The elliptic Hall algebra, Cherednick Hecke algebras and Macdonald polynomials
aa r X i v : . [ m a t h . QA ] F e b THE ELLIPTIC HALL ALGEBRA, CHEREDNIK HECKEALGEBRAS AND MACDONALD POLYNOMIALS.
O. SCHIFFMANN, E. VASSEROT
Contents
0. Introduction 11. Elliptic Hall algebra 52. Double affine Hecke algebras 83. The projection map 124. Stable limits of DAHA’s 145. Macdonald Polynomials 186. Eisenstein series 217. Geometric construction of Macdonald polynomials 27Appendix A. Proof of Proposition 3.2. 32Appendix B. Proof of Proposition 3.3. 39Appendix C. Proof of Theorem 6.3 44Appendix D. Proof of Lemma 7.4. 45References 470.
Introduction . In this paper we continue the study of the Hall algebra H X of an elliptic curve X defined over a finite field F l , started in [BS], [S]. Here we exhibit some strong linkbetween the Hall algebra H X , or more precisely its composition subalgebra U X ,and Cherednik’s double affine Hecke algebras ¨ H n of type GL ( n ), for all n . Thisallows us to obtain a geometric construction of Macdonald polynomials P λ ( q, t )in terms of certain functions (Eisenstein series) on the moduli space of semistablevector bundles on the elliptic curve X .Let us describe our results in more details. The spherical affine Hecke algebra S ˙ H G of a reductive algebraic group G is the convolution algebra of G ( O )-invariantfunctions on the affine Grassmanian c Gr = G ( K ) /G ( O ), where K = F l (( z )) and O = F l [[ z ]], see [IM]. The Satake isomorphism identifies S ˙ H G with the representationring Rep ( G L ) of the dual group of G . Now let us assume that G = G L = GL ( n ),so that the set of F l -points of c Gr is equal to { L ⊂ F nl (( z )); L is a free F l [[ z ]] − module of rank n } and Rep ( G ) ≃ C [ x ± , . . . , x ± n ] S n . Lusztig [Lu] embeds the nilpotent cones N k ⊂ gl ( k ), k ≥
1, into the positive Schubert variety c Gr + = { L ⊂ F nl [[ z ]]; L is a free F l [[ z ]] − module of rank n } of c Gr . This yields a surjective algebra homomorphism(0.1) Θ + n : H cl = M k ≥ C GL ( k ) [ N k ] ։ S ˙ H + n ≃ C [ v ± ][ x , . . . , x n ] S n . See [M1, Chap. II] or [Lu]. Here H cl is the classical Hall algebra, and v = l − / .Since the dependence on v is polynomial, we may treat it as a formal parameter.Letting n tend to infinity in (0.1) yields an isomorphism in the stable limit(0.2) Θ + ∞ : H cl ∼ → S ˙ H + ∞ = lim ←− S ˙ H + n ≃ C [ v ± ][ x , x , . . . ] S ∞ . The first main result of this paper provides an affine version of (0.1) and (0.2).In [BS] it was found that the Hall algebra H X of the category of coherent sheaveson an elliptic curve X defined over F l contains a natural “composition” subalgebra U X which is a two-parameter deformation of the ring of diagonal invariants R + n = C [ x , . . . , x n , y ± , . . . , y ± n ] S n where S n acts simultaneously on the x -variables and the y -variables. The twodeformation parameters are v = l − / and t = σv , where σ is a Frobenius eigenvalueof H ( X , Q p ) (viewed as a complex number). The dependence on v , t is polynomialand we may treat them as formal variables.Let ¨ H n denote Cherednik’s double affine Hecke algebra of type GL ( n ), and let S ¨ H n = S · ¨ H n · S stand for its spherical subalgebra. Here S is the completeidempotent associated to the finite Hecke algebra H n ⊂ ¨ H n . The algebra S ¨ H n isa deformation of the ring R n = C [ x ± , . . . , x ± n , y ± , . . . , y ± n ] S n depending on two parameters v and q . Let S ¨ H + n be the positive part of S ¨ H n , seeSection 2.1. In Theorem 3.1 we prove the following. Theorem. If q = vt then for any n there exists a surjective algebra homomorphism Ψ + n : U X ։ S ¨ H + n . This map extends to a surjective algebra homomorphism Ψ n : DU X ։ S ¨ H n . Here DU X is the Drinfeld double of U X . It is equipped with an action of SL (2 , Z ) coming from the group of derived autoequivalences of D b ( Coh ( X )). Chered-nik has defined an action of SL (2 , Z ) on S ¨ H n , see [C], [I]. The map Ψ n is definedso as to intertwine these two actions. The maps Ψ + n behave well with respect tothe stable limit, see Theorem 4.6. Theorem.
The maps Ψ + n induce an algebra isomomorphism Ψ + ∞ : U X ∼ → S ¨ H + ∞ = lim ←− S ¨ H + n . Note that our approach doesn’t use any affinization of the affine Grassmanian.One of the essential features of the construction of the spherical affine Heckealgebras as convolution algebra of functions (on the affine Grassmanian or on thenilpotent cones) is that it lifts to a tensor category of perverse sheaves (see e.g. [Gi],[MV]). Such a geometric lift also makes sense here, and fits in Laumon’s theory ofautomorphic sheaves. We refer to [S] and Section 4.3. for more details.In the second part of this paper, we give an application of the above geometricconstruction of S ¨ H n to Macdonald polynomials.The Hall algebra H vec X of the category of vector bundles on X (or on any smoothprojective curve) can be viewed as the algebra of (unramified) automorphic formsfor GL ( n ), for all n ≥
1, over the function field of X . The product is given by ALL, CHEREDNIK, EISENSTEIN, MACDONALD 3 the functor of parabolic induction, see [K1]. To obtain the whole Hall algebra H X one needs to take into account the torsion sheaves as well. The Hall algebra H tor X of the category of torsion sheaves on X acts on H vec X by the adjoint action and H X is isomorphic to the semi-direct product H vec X ⋊ H tor X . The action of torsionsheaves is interpreted in the language of automorphic forms as Hecke operators . Forinstance, the skyscraper sheaf O x at a point x ∈ X corresponds to the elementarymodification at x .Under the map Ψ + ∞ , the element (0 , ∈ U tor X responsible for the Hecke operator of rank one is sent to
Macdonald’s element ∆ = S P i Y i S ∈ S ¨ H + ∞ , see Section 2.The importance of this element stems from the fact that, in the polynomial repre-sentation of S ¨ H + ∞ , the operator ∆ has distinct eigenvalues and the correspondingeigenvectors are the Macdonald polynomials P λ ( q, v ). Thus the map Ψ + ∞ allowsus to relate Hecke eigenvectors on the Hall or automorphic side to Macdonald poly-nomials on the Hecke algebra side. In particular, we are naturally led to find Heckeeigenvectors in U X whose eigenvalues match those of the P λ ( q, v ).Eisenstein series yield a way to produce new Hecke eigenvectors from old onesvia parabolic induction. In the present situation, it so happens that the simplestEisenstein series, i.e., those induced from trivial characters of parabolic subgroups,already have the good eigenvalues under the Hecke operator. Unfortunately, weare unable to construct the polynomial representation of S ¨ H + ∞ in a geometric way,see Remark 5.1, and thus we cannot obtain directly a geometric construction of P λ ( q, v ). To remedy this, we manage to lift the Macdonald polynomials from thepolynomial representation and view them inside the Hecke (or Hall) algebra itself.More precisely, it is shown in [BS] that the subalgebra of U X consisting of functionssupported on the set Coh ( X ) (0) of semistable sheaves of zero slope is canonicallyisomorphic to the algebraΛ + v,t = C [ v ± , t ± ][ x , x , . . . ] S ∞ . In a few words, under Fourier-Mukai transform the set
Coh ( X ) (0) is identified withthe set of torsion sheaves on X , and any function on the set of torsion sheaves witha fixed ponctual support in X can be viewed as an element of the classical Hallalgebra. See [P], Theorem 14.7 for details. If f is any function in U X we let f (0) be its restriction to Coh ( X ) (0) , viewed as an element of Λ + v,t . For any l ∈ N + put(0.3) E l ( z ) = X d ∈ Z ( l,d ) v d ( l − z d ∈ b U X [[ z, z − ]]where ( l,d ) is the characteristic function of the set of all coherent sheaves on X ofrank l and degree d , and b U X is a certain completion of U X . For ( l , . . . , l n ) ∈ N n we form the Eisenstein series(0.4) E l ,...,l n ( z , . . . , z n ) = E l ( z ) · E l ( z ) · · · E l n ( z n ) ∈ b U X [[ z ± , . . . , z ± n ]] . By a theorem of Harder, this is a rational function in z , . . . , z n . Our second mainresult (Theorem 7.1) reads as follows. Theorem.
Let ( l , . . . , l n ) ∈ N n . We have E l ,...,l n ( z, q − z, . . . , q − n z ) = 0 unless ( l , . . . , l n ) is dominant, i.e., unless ( l , . . . , l n ) is a partition. In that case we have E l ,...,l n ( z, q − z, . . . , q − n z ) (0) = ωP λ ( q, v ) where λ = ( l , . . . , l n ) ′ is the conjugate partition and where ω stands for the standardinvolution on symmetric functions. O. SCHIFFMANN, E. VASSEROT
We also give a similar construction of skew Macdonald polynomials P λ/µ ( q, v ).Note that the above Eisenstein series can be lifted to some constructible sheaves viathe theory of Eisenstein sheaves , see [La] and [S]. Hence the Macdonald polynomials P λ ( q, v ) may be realized as Frobenius traces of certain canonical constructiblesheaves on the moduli stack of semistable sheaves of zero slope on X . We hope tocome back to this point in the future.There is a well-known and important geometric approach to Macdonald polyno-mials, which is based on the equivariant K-theory of the Hilbert schemes ( C ) [ n ] of points on C . There the polynomials P λ ( q, v ) are realized as the classes ofcertain canonical coherent sheaves on ( C ) [ n ] , see [Hai]. It would be interestingto relate this “coherent sheaf” picture with our “constructible functions” (or “per-verse sheaf”) picture in a precise fashion, and to understand this relation in theframework of Langlands duality, see e.g. [B]. Note that the Hall algebra H X isalready involved in a Langlands type duality (the geometric Langlands duality forthe elliptic curve X ), with the category of coherent sheaves on the moduli stack oflocal systems on X .The structure of the paper is as follows : Sections 1 and 2 contain some rec-ollections on elliptic Hall algebras H X and U X , taken from [BS], and Cherednikdouble affine Hecke algebra ¨ H n and S ¨ H n respectively; in Section 3 we constructthe algebra morphism Ψ n : DU X ։ S ¨ H n ; in Section 4 we study and define thestable limit S ¨ H + ∞ of the spherical Cherednik algebra and establish the isomorphismΨ + ∞ : U X ∼ → S ¨ H + ∞ . This is the first main result of this paper. A comparison tablewith the picture of the classical Hall algebra and the “finite” spherical affine Heckealgebra is found in Section 4.3. Section 5 deals with Macdonald polynomials : werecall the definition and provide a characterization of the family of all (possiblyskew) Macdonald polynomials which we use later. In Section 6 we introduce theEisenstein series which are relevant to us, and study some of their specializations.Finally, our second main theorem, which gives a geometric construction of (possiblyskew) Macdonald polynomials from Eisenstein series, is given in Section 7. Severalproofs in the paper require some lengthy computations; these are written up indetails in Appendices A through D.A final word of warning concerning the notation. There is an unfortunate clashbetween the conventional notations used in the quantum group/Hall algebra litera-ture and those used in the Macdonald polynomials literature : the letter q generallydenotes the size of the finite field in the first case whereas it denotes the modu-lar parameter in the second case. We have opted to comply with the Macdonaldpolynomials conventions. ALL, CHEREDNIK, EISENSTEIN, MACDONALD 5 Elliptic Hall algebra
