a system of (birational, ´etale) coordinates on ◦ Z α . The purpose of the present note is to compute how these structures look like in theabove coordinates. In particular, it turns out that the Gaiotto-Witten superpotential [9]admits a natural interpretation in terms of Whittaker D -modules of [8].1.2. Quasi-maps.
The scheme ◦ Z α has a natural partial compactification Z α . It can berealized as the space of based quasi-maps of degree α ; set-theoretically it can be describedin the following way: Z α = G ≤ β ≤ α ◦ Z β × A α − β , here for γ ∈ Λ + we denote by A γ the space of all colored divisors P γ i x i with x i ∈ A , γ i ∈ Λ + such that P γ i = γ .1.3. A “symmetric” definition of the Zastava space.
Fix λ, µ ∈ Λ. Let us denote by ◦ Z λ,µ the scheme classifying the following data:1) A G -bundle F on P with a trivialization at ∞ ∈ P .2) A B -structure F B on F such that the induced T -bundle F T, + is of degree λ . We requirethat F B is equal to B at ∞ .3) A B − -structure F B − on F such that the induced T -bundle F T, − is of degree µ . Werequire that F B − is equal to B − at ∞ .It is easy to see that this is indeed a scheme. Moreover, we claim that ◦ Z λ,µ is naturallyisomorphic to ◦ Z λ − µ (Section 6).1.4. Structures on the Zastava space.
It is easy to see that the space ◦ Z α is endowedwith the following structures (the precise constructions are given in the main body of thepaper):(1) The scheme ◦ Z α possesses a natural symplectic structure [6].(2) There is a natural morphism π α : ◦ Z α → A α . Moreover, given β, γ ∈ Λ + let ( A β × A γ ) disj denote the space of pairs of colored divisors of degrees β and γ which aremutually disjoint. If α = β + γ then we have a natural ´etale map ( A β × A γ ) disj → A α .The factorization is a canonical isomorphism f β,γ : ( A β × A γ ) disj × A α Z α ∼ −→ ( A β × A γ ) disj × A β × A γ ( Z β × Z γ ) . We shall refer to the latter as the factorization property of Zastava.(3) The Cartan involution on G (which interchanges B and B − and induces the map t t − on T ) induces an involution ι on ◦ Z α (this is clear from the point of view ofthe definition of ◦ Z α given in Section 1.3).(4) Let ∂Z α = Z α \ ◦ Z α . Then ∂Z α is a Cartier divisor and moreover it is the divisor ofzeros of some function F α on Z α which is invertible on ◦ Z α (this function is uniqueup to a multiplicative scalar).(5) Fix λ, µ ∈ Λ such that λ − µ = α . Then for every simple root ˇ α i of G we have canon-ical maps E αλ, + ,i : ◦ Z α → H ( P , O ( h− ˇ α i , λ i ), E αµ, − ,i : ◦ Z α → H ( P , O ( h− ˇ α i , µ i ).The precise definition is given in Section 6, so let us just explain the definition for G = SL (2) here. In this case Z λ,µ ≃ Z α just classifies rank 2 vector bundles F on P with trivialized determinant together with two short exact sequences 0 → L + → F → L − → → L − → F → L − − → L + = − λ, deg L − = − µ ,where we identify the lattice Λ with Z in a natural way. In addition F is endowedwith a trivialization at ∞ , which is compatible with L + and L − ; in particular L + and L − also get a trivialization at ∞ which allows us to identify them canonicallywith O ( − λ ) and O ( − µ ) (here we use a notation O ( n ) , n ∈ Z , for a line bundle on trivialized at ∞ ∈ P ). Hence the above short exact sequences define elementsin H ( P , O ( − λ )) and H ( P , O ( − µ )).Let χ λi, + : ◦ Z λ,µ × H ( P , O ( h λ, ˇ α i i − → C be the composition of E αλ, + ,i andthe natural pairing H ( P , O ( h λ, ˇ α i i − × H ( P , O ( −h λ, ˇ α i i )) → C . Note that anelement of H ( P , O ( h λ, ˇ α i i −
2) can be regarded as a polynomial K i of one variable z of degree ≤ h λ, ˇ α i i −
2. Similarly, we let χ µi, − : ◦ Z λ,µ × H ( P , O ( h µ, ˇ α i i − → C be the corresponding function (obtained by replacing E αλ, + ,i with E αµ, − ,i ). We set E αλ, + to be the direct sum of all the E αλ, + ,i and similarly for E αµ, − (sometimes we shalldrop the indices λ, µ and α when it does not lead to a confusion). Obviously themaps E + and E − are interchanged by the involution ι .1.5. Coordinates on Zastava.
A system of ´etale birational coordinates on ◦ Z α is intro-duced in Section 2.2. Let us recall the definition for G = SL (2). In this case ◦ Z α consistsof all maps P → P of degree α which send ∞ to 0. We can represent such a map bya rational function RQ where Q is a monic polynomial of degree α and R is a polynomialof degree < α . Let w , . . . , w α be the zeros of Q . Set y r = R ( w r ). Then the functions( y , . . . , y α , w , . . . , w α ) form a system of ´etale birational coordinates on ◦ Z α .For general G the definition of the above coordinates is quite similar. In this case givena point in ◦ Z α we can define polynomials R i , Q i where i runs through the set of vertices ofthe Dynkin diagram of G and(1) Q i is a monic polynomial of degree h α, ˇ ω i i (2) R i is a polynomial of degree < h α, ˇ ω i i .Hence, we can define (´etale, birational) coordinates ( y i,r , w i,r ) where i is as above and r = 1 , . . . , h α, ˇ ω i i . It will be convenient for us to use slightly modified coordinates y i,r := y i,r Q j = i Q h α j , ˇ α i i / j ( w i,r ). Then the main result of this note is the following Theorem 1.6. (1)
The Poisson brackets of the modified coordinates (with respect tothe symplectic structure defined in [6] ) are as follows: { w i,r , w j,s } = 0 , { w i,r , y j,s } = ˇ d i δ ij δ rs y j,s , { y i,r , y j,s } = 0 . (2) (Recall that the boundary equation F α is defined up to a multiplicative constant.)We have F α = Q i,r y d i i,r = Q i,r y d i i,r Q j = i Q α j · α i / j ( w i,r ) . (3) Let us introduce yet another modified system of rational ´etale coordinates on ◦ Z α :we define y i,r := y i,r Q ′ i ( w i,r ) . (1.1) where Q ′ i stands for the derivative of the polynomial Q i ( z ) . Then we have f β,γ ( w i,r , y i,r ) ≤ r ≤ a i i ∈ I = (cid:16) ( w i,r , y i,r ) ≤ r ≤ b i i ∈ I , ( w i,r , y i,r ) b i +1 ≤ r ≤ a i i ∈ I (cid:17) . (1.2)(4) The involution ι sends ( w i,r , y i,r ) to ( w i,r , y − i,r ) . We have χ λi, + ( w, y, z ) = a i X r =1 y − i,r Q j = i Q −h α j , ˇ α i i j ( w i,r ) Q ′ i ( w i,r ) K i ( w i,r ) = a i X r =1 y − i,r Q j = i Q −h α j , ˇ α i i / j ( w i,r ) Q ′ i ( w i,r ) K i ( w i,r ) . (1.3) Similarly, χ µi, − ( w, y, z ) = a i X r =1 y i,r Q j = i Q −h α j , ˇ α i i / j ( w i,r ) Q ′ i ( w i,r ) K i ( w i,r ) . (1.4) Remark . The set of irreducible components Irr α of the central factorization fiber π − α ( α · ⊂ Z α is in a natural bijection with the weight α component of the Kashiwara crystal B ˇ g ( ∞ ), [2, Section 14]. The involution induced by ι on F α Irr α is nothing but the involution ∗ : B ˇ g ( ∞ ) → B ˇ g ( ∞ ) of [12, 8.3].1.8. Relation with the works of Gaiotto-Witten and Gaitsgory.