We will use the standard v -integers and v -factorials[ i ] = [ i ] v = v i − v − i v − v − , [ i ]! = [2] · · · [ i ] , as well as some positive and negative variants[ i ] + = v i − v − , [ i ] + ! = [2] + · · · [ i ] + , [ i ] − = v − i − v − − , [ i ] − ! = [2] − · · · [ i ] − . Let us denote by Λ + v Macdonald’s ring of symmetric functions ([M1])Λ + v = C [ v ± ][ x , x , . . . ] S ∞ defined over C [ v ± ]. We will denote by e λ , p λ , m λ the elementary, the power-sum,and the monomial symmetric functions respectively. This ring is equipped with anatural bialgebra structure ∆ : Λ + v → Λ + v ⊗ Λ + v defined by ∆( p r ) = p r ⊗ ⊗ p r for r ≥ . Let X be a smooth elliptic curve over some finite field F l , and let Coh ( X )stand for the category of coherent sheaves on X . If F is a sheaf sheaf in Coh ( X )we call the pair F = ( rk ( F ) , deg ( F )) the class of F . The set of possible classesof sheaves in Coh ( X ) is equal to Z , + = { ( r, d ) ∈ Z ; r ≥ r = 0 , d ≥ } . Webriefly recall the definition of the Hall algebra of Coh ( X ). See [BS] for more details.Let I ( X ) stand for the set of isomorphism classes of objects in Coh ( X ). Follow-ing Ringel [R], the C -vector space of finitely supported functions H X = { f : I ( X ) → C ; | supp ( f ) | < ∞} may be equipped with the convolution product( f · g )( M ) = X N ⊆ M ν −h M/N,N i f ( M/N ) g ( N ) , where ν = l − and h P, Q i = dim Hom(
P, Q ) − dim Ext(
P, Q ) is the Euler form.Here we write Ext(
P, Q ) for Ext ( P, Q ). The sum on the right hand side is finitefor any M since f and g have finite support and, for any N, M ∈ Coh ( X ), thegroup Hom( N, M ) is finite. The above formula indeed defines an element in H X asfor any P, Q ∈ Coh ( X ), the group Ext( P, Q ) is also finite. By the Riemann-Rochtheorem, we have(1.1) h P, Q i = rk ( P ) deg ( Q ) − deg ( P ) rk ( Q ) . By [Gr] the algebra H X also has the structure of a bialgebra, with coproduct(∆( f ))( P, Q ) = 1 | Ext(
P, Q ) | X ξ ∈ Ext(
P,Q ) f ( M ξ )where M ξ is the extension of P by Q corresponding to ξ . The product and thecoproduct are related by the pairing H X ⊗ H X → C , ( f, g ) = X M f ( M ) g ( M ) | Aut( M ) | which is a Hopf pairing, i.e., which satisfies the identity ( f g, h ) = ( f ⊗ g, ∆( h )) forany f, g, h . O. SCHIFFMANN, E. VASSEROT
Remarks. i) In our situation, as opposed to [Gr], it is not necessary to twist theproduct in H X ⊗ H X in order to obtain a bialgebra, because the Euler form h , i is antisymmetric.ii) The coproduct ∆ only takes values in a certain formal completion of H X ⊗ H X ,see [BS, Section 2.2] for details.The characteristic functions { M ; M ∈ I ( X ) } form a basis H X . Assigning thedegree ( rk ( M ) , deg ( M )) to the element M yields a Z -grading on H X which iscompatible with the (co)multiplication. Let µ ( M ) = deg ( M ) /rk ( M ) ∈ Q ∪ {∞} be the slope of a sheaf M ∈ Coh ( X ),and for µ ∈ Q ∪ {∞} let C µ stand for the category of semistable sheaves of slope µ .For instance, C ∞ is the category of torsion sheaves on X . The following fundamentalresult on the structure of Coh ( X ) is due to Atiyah. Theorem 1.1 (Atiyah, [A]) . The following hold :i) for any µ , µ ′ there is an equivalence of abelian categories ǫ µ,µ ′ : C µ ′ ∼ → C µ , ii) any coherent sheaf F decomposes uniquely as a direct sum F = F ⊕ · · · ⊕ F s ofsemistable sheaves F i ∈ C µ i with µ < · · · < µ s . By a standard property of semistable sheaves we have Hom( C µ , C µ ′ ) = { } for µ > µ ′ . By Serre duality, this implies that Ext( C µ ′ , C µ ) = { } whenever µ > µ ′ .Hence any extension 0 → F → G → H → F ∈ C µ , H ∈ C µ ′ is split. Fromthe above two facts, it follows that in H X we have(1.2) H · F = ν −hH , Fi F⊕H if F ∈ C µ , H ∈ C µ ′ and µ > µ ′ .For µ ∈ Q ∪{∞} let H ( µ ) X stand for the subspace consisting of functions supportedon the set of semistables sheaves of slope µ . Since C µ is stable under extensions, H ( µ ) X is a subalgebra of H X . By Theorem 1.1 i), all these subalgebras are isomorphic. Let ~ N µ H ( µ ) X denote the ordered tensor product of spaces H ( µ ) X with µ ∈ Q ∪ {∞} , i.e.,the vector space spanned by elements of the form a µ ⊗ · · · ⊗ a µ r with a µ i ∈ H ( µ i ) X and µ < · · · < µ r . From (1.2) and Theorem 1.1 ii) we deduce the following, see[BS, Lemma 2.4.]). Corollary 1.2.
The multiplication map induces an isomorphism of vector spaces ~ N µ H ( µ ) X ∼ → H X . We will mainly be interested in a certain subalgebra U + X ⊂ H X which we nowdefine. For any class α ∈ Z , + we set ssα = X F = α ; F ∈C µ ( α ) F ∈ H X . This sum is finite. Indeed, by Theorem 1.1 i) it is enough to check this for µ ( α ) = ∞ .Then, this follows from the fact that there are only finitely many closed points on X which are rational over a fixed finite extension of F l . Let U + X be the subalgebragenerated by ssα for α ∈ Z , + . It will be useful to consider a different set ofgenerators T α of U + X , uniquely determined by the collection of formal relations(1.3) 1 + X l ≥ sslα s l = exp X l ≥ T lα [ l ] s l , ALL, CHEREDNIK, EISENSTEIN, MACDONALD 7 for any α = ( r, d ) with r and d relatively prime.To a slope µ ∈ Q ∪ {∞} is naturally associated the subalgebra U + , ( µ ) X ⊂ U + X generated by { ssα ; µ ( α ) = µ } . Of course, we have U + , ( µ ) X ⊂ H ( µ ) X . Proposition 1.3 ([BS, Theorem 4.1]) . The following hold : i) The multiplication map induces an isomorphism ~ N µ U + , ( µ ) X ∼ → U + X , ii) For any x = ( r, d ) ∈ Z , + for which r and d are relatively prime, the as-signment T l x / [ l ] p l /l extends to an isomorphism of algebras U + , ( µ ( x )) X ∼ → (Λ + v ) | v = ν . In particular, U + , ( µ ( x )) X is a free commutative polynomial algebrain the generators { T l x ; l ≥ } . We now wish to give a presentation of U + X by generators and relations. In fact,we will give such a presentation for the Drinfeld double of U + X , which is a moresymmetric object. Recall that if H is a bialgebra equipped with a Hopf pairing( , ) then its Drinfeld double DH is the algebra generated by two copies H + and H − of H subject to the collection of relations(1.4) X i,j ( h + ) (1) i ( g − ) (2) j ( h (2) i , g (1) j ) = X i,j ( g − ) (1) j ( h + ) (2) i ( h (1) i , g (2) j )for any g, h ∈ H , where we write h + ∈ H + and g − ∈ H − for the correspondingelements, and where use Sweedler’s notation ∆( x ) = P i x (1) i ⊗ x (2) i .By [BS, Proposition 4.2], the algebra U + X is a subbialgebra of H X , and we denoteby U X its Drinfeld double.Set Z , ∗ equal to Z \ { (0 , } . For x , y ∈ Z , ∗ let ∆ x , y stand for the trianglewith vertices o , x , x + y , where o denotes the origin in Z . If x = ( r, d ) ∈ Z , ∗ wewrite d ( x ) = gcd ( r, d ). For a pair of non-colinear vectors ( x , y ) ∈ Z , ∗ we set ǫ x , y equal to sign ( det ( x , y )).We set A = C [ v ± , t ± ] and K = C ( v, t ). Definition.
For i ∈ N , put c i = ( v i + v − i − t i − t − i )[ i ] /i ∈ A . Let A K be theunital K -algebra generated by elements t x for x ∈ Z , ∗ bound by the following setof relations.i) If x , x ′ belong to the same line in Z then [ t x , t x ′ ] = 0.ii) Assume that x , y are such that d ( x ) = 1 and that ∆ x , y has no interiorlattice point. Then [ t y , t x ] = ǫ x , y c d ( y ) θ x + y v − − v where the elements θ z , z ∈ Z are obtained by equating the Fourier coeffi-cients of the collection of relations X i θ i x s i = exp (( v − − v ) X i ≥ t i x s i ) , for any x ∈ Z such that d ( x ) = 1.The algebra A K is Z -graded by deg ( t x ) = x . Put ˜ t x = t x / [ d ( x )] and let A A be the unital A -subalgebra of A K generated by { ˜ t x ; x ∈ Z , ∗ } . We will write A ±A for the subalgebra of A generated by { ˜ t x ; x ∈ ± Z , + } . By [BS], Proposition 5.1,the multiplication yields an isomorphism A −A ⊗ A A + A ≃ A A . Let A ++ A be thesubalgebra generated by { ˜ t x ; x ∈ Z , ++ } , where Z , ++ = { ( r, d ); r ≥ , d ≥ } . We O. SCHIFFMANN, E. VASSEROT have A + A = M x ∈ Z , + A A [ x ] , A ++ A = M x ∈ Z , ++ A A [ x ] . The algebra A A has an obvious symmetry : the group SL (2 , Z ) acts by automor-phisms such that g · ˜ t x = ˜ t g ( x ) .Let σ , σ be the two eigenvalues of the Frobenius endomorphism acting on thevector space H ( X ⊗ F l , Q p ), with p prime to l . We’ll fix once for all a fieldisomorphism C ≃ Q p . This allows us to view σ , σ as complex numbers. Let A X stand for the specialization of A A at v = ν = l − / and t = σν . Theorem 1.4 ([BS, Theorem 5.1]) . The assignment ˜ t x T x / [ deg ( x )] , x ∈ Z , ∗ ,extends to an isomorphism A X ∼ → U X . It restricts to an isomorphism A + X ∼ → U + X . To a slope µ ∈ Q ∪ {∞} is naturally associated the subalgebra A + , ( µ ) A ⊂ A + A generated by { T α ; µ ( α ) = µ } .2. Double affine Hecke algebras
We set A ′ = C [ v ± , q ± ] and K ′ = C ( v, q ). The double affine Hecke algebra¨ H n of GL ( n ), abreviated DAHA, is the K ′ -algebra generated by elements T ± i , X ± j and Y ± j for 1 ≤ i ≤ n − ≤ j ≤ n , subject to the following relations :(2.1) ( T i + v − )( T i − v ) = 0 , T i T i +1 T i = T i +1 T i T i +1 (2.2) T i T k = T k T i if | i − k | > X j X k = X k X j , Y j Y k = Y k Y j (2.4) T i X i T i = X i +1 , T − i Y i T − i = Y i +1 (2.5) T i X k = X k T i , T i Y k = Y k T i if | i − k | > Y X · · · X n = qX · · · X n Y (2.7) X − Y = Y X − T − The subalgebra H n generated by { T i } is the usual Hecke algebra of the symmetricgroup S n , while the subalgebras ˙ H n,X and ˙ H n,Y respectively generated by H n and { X ± i } , and H n and { Y ± i } , are both isomorphic to the Hecke algebra of the affineWeyl group b S n ≃ S n ⋉ Z n . We define a Z -grading on ¨ H n by giving T i , X i and Y i degrees 0 , (1 ,
0) and (0 ,
1) respectively.Let s i ∈ S n denote the transposition ( i, i + 1), and let l : S n → N be thestandard length function. If w = s i · · · s i r is a reduced decomposition of w ∈ S n ,we set T w = T i · · · T i r . We put ˜ S = P w ∈ S n v l ( w ) T w . We have ˜ S = [ n ] + ! ˜ S , sothat the element S = ˜ S/ [ n ] + ! is idempotent. For any i we have T i S = ST i = vS . ALL, CHEREDNIK, EISENSTEIN, MACDONALD 9
We will mainly be interested in the spherical
DAHA of ¨ H n equal to S ¨ H n = S ¨ H n S . Before we can give some bases for S ¨ H n we need a few notations. Set R n , F n equal to the algebras C [ x ± , . . . , x ± n , y ± , . . . y ± n ] S n , C h x ± , . . . , x ± n , y ± , . . . y ± n i S n . The algebra R n consists of the symmetric Laurent polynomials where the x ’s andthe y ’s commute between themselves and with each other. The algebra F n consistsof the symmetric Laurent polynomials where the x ’s and the y ’s commute betweenthemselves, but not with each other. Here the symmetric group S n acts by simul-taneous permutation on the x ’s and the y ’s. There is an obvious projection map Com : F n → R n . The size of the spherical DAHA is described in the followingresult.
Proposition 2.1.
Let { E t } be any collection of elements of F n such that { Com ( E t ) } forms a basis of R n . Then { SE t S } is a K ′ -basis of S ¨ H n .Proof. Let ¨ H A ,n stand for the A ′ -subalgebra of ¨ H n generated by T i for i = 1 , . . . , n − X ± j , Y ± j for j = 1 , . . . , n . We also set S ¨ H A ,n = S ¨ H A ,n S . Both ¨ H A ,n and S ¨ H A ,n are free A -modules, see [C]. Write C for the one-dimensional complexrepresentation of A ′ in which v and q act as 1. The assignment SP ( X i , Y i ) S ⊗ n ! X σ ∈ S n σ · P ( x i , y i )give rise to a C -algebra isomorphism π : S ¨ H A ,n ⊗ C ∼ → R n . Now let { E t } be as in the hypothesis of the proposition. Then SE t S ∈ S ¨ H A ,n forall t and { π ( SE t S ) } forms a C -basis of R n . It follows that { SE t S } are K ′ -linearlyindependent and generate S ¨ H n over K ′ . ⊓⊔ There is an action of the braid group B on three strands by automorphisms on¨ H n , explicitly given by the following operators. ρ : T i T i ,X i X i Y i ( T i − · · · T i )( T i · · · T i − ) ,Y i Y i ,ρ : T i T i ,Y i Y i X i ( T − i − · · · T − i )( T − i · · · T − i − ) ,X i X i . These operators preserve S ¨ H n , and the corresponding B -action factors throughan SL (2 , Z )-action ρ : SL (2 , Z ) → Aut( S ¨ H n ) satisfying ρ ( A ) = ρ , ρ ( A ) = ρ ,where A = (cid:16) (cid:17) and A = (cid:16) (cid:17) . The following technical lemma will be often used.
Lemma 2.2.
For each l ≥ we put α l = T − l − · · · T − T − T · · · T l − . The followingrelations hold (2.8) X − l Y X l = α l Y , (2.9) Y l X = X Y l + ( v − − v ) T − l − · · · T − T − T − · · · T − l − Y X , (2.10) qX Y = T − · · · T − n − T − n − T − n − · · · T − Y X . (2.11) α · · · α l = T − · · · T − l − T − l − T − l − · · · T − , Proof.
By definition, we have T − Y X − = X − Y , which is relation (2.8) for l = 2.Multiplying on the left and on the right by T − and using the fact that [ T , Y ] = 0we obtain T − T − T Y · T − X − T − = T − X − T − Y = Y X − Since T − X − T − = X − , we get T − T − T Y X − = X − Y which is (2.8) for l = 3. A similar reasoning with multiplying on the left and onthe right by T − yields (2.8) for l = 4, etc.We now prove (2.9). From the defining relations of ¨ H n we have Y X = X Y X − T − X = X Y + ( v − − v ) X Y X − T − X = X Y + ( v − − v ) X Y X − T − X T T − = X Y + ( v − − v ) X Y X − T − X T − = X Y + ( v − − v ) Y X T − = X Y + ( v − − v ) T − Y X , which is (2.9) for l = 2. Now we multiply on the left and on the right by T − andusi the fact that [ T , X ] = [ T , Y ] = 0 to get T − Y T − X = X T − Y T − + ( v − − v ) T − T − T − Y X which, by virtue of the relation T − Y T − = Y , gives (2.9) for l = 3. To obtain(2.9) for l = 3 , T − , T − , etc.We turn to (2.10). Recall that by definition( X − n · · · X − Y X · · · X n ) X = qX Y . By (2.8) above we have X − Y X = α Y . Since [ α l , X k ] = 0 if k > l , we obtainafter conjugation by X , X − X − Y X X = α X − Y X = α α Y . Continuing in this manner yields in the end X − n · · · X − Y X · · · X n = α · · · α n Y . Thus (2.10) will follow from (2.11) which we now prove. It is easy to check (2.11)for l = 2 and l = 3. We argue by induction on l . So we fix l and assume that α · · · α l = T − · · · T − l − T − l − T − l − · · · T − . Then α · · · α l +1 == T − · · · T − l − T − l − T − l − · · · T − · T − l · · · T − T − T · · · T l = T − · · · T − l − T − l − T − l − · · · T − · T − l · · · T − T − T − T − T T · · · T l = T − · · · T − l − T − l − T − l − · · · T − · T − l · · · T − T − T − T − T − T T · · · T l = T − · · · T − l − T − l − T − l − · · · T − · T − l · · · T − T − T − T · · · T l = T − · · · T − l − T − l − T − l − · · · T − · T − l · · · T − T − T · · · T l T − . This last expression is of the form T − ZT − where Z is (using the induction hy-pothesis) equal to α . . . α l for the subalgebra of H n generated by T , . . . , T l . In ALL, CHEREDNIK, EISENSTEIN, MACDONALD 11 particular, T is not involved in Z . By our induction hypothesis again, we deducethat Z = T − · · · T − l − T − l T − l − · · · T − from which α · · · α l +1 = T − · · · T − l − T − l T − l − · · · T − of course follows. The lemma is proved. ⊓⊔ For e > P n (0 ,e ) = S P i Y ei S. More generally, if ( r, d ) = g · (0 , e ) weput P n ( r,d ) = ρ ( g ) P n (0 ,e ) . If the element g ′ ∈ SL (2 , Z ) fixes the couple (0 , e ) then ρ ( g ′ ) = ρ l for some l hence ρ ( g ′ ) P n (0 ,e ) = P n (0 ,e ) . Therefore the above definitionmakes sense, and it yields an element P n x ∈ S ¨ H n for each x ∈ Z , ∗ , such that ρ ( g ) P n x = P ng ( x ) for any g ∈ SL (2 , Z ).As an illustration, let us give the expression for certain elements P n ( r,d ) when r, d are relatively prime. To unburden the notation, we drop the exponent n in P n ( r,d ) . Lemma 2.3.
For any l ∈ Z we have (2.12) P ( l, = [ n ] − SY X l S, (2.13) P (1 ,l ) = q [ n ] + SX Y l S, (2.14) P (0 , − = q [ n ] + SY − S, (2.15) P ( l, − = q [ n ] + SX l Y − S. Proof.
Observe that since SY i +1 S = ST − i Y i T − i S = v − SY i S , we have P (0 , = S P i Y i S = [ n ] − SY S . Equation (2.12) follows from this and an application of ρ l .In particular, we have P (1 , = [ n ] − SY X S . Using Lemma 2.2, (2.10), we obtain P (1 , = qv n − [ n ] − SX Y S = q [ n ] + SX Y S . Application of ρ l yields (2.13).The last two equalities are proved using similar techniques. ⊓⊔ The values of P ( r,d ) when r and d are not relatively prime is usually harder tocompute. We give a few examples, which will be important for us. Lemma 2.4.
For any l ≥ , we have (2.16) P ( − l, = S X i X − li S, (2.17) P (0 , − l ) = q l S X i Y − li S. (2.18) P ( l, = q l S X i X li S, Proof.
Set A = (cid:16) −
11 0 (cid:17) and A = (cid:16) − − (cid:17) . We have ρ ( A ) Y = ρ ρ − · Y = X − and hence ρ ( A ) Y i = X − i for all i . The first relation (2.16) immediately follows.The proof of the second and the third relations are identical, and we only treat(2.17). We have, using Lemma 2.2, (2.10) ρ ( A ) Y = ρ − ρ ρ − · Y = X − Y − X = qY − T · · · T n − T n − T n − · · · T = qT − · · · T − n − Y − n T n − · · · T , where in the last equation we have the relations Y − i T i = T − i Y i +1 . It follows that(2.19) ρ ( A )( Y l ) = q l T − · · · T − n − Y − ln T n − · · · T for any l . Hence ρ ( A )( SY l S ) = q l SY − ln S . It is well-known and easy to show thatthe elements SY l S for l = 1 , . . . , n freely generate the ring S C [ Y , . . . , Y n ] S . Let θ : C [ SY S, . . . , SY n S ] ∼ → S C [ Y , . . . , Y n ] S = S C [ Y , . . . , Y n ] S n S and let θ ′ : C [ SY − n S, . . . , SY − nn S ] ∼ → S C [ Y − , . . . , Y − n ] S = S C [ Y − , . . . , Y − n ] S n S be defined in a similar fashion. Equation (2.17) is a consequence of the followingresult. Sublemma 2.5.
The composition u = θ ′ ◦ ρ ( A ) ◦ θ − satisfies u ( SP ( Y , . . . , Y n ) S ) = q l SP ( Y − , . . . , Y − n ) S for any symmetric polynomial P ( t , . . . , t n ) .Proof of sublemma. Let ˙ H + n,Y (resp. ˙ H − n,Y ) be the subalgebras generated by H n and the elements Y , . . . , Y n (resp. by H n and the elements Y − , . . . , Y − n ). Theassignment T i T n − i , Y i Y − n +1 − i gives rise to an isomorphism of algebrasΘ : ˙ H + n,Y ∼ → ˙ H − n,Y . It restricts to an isomorphism of spherical algebras S Θ : S ˙ H + n,Y ∼ → S ˙ H − n,Y . This last map clearly satisfies S Θ( SP ( Y , . . . , Y n ) S ) = SP ( Y − , . . . , Y − n ) S for any symmetric polynomial P ( t , . . . , t n ). It remains to observe that u coincideswith q l S Θ on the elements SY l S , and that these elements generate S ˙ H + n,Y S , sothat in fact u = q l S Θ. The sublemma is proved. ⊓⊔ Proposition 2.6.
The elements { P n x ; x ∈ Z , ∗ } generate S ¨ H n as a K ′ -algebra.Proof. As in the proof of Proposition 2.1, let S ¨ H A ,n be the integral form of S ¨ H n and let π : S ¨ H A ,n → R n be the specialization at q = v = 1. It is easy to see that π ( P n ( r,d ) ) = P i x ri y di . By Weyl’s theorem, the elements { π ( P n ( r,d ) ); ( r, d ) ∈ Z , ∗ } generate the ring R n . It follows that { P n ( r,d ) ; ( r, d ) ∈ Z , ∗ } generate the algebra S ¨ H n over the field K ′ . ⊓⊔ The projection map
Recall that K = C ( v, t ), K ′ = C ( v, q ), and A K = A A ⊗ K . The first mainTheorem of this paper is the following. Theorem 3.1.
For any n > , the assignment t qv − , ˜ t x d ( x ) ( q − d ( x ) v d ( x ) − v − d ( x ) ) P n x for x ∈ Z , ∗ extends to a surjective C -algebra homomorphism Ψ n : A K ։ S ¨ H n .Proof. Fix an integer n . For simplicity we drop the index n in all notations. Wehave to show that the elements [ d ( x )]( q − d ( x ) v d ( x ) − v − d ( x ) ) P n x /d ( x ) satisfy relationsi) and ii) of Section 1.4. Relation i) is clear for x = (0 , r ) , x ′ = (0 , r ′ ) with r, r ′ ofthe same sign, and follows from Lemma 2.4, (2.17) for r, r ′ of different signs. By ALL, CHEREDNIK, EISENSTEIN, MACDONALD 13 applying a suitable automorphism ρ g for g ∈ SL (2 , Z ), we deduce relation i) forany other line in Z through the origin.The proof of relation ii) is much more involved. We reduce it to two sets ofequalities (3.3) and (3.9) which are dealt with in Appendices A and B respectively.Note that the assignment Ψ respects the SL (2 , Z ) action on both sides. Hence itis enough to check the relation ii) for one pair in each orbit under this SL (2 , Z )-action. By the same argument (based on Pick’s formula) as in [BS], Theorem 5.1,we may reduce ourselves to the cases where x = (1 , , y = (0 , l ) with l ∈ Z ∗ , or x = (0 , , y = ( l, −
1) with l ≥ Case a1.
Assume that x = (1 ,
0) and y = (0 , l ) with l >
0. We have to show that(3.1) [Ψ( t (1 , ) , Ψ( t (0 ,l ) )] = − c l Ψ( t (1 ,l ) ) , which we may rewrite as(3.2) [ l ] l ( q − v − v − )( q − l v l − v − l )[ P (1 , , P (0 ,l ) ] = − c l ( q − v − v − ) P (1 ,l ) . By Lemma 2.3, we have P (1 , = q [ n ] + SX S, P (0 ,l ) = X i SY li S, P (1 ,l ) = q [ n ] + SX Y l S. Using this and the identity [ l ](1 − q l )( q − l v l − v − l ) /l = − c l , we see that (3.2) isequivalent to the following proposition. Proposition 3.2.
For any l > we have (3.3) (cid:20) SX S, X i SY li S (cid:21) = S (cid:20) X , X i Y li (cid:21) S = (1 − q l ) SX Y l S. Case a2.
Let us now assume that x = (1 ,
0) and y = (0 , − l ) with l >
0. We haveto show that(3.4) [Ψ( t (1 , ) , Ψ( t (0 , − l ) )] = c l Ψ( t (1 , − l ) ) , which, after using the definitions and Lemmas 2.3 and 2.4, reduces to(3.5) (cid:20) SX S, X i SY − li S (cid:21) = (1 − q − l ) SX Y − l S. Consider the C -algebra isomorphism σ : ¨ H n → ¨ H n given by T i T − i , X i Y i , Y i X i , v v − , q q − , see [C]. Applying σ to (3.5) gives the equation (cid:20) SY S, X i SX − li S (cid:21) = (1 − q l ) SY X − l S, which, once transformed by the automorphism ρ ( A ), A = (cid:16) − (cid:17) , is nothingelse than (3.3). Thus this case also follows from Proposition 3.2. Case b.