We keep the no-tation from Theorem 1.6. Let us observe that a monic polynomial K ( z ) of degree d is thesame as a point in A ( d ) . Thus if all K i are monic, together they form an point in A λ − ρ .Thus we may regard χ λ ± := P i ∈ I χ λi, ± as functions on ◦ Z α × A λ − ρ .Let Λ = ( λ , . . . , λ n ) be an unordered collection of dominant coweights whose sum isequal to λ − ρ . Then Λ defines a locally closed subvariety ◦ A Λ in A λ − ρ (namely, themoduli space of configurations of distinct colored points z = ( z , . . . , z n ) so that the colorof z i is λ i ) and we denote by χ Λ ± the restriction of χ λ ± to ◦ Z α × ◦ A Λ . We now define the(multivalued) superpotentials W Λ ,α ± : h ∨ × ◦ Z α × ◦ A Λ → A by setting W Λ ,α ± = X ≤ n ≤ N h λ n , h ∗ i z n − X ( i,r ) h α i , h ∗ i w i,r ± log F α + χ Λ ± + X ≤ m Let M κ ,α, Λ − denote the D -module on ◦ Z α × ◦ A Λ generated by the function exp( κ W Λ ,α − ) . Let π α, Λ : h ∨ × ◦ Z α × ◦ A Λ → h ∨ × A α × ◦ A Λ be the corresponding morphism. Then e have π α, Λ! ( M κ ,α, Λ − ) = π α, Λ ∗ ( M κ ,α, Λ − ) and it is isomorphic to the the minimal extension ofthe D -module on the open stratum generated by the function Y ≤ n ≤ N exp( h λ n , κ h ∗ i z n ) × Y ( i,r ) exp( −h α i , κ h ∗ i w i,r ) ×× Y ( i,r ) =( j,s ) ( w i,r − w j,s ) κ α i · α j / × Y ( i,r ) , ≤ n ≤ N ( z n − w i,r ) − κ α i · λ n × Y ≤ m The above Theorem is essentially due to Gaiotto and Witten when re-stricted to the open stratum (in this case it is not difficult to deduce it from the coordinatedescription of the superpotential). Interpreting the superpotential in terms of (1.5) allowsone to extend this statement to all of A α × ◦ A Λ using the work of Gaitsgory. It would be in-teresting to find an interpretation of this refined statement in terms of the Landau-Ginzburgmodel studied by Gaiotto and Witten.1.11. Acknowledgments. We are grateful to R. Bezrukavnikov, B. Feigin, D. Gaiotto,D. Gaitsgory, M. Gekhtman, S. Oblezin, L. Rybnikov, V. Schechtman and A. Uteshevfor the useful discussions. A.B. was partially supported by the NSF and by the SimonsFoundation. The financial support from the Government of the Russian Federation withinthe framework of the implementation of the 5-100 Programme Roadmap of the NationalResearch University Higher School of Economics, AG Laboratory is acknowledged by M.F.2. Recollections about zastava Notations. G is an almost simple simplyconnected complex algebraic Lie group. Wefix its Cartan and Borel subalgebras T ⊂ B ⊂ G with the Lie algebras h ⊂ b ⊂ g . Theset of simple roots is denoted I ; the simple roots (resp. coroots) are denoted ˇ α i (resp. α i ), i ∈ I . We fix a Weyl group invariant symmetric bilinear form ? · ? on the Cartan Lie algebra h such that the square length of a short coroot is α i · α i = 2. This bilinear form gives riseto an isomorphism h ∨ ∼ −→ h so that the root lattice X generated by { ˇ α i } i ∈ I embeds into h .We then have ˇ α i · ˇ α i ∈ { , , } , and α i · α i ∈ { , , } . We set d i = α i · α i . Let d be the ratioof the square lengths of the long and short coroots, so that d ∈ { , , } . We set ˇ d i = d/d i .Then h α i , ˇ α j i = α i · α j d j = d i ˇ α i · ˇ α j = d ˇ α i · ˇ α j ˇ d i .For α = P i ∈ I a i α i , a i ∈ N , we consider the corresponding zastava space Z α (see e.g. [3])with an open smooth subvariety Z α ⊃ ◦ Z α : the moduli space of degree α based mapsfrom C = P to the flag variety B = G/B (also known as the moduli space of framed G -monopoles on R of topological charge α with the maximal symmetry breaking at infinity).The complementary boundary divisor is denoted ∂Z α := Z α \ ◦ Z α .2.2. Coordinates on zastava. Let z be a coordinate on C = P . We think of the zastavaspace Z α in its Pl¨ucker embedding as of collections of degree h α, ˇ λ i V ˇ λ -valued polynomials(here ˇ λ is a dominant weight, and V ˇ λ is the corresponding irreducible representation) suchthat the highest weight component is of the form z h α, ˇ λ i + . . . (the smaller powers of z ), andall the other weight components are of degree strictly smaller than h α, ˇ λ i . In particular, ifˇ λ = ˇ ω i , a fundamental weight, then the highest weight component is denoted Q i (a monic olynomial of degree a i = h α, ˇ ω i i ), and the prehighest weight (= ˇ ω i − ˇ α i ) component isdenoted R i (a polynomial of degree < a i ). The polynomial Q i is determined uniquely bythe (unordered) set of its roots w i,r , ≤ r ≤ a i . The ramified cover ̟ : b Z α → Z α isformed by all the orderings of the roots of all the polynomials Q i , i ∈ I . We have regularfunctions y i,r := R i ( w i,r ) on b Z α . According to [6, Remark 2], on the open subset whereall the roots w i,r , i ∈ I are distinct (and ̟ is unramified), { w i,r , y i,r } form a coordinatesystem (an open embedding into A h α, ρ i ).2.3. A symplectic form and modified coordinates. The main result of [6] is a con-struction of a symplectic form on ◦ Z α which extends as a Poisson structure to Z α . Accordingto [6, Proposition 2], the Poisson brackets of the coordinates of Section 2.2 are as follows: { w i,r , w j,s } = 0 , { w i,r , y j,s } = ˇ d i δ ij δ rs y j,s , { y i,r , y j,s } = d ˇ α i · ˇ α j y i,r y j,s w i,r − w j,s for i = j , and finally { y i,r , y i,s } = 0.Following the private communications of S. Oblezin and L. Rybnikov, we consider themodified rational ´etale coordinates y i,r := y i,r Q j = i Q h α j , ˇ α i i / j ( w i,r ) (they are regular onlyon the open subset where all the roots w i,r , i ∈ I are distinct). Lemma 2.4. The Poisson brackets of the modified coordinates are as follows: { w i,r , w j,s } =0 , { w i,r , y j,s } = ˇ d i δ ij δ rs y j,s , { y i,r , y j,s } = 0 . Proof. Straightforward. (cid:3) Note that this is exactly the statement of Theorem 