The final case to consider is that of x = (0 ,
1) and y = ( − , l ) with l > t (1 , ) , Ψ ( t ( − ,l ) )] = c Ψ( θ (0 ,l ) ) v − − v , which reduces to(3.7) (1 − v n )(1 − qv − ) q − (cid:20) X i SY i S, SX l Y − S (cid:21) = Ψ( θ (0 ,l ) ) . Forming a generating series, we may write this as(3.8) 1 + X l ≥ (1 − v n )(1 − qv − ) q − S (cid:20) X i Y i , X l Y − (cid:21) Ss l = 1 + X l ≥ Ψ( θ (0 ,l ) ) s l . Given the definition of θ (0 ,l ) , we finally obtain that (3.6) for all l > Proposition 3.3.
The following holds : exp X l ≥ ( v − l − v l )( v l − q l v − l ) l X i SX li Ss l == 1 + X l ≥ (1 − qv − )(1 − v n ) q − S (cid:20) X i Y i , X Y − (cid:21) Ss l . (3.9)This completes the proof of the Theorem. ⊓⊔ Stable limits of DAHA’s
In considering stable limits of DAHA’s we will be concerned with the gradedsubalgebras S ¨ H + m , S ¨ H ++ m of S ¨ H m generated by the elements P m x for x ∈ Z , + , Z , ++ respectively. We have S ¨ H + m = M x ∈ Z , + S ¨ H + m [ x ] , S ¨ H ++ m = M x ∈ Z , ++ S ¨ H ++ m [ x ] . Proposition 4.1.
The assignment P m x P m − x for each x ∈ Z , + extends to aunique K ′ -algebra morphism Φ m : S ¨ H + m ։ S ¨ H + m − . A similar statement holds for S ¨ H ++ m .Proof. The proof is based on the realization of Cherednik algebras as certain alge-bras of difference operators. Let D m stand for the algebra of q -difference operators K ′ [ x ± , ∂ ± i , . . . , x ± m , ∂ ± m ] with defining relations[ x i , x j ] = [ ∂ i , ∂ j ] = 0 , ∂ i x j = q δ ij x j ∂ i . We also denote by D m,loc the localization of D m with respect to the elements { x i − v l q n x j ; l, n ∈ Z , i, j = 1 , . . . , m, } . The symmetric group S m acts in anobvious fashion on D m,loc and we may form the semidirect product D m,loc ⋊ S m .The following lemma is due to Cherednik. Lemma 4.2 (Cherednik, [C]) . Set ω = s m − · · · s ∂ . There is a unique embeddingof algebras ϕ m : ¨ H m ֒ → D m,loc ⋉ S m satisfying ϕ m ( X i ) = x i ,ϕ m ( T i ) = vs i + v − v − x i /x i +1 − s i − ,ϕ m ( Y i ) = ϕ m ( T i ) · · · ϕ m ( T m − ) ωϕ m ( T − ) · · · ϕ m ( T − i − ) . It is known that ϕ m ( S ¨ H m ) ⊂ D S m m,loc ⋊ S m . Composing the restriction of ϕ m to S ¨ H m with the projection D S m m,loc ⋊ S m → D S m m,loc , P ( x ± i , ∂ ± i ) σ P ( x ± i , ∂ ± i ) ALL, CHEREDNIK, EISENSTEIN, MACDONALD 15 provides us with an embedding ψ m : S ¨ H m ֒ → D S m m,loc . Set D ++ m,loc equal to thealgebra K ′ [ x , ∂ , . . . , x m , ∂ m ] loc . Lemma 4.3.
We have ψ m ( S ¨ H ++ m ) ⊂ ( D ++ m,loc ) S m .Proof. It is easy to see that A ++ K is generated by { t (0 ,l ) , t ( l, ; l ≥ } . Hence, bytheorem 3.1, S ¨ H ++ m is generated by { P m (0 ,l ) , P m ( l, ; l ≥ } , and it suffices to checkthe veracity of the lemma for these elements, for which it is obvious. ⊓⊔ We now consider the map π m : ( D ++ m,loc ) S m → ( D m − ,loc ) S m − defined by sending x l , ∂ l to x l , v − ∂ l if l < m , and x m , ∂ m to zero. This is a well-defined algebra homomorphism. We may summarize the situation in the followingdiagram of algebra homomorphisms S ¨ H ++ m ψ m / / ( D ++ m,loc ) S m π m (cid:15) (cid:15) S ¨ H ++ m − ψ m − / / ( D m − ,loc ) S m − in which ψ m and ψ m − are embeddings. Therefore, Proposition 4.1 will be provedfor the algebra S ¨ H ++ m once we show that(4.1) π m ◦ ψ m ( P m x ) = ψ m − ( P m − x ) , x ∈ Z , ++ . Lemma 4.4.
For any x ∈ Z , ++ there exists a polynomial Q x ∈ F ∞ such that thefollowing formula holds in S ¨ H m for any mQ x ( P m (0 , , P m (0 , , . . . , P m (1 , , P m (2 , , . . . ) = P m x . Proof.
Since A ++ K is generated by the elements t (0 ,l ) , t ( l, for l ≥ x ∈ Z , ++ , a polynomial R x such that R x ( t (0 , , t (0 , , . . . , t (1 , , t (2 , , . . . ) = t x . By theorem 3.1, we may take as Q x the polynomial defined by Q x ( u (0 , ,u (0 , , . . . , u (1 , , u (2 , , . . . )= 1 o ( d ( x )) R x ( o (1) u (0 , , o (2) u (0 , , . . . , o (1) u (1 , , o (2) u (2 , , . . . )where we have set o ( l ) = ( q − l v l − v − l ) /l . ⊓⊔ Lemma 4.4 implies that it is enough to show that (4.1) holds for x = ( l, x = (0 , l ). This is obvious by definition for x = ( l, π m ◦ ψ m ( Sf r ( Y , . . . , Y m ) S ) = ψ m − ( Sf r ( Y , . . . , Y m − , S ) for any family of symmetric polynomials { f r } whichgenerates the ring C [ Y , . . . , Y m ] S m . We may in particular take for f r the monomialsymmetric function m r ( Y , . . . , Y m ) = X ≤ i < ···
1, give rise in the limit tosome elements P x of these projective limits. Let S ¨ H + ∞ , S ¨ H ++ ∞ stand for the sub-algebras generated by P x for x ∈ Z , + , Z , ++ respectively. We may view lim ←− S ¨ H + m and lim ←− S ¨ H ++ m as some completions of S ¨ H + ∞ and S ¨ H ++ ∞ .By construction the map Ψ m : A K ։ S ¨ H m sends A + K and A ++ K onto S ¨ H + m and S ¨ H ++ m respectively. Let us call Ψ + m and Ψ ++ m the restrictions of Ψ m to A + K and A ++ K . The collection of maps Ψ + m and Ψ ++ m give rise, in the limit, to algebrahomomorphisms Ψ + ∞ : A + K → S ¨ H + ∞ , Ψ ++ ∞ : A ++ K → S ¨ H ++ ∞ . Theorem 4.6.
The maps Ψ + ∞ and Ψ ++ ∞ are isomorphisms.Proof. Both Ψ + ∞ and Ψ ++ ∞ are surjective by construction and we have to show theirinjectivity. The subgroup G = (cid:26)(cid:18) n (cid:19) ; n ∈ Z (cid:27) ⊂ SL (2 , Z ) ALL, CHEREDNIK, EISENSTEIN, MACDONALD 17 preserves Z , + and for any x ∈ Z , + there exists g ∈ G such that g · x ∈ Z , ++ .Since the map Ψ + ∞ is clearly compatible with the action of G on A + K and S ¨ H + ∞ , wesee that it is in fact enough to prove the injectivity of Ψ ++ ∞ .Now fix ( r, d ) ∈ Z , ++ . By Section 1.5., the dimension of the weight space A ++ K [ r, d ] is equal to the number of convex paths p = ( x , . . . , x r ) for which x i ∈ Z , ++ for all i and for which P i x i = ( r, d ). By Proposition 2.1, the dimension ofthe weight space S ¨ H ++ n [ r, d ] is equal to the dimension of the space of polynomialdiagonal invariants R ++ n = C [ x , . . . , x n , y , . . . , y n ] S n of x -degree r and y -degree d .The latter dimension is equal to the number of orbits under S n of monomials x g · · · x g n n y h · · · y h n n with g i , h i ∈ N satisfying P i g i = r and P i h i = d ; equiva-lently, it is equal to the number of n -tuples of pairs { ( g , h ) , . . . , ( g n , h n ) } satisfyingagain P i g i = r and P i h i = d , or to the number of convex paths p = ( x , . . . , x r )in Z , ++ of length r ≤ n satisfying P i x i = ( r, d ). It remains to observe that forany given ( r, d ), the length of convex paths p = ( x , . . . , x r ) in Z , ++ for which P i x i = ( r, d ) is bounded above by, say, n ( r, d ). Hencedim A ++ K [ r, d ] = dim S ¨ H ++ n [ r, d ]whenever n ≥ n ( r, d ), and finallydim A ++ K [ r, d ] = dim S ¨ H ++ ∞ [ r, d ] . The injectivity of the map Ψ ++ ∞ follows. Theorem 4.6 is proved. ⊓⊔ Remark.
Theorem 4.6 allows us to transport the PBW basis { β p ; p ∈ Conv + } and the canonical basis { b p ; p ∈ Conv + } of A + K defined in [S, Section 2.3] to bases { γ p ; p ∈ Conv + } and { c p ; p ∈ Conv + } of S ¨ H + ∞ such that γ p = Ψ + ∞ ( β p ) and c p = Ψ + ∞ ( b p ). The element c p belongs to the completion d S ¨ H + ∞ of S ¨ H + ∞ equal tothe sum L ( r,d ) d S ¨ H + ∞ [ r, d ] over all couples ( r, d ) ∈ Z , + of the vector spaces d S ¨ H + ∞ [ r, d ] = Y p K ′ γ p . Here p runs among all paths p = ( x , . . . , x r ) in Conv + satisfying P i x i = ( r, d ). Theorem 4.6 should be put in perspective with the theory of the classical Hallalgebra H cl of a discrete valuation ring O . See [M1, Chap.II]. Recall that H cl iscanonically isomorphic to the algebra Λ + v , and that this isomorphism naturally fitsin a chain(4.2) H cl ≃ S ˙ H + ∞ ≃ Λ + v where S ˙ H + ∞ is the stable limit of the positive spherical affine Hecke algebra of type GL ( n ) as n tends to infinity. Hence Theorem 4.6 may be interpreted as an affineversion of (4.2). Observe that S ˙ H + ∞ is a trivial one-parameter deformation Λ + v ofΛ + , while S ¨ H + ∞ is a nontrivial two-parameter deformation of the ring R + = C [ x , x , . . . , y ± , y ± , . . . ] S ∞ . The analogy may be summarized in the following table.
Classical Hall algebra H cl Elliptic Hall algebra H el = A + K O - Mod
Coh ( X )Λ + = C [ x , x , . . . ] S ∞ R + = C [ x , x , . . . , y ± , y ± , . . . ] S ∞ Θ + ∞ : H cl ∼ → S ˙ H + ∞ Ψ + ∞ : H el ∼ → S ¨ H + ∞ Π = F n ( Z + ) n / S n Conv + = F n ( Z , + ) n / S n O λ = v − n ( λ ) P λ ( v ) PBW-basis β p N n , n ∈ N Coh r,d ( X ) , ( r, d ) ∈ Z , + F n P erv GL ( n ) ( N n ) F r,d U r,d IC ( O λ ) , λ ∈ Π P p , p ∈ Conv + Θ + ∞ ( tr ( IC ( O λ )) = s λ Ψ + ∞ ( tr ( P p )) = c p K λ,µ ( v ) ∈ N [ v ] k p , q ∈ N [ v, − t ± ]Affine Grassmanian c Gr ??Geometric Satake isomorphism ?? F n P erv GL ( n ) ( N n ) ≃ Rep + GL ( ∞ )Here P λ is the Hall-Littlewood polynomial, s λ is the Schur polynomial, and K λ,µ is the Kostka polynomial.The second part of the table is based on the geometric version of the ellipticHall algebra which involves the theory of automorphic sheaves defined in [La] andstudied in details for an elliptic curve in [S]. We refer to that paper for notations.Lastly, in the third part of the table we mention two important features of theclassical picture, for which we don’t know of any analog in the setting of the ellipticHall algebra : functions on the nilpotent cone N n may be lifted to functions on someSchubert variety of the affine Grassmanian c Gr of type GL ( n ), and the category ofperverse sheaves F n P erv GL ( n ) ( N n ) is equivalent to the category Rep + ( GL ( ∞ ))of finite-dimensional polynomial representations of GL ( ∞ ), see [Gi], [MV].5. Macdonald Polynomials
Macdonald discovered in the late 80’s in [M2] a remarkable family of symmetricpolynomials P λ ( q, t ) depending on two parameters, and from which many of theclassical symmetric functions may be obtained by specializations. We will use the ALL, CHEREDNIK, EISENSTEIN, MACDONALD 19 variable v rather than the conventional t to comply with the notation in force inthe rest of this paper.The Macdonald polynomials are defined as eigenfunctions of certain differenceoperators acting on the spaces of symmetric functionsΛ m ( q,v ) = K ′ [ x , . . . , x m ] S m . Recall the embedding ψ m : S ¨ H + m ֒ → D S m m,loc , which gives rise to an action ρ m of S ¨ H + m on Λ m ( q,v ) . Consider the following linear operator on Λ m ( q,v ) D m = ρ m ( S ( Y + · · · + Y m ) S ) = m X i =1 (cid:18) Y j = i vx i − v − x j x i − x j (cid:19) ∂ i . By [M1], VI, (3.10) the operator D m is upper triangular with respect to the basis { m λ } of monomial symmetric functions and has distinct eigenvalues.We are interested in the stable limit as m goes to infinity of the correspondingeigenfunctions. Let θ m : Λ m ( q,v ) → Λ m − q,v ) be the specialization x m = 0. It is not truethat θ m ◦ D m = D m − ◦ θ m . However, the operator E m = v − m ( D m − [ m ]) doessatisfies θ m ◦ E m = E m − ◦ θ m . Recall that the spaceΛ ( q,v ) = K ′ [ x , x , . . . ] S ∞ of symmetric functions is the projective limit of (Λ m ( q,v ) , θ m ) in the category ofgraded rings. See [M1], Remark 1.2.1. Hence the operators E m , m ≥
1, give riseto a linear operator E on the space Λ ( q,v ) . This operator is still upper triangularwith respect to the basis { m λ } and has distinct eigenvalues { α λ } given by(5.1) α λ = X i ≥ ( q λ i − v − i − . The Macdonald polynomial is defined to be the unique α λ -eigenvector of E suchthat P λ ( q, v ) ∈ m λ ⊕ M µ<λ K ′ m µ . For a pair of partitions µ ⊂ λ , the skew Macdonald polynomial P λ/µ ( q, v ) isdetermined by the coproduct formula∆( P λ ( q, v )) = X µ ⊂ λ P µ ( q, v ) ⊗ P λ/µ ( q, v ) . Examples. i) We have P (1 r ) ( q, v ) = e r .ii) We have P ( r ) ( q, v ) = r − Y l =0 − q l +1 − v q l · X λ ⊢ r z λ ( q, v ) − p λ , where z λ ( q, v ) = z λ l ( λ ) Y i =1 − q λ i − v λ i , z (1 m m ··· ) = Y i i m i m i ! . In particular, P (2) ( q, v ) = (1 − q )(1 + v )2(1 − qv ) p + (1 + q )(1 − v )2(1 − qv ) p . Remark.
The representations ρ m : S ¨ H + m → End(Λ m ( q,v ) ) lift, after a suitablerenormalization, to a stable limit representation ρ ∞ : S ¨ H ∞ → End(Λ ( q,v ) ) in which P (0 , = S ( P i Y i ) S acts as the operator E . Composing with the isomorphismΨ + ∞ : A + K ≃ S ¨ H + ∞ we obtain a representation of the Hall algebra A + K on Λ ( q,v ) inwhich the element t (0 , / c , i.e., the so-called Hecke operator , acts as Macdonald’soperator E/ ( q − There are many different characterizations of Macdonald polynomials. See[Hai] for instance. The one which fits our needs best treats the polynomials P λ ( q, v )and P λ/µ ( q, v ) at the same time. We first recall some standard notations from [M1].Let µ ⊂ λ be two partitions. Put | λ/µ | = | λ | − | µ | . The skew partition λ/µ issaid to be a vertical strip if λ i − µ i ≤ i , i.e., if the corresponding diagramcontains at most one box per row. A skew partition λ/µ is a horizontal strip if itsconjugate λ ′ /µ ′ is a vertical strip. If λ/µ is a horizontal strip, we put ψ λ/µ ( q, v ) = Y (1 − v µ ′ i − µ ′ j ) q j − i − )(1 − v µ ′ i − µ ′ j − q j − i +1 )(1 − v µ ′ i − µ ′ j ) q j − i )(1 − v µ ′ i − µ ′ j − q j − i ) , where the sum ranges over all pairs ( i, j ) with i < j such that µ ′ i = λ ′ i but µ ′ i = λ ′ i − ψ λ/µ ( q, v ) = 1 if λ/µ is a horizontal strip containing noempty columns. Proposition 5.1.
The family { P λ/µ ( q, v ); µ ⊂ λ } is uniquely determined by thefollowing set of properties : i) P λ/µ ( q, v ) is homogeneous of degree | λ/µ | , ii) we have ∆( P λ/µ ( q, v )) = X µ ⊆ ν ⊆ λ P ν/µ ( q, v ) ⊗ P λ/ν ( q, v ) , iii) if λ/µ is not a horizontal strip then P λ/µ ( q, v ) ∈ M ν< ( r ) K ′ m ν , where r = | λ/µ | , iv) if λ/µ is a horizontal strip then P λ/µ ( q, v ) ∈ ψ λ/µ ( q, v ) m r ⊕ M ν< ( r ) K ′ m ν , where r = | λ/µ | .Proof. Properties i) through iv) are all known to hold for Macdonald polynomials:statement ii) follows from [M1, VI.7, (7.9’)], while statements iii) and iv) are conse-quences of [M1, VI.7, (7.13’)]). We now prove the unicity of polynomials satisfyingi) through iv). Let Q λ/µ ( q, v ) be such a family. When | λ/µ | = 1 we have, by iv) Q λ/µ ( q , v ) = ψ λ/µ ( q, v ) m = P λ/µ ( q, v ) . Let r > Q η/ν ( q, v ) = P η/ν ( q, v ) for all η/ν satisfying | η/ν | < r .Let λ/µ be a skew partition with | λ/µ | = r . By ii) and the induction hypothesis∆( Q λ/µ ( q, v )) == Q λ/µ ( q, v ) ⊗ ⊗ Q λ/µ ( q, v ) + X µ ⊂ ν ⊂ λ Q ν/µ ( q, v ) ⊗ Q λ/ν ( q, v )= Q λ/µ ( q, v ) ⊗ ⊗ Q λ/µ ( q, v ) + X µ ⊂ ν ⊂ λ P ν/µ ( q, v ) ⊗ P λ/ν ( q, v ) . ALL, CHEREDNIK, EISENSTEIN, MACDONALD 21
It follows that Q λ/µ ( q, v ) − P λ/µ ( q, v ) is contained inKer (cid:0) ∆ − Id ⊗ − ⊗ Id (cid:1) = K ′ p | λ/µ | . But then the coefficient of p | λ/µ | in Q λ/µ ( q, v ) is uniquely determined by iii) oriv), and Q λ/µ ( q, v ) = P λ/µ ( q, v ). ⊓⊔ Eisenstein series
We return to the setting of Section 1, i.e., X is a smooth elliptic curve over F l , H X is its Hall algebra, and U X ⊂ H X is the subalgebra introduced in Section 1.4.For simplicity, we drop the exponent in U + X . Recall that H X and U X are Z -gradedin the following way : H X = M ( r,d ) H X [ r, d ] , U X = M ( r,d ) U X [ r, d ] . The Eisenstein series which we will need to consider are certain elements of acompletion of the Hall algebra, which we now define in details. Let b H X [ r, d ] standfor the space of all functions f : I ( X ) r,d → C on the set of coherent sheaves of rank r and degree d , and put b H X = L ( r,d ) b H X [ r, d ]. By [BS, Proposition 2.1], the space b H X is still a bialgebra. Recall, see Section 1.3, that as a vector space we have H X [ r, d ] = M α ,...,α n H ( µ ( α )) X [ α ] ⊗ · · · H ( µ ( α n )) X [ α n ] , where the sum ranges over all tuples ( α , . . . , α n ) of elements in Z , + satisfying µ ( α ) < · · · < µ ( α n ) and P α i = ( r, d ). Then b H X [ r, d ] is simply b H X [ r, d ] = Y α ,...,α n H ( µ ( α )) X [ α ] ⊗ · · · H ( µ ( α n )) X [ α n ] . In a similar fashion, we define the subalgebra b U X of b H X as b U X = L ( r,d ) b U X [ r, d ]where b U X [ r, d ] = Y α ,...,α n U ( µ ( α )) X [ α ] ⊗ · · · U ( µ ( α n )) X [ α n ] . For instance, for any ( r, d ) the element ( r,d ) = X FF =( r,d ) F is a function with infinite support belonging to b U X [ r, d ] since it may be written asthe infinite sum (see [BS, Equation (4.4)])(6.1) r,d = ssr,d + X µ ( α ) < ··· <µ ( α n ) α + ··· + α n =( r,d ) ν P i Theorem 6.1 (Harder, [Har]) . The series E r ,...,r n ( z , . . . , z n ) converges in theregion | z | ≪ · · · ≪ | z n | to a rational function in b U X ( z , . . . , z n ) with at mostsimple poles along the hyperplanes z i /z j ∈ { , ν , . . . , ν r } where r = P r i . In other words, for each F the series (6.2) is the expansion in the region | z | ≪· · · ≪ | z n | of some rational function in the variables z , . . . , z n . When r = · · · = r n = 1 the series E ,..., ( z , . . . , z n ) is the Eisenstein series attached to the cuspform of rank one corresponding to the trivial character P ic ( X ) → C ∗ taken n times. See [K1, Section (2.4)] for details. For other values of r , . . . , r n the series E r ,...,r n ( z , . . . , z n ) is the Eisenstein series attached to the trivial character of theparabolic subgroup GL r ( k X ) × · · · × GL r n ( k X ) of GL P r i ( k X ), where k X is thefunction field of X .The Eisenstein series behave well with respect to the coproduct. Proposition 6.2. For nonnegative integers r , . . . , r n we have ∆( E r ,...,r n ( z , . . . , z n )) == X ≤ s i ≤ r i E s ,...,s n ( z , . . . , z n ) ⊗ E r − s ,...,r n − s n ( ν s z , . . . , ν s n z n ) . In particular, we have ∆( E r ( z )) = r X s =0 E s ( z ) ⊗ E r − s ( ν s z ) . Proof. This is a consequence of the fact that b U X is a bialgebra and that, by [BS,Equation (4.5)] we have∆( ( r,d ) ) = X r + r = rd + d = d ν r d − r d ( r ,d ) ⊗ ( r ,d ) . ⊓⊔ One of the most crucial properties of Eisenstein series for us is the fact thatthey are eigenvectors for the adjoint action of the element T (0 , = P x ∈X ( F l ) O x , ALL, CHEREDNIK, EISENSTEIN, MACDONALD 23 and more generally of the elements T (0 ,d ) for d ≥ 1. These are the so-called Heckeoperators in the theory of automorphic forms on function fields. Let ζ ( z ) = (1 − σz )(1 − σz )(1 − z )(1 − ν − z )be the zeta function of X . Theorem 6.3. For any r ≥ the following holds : (6.3) [ T (0 , , E r ( z )] = ν X ( F l ) ν − r − ν − − z − E r ( z ) , (6.4) E ( z ) E r ( z ) = r − Y i =0 ζ (cid:18) ν − i z z (cid:19) · E r ( z ) E ( z ) . In particular, we have E ( z ) E ( z ) = ζ (cid:16) z z (cid:17) E ( z ) E ( z ) .Proof. Both statements are well-known (maybe in a different