1.6(1). Definition 2.5. We define the logarithmic coordinates y i,r := log y i,r on an appropriate Z | α | -cover of the open subset of b Z α where all the roots w i,r , i ∈ I are distinct, and y i,r = 0 . A version of zastava. Given λ, µ ∈ X ∗ ( T ), we consider the moduli stack ◦ Z λ,µ ofthe following data: (a) a G -bundle F G on C trivialized at ∞ ∈ C ; (b) a reduction of F G to a B -bundle (a B -structure on F G ) such that the induced T -bundle has degree λ , andthe fiber of the B -structure at ∞ ∈ C is B ⊂ G ; (c) a reduction of F G to a B − -bundle(a B − -structure on F G ) such that the induced T -bundle has degree µ , and the fiber of the B − -structure at ∞ ∈ C is B − ⊂ G .According to [1, Section 2], ◦ Z λ,µ is representable by a scheme. More precisely, α := λ − µ is automatically a nonnegative combination of positive coroots, and ◦ Z λ,µ is isomorphic tothe zastava scheme ◦ Z α .The Cartan involution of G interchanging B and B − and acting on T as t t − inducesan isomorphism ι : ◦ Z λ,µ ∼ −→ ◦ Z − µ, − λ . The composition ◦ Z α ≃ ◦ Z λ,µ ι −→ ◦ Z − µ, − λ ≃ ◦ Z α is a well defined involution ι : ◦ Z α ∼ −→ ◦ Z α (independent of the choice of a presentation α = λ − µ : the independence is clear from the description of the identification ◦ Z λ,µ ∼ −→ ◦ Z α of [1, Section 2]). 3. Factorization (Proof of Theorem 1.6(3)) .1. Factorization in coordinates. Recall the fundamental factorization property of zas-tava spaces. For α = β + γ we have a natural morphism a : A β × A γ → A α . An open subset( A β × A γ ) disj ⊂ ( A β × A γ ) is formed by the pairs ( D β , D γ ) of disjoint divisors D β , D γ ∈ A .The factorization is a canonical isomorphism f β,γ : ( A β × A γ ) disj × A α Z α ∼ −→ ( A β × A γ ) disj × A β × A γ ( Z β × Z γ ) . We introduce yet another modified system of rational ´etale coordinates on Z α : we define y i,r := y i,r Q ′ i ( w i,r ) . (3.1)Let β = P i ∈ I b i α i , γ = P i ∈ I c i α i , so that a i = b i + c i . Proposition 3.2. f β,γ ( w i,r , y i,r ) ≤ r ≤ a i i ∈ I = (cid:16) ( w i,r , y i,r ) ≤ r ≤ b i i ∈ I , ( w i,r , y i,r ) b i +1 ≤ r ≤ a i i ∈ I (cid:17) . Proof. We recall the construction of the factorization isomorphism. Let U stand for theunipotent radical of the Borel B , and let U − be the unipotent radical of the oppositeBorel (with the same Cartan torus T ) B − . Let G/U stand for the affinization of the baseaffine space. The quotient stack U − \ G/U /T has an open dense point; and the complementis a Cartier (Schubert) divisor D . Now Z α is the moduli space of degree α maps C → U − \ G/U /T (i.e. such that the induced T -bundle on C has degree α ) such that ∞ ∈ C goesto the complement of the Schubert divisor, see e.g. [1].For φ ∈ Z α , the pullback of the Schubert divisor φ ∗ D is nothing but π α ( φ ) ∈ ( C \ ∞ ) α .Given φ ∈ Z β , φ ∈ Z γ with disjoint π β ( φ ) , π γ ( φ ), we construct the corresponding φ ∈ Z α as follows. Note that the disjointness condition guarantees that U := C \ φ ∗ D and U := C \ φ ∗ D cover C , and φ | U ∩ U = φ | U ∩ U (the constant map to the point). So wedefine φ by gluing φ and φ over U ∩ U .Now let us replace G, U, U − , T by SL i , U i , U i − , T i corresponding to the i -th root. Then SL i /U i is isomorphic to a 2-dimensional vector space V i ; the right action of T i is isomor-phic to the scalar action of C ∗ ; the left action of U i − is isomorphic to the one coming fromthe natural left action of SL i . We have the canonical homomorphisms χ i : U − ։ U i − ,and ˇ α i : T → T i . We also have a natural projection pr i : G/U ։ SL i /U i . In effect, G/U in Pl¨ucker realization consists of collections of vectors in the irreducible G -modules.In particular, each collection contains a vector v ˇ ω i ∈ V ˇ ω i . So we set pr i ( v ˇ λ ) ˇ λ ∈ X ∗ ( T ) + :=pr i ( v ˇ ω i ) ∈ V Rad P i ˇ ω i = V i . It is straightforward to check that pr i is χ i : U − ։ U i − -equivariant, and ˇ α i : T → T i -equivariant. In other words, we have a morphism ofstacks pr i : U − \ G/U /T → U i − \ SL i /U i /T i , and the inverse image of the Schubert di-visor D i ⊂ U i − \ SL i /U i /T i lies inside the Schubert divisor D ⊂ U − \ G/U /T (in fact, thisinverse image coincides with the corresponding irreducible component of D ). Hence weobtain the same named projection pr i : Z α g → Z a i sl i , and the following diagram commutes: Z α g pr i −−−−→ Z a i sl i π α y π ai y A α pr i −−−−→ A ( a i ) (3.2) oreover, the following diagram commutes as well:( A β × A γ ) disj × A α Z α g f β,γ −−−−→ ( A β × A γ ) disj × A β × A γ ( Z β g × Z γ g ) pr i y pr i y ( A ( b i ) × A ( c i ) ) disj × A ( ai ) Z a i sl i f bi,ci −−−−→ ( A ( b i ) × A ( c i ) ) disj × A ( bi ) × A ( ci ) ( Z b i sl i × Z c i sl i ) (3.3)Hence the proposition is reduced to the case of g = sl i that will be dealt with in the nextsection.3.3. Factorization for SL . In this section G = SL i , and to unburden the notationswe will write G, U, U − , T for SL i , U i , U i − , T i . We will use another point of view on thefactorization. Namely, we will think of Z a ∋ φ : C → U − \ G/U /T as of a G -bundle F on C with a generalized B -structure, and a U − -structure transversal to the B -structure at ∞ ∈ C . These generically transversal structures define a generic trivialization of F , i.e. apoint of the Beilinson-Drinfeld Grassmannian Gr BD . Moreover, since any U − -bundle over C is trivial, F is trivial too, and its trivialization at ∞ ∈ C extends to a canonical globaltrivialization. Thus the above trivialization (coming from two transversal structures) maybe viewed as a rational function C → G ; more precisely, as a rational function f : C → U − (because of the reduction to U − ) sending ∞ ∈ C to the neutral element of U − . Now recallthat G = SL i , and U − = G a = A . Then in the elementary terms f is nothing but R i Q i .Back to factorization, it arises from the factorization of the Beilinson-Drinfeld Grass-mannian. Given G -bundles F , F with trivializations σ , σ defined on the open subsets U , U ⊂ C such that U ∪ U = C we construct a new bundle F with trivialization σ on U = U ∩ U by gluing F | U and F | U over U where they are both trivialized.Given Z b ∋ φ (resp. Z c ∋ φ ) corresponding to ( F , U , σ ) (resp. ( F , U , σ )) and f = R Q (resp. f = R Q ) we want to compute the result of gluing Z a ∋ φ correspondingto ( F , U , σ ) and f = RQ . Note that by the construction, the principal part of f at C \ U (resp. C \ U ) coincides with the principal part of f at C \ U (resp. with that of f at C \ U ). On the other hand, the rational function f of degree a vanishing at ∞ ∈ C isuniquely determined by its principal parts at ( C \ U ) ∪ ( C \ U ). We conclude f = f + f .This is equivalent to the desired formula of (3.1) and Proposition 3.2 (since the principalpart of f at w i,r , i.e. the residue of f dz at w i,r , is given by the formula (3.1)).This completes the proof of the proposition. (cid:3) Another factorization. Recall from Section 3.3 that the factorization isomorphism f β,γ : ( A β × A γ ) disj × A α ◦ Z α ∼ −→ ( A β × A γ ) disj × A β × A γ ( ◦ Z β × ◦ Z γ )(Section 3.1) is induced by the embedding ◦ Z α ֒ → Gr BD ( U − ) ֒ → Gr BD ( G ). Given x = P m α m · x m ∈ A α the fiber π − α ( x ) goes under this embedding to Q m ( T ∩ S α m ) ⊂ Gr BD ( G ).Here T ⊂ Gr G,x m (resp. S α m ⊂ Gr G,x m ) is the semiinfinite orbit U − ( K x m ) · U ( K x m ) · α m ), and K x m ⊃ O x m is the local field (resp. ring) around the point x m ∈ C , and α m ∈ Gr G is a T -fixed point. Note that T ⊂ Gr G is canonically isomorphic to Gr U − ⊂ Gr G . e also have a natural embedding ◦ Z , − α ֒ → Gr BD ( G ) sending the fiber over x to Q m ( T − α m ∩ S ) ⊂ Gr BD ( G ). Note that S ⊂ Gr G is canonically isomorphic to Gr U ⊂ Gr G .Under the identification ◦ Z α ≃ ◦ Z , − α the factorization of Gr U induces the factorization f + β,γ : ( A β × A γ ) disj × A α ◦ Z α ∼ −→ ( A β × A γ ) disj × A β × A γ ( ◦ Z β × ◦ Z γ ) . Recall the Cartan involution ◦ Z α ≃ ◦ Z α, ι −→ ◦ Z , − α ≃ ◦ Z α of Section 2.6. The followinglemma is used in the next Section 4. Lemma 3.5. The following diagram commutes: ( A β × A γ ) disj × A α ◦ Z α f β,γ −−−−→ ( A β × A γ ) disj × A β × A γ ( ◦ Z β × ◦ Z γ ) Id × ι y Id × ι × ι y ( A β × A γ ) disj × A α ◦ Z α f + β,γ −−−−→ ( A β × A γ ) disj × A β × A γ ( ◦ Z β × ◦ Z γ ) (3.4) Proof. Obvious. (cid:3) Cartan involution (Proof of Theorem 1.6(4)) Involution in coordinates. Recall the modified coordinates y i,r of Section 2.3, andthe Cartan involution ι : ◦ Z α → ◦ Z α of Section 2.6. Proposition 4.2. The involution ι : ◦ Z α → ◦ Z α in coordinates acts as follows: ι : ( w i,r , y i,r ) ( w i,r , y − i,r ) (equivalently, ( w i,r , y i,r ) ( w i,r , y − i,r Q j = i Q −h α j , ˇ α i i j ( w i,r )) ). Proof. Recall that a B -structure on F G is encoded in a collection κ ˇ λ : L ˇ λ ֒ → V ˇ λ F G ofline subbundles satisfying the Pl¨ucker relations. Equivalently, we can consider a collection κ ∗− w ˇ λ : V ˇ λ F G ։ ′ L ˇ λ of the quotient line bundles satisfying the Pl¨ucker relations (wehave ′ L ˇ λ = L ∗− w ˇ λ ). Similarly, a B − -structure on F G is encoded in a collection of linesubbundles κ − ˇ λ : L − ˇ λ ֒ → V ˇ λ F G or equivalently, a collection of the quotient line bundles κ −∗− w ˇ λ : V ˇ λ F G ։ ′ L − ˇ λ . Let P i (resp. P − i ) be the i -type subminimal parabolic subgroupcontaining B (resp. B − ). Then a B -structure on F G induces a P i -structure on F G thatgives rise to a 2-dimensional subbundle V i ֒ → V ˇ ω i F G (associated to the 2-dimensional subspaceof invariants V Rad P i ˇ ω i ⊂ V ˇ ω i ). Similarly, a B − -structure on F G induces a P − i -structure on F G that gives rise to a 2-dimensional quotient bundle V ˇ ω i F G ։ ′ V − i .We have the natural embedding L ˇ ω i ֒ → V i and the natural projection ′ V − i ։ ′ L ˇ ω i . Wedefine the line bundle M i := V i / L ˇ ω i so that we have a short exact sequence 0 → L ˇ ω i → V i → M i → 0. We define the line bundle ′ M i as the kernel of ′ V − i ։ ′ L ˇ ω i so that we havea short exact sequence 0 → ′ M i → ′ V − i → ′ L ˇ ω i → 0. We also consider the composition L ˇ ω i ֒ → V ˇ ω i F G ։ ′ V − i . We define N i as the cokernel of this composed map. Note thatgenerically over ◦ Z λ,µ this composed map is an embedding of the line bundle L ˇ ω i , so that N i is a line bundle as well, and we have a short exact sequence 0 → L ˇ ω i → ′ V − i → N i → iven a general ( F G , κ ˇ λ , κ −∗ ˇ λ ) ∈ ◦ Z λ,µ such that N i is a line bundle, we consider the followingdiagram: L ˇ ω i −−−−→ V i −−−−→ M i (cid:13)(cid:13)(cid:13) y Q y L ˇ ω i −−−−→ ′ V − i −−−−→ N i (4.1)Here the rows are the above short exact sequences, the middle vertical map is defined as thecomposition V i ֒ → V ˇ ω i F G ։ ′ V − i , and the right vertical map Q is defined as follows. Note thatthe trivialization of F G at ∞ ∈ C compatible with the B, B − -structures gives rise to thetrivializations of L ˇ ω i , M i , N i at ∞ ∈ C . For degree reasons, M i is canonically isomorphic to O C ( h λ, − ˇ ω + ˇ α i i ), and N i is canonically isomorphic to O C ( h α, ˇ ω i i + h µ, − ˇ ω i + ˇ α i i ). Finally Q ∈ Hom( M i , N i ) = Γ( C, O ( h α, ω i − ˇ α i i )) is defined as Q j = i Q −h α j , ˇ α i i j . Lemma 4.3. The diagram (4.1) commutes. Proof. Straightforward. (cid:3) Now given a general ( F G , κ ˇ λ , κ −∗ ˇ λ ) ∈ ◦ Z λ,µ ≃ ◦ Z α , the coordinates w i,r are nothing but thepoints of C where the line subbundles L ˇ ω i ֒ → ′ V − i and ′ M i ֒ → ′ V − i are not transversal.The trivialization of L ˇ ω i , ′ M i at ∞ ∈ C gives rise to a canonical trivialization of these linebundles restricted to A = C \ {∞} . Hence at a nontransversality point w i,r ∈ A we havetwo collinear vectors in the fiber ′ V − i | w i,r , and the coordinate y i,r is nothing but their ratio.Since the Cartan involution ◦ Z α ≃ ◦ Z λ,µ ι −→ ◦ Z − µ, − λ ≃ ◦ Z α takes ( F G , κ ˇ λ , κ −∗ ˇ λ ) to( F G , κ −∗ ˇ λ , κ ˇ λ ), and interchanges the line bundles L ˇ ω i , M i with and without primes, theproposition follows. (cid:3) An equation of the boundary (Proof of Theorem 1.6(2)) An equation in modified coordinates. A regular function F α on Z α was con-structed in [3, Section 4] such that the divisor of F α is the boundary divisor ∂Z α (the mul-tiplicities of various irreducible components of the boundary are 1 or d ), see [3, Lemma 4.2].Recall the modified coordinates y i,r of Section 2.3. Theorem 5.2. There is c α ∈ C ∗ such that c α F α = Q i,r y d i i,r = Q i,r y d i i,r Q j = i Q α j · α i / j ( w i,r ) . The rest of the section is devoted to the proof of the theorem.5.3. Invertible functions on zastava. Let us denote the RHS of Theorem 5.2 by F α .If we can prove that Y α is a regular function on Z α invertible on ◦ Z α with a correct orderof vanishing at ∂Z α , then F α /F α is a rational function on Z α regular and nonvanishingat ◦ Z α and at the generic points of the irreducible components of the divisor ∂Z α . Due tonormality of Z α [3, Corollary 2.10], the ratio F α /F α is a regular invertible function on Z α .Then according to the following lemma, the ratio F α /F α is a nonzero constant c α . Lemma 5.4. Γ( Z α , O ∗ Z α ) = C ∗ . roof. Recall the factorization morphism π α : Z α → A α . Let ∆ ⊂ A α be the diagonaldivisor. For an off-diagonal configuration D ∈ A α the fiber π − α ( D ) is isomorphic to the h α, ˇ ρ i -dimensional affine space. Hence for f ∈ Γ( Z α , O ∗ Z α ) the restriction of f to any off-diagonal fiber of π α is constant. Hence f = ¯ f ◦ π α for a certain (invertible) function ¯ f on A α . Such ¯ f is necessarily constant. (cid:3) Codimension one: A and A × A . The order of vanishing of F α at the genericpoints of the irreducible components of ∂Z α clearly coincides with that of F α : see [3,Lemma 4.2]. We prove the regularity of F α . Due to normality of Z α it suffices to checkthe regularity at the generic points of divisors w i,r = w j,s . By the factorization property, itsuffices to consider the case α = α i + α j . The case when α i · α j = 0 being evident, we startwith i = j . Then we can assume g = sl , so that Z sl ≃ A = { ( Q i = z + a z + a , R i = b z + b ) } . We have Q i = ( z − w )( z − w ) , R i = ( y ( z − w ) − y ( z − w )) / ( w − w ), so that y y is the resultant R ( Q i , R i ): a regular function on Z sl , an equation of the boundary.5.6. Codimension one: A . Next assume i = j , and α i · α j = 0, and d i = d j . Then wecan assume g = sl . Both fundamental representations V ˇ ω i , V ˇ ω j of sl are 3-dimensional.The zastava space Z α i + α j sl is formed by the polynomials with values in V ˇ ω i , V ˇ ω j of the form( z − w i , y i , u ) , ( z − w j , y j , − u ) such that y i y j + ( w i − w j ) u = 0. We have y i √ w j − w i y j √ w i − w j = √− u : a regular function on Z α i + α j sl , an equation of the boundary.5.7. Codimension one: B . Next assume i = j , and α i · α j = 0, and d i = 2 , d j = 1. Thenwe can assume g = sp . The fundamental representation V ˇ ω i (resp. V ˇ ω j ) is 4-dimensional(resp. 5-dimensional). The Pl¨ucker coordinates for Z α i + α j sp are as follows: c b z + A b c z + A c b b . Here the boxed coordinates are the weight components of V ˇ ω i , and the remaining ones arethe weight components of V ˇ ω j . They are placed in the weight lattice of Sp (4). The originof the weird notation is in [7, Example 2.3.2]. The Pl¨ucker equations are as follows. First,we have the natural pairing V ˇ ω j ⊗ V ˇ ω j → C coming from V ˇ ω j ⊂ Λ V ˇ ω i , and Λ V ˇ ω i = C .The V ˇ ω j -valued polynomial must be selforthogonal. The vanishing of the leading coefficientof the selfpairing is c = 0. The vanishing of the degree zero coefficient of the selfpairingis 2 c b − c = 0. Second, we have the projection V ˇ ω i ⊗ V ˇ ω j → Λ V ˇ ω i which mustvanish on our polynomials. The vanishing of the leading coefficient of the projection is c = b , c = b . The vanishing of the degree zero coefficient of the projection is − c b + cA + b b = 0 , A b − A c − b b = 0 , b c − b c − b c = 0 , A c − A b + b c = 0. Note that the former quadratic equation is equivalent to the first quadratic Pl¨ucker equation. All in all, we can take A , A , b , b , b , b as independentcoordinates, and we will have three quadratic equations: b ( A − A ) = b b , b ( A − A ) = b b , b = b b (noncomplete intersection of three quadrics). To compare withthe coordinates of Section 2.2: w i = − A , w j = − A , y i = b , y j = b . e have ( y i √ w j − w i ) y j w i − w j = − b b ( w i − w j ) = − b b w i − w j = − b : a regular function on Z α i + α j sp ,an equation of the boundary.5.8. Codimension one: G . Next assume i = j , and α i · α j = 0, and d i = 3 , d j = 1.Then g is of type G . We have the regular functions w i , w j , y i , y j on Z α i + α j g . We have toshow that ( y i √ w j − w i ) y j √ ( w i − w j ) = √− y i y j ( w i − w j ) is a regular function on Z α i + α j g . Accordingto the formulas of Section 2.3, the Poisson bracket { y i , y j } = − y i y j w i − w j is a regular function.Furthermore, { y i , y i y j w i − w j } = y i { y i ,y j } w i − w j + y i y j { y i , w i − w j } = − y i y j ( w i − w j ) + y i y j ( w i − w j ) = − y i y j ( w i − w j ) is a regular function. Finally, { y i , y i y j ( w i − w j ) } = − y i y j ( w i − w j ) is a regular function on Z α i + α j g .5.9. Invertibility. The last thing to check is the invertibility of F α on ◦ Z α . To this end recallthe Cartan involution ι : ◦ Z α → ◦ Z α of Section 2.6 and note that according to Proposition 4.2we have F α ◦ ι = F − α .The theorem is proved. (cid:3) Remark . Here is an alternative way to prove the invertibility of F α on ◦ Z α : much shorterbut less elementary. According to [3, Proposition 4.4], the weight of F α with respect to theloop rotations in all the examples of Section 5.6, Section 5.7, Section 5.8 is equal to one. If F α were not regular on Z α , for certain m > w i − w j ) m F α would be regular on Z α and invertible on ◦ Z α . Thus, the ratio ( w i − w j ) m F α /F α would be invertible on Z α (sincethe numerator and denominator have the same order of vanishing at the boundary ∂Z α ) andhence constant by Lemma 5.4. Since the weight of both ( w i − w j ) and F α with respect to theloop rotations is also equal to one, we conclude m = 0: a contradiction with our assumption m > 0. Hence we have proved the regularity of F α on Z α and simultaneously the invertibilityof F α on ◦ Z α . The general case is reduced to the above examples by factorization.This completes the proof of Theorem 5.2. (cid:3) An Ext calculation (Proof of Theorem 1.6(5)) P GL -bundles. A P GL -bundle with a flag on C = P can be viewed as a shortexact sequence 0 → L → V → M → L and M are the line bundles, and V is a ranktwo vector bundle) modulo the twistings by the line bundles. In particular, the line bundle M − ⊗ L = H om( M , L ) is well defined: this is nothing but the induction of the Borelbundle to the Cartan bundle. We consider the moduli stack F of P GL -bundles witha flag on C equipped with a trivialization at ∞ ∈ C of the corresponding line bundle M − ⊗ L . The connected components of F are numbered by the integers deg M − deg L .On a connected component F n , we have a canonical isomorphism H om( M , L ) = O C ( − n ),and so Ext ( M , L ) = H ( C, O ( − n )) = H ( C, O ( n − ∨ . Thus we have a morphism E : F n → H ( C, O ( n − ∨ . .2. A map to Ext. By Pl¨ucker, we may view a B -structure on F G as a collection ofline subbundles L ˇ λ ⊂ V ˇ λ F G satisfying the Pl¨ucker relations (here ˇ λ runs through the coneof dominant weights of G , and V ˇ λ F G is the vector bundle associated to the irreducible G -module V ˇ λ with the highest weight ˇ λ ). For a B -structure coming from a point of ◦ Z λ,µ wehave deg L ˇ λ = −h λ, ˇ λ i . The trivialization at ∞ ∈ C extends to a canonical isomorphism L ˇ λ = O C ( −h λ, ˇ λ i ). Since the assignment ˇ λ L ˇ λ is multiplicative in ˇ λ : L ˇ µ +ˇ ν = L ˇ µ ⊗ L ˇ ν ,we can extend the notion of L ˇ λ for arbitrary weights ˇ γ ∈ X ∗ ( T ). We have a canonicalisomorphism L ˇ γ = O C ( −h λ, ˇ γ i ).For i ∈ I we have a morphism P i : ◦ Z λ,µ → F h λ, ˇ α i i defined as follows. Let G ⊃ P i ⊃ B be the i -type subminimal parabolic subgroup. We have the projection P i → L i to the corresponding Levi, also we have the projection L i → P GL , and so the composedprojection B ֒ → P i ։ L i ։ P GL . Given a B -reduction of a G -bundle on F G trivializedat ∞ ∈ C , we consider the induced P GL -bundle. It comes equipped with a flag andtrivialization at ∞ ∈ C . This is our desired P i applied to F G with the B -structure (notethat we have not used the B − -structure).If we specialize to the case ( λ, µ ) = ( α, µ = 0 guarantees that the B − -structure further reduces to the U − -structure (where U − is the radical of B − ). Sinceany U − -bundle on C = P is automatically trivial, the ambient G -bundle F G is trivial too,and its trivialization at ∞ ∈ C extends to a canonical trivialization over C . Thus we arriveat an identification ◦ Z α, ≃ ◦ Z α with the usual zastava space, i.e. the moduli space of degree α based maps from ( C, ∞ ) to ( B , B ). For φ ∈ ◦ Z α let us describe explicitly two particularrepresentatives (2-dimensional bundles with a flag) of P i ( φ ).We have a projection p i : B = G/B → G/P i =: B i . We define B i := B × B i B , and p i : B i → B (the first projection). By construction, p i is a P -bundle over B equipped with acanonical (diagonal) section ∆ i : B → B i . We define V ′ i := p i ∗ O B i (∆ i ) ⊃ L ′ i := p i ∗ O B i = O B .Thus we get a short exact sequence 0 → L ′ i → V ′ i → M ′ i → B ∈ B ; here M ′ i = O B ( ˇ α i ). Finally, P i ( φ ) = { → φ ∗ L ′ i → φ ∗ V ′ i → φ ∗ M ′ i → } .Alternatively, let V ˇ ω i be the trivial vector bundle over B associated with the fundamental G -module V ˇ ω i . It has a line subbundle L i : the fiber L i | B ′ is the B ′ -highest line V Rad B ′ ˇ ω i .If P ′ i is the i -type subminimal parabolic containing B ′ , then the invariants V Rad P ′ i ˇ ω i are2-dimensional (the highest and next highest lines), and as B ′ varies in B , we obtain a 2-dimensional subbundle V i ⊂ V ˇ ω i . Thus we have a short exact sequence 0 → L i → V i → M i → B ∈ B ; here L i = O B ( − ˇ ω i ), and M i = O B ( − ˇ ω i + ˇ α i ). Again, we have P i ( φ ) = { → φ ∗ L i → φ ∗ V i → φ ∗ M i → } .Finally, we define E i : ◦ Z λ,µ → Ext ( O C , L ˇ α i ) = Ext ( O C , O C ( h µ − λ, ˇ α i i )) as the com-position ◦ Z λ,µ ≃ ◦ Z λ − µ, P i −→ F h λ − µ, ˇ α i i E −→ Ext ( O C , O C ( h µ − λ, ˇ α i i )).6.3. Recollections of [8] . We recall some of the constructions of [8] in the particular caseof a curve of genus 0 (projective line C ). In this case the canonical bundle ω C is isomorphicto O C ( − ω / C ≃ O C ( − λ , . . . , λ N ) be an orderedcollection of dominant coweights. Let ◦ A Λ be the moduli space of ordered configurations of istinct points ( z , . . . , z N ∈ A ). Let ◦ A α, Λ ⊂ A α × ◦ A Λ be the open subspace formed bythe configurations of pairwise distinct points. For i ∈ I and z ∈ ◦ A Λ we define a monicpolynomial K i ( z ) := Q ≤ n ≤ N ( z − z n ) h λ n , ˇ α i i .Given a point z ∈ ◦ A Λ , we consider a moduli stack W z, Λ classifying the following data:(a) A G -bundle F G on C ; (b) For each dominant weight ˇ λ a nonzero map κ ˇ λ : ω h ρ, ˇ λ i C → V ˇ λ F G having the poles of order exactly λ i at z i , and regular nonvanishing at C \ { z } . Here V ˇ λ F G isthe vector bundle associated to F G and the irreducible G -module V ˇ λ , and ω h ρ, ˇ λ i C stands for( ω / C ) ⊗h ρ, ˇ λ i . The collection of maps κ ˇ λ must satisfy the Pl¨ucker relations (cf. [8, 2.1, 2.6]).Alternatively, note that κ ˇ λ | ω h ρ, ˇ λ i C ( − P Nn =1 h λ n , ˇ λ i· z n ) is a regular embedding, and the im-age is a line subbundle L ˇ λ ⊂ V ˇ λ F G . So W z, Λ is the moduli stack of the collections ofline subbundles L ˇ λ ⊂ V ˇ λ F G satisfying the Pl¨ucker relations plus the identifications L ˇ λ = ω h ρ, ˇ λ i C ( − P Nn =1 h λ n , ˇ λ i · z n ).Since the assignment ˇ λ L ˇ λ is multiplicative in ˇ λ : L ˇ µ +ˇ ν = L ˇ µ ⊗ L ˇ ν , we can extendthe notion of L ˇ λ for arbitrary weights ˇ γ ∈ X ∗ ( T ). The construction of Section 6.2 definesa morphism E i : W z, Λ → Ext ( O C , L ˇ α i ). The canonical embedding L ˇ α i ֒ → ω h ρ, ˇ α i i C = ω C gives rise to the projection Ext ( O C , L ˇ α i ) → Ext ( O C , ω C ) = A . Composing it with E i weobtain a function χ i : W z, Λ → A .Following [8, 4.3] we consider the moduli stack ◦ Z α → W z, Λ classifying the same dataas W z, Λ plus (a) a trivialization of F G at ∞ ∈ C such that the B -structure (given by thecollection { κ ˇ λ } ) at ∞ coincides with B ⊂ G ; (b) an additional B − -structure on F G of degree − ρ − α equal at ∞ ∈ C to B − ⊂ G . By an abuse of notation we preserve the notation χ i : ◦ Z α → A for the composition of χ i : W z, Λ → A and the projection ◦ Z α → W z, Λ .According to [8, 4.5, 4.6], the stack ◦ Z α is actually a scheme; moreover, we have a canonicalisomorphism ◦ Z α = ◦ Z − ρ, − ρ − α = ◦ Z α . Thus we obtain a function χ i : ◦ Z α → A . If we allow z to vary in ◦ A Λ , we obtain the same named function χ i on ◦ Z α × ◦ A Λ . Theorem 6.4. The function χ i on ◦ Z α × ◦ A Λ in the coordinates ( w, y, z ) is given by χ i ( w, y, z ) = a i X r =1 y − i,r Q j = i Q −h α j , ˇ α i i j ( w i,r ) Q ′ i ( w i,r ) K i ( w i,r ) = a i X r =1 y − i,r Q j = i Q −h α j , ˇ α i i / j ( w i,r ) Q ′ i ( w i,r ) K i ( w i,r ) . Proof. Recall the involution ι : ◦ Z α ∼ −→ ◦ Z α of Section 2.6. We have to prove that χ i ◦ ι ( w, y, z ) = P a i r =1 y i,r K i ( w i,r ) /Q ′ i ( w i,r ). Recall also the modified coordinates y i,r := y i,r Q ′ i ( w i,r ) of (3.1). Thus we have to prove χ i ◦ ι ( w, y , z ) = a i X r =1 y i,r K i ( w i,r ) (6.1) ecall the map E i : ◦ Z λ,µ → Ext ( O C , L ˇ α i ) of Section 6.2. We have to prove its factorizationproperty, i.e. the commutativity of the following diagram:( A β × A γ ) disj × A α ◦ Z α f + β,γ −−−−→ ( A β × A γ ) disj × A β × A γ ( ◦ Z β × ◦ Z γ ) E i y E i × E i y Ext ( O C , L ˇ α i ) + ←−−−− Ext ( O C , L ˇ α i ) × Ext ( O C , L ˇ α i ) (6.2)Unraveling the definition of E i and using the compatibility of factorizations (3.3) wereduce the problem to G = SL i . This problem is formulated as follows. For a point x ∈ C and a line bundle E on C we have a natural map ϕ x : ( E ⊗ O C K x ) / ( E ⊗ O C O x ) → H ( C, E )arising from the identification ( E ⊗ O C K x ) / ( E ⊗ O C O x ) = j ∗ E / E and the boundary map inthe long exact cohomology sequence coming from 0 → E → j ∗ E → j ∗ E / E → j isthe open embedding C \ { x } ֒ → C ). Given a short exact sequence 0 → L → V → M → T over E = H om( M , L ) with the class e ( T ) ∈ H ( C, E ). Given a generic splitting of this exact sequence we obtain a generic section s of T . Choosing a local splitting around x we obtain s x ∈ E ⊗ O C K x whose principal partin ( E ⊗ O C K x ) / ( E ⊗ O C O x ) is well defined, i.e. independent of the choice of a local splitting.Then clearly e ( T ) = P x ∈ C ϕ x ( s x ).This completes the proof of Theorem 6.4. (cid:3) More recollections (Proof of Theorem 1.9). Recall the sequence of morphisms ◦ Z α ≃ ◦ Z − ρ, − ρ − α = ◦ Z α → W z, Λ → Bun G ( C ) of Section 6.3. The line bundle P on ◦ Z α isdefined as the inverse image of the determinant line bundle on Bun G ( C ), cf. [8, 2.2]. Infact, P is the restriction of the same named line bundle on Z α with the canonical section F α , see [3, 4.9]. Hence F α gives rise to a canonical trivialization of P on ◦ Z α . Given alevel κ ∈ C , Gaitsgory constructs a certain P κ -twisted D -module F κ z, Λ on ◦ Z α × ◦ A Λ (asa lift from W z, Λ × ◦ A Λ ) [8, 2.7]. It is smooth of rank 1 on ◦ Z α × ◦ A Λ but has irregularsingularities at ∂Z α × ◦ A Λ . In case κ is irrational, F κ z, Λ is clean. The trivialization of P on ◦ Z α gives rise to the identification of P κ -twisted D -modules with the usual D -modules,and then the corresponding D -module F κ , triv z, Λ on ◦ Z α × ◦ A Λ is generated by the function F κ α · exp( χ i ) · Q ≤ m The (multival-ued) Master function [5, Section 3] on h ∨ × ◦ A α, Λ is defined as follows:Φ( h ∗ , w, z ) := X ≤ n ≤ N h λ n , h ∗ i z n − X ( i,r ) h α i , h ∗ i w i,r + X ( i,r ) =( j,s ) α i · α j w i,r − w j,s ) −− X ( i,r ) , ≤ n ≤ N α i · λ n log( z n − w i,r ) + X ≤ m Straightforward. (cid:3) Appendix In this appendix we give another (elementary) derivation of a particular case of Theo-rem 6.4 for G = SL (2).7.1. Zastava for SL (2) . For G = SL (2) the coroot lattice is just Z , and given a ∈ N the moduli space of based maps ◦ Z a is identified with the moduli space of extensions 0 → O C ( − a ) → O C ⊕ O C → O C ( a ) → ∞ ∈ C . So we have a map E : ◦ Z a → Ext ( O C ( a ) , O C ( − a )) = Γ( C, O C (2 a − ∨ . Proposition 7.2. For a polynomial K ∈ H ( C, O C (2 a − we have h K, E i = a X r =1 y − r K ( w r ) Q ′ ( w r ) . Proof. We denote H ( C, O C (1)) ∨ by V (a 2-dimensional vector space with a base formedby the highest vector x and the lowest vector t ). We have Ext ( O ( a ) , O ( − a )) = Γ( C, O (2 a − ∨ = Sym a − V . We will write down an element of Sym a − V in the basis of products ofdivided powers of x, t : c x (2 n − + . . . + c k x (2 a − − k ) t ( k ) + . . . + c n − t (2 a − .For a point φ ∈ ◦ Z a , the first map in the corresponding exact sequence 0 → O C ( − a ) → O C x ⊕ O C t → O C ( a ) → Q, R ), and the second oneis given by ( − R, Q ). In the corresponding long exact sequence 0 = H ( O C ( − a )) → H ( O C ⊕ O C ) → H ( O C ( a )) → H ( O C ( − a )) → . . . the boundary map is given by the cupproduct with our desired Ext -class E ( φ ) in Sym a − V . Note that H ( O C ( a )) = Sym a V ∨ ,and H ( O C ( − a )) = Sym a − V . So the boundary map is the contraction Sym a V ∨ → Sym a − V with the desired class E ( φ ) ∈ Sym a − V . Since the composition H ( O C ⊕ O C ) → H ( O C ( a )) → H ( O ( − a )) is 0, and the first map is given by ( − R, Q ), we con-clude that the contraction of E ( φ ) and Q equals 0, as well as the contraction of E ( φ ) and R . This is a system of linear equations on E ( φ ) which defines it up to proportionality.To write down the formula for contraction, we think of Q, R as of differential operators Q = ∂ ax + . . . + a k ∂ a − kx ∂ kt + . . . + a a ∂ at , R = b ∂ a − x + . . . + b k ∂ a − − kx ∂ kt + . . . + b a ∂ a − t ,and then the contraction is nothing but the application of differential operators Q, R to thepolynomial E := c x (2 a − + . . . + c k x (2 a − − k ) t ( k ) + . . . + c a − t (2 a − .Note that the matrix of this system of equations is (up to proportionality) exactly theSylvester matrix S = a . . . a a . . . . . . a . . . a a b b . . . b a − . . . . . . b b . . . b a − with the middle row (thefirst one with b ’s) removed. Solving it via the Cramer rule we obtain c k = ( − k det S − imes the (2 a − × (2 a − k -th column. Note also that the resultant R ( Q, R ) is nothing but det S , and R ( Q, R ) = 0 under our assumptions: ( O C ⊕ O C ) / O C ( − a ) torsionless.Equivalently, if we think of Q, R as of two (relatively prime) polynomials in z = ∂ x /∂ t (as in Section 2.2), then the equation RD − QF = 1 has a unique solution such that D is a polynomial in z of degree a − 1, and F is a polynomial in z of degree a − 2. Theprincipal part at ∞ ∈ C of the ratio D ( z ) Q ( z ) is nothing but c z + c z + . . . + c a − z a − + . . . ( c k from the previous paragraph). By the Lagrange interpolation we find c k = P ar =1 w kr D ( w r ) Q ′ ( w r ) = P ar =1 w kr R ( w r ) − Q ′ ( w r ) = P ar =1 w kr y − r Q ′ ( w r ) . The desired formula for h K, E i follows. (cid:3) Remark . We keep the notations introduced in the proof of Proposition 7.2. Let usdefine ˜ c , . . . , ˜ c a − by R ( z ) Q ( z ) = ˜ c z + ˜ c z + . . . + ˜ c a − z a − + . . . Then by the Lagrange interpolation˜ c k = P ar =1 w kr y r Q ′ ( w r ) . According to L. Kronecker [13], the resultant R ( Q, R ) = det ˜ L where ˜ L isa Hankel matrix ˜ L := ˜ c ˜ c ˜ c . . . ˜ c a − ˜ c ˜ c ˜ c . . . ˜ c a ˜ c ˜ c ˜ c . . . ˜ c a +1 ... ... ... . . . ...˜ c a − ˜ c a ˜ c a +1 . . . ˜ c a − . We obtain R ( Q, R ) = R ( D, R ) − =det L − where L is a Hankel matrix L := c c c . . . c a − c c c . . . c a c c c . . . c a +1 ... ... ... . . . ... c a − c a c a +1 . . . c a − . This identity R ( Q, R ) = det L − was independently obtained by A. Uteshev (private communication).Note that the equation det L = 0 is the equation of the locus in Ext ( O C ( a ) , O C ( − a ))formed by the extensions with the middle term a nontrivial C [4]. References [1] A. Braverman, M. Finkelberg, D. Gaitsgory, I. Mirkovi´c, Intersection cohomology of Drinfeld’s com-pactifications. Selecta Math. (N.S.) (2002), no. 3, 381–418.[2] A. Braverman, M. Finkelberg, D. Gaistgory, Uhlenbeck spaces via affine Lie algebras , Progress in Math. (2006), 17–135.[3] A. Braverman, M. Finkelberg, Semi-infinite Schubert varieties and quantum K -theory of flag manifolds ,J. Amer. Math. Soc. (2014), 1147–1168.[4] G. Comas, M. Seiguer, On the rank of a binary form , Found. Comput. Math. (2011), no. 1, 65–78.[5] G. Felder, Y. Markov, V. Tarasov, A. Varchenko, Differential equations compatible with KZ equations ,Math. Phys. Anal. Geom. (2000), 139–177.[6] M. Finkelberg, A. Kuznetsov, N. Markarian, I. Mirkovi´c, A note on a symplectic structure on the Spaceof G -monopoles , Commun. Math. Phys. (1999), 411–421. Erratum , Commun. Math. Phys. (2015), 1153–1155; arXiv:math/9803124, v6.[7] M. Finkelberg, L. Rybnikov, Quantization of Drinfeld zastava in type C , Journal Algebraic Geometry (2014), no. 2, 166–180. 8] D. Gaitsgory, Twisted Whittaker model and factorizable sheaves , Selecta Math. (N.S.) (2008), no. 4,617–659.[9] D. Gaiotto, E. Witten, Knot invariants from four-dimensional gauge theory , Adv. Theor. Math. Phys. (2012), no. 3, 935–1086.[10] S. Jarvis, Euclidean monopoles and rational maps , Proc. Lond. Math. Soc. (3) (1998), no.1, 170–192.[11] S. Jarvis, Construction of Euclidean monopoles , Proc. Lond. Math. Soc. (3) (1998), no.1, 193–214.[12] M. Kashiwara, On crystal bases , CMS Conf. Proc. (1995), 155–197.[13] L. Kronecker, Zur Theorie der Elimination einer Variabeln aus zwei algebraischen Gleichungen , Werke,Bd. 2 (1897), 113–192, Teubner, Leipzig. A.B. : Department of Mathematics, Brown University, 151 Thayer st, Providence RI 02912, USA; [email protected] G.D. : Department of Mathematics, Columbia University, New York, NY 10027, USA; [email protected] M.F. : National Research University Higher School of Economics, Math. Dept.,20 Myasnitskaya st, Moscow 101000 Russia; IITP [email protected]@gmail.com