form) in the theoryof automorphic forms. For the reader’s convenience, we have included a proof inthe spirit of Hall algebras in the Appendix C. ⊓⊔ We finish with the so-called functional equation for rank one Eisenstein series. Theorem 6.4 (Harder, [Har]) . The rational function E ,..., ( z , . . . , z n ) is sym-metric in variables z , . . . , z n . Remark. Strictly speaking, the Eisenstein series most often considered in the the-ory of automorphic forms are given by expressions like (6.2) but in which onerequires in addition each factor F i / F i − to be a vector bundle. In other words, ifone sets vec ( r,d ) = X V vec. bdle V =( r,d ) V , E vecr ( z ) = X d ∈ Z vec ( r,d ) ν ( r − d z d then the corresponding product would be E vecr ,...,r n ( z , . . . , z n ) = E vecr ( z ) · · · E vecr n ( z n ) . The two series, when restricted to vector bundles, are related by a global rationalfactor. It is the so-called L -factor . Indeed there is an obvious factorisation E r ( z ) = E vecr ( z ) E ( ν r z ) . Therefore, by Theorem 6.3 we have, after restricting to the set of vector bundles E r ,...,r n ( z , . . . , z n ) = L r ,...,r n ( z , . . . , z n ) E vecr ,...,r n ( z , . . . , z n ) , where L r ,...,r n ( z , . . . , z n ) = Y i 6∈ {L , L ′ }{ } if d < . Hence we get(6.6) E , ( z , z )( F ) = z z (1 + ν − ) X ( F l )( z − ν − z )( ν − z − z ) + 2 . From (6.5) and (6.6) we deduce that the semistable component of E , ( z , z ) isequal to E , ( z , z ) (0) = z z (1 + ν − ) X ( F l )( z − ν − z )( ν − z − z ) ( T (2 , [2] + T , ) + T , . Finally, to compute the unstable component of E , ( z , z ) we use the coprod-uct. Observe that since Ext( L − d , L d ) = { } the component of bidegree (1 , − d ),(1 , d ) of ∆( L − d ⊕L d ) is equal to ν d L − d ⊗ L d , and no other term may contribute to L − d ⊗ L d . Hence E , ( z , z )( L − d ⊕ L d ) = ν − d ∆( E , ( z , z ))( L − d , L d )By Proposition 6.2 and Theorem 6.3 we have∆ , ( E , ( z , z )) == E ( z ) E ( z ) ⊗ E ( z ) E ( ν z ) + E ( z ) E ( z ) ⊗ E ( ν z ) E ( z )= ζ (cid:18) z z (cid:19) E ( z ) E ( z ) ⊗ E ( z ) E ( ν z ) + ζ (cid:18) z z (cid:19) E ( z ) E ( z ) ⊗ E ( z ) E ( ν z ) ALL, CHEREDNIK, EISENSTEIN, MACDONALD 25 from which we eventually obtain E , ( z , z )( L − d ⊕ L d ) = ν − d (cid:20) ζ (cid:18) z z (cid:19) z d z − d + ζ (cid:18) z z (cid:19) z − d z d (cid:21) . We have so far considered the Eisenstein series E r ,...,r n ( z , . . . , z n ) for a fixed elliptic curve X only. Recall from Section 1.5 that there exists an algebra A + A defined over the ring A = C [ v ± , t ± ] whose specialization at v = ν = l − / and t = σν for any X is isomorphic to U + X . Using the formulas (1.3) and (6.1) we seethat the generating series E r ( z ) and hence the Eisenstein series E r ,...,r n ( z , . . . , z n )may naturally be lifted to elements A E r ( z ) ∈ b A + A [[ z, z − ]] , A E r ,...,r n ( z , . . . , z n ) ∈ b A + A [[ z ± , . . . , z ± n ]] . Proposition 6.5. The series A E r ,...,r n ( z , . . . , z n ) converges in the region | z | ≪· · · ≪ | z n | to a rational function in b A + A ( z , . . . , z n ) with at most simple poles alongthe hyperplanes z i /z j ∈ { , v , . . . , v r } , where r = P r i .Proof. The coefficient of A E r ,...,r n ( z , . . . , z n ) on any basis element of A + A is givenby a Laurent series of the form P ( z , . . . , z n ) X d ,...,d n ≥ α d ,...,d n (cid:18) z z (cid:19) d · · · (cid:18) z n − z n (cid:19) d n , where P ( z , . . . , z n ) ∈ A [ z ± , · · · , z ± nn ] and α d ,...,d n ∈ A . By Harder’s Theorem,the evaluation at v = ν and t = σν for any elliptic curve X of the expression (cid:18) r Y l =1 Y i,j ( z i − v − l z j ) (cid:19) · P ( z , . . . , z n ) X d ,...,d n α d ,...,d n (cid:18) z z (cid:19) d · · · (cid:18) z n − z n (cid:19) d n is a Laurent polynomial (of fixed degree). This is equivalent to the vanishing ofcertain A -linear combinations of the α d ,...,d n ’s. Of course, if such a linear combi-nation vanishes when evaluated at all (i.e., infinitely many) elliptic curves X thenit must already vanish in A . We are done. ⊓⊔ Motivated by the analogy between the Hecke operator T (0 , and Macdonald’soperator (see Section 5.1 and Remark 5.1) we introduce, for every partition λ =( λ , . . . , λ n ) the following specialization of Eisenstein series E λ ( z ) = A E λ ,...,λ n ( z, q − z, . . . , q − n z )where q = vt . By Proposition 6.5, the line ( z, q − z, . . . , q − n z ) is not contained inthe pole locus of A E λ ,...,λ n ( z , . . . , z n ) and hence E λ ( z ) belongs to b A A ⊗ A K ( z ).More generally, for any pairs of partitions µ ⊂ λ we put E λ/µ ( z ) = A E λ − µ ,...,λ n − µ n ( v µ z, v µ q − z, . . . , v µ n q − n z ) . Observe that by Theorem 6.3 the series E λ ( z ) are eigenvectors for the adjoint actionof the Hecke operator T (0 , , whose eigenvalues β λ are (up to a global factor) equalto that of the Macdonald polynomials, namely we have β λ = z − c ( v, t ) X i v − λ i − v − − q i − = z − c ( v, t ) q − α λ ′ where α λ ′ is given by formula (5.1) and λ ′ is the conjugate partition to λ . It would seem natural to define more generally the specialization E l ( z ) = A E l ,...,l n ( z, q − z, . . . , q − n z )for any sequence of nonnegative integers l , . . . , l n . However we have the followingvanishing result. Lemma 6.6. If l = ( l , . . . , l n ) is not dominant, i.e., if l k > l k − for some k , then E l ( z ) = 0 .Proof. One may check that the L -factor L l ,...,l n ( z , . . . , z n ) vanishes on the line( z, q − z, . . . , q − n z ) whenever l is not dominant. Hence Lemma 6.6 would followfrom the fact that the unnormalized Eisenstein series E vecl ,...,l n ( z , . . . , z n ) is regularon that line. Rather than appealing to this fact, we provide a direct proof. Tounburden the notation, we drop the subscript A throughout. By Proposition 6.2we have ∆ (1 ,..., ( E r ( z )) = E ( z ) ⊗ E ( v z ) ⊗ · · · ⊗ E ( v r − z ) , and more generally given integers ǫ k ∈ { , } with ǫ i = · · · = ǫ i r = 1 while ǫ k = 0if k 6∈ { i , . . . , i r } we have(6.7) ∆ ( ǫ ,...,ǫ n ) ( E r ( z )) = E ǫ ( z ) ⊗ · · · ⊗ E ǫ k ( v s k z ) ⊗ · · · ⊗ E ǫ n ( v s n z )where s k = { l ; i l < k } . Now let l = ( l , . . . , l n ) ∈ N n and set l = P l i . We maycompute ∆ (1 ,..., ( E l ( z )) using (6.7). It is equal to a sum, indexed by the set ofmaps φ : { , . . . , l } → { , . . . , n } of terms a φ = ∆ ( ǫ ,...,ǫ n ) ( E l ( z )) · · · ∆ ( ǫ k ,...,ǫ kn ) ( E l k ( q − k z )) · · · ∆ ( ǫ n ,...,ǫ nn ) ( E l n ( q − n z )) , where ǫ ki ∈ { , } is defined by ǫ ki = ( if φ ( i ) = k, if φ ( i ) = k. In other terms, the map φ describes the way the coproducts (6.7) of the E l k ( q − k z )have been distributed among the l components of the tensor product. We claimthat if l is not dominant then each term a φ vanishes. Indeed, suppose that l k > l k − for some k . Then a φ is divisible by a term of the form∆ ( ǫ ,...,ǫ n ) ( E l k − ( q − k z )) · ∆ ( ǫ ′ ,...,ǫ ′ n ) ( E l k ( q − k z )) == E ǫ ( q − k z ) E ǫ ′ ( q − k z ) ⊗ · · · ⊗ E ǫ n ( v s n q − k z ) E ǫ ′ n ( v s ′ n q − k z ) . (6.8)Of course, in the above if ǫ i = 1 then ǫ ′ i = 0 and vice versa. As s = s ′ = 0 while s n ∈ { l k − , l k − − } and s ′ n ∈ { l k , l k − } it is easy to see that there exists an index j for which ǫ j = 0 , ǫ ′ j = 1 and s j = s ′ j . But then the j th component of (6.8) isequal to E ( v s j q − k z ) E ( v s j q − k z ) = ζ ( q ) E ( v s j q − k z ) E ( v s j q − k z ) = 0since ζ ( q ) = 0. Hence a φ = 0 as wanted and ∆ (1 ,..., ( E l ( z )) = 0. It remains toshow that the map∆ (1 ,..., : b U [ r, d ] → Y d + ··· + d r = d b U [1 , d ] ⊗ · · · ⊗ b U [1 , d r ]is injective. This in turn follows from the fact that b U is equipped with a nonde-generate Hopf pairing, and that it is generated by elements of degree zero and one,see [BS, Cor. 5.1]. ⊓⊔ ALL, CHEREDNIK, EISENSTEIN, MACDONALD 27 Proposition 6.7. For any partition λ we have (6.9) ∆( E λ ( z )) = X µ ⊆ λ E µ ( z ) ⊗ E λ/µ ( z ) . More generally, for any skew partition λ/µ we have (6.10) ∆( E λ/µ ( z )) = X µ ⊆ ν ⊆ λ E ν/µ ( z ) ⊗ E λ/ν ( z ) . Proof. We prove the first statement. By Proposition 6.2 it holds∆( E λ ( z, . . . , q − n z )) == X s ,...,s n ≤ s i ≤ λ i A E s ,...,s n ( z, . . . , q − n z ) ⊗ A E λ − s ,...,λ n − s n ( v s z, . . . , v s n q − n z ) . (6.11)By Lemma 6.6 we have A E s ,...,s n ( z, q − z, . . . , q − n z ) = 0 if ( s , . . . , s n ) is not apartition. Therefore the r.h.s. of (6.11) reduces to (6.9). The proof of the secondstatement of the Proposition is similar. ⊓⊔ Geometric construction of Macdonald polynomials In this section, we make explicit the link between Macdonald polynomials P λ ( q, v )and the Eisenstein series E λ ( z ). For any skew partition λ/µ we denote by E (0) λ/µ the restriction of E λ/µ ( z ) tothe set of semistable vector bundles of degree zero. Notice that by homogeneity thisis independent of z . This is therefore an element of the subalgebra A + , (0) A of theuniversal Hall algebra A + A generated by elements e t ( r, for r ≥ 0. See Section 1.5for details. By Proposition 1.3 this last subalgebra is canonically identified withthe algebra of symmetric functions Λ ( q,v ) , where we have set q = tv . Explicitly, theisomorphism is given by ˜ t ( r, = p r /r .For instance, from Example 6.3 we see that E (0)1 , = q − (1 + v − )(1 − v − q )(1 − q − )( q − − v − )( q − v − − (cid:18) p p (cid:19) + p = (1 + v − )( q − v − − q p v − − q + 1) v − − q p . We let ω stand for the standard involution on symmetric functions, defined by ω ( p r ) = ( − r − p r . We are now ready to state the second main Theorem of this paper. Theorem 7.1. For any partition λ we have E (0) λ = ωP λ ′ ( q, v ) , and for any skew partition λ/µ we have E (0) λ/µ = ωP λ ′ /µ ′ ( q, v ) . The rest of this section is devoted to the proof of this theorem. We will use thecharacterization of the polynomials P λ/µ ( q, v ) given in Proposition 5.1. It is clearfrom the definitions that ω ( E (0) λ ′ /µ ′ ) is of degree | λ/µ | . Property ii) of Proposition 5.1is shown for ω ( E (0) λ ′ /µ ′ ) in Proposition 6.7. Thus it only remains to check that thecoefficient of m r in ω ( E (0) λ ′ /µ ′ ) for r = | λ/µ | is given by Proposition 5.1, iii) and iv).To this aim we introduce the following family of elements in A + , (0) : g r = X λ ⊢ r z − λ Y i v − λ i − v λ i c λ i ( v, t ) t λ i = X λ ⊢ r z − λ Y i v − λ i − − q − r )(1 − ( qv − ) r ) p λ . Alternatively, these may be defined by the formula1 + X r> g r s r = exp (cid:18) ( v − − v ) X r ≥ t ( r, v r + v − r − t r − t − r s r (cid:19) . Recall that A + is equipped with a nondegenerate Hopf scalar product. By [BS,Lemma 5.2] it satisfies(7.1) h t ( r, , t ( s, i = δ r,s [ r ] X ( F l r ) r ( v − r − 1) = δ r,s c r ( v, t ) v − − v , so that, after identification with Λ ( q,v ) it reads(7.2) h p r , p s i = δ r,s r (1 − q − r )(1 − ( qv − ) r ) v − r − . Using [M1, Chap VI.2] we deduce that g r is dual to m r with respect to the basis { m λ } , i.e., that h g r , m r i = 1 , h g r , m λ i = 0 if | λ | = r and λ < ( r ) . Therefore the proof of Theorem 7.1 will be complete once we have shown that(7.3) h g r , ω ( E (0) λ ′ /µ ′ ) i = 0if λ/µ is not a horizontal strip while(7.4) h g r , ω ( E (0) λ ′ /µ ′ ) i = ψ λ/µ ( q, v )if λ ′ /µ ′ is a horizontal strip. As we will see, these equations essentially amount tocertain relations between the factors ψ λ/µ ( q, v ) and the L -factors appearing in theEisenstein series.Observe that as g r is itself semistable of degree zero, i.e., we have g r ∈ A + , (0) ,and the subalgebras A + , ( µ ) are all mutually orthogonal, we may as well replace E (0) λ ′ /µ ′ by E λ ′ /µ ′ ( z ) in equations (7.3) and (7.4). Note also that ω is an orthogonalinvolution for h , i . The basic idea is to find a factorization of g r and to use the Hopf propertyof the scalar product h , i to reduce (7.3) and (7.4) to a lower rank. Of course,since g r is dual to m r and m r is primitive, this is not directly feasable. However, itbecomes possible as soon as we step out of the subalgebra A + , (0) . More precisely,put g (1) r = [ t (0 , , g r ] and ωg (1) r = [ t (0 , , ωg r ] . Lemma 7.2. For any r ≥ we have ωg (1) r +1 = v c ( v, t ) [ t (1 , , ωg (1) r ] + ( v − − v ) ωg r t (1 , . ALL, CHEREDNIK, EISENSTEIN, MACDONALD 29 Proof. An essentially direct computation, based on the relation [ t ( s, , t ( u, ] = c u t ( s + u, for any s, u , yields(7.5) ωg (1) r = ( − r r X s =1 v s (1 − v − )( − s ωg r − s t ( s, . The recursion formula in the lemma is an easy consequence of (7.5). ⊓⊔ Lemma 7.3. For any skew partition λ/µ we have h ωg (1) r , E λ ′ /µ ′ ( z ) i = c ( v, t )1 − v (cid:0) X i ( v µ ′ i − v λ ′ i ) q − i z (cid:1) h ωg r , E λ ′ /µ ′ ( z ) i . Proof. Because h , i is a Hopf pairing we have h ωg (1) r , E λ ′ /µ ′ ( z ) i = h t (0 , · ωg r − ωg r · t (0 , , E λ ′ /µ ′ ( z ) i = h t (0 , ⊗ ωg r , ∆ ,r ( E λ ′ /µ ′ ( z )) i − h ωg r ⊗ t (0 , , ∆ r, ( E λ ′ /µ ′ ( z )) i . Using Proposition 6.2 the coproducts are computed to be∆ ,r ( E λ ′ /µ ′ ( z )) = E ( v µ ′ z ) · · · E ( v µ ′ n q − n z ) ⊗ E λ ′ /µ ′ ( z )= (1 + X i v µ ′ i − q − i zt (0 , + · · · ) ⊗ E λ ′ /µ ′ ( z )and ∆ r, ( E λ ′ /µ ′ ( z )) = E λ ′ /µ ′ ( z ) ⊗ E ( v λ ′ z ) · · · E ( v λ ′ n q − n z )= E λ ′ /µ ′ ( z ) ⊗ (1 + X i v λ ′ i − q − i zt (0 , + · · · ) . The lemma follows. ⊓⊔ We now proceed with the proof of (7.3) and (7.4). We argue by induction on | λ/µ | . Assume first that | λ/µ | = 1. This means that λ ′ i = µ ′ i for all i except for onevalue, say j , for which λ ′ j = µ ′ j + 1. Then on the one hand P λ/µ ( q, v ) = ψ λ/µ m = Y i For any r ≥ , the following identity holds over the field of rationalfunctions K ′ ( α , . . . , α r ) : r X i =1 α i = 1 v − − r X j =1 (cid:18) Y l = j ζ (cid:18) α l α j (cid:19) − Y l = j ζ (cid:18) α j α l (cid:19) (cid:19) · (cid:18) X l = j α l (cid:19) + r X j =1 (cid:18) Y l = j ζ (cid:18) α j α l (cid:19) (cid:19) α j . (7.7)Next, let us assume that λ ′ /µ ′ has exactly one part of length two and r − L -factor, we see that (7.3) isequivalent to h ωg r , E ( v µ ′ j q − j z ) · · · E ( v µ ′ k q − k z ) · · · E ( v µ ′ jr − q − j r − z ) i = 0 . Again, we claim that in fact h ωg r , E ( α ) · · · E ( α k ) · · · E ( α r − ) i = 0 for any α , . . . , α r − . This may be checked directly using (7.6).In all the remaining cases λ ′ /µ ′ has at least two parts of length at least two, i.e.,if λ ′ i > µ ′ i + 1 for more than one value of i . But then no sub skew partition of size r − λ ′ /µ ′ may be vertical. By the induction hypothesis this implies that allterms on the r.h.s. vanish and thus h ωg r , E λ ′ /µ ′ ( z ) i = 0 as wanted. Theorem 7.1 isproved. ⊓⊔ ALL, CHEREDNIK, EISENSTEIN, MACDONALD 31 Remarks. i) A factorization similar to (7.5) involving rank one difference operatorsin the context of Pieri rules for skew Macdonald polynomials appears in [BGHT].We thank Mark Haiman for this remark.ii) In addition to Macdonald’s operator ∆ , one defines an operator ∇ acting onsymmetric polynomials in Λ ( q,v ) (see [BGHT]), which has distinct eigenvalues andwhose eigenvectors are the Macdonald polynomials. Namely ∇ is defined by ∇ ( P λ ( q, v )) = v − n ( λ ) q n ( λ ′ ) P λ ( q, v ) . Our conventions, taken from [M1], differ slightly from [BGHT]. In our picture,this operator ∇ is simply given by the action of the element A ∈ SL (2 , Z ) byautomorphism on the Hall algebra, i.e., it is the tensor product with a line bundleof degree one. Thus we have ρ ( A )( E λ ( z )) = v − n ( λ ′ ) q n ( λ ) E λ ( z ) . iii) Laumon defined and studied in [La] a “geometric lift” of Eisenstein series tocertain perverse sheaves (or more precisely, constructible complexes) on the stacks Coh r,d ( X ) called Eisenstein sheaves . The Eisenstein series themselves are recov-ered from the Eisenstein sheaves via the faisceaux-function correspondence. In thespecial case of an elliptic curve simple Eisenstein sheaves are determined in [S]. Theconstruction of the (non simple) Eisenstein sheaves relevant to Macdonald polyno-mials may be easily translated from Theorem 3.1. Let us denote by ( Q p ) r,d thetrivial rank one constructible sheaf on Coh r,d ( X ), and let us consider the formalseries whose coefficients are semisimple constructible complexes E r ( q − l z ) = M d ∈ Z ( Q p ) r,d [( r − d ]( ld ) X z d . Here [ n ] is the standard shift of complexes and ( m ) X denotes the Tate twist by theFrobenius eigenvalue σ in H ( X , Q p ). Note that there is a choice of one Frobeniuseigenvalue σ involved here, but of course choosing the other eigenvalue σ wouldgive a similar result. Using the induction functor of Laumon [La] we may form theproduct E λ ( z ) = E λ ( z ) ⋆ E λ ( q − z ) · · · ⋆ E λ l ( q − l z ) . It is still a series with coefficients in semisimple constructible complexes. Thesewill usually be of infinite rank. Restricting to the open substack parametrizingsemistable sheaves of zero slope we finally obtain a semisimple constructible com-plex E (0) λ . Using [S], Proposition 6.1, one can show that the Frobenius eigenvalues of E λ ( z ) and E (0) λ all belong to v Z q Z . Hence the Frobenius trace T r ( E (0) λ ) is a Laurentseries in v and q . Recall that we have fixed an isomorphism C ≃ Q p . By Harder’stheorem the series T r ( E (0) λ ) converges (in a suitable domain) to E (0) λ and hence byTheorem 3.1 we have T r ( E (0) λ ) = ωP λ ′ ( q, v ) . iv) Pick a F l -rational closed point x ∈ X ( F l ). Let i : D x → X be the embeddingof the formal neighborhood of x in X . Given an ´etale coordinate at x we get anisomorphism D x ≃ Spec( F l (( ̟ ))), where ̟ is a formal variable. Thus the set ofisomorphism classes of torsion sheaves on D x is equal to the set of conjugacy classesof nilpotent matrices. Invariant functions on the nilpotent cone N d , d ≥ 1, arecanonically identified with elements of the ring Λ + of symmetric functions. Therestriction of coherent sheaves on X to D x yields a map I ( X ) ,d → ` d ′ ≤ d N d ′ . Itfactors to an algebra isomorphism Λ + ≃ U + , ( ∞ ) X . Fourier-Mukai transform yields an algebra isomorphism F M : U + , (0) X → U + , ( ∞ ) X . The composed map U + , (0) X → Λ + coincides with the isomorphism in Proposition 1.3.The involution ω in Theorem 7.1 can be removed as follows. We’ll give anotherisomorphism U + , ( ∞ ) X ≃ Λ + which takes the Laurent series F M ( E (0) λ ) to P λ ′ ( q, v ).Let X ( d ) be the d -th symmetric power of E , and g Coh ,d ( X ) be the stacks of flags M d → M d − → · · · M , where each M i is a coherent sheaf on X of length i . Consider the Cartesian square X d ι d / / r d (cid:15) (cid:15) g Coh ,d ( X ) π d (cid:15) (cid:15) X ( d ) ι ( d ) / / Coh ,d ( X )in which π d is the Springer map, r d is the ramified finite cover ( x , x , . . . x d ) x + x + · · · x d , and ι ( d ) takes a divisor D to the sheaf O D . According to Laumon,the complex F = R ( π d ) ∗ ( Q p ) is the intermediate extension of its restriction F | U d tothe dense open subset U d = ι ( d ) ( X ( d ) ). We have ι ∗ ( d ) F = ( r d ) ∗ ( Q p ) by base change.Thus the symmetric group S d acts on F | U d . For each irreducible character φ of S d let F φ be the intermediate extension of the constructible sheaf Hom S d ( φ, F | U d ).Each F φ is a simple constructible complex on Coh ,d ( X ). The representation ring of S d is canonically identified with a subring of Λ + . We claim that there is an uniqueisomorphism U + , ( ∞ ) X ≃ Λ + taking T r ( F φ ) to the symmetric function associated to φ . This is the map we want. APPENDICES Appendix A. Proof of Proposition 3.2. A.1. We start the proof of Proposition 3.2 with a sequence of lemmas. Lemma A.1. We have (A.1) S (cid:20) X , X i Y i (cid:21) S = (1 − q ) SX Y S. Proof. Using Lemma 2.2, equations (2.8) and (2.9), we get S X i Y i X S = SX ( X i ≥ Y i ) S + qv n − SX Y S + ( v − − SY X S + ( v − − v − ) SY X S + · · · + ( v − n − − v − n − ) SY X S = SX ( X i ≥ Y i ) S + qv n − SX Y S + qv n − ( v − n − − SX Y S = SX ( X i Y i X ) S + ( q − SX Y S. ⊓⊔ ALL, CHEREDNIK, EISENSTEIN, MACDONALD 33 Lemma A.2. For any indices ≤ j < j < · · · < j l ≤ n we have (A.2) SY Y j · · · Y j l X = qv n − l ) SX Y Y j · · · Y j l . Proof. We have SY X = qv n − SX Y by Lemma 2.2, equation (2.8). By (2.9)we have Y j X = X Y j + ( v − − v ) T − j − · · · T − · · · T − j − Y X . Multiplying by Y j · · · Y j l and using the fact that [ T k , Y h ] = 0 if h > k − Y j Y j · · · Y j l X = Y j · · · Y j l X Y j ++ ( v − − v ) T − j − · · · T − · · · T − j − Y Y j · · · Y j l X . Multiplying now by Y and using the relation Y T − · · · Y − j − = T · · · T j − Y j yields Y Y j · · · Y j l X = Y Y j · · · Y j l X Y j ++ ( v − − v ) T − j − · · · T − T · · · T j − Y Y j · · · Y j l X , from which it follows in turn that SY Y j · · · Y j l X = SY Y j · · · Y j l X Y j + (1 − v ) SY Y j · · · Y j l X and thus that v SY Y j · · · Y j l X = SY Y j · · · Y j l X Y j . By the same argument, v SY Y j · · · Y j l X Y j = SY Y j · · · Y j l X Y j Y j , and continuing in this manner we finally arrive at SY Y j · · · Y j l X = v − l − SY X Y j · · · Y j l = qv n − l ) SX Y Y j · · · Y j l as expected. ⊓⊔ Lemma A.3. For any indices < j < j · · · < j l ≤ n we have SY j · · · Y j l X == SX Y j · · · Y j l + l X u =1 q (1 − v ) v n − l − j u + u ) SX Y Y j · · · c Y j u · · · Y j l . (A.3) Here the symbol b x means that we omit the term x in the product.Proof. First of all, we have, again by Lemma 2.2, (2.9) Y j X = X Y j + ( v − − v ) β j Y j X , for all j > β j = T − j − · · · T − · · · T − j − . Define elements A ( j , . . . , j l ) = SY j · · · Y j l X , B ( j , . . . , j l ) = SY Y j · · · Y j l X . We have, by the same arguments as in the previous lemma, A ( j , . . . , j l ) = A ( j , . . . , j l ) Y j + ( v − − v ) Sβ j Y Y j · · · Y j l X = A ( j , . . . , j l ) Y j + ( v − − v ) v − j − B ( j , . . . , j l ) . By Lemma A.2, B ( j , . . . , j l ) = qv n − l ) SX Y Y j · · · Y j l , therefore A ( j , . . . , j l ) = A ( j , . . . , j l ) Y j + q (1 − v ) v n − l − j +1) SX Y Y j · · · Y j l = (cid:16) A ( j , . . . , j l ) Y j + q (1 − v ) v n − l − j +2) SX Y Y j · · · Y j l (cid:17) Y j + q (1 − v ) v n − l − j +1) SX Y Y j · · · Y j l = · · · = SX Y j · · · Y j l + l X u =1 q (1 − v ) v n − l − j u + u ) SX Y j · · · c Y j u · · · Y j l which is what we wanted to prove. ⊓⊔ Lemma A.4. The following holds : (A.4) S (cid:20) X , X j < ··· Using the previous two lemmas, we compute X j < ··· We are finally ready to give the proof of Proposition 3.2. We will argue byinduction, with Lemma A.1 being the case l = 1. So we fix l ∈ N and assume thatProposition 3.2 has been proved for all l ′ < l . It is necessary to distinguish twocases : Case 1. Let us assume that l ≤ n . We will use the formula X i Y li = (cid:18) X i Y l − i · X i Y i (cid:19) − (cid:18) X i Y l − i · X i Collecting terms we get S (cid:20) X , X i Y li (cid:21) S = SX Y l S (cid:26) (1 − q ) + (1 − q l − ) + (1 − q l − )( q − (cid:27) + SX Y l − X 1) + (1 − q l − ) − (1 − q l − )( q − − (1 − q l − ) (cid:27) + SX Y l − X 1) + ( − l +1 l + ( − l (cid:19)(cid:27) + X j> l − X t =1 SX Y Y l − tj X Case 2. Let us deal with the situation when l > n . The method is very similar tothe one used in the proof of Case 1 above. This time we use the following identity : X i Y li = X i Y l − i · X i Y i − X i Y l − i · X j 1) + ( − t ( q − (cid:27) = (1 − q l ) SX Y l S as desired. This concludes the proof of Case 2 as well as that of Proposition 3.2. ⊓⊔ ALL, CHEREDNIK, EISENSTEIN, MACDONALD 39 Appendix B. Proof of Proposition 3.3. We will start by giving a closed expression for the commutator S (cid:20) P i Y i , X l Y − (cid:21) S . Lemma B.1. For any l ≥ we have S (cid:20) X i Y i , X l Y − (cid:21) S =( v − n ) − S (cid:26) qX l − X n + q X l − X n + · · · + q l X ln (cid:27) S + q l SX ln S − SX l S (B.1) Proof. First of all, by (2.10) Y X Y − = qT · · · T n − T n − T n − · · · T X = qT · · · T n − T n − X n T − n − · · · T − , from which it follows that(B.2) Y X l Y − = q l T · · · T n − T n − X ln T − n − · · · T − and hence that SY X l Y − S = q l SX ln S . Now we compute S (cid:20) X i Y i , X l Y − (cid:21) S = SY X l Y − S − SX l S + n X m =2 S (cid:20) Y m , X l Y − (cid:21) S = q l SX ln S − SX l S + n X m =2 S (cid:20) Y m , X l Y − (cid:21) S. The lemma will thus be proved once we have shown that(B.3) S (cid:20) Y m , X l Y − (cid:21) S = (1 − v ) v − m ) S (cid:26) qX l − X n + q X l − X n + · · · + q l X ln (cid:27) S. For this, we need some preparatory result Sublemma B.2. For any m , the following identity holds : T − m − · · · T − T − T − · · · T − m − T · · · T n − T n − T n − · · · T = T − · · · T − m − T m T m +1 · · · T n − T n − T n − · T . (B.4) Proof. We argue by induction. The relation can easily be checked directly for m = 2. Fix m and assume that (B.4) holds for m − 1. We have T − m − · · · T − T − T − · · · T − m − T · · · T n − T n − T n − · · · T = T − m − · · · T − T − T − · · · T − m − T · · · T n − T n − T n − · · · T = T − T − m − · · · T − T − T − · · · T − m − T − T · · · T n − T n − T n − · · · T = T − (cid:18) T − m − · · · T − T − T − · · · T − m − T · · · T n − T n − T n − · · · T (cid:19) T . Using the induction hypothesis applied to the set of indices 2 , , . . . , n instead of1 , , . . . , n , we may simplify the expression in parenthesis to get T − m − · · · T − T − T − · · · T − m − T · · · T n − T n − T n − · · · T = T − (cid:18) T − · · · T − m − T m T m +1 · · · T n − T n − T n − · · · T (cid:19) T which proves (B.4) for the integer m . This finishes the induction step and the proofof the sublemma. ⊓⊔ We may now prove Lemma B.1. We argue once again by induction. Fix m andset u l = Y m X l Y − . We compute u directly, using (2.9) and (B.4) : u = X Y m Y − + ( v − − v ) T − m − · · · T − T − T − · · · T − m − Y X Y − = X Y m Y − + q ( v − − v ) T − m − · · · T − · · · T − m − T · · · T n − · · · T X Y Y − = X Y m Y − + q ( v − − v ) T − · · · T − m − T m · · · T n − · · · T X = X Y m Y − + q ( v − − v ) T − · · · T − m − T m · · · T n − X n T − n − · · · T − . We will now prove, by induction on l the following formula :(B.5) u l = X l Y − Y m + ( v − − v ) T − · · · T − m − T m · · · T n − e l T − n − · · · T − where e l = qX n X l − + q X n X l − + · · · + q l X ln . The case l = 1 is proved above. Let us assume that formula (B.5) holds for theinteger l . We have u l +1 = X u l + q ( v − − v ) T − · · · T − m − T m · · · T n − X n T − n − · · · T − Y X l Y − = X u l + q l +1 ( v − − v ) T − · · · T − m − T m · · · T n − X n T − n − · · · T − ×× T · · · T n − X ln T − n − · · · T − = X u l + q l +1 ( v − − v ) T − · · · T − m − T m · · · T n − X l +1 n T − n − · · · T − (B.6)from which (B.5) follows for l + 1 by the induction hypothesis.Equation (B.3) is obtained simply by multiplying (B.5) by S on both sides.Lemma B.1 is now proved. ⊓⊔ We can now start the proof of Proposition 3.3. Let us form the generating seriesfor S (cid:20) P i Y i , X l Y − (cid:21) S . By Lemma B.1, we find X r ≥ S (cid:20) X i Y i , X r Y − (cid:21) Su r = S (cid:26) − X u − X u + ( v − n ) − P i ≥ q i X in u i · X u − X u + v − n ) qX n u − X n u (cid:27) = S − X u + v − n ) qX n u (1 − X u )(1 − qX n u ) S. ALL, CHEREDNIK, EISENSTEIN, MACDONALD 41 On the other hand, we have exp (cid:18) X r ≥ ( v − r − v r )( v r − q r v − r ) r X i X ri u r (cid:19) = exp (cid:18) P r ≥ r P i X ri u r (cid:19) exp (cid:18) P r ≥ q r r P i X ri u r (cid:19) exp (cid:18) P r ≥ v r r P i X ri u r (cid:19) exp (cid:18) P r ≥ q r v − r r P i X ri u r (cid:19) = n Y i =1 exp (cid:18) P r ≥ r X ri u r (cid:19) exp (cid:18) P r ≥ q r r X ri u r (cid:19) exp (cid:18) P r ≥ v r r X ri u r (cid:19) exp (cid:18) P r ≥ q r v − r r X ri u r (cid:19) = n Y i =1 (1 − v X i u )(1 − qv − X i u )(1 − qX i u )(1 − X i u ) . Hence we are reduced to proving the following relation : S n Y i =1 (1 − v X i u )(1 − v − qX i u ) S = S n Y i =1 (1 − X i u )(1 − qX i u ) S + (1 − v n )(1 − v − q )1 − q S (cid:26) ( X u − v − n ) qX n u ) n Y (1 − X i u ) n − Y i =1 (1 − qX i u ) (cid:27) S (B.7)Of course we may, by homogeneity, drop the dummy variable u in this formula. Abrute force approach based on the equalities : SX S = 11 + v S ( X + X ) S = v − SX S,SX S = 11 + v S ( X + X + (1 − v ) X X ) S,SX S = v v S ( X + X + (1 − v − ) X X ) S allows one to check (B.7) directly for n = 2. We will now prove (B.7) by inductionon n . So let us fix n and assume that (B.7) holds for the integer n − 1, with n − ≥ 2. For any subset { i , . . . , i r } of { , . . . , n } we denote by S i ,...,i r the partialsymmetrizer with respect to the indices { i , . . . , i r } .Using the relation S ( X − v − n ) qX n )(1 − X ) S = v − S ( X − v − n ) qX n )(1 − v X ) S we get(1 − v n )(1 − v − q )1 − q S (cid:26) ( X − v − n ) qX n ) n Y (1 − X i ) n − Y i =1 (1 − qX i ) (cid:27) S = (1 − v n )(1 − v − q )1 − q v − S (cid:26) (1 − qX )(1 − v X )( X − v − n ) qX n ) n Y (1 − X i ) n − Y i =2 (1 − qX i ) (cid:27) S = 1 − v n − v n − v − S (cid:26) (1 − qX )(1 − v X ) (cid:18) n Y i =2 (1 − v X i )(1 − v − qX i ) − n Y i =2 (1 − X i )(1 − qX i ) (cid:19)(cid:27) S Next, we use the formulas S (1 − qX ) n Y i =2 (1 − v − qX i ) S = v n Y i =1 (1 − v − qX i ) + (1 − v ) S n Y i =2 (1 − v − qX i ) S,S (1 − v X ) n Y i =2 (1 − X i ) S = v n Y i =1 (1 − X i ) + (1 − v ) S n Y i =2 (1 − X i ) S to simplify (B.7) to the following relation :( v n − v n − ) S (cid:26) n Y i =1 (1 − v X i )(1 − v − qX i ) − n Y i =1 (1 − X i )(1 − qX i ) (cid:27) S = (1 − v n )( v − − (cid:18) n Y i =1 (1 − v X i ) S n Y i =2 (1 − v − qX i ) S − n Y i =1 (1 − qX i ) S n Y i =2 (1 − X i ) S (cid:19) (B.8)Let as usual m λ ( z , . . . , z t ) = X σ ∈ S t z λ σ (1) · · · z λ t σ ( t ) stand for the monomial symmetric function. The computation of the ( v )-symmetrizationof a monomial symmetric function m (1 r ) is an easy exercise which we leave to thereader : Sublemma B.3. For any ≤ r ≤ n , we have Sm (1 r ) ( X , . . . X n ) S = v r (cid:20) n − r (cid:21) + (cid:20) nr (cid:21) + Sm (1 r ) ( X , . . . , X n ) S. In the above we take the convention that (cid:20) n − r (cid:21) + = 0 if r = n . Using Sub-lemma B.3, we may now write down closed and symmetric expressions for all termsinvolved in (B.8) : n Y i =1 (1 − v X i ) = n X r =0 ( − r v r m (1 r ) ( X , . . . , X n ) , n Y i =1 (1 − qX i ) = n X r =0 ( − r q r m (1 r ) ( X , . . . , X n ) ,S n Y i =2 (1 − X i ) S = n X r =0 ( − r v r (cid:20) n − r (cid:21) + (cid:20) nr (cid:21) + Sm (1 r ) ( X , . . . , X n ) S,S n Y i =2 (1 − v − qX i ) S = n X r =0 ( − r q r (cid:20) n − r (cid:21) + (cid:20) nr (cid:21) + Sm (1 r ) ( X , . . . , X n ) S. This allows us to write ALL, CHEREDNIK, EISENSTEIN, MACDONALD 43 n Y i =1 (1 − v X i ) S n Y i =2 (1 − v − qX i ) S = X r,t ( − r (cid:18) r X u =0 (cid:18) ru (cid:19) q u + t (cid:20) n − u + t (cid:21) + (cid:20) nu + t (cid:21) + v r − u + t ) (cid:19) Sm (1 r t ) ( X , . . . , X n ) S and n Y i =1 (1 − qX i ) S n Y i =2 (1 − X i ) S = X r,t ( − r (cid:18) r X u =0 (cid:18) ru (cid:19) v u + t ) (cid:20) n − u + t (cid:21) + (cid:20) nu + t (cid:21) + q r − u + t (cid:19) Sm (1 r t ) ( X , . . . , X n ) S Hence, n Y i =1 (1 − v X i ) S n Y i =2 (1 − v − qX i ) S − n Y i =1 (1 − qX i ) S n Y i =2 (1 − X i ) S = X r,t ( − r r X u =0 (cid:18) ru (cid:19) (cid:20) n − u + t (cid:21) + (cid:20) nu + t (cid:21) + (cid:26) q u + t v r − u + t ) − v u + t ) q r − u + t (cid:27) Sm (1 r t ) ( X , . . . , X n ) S (B.9)while of course S (cid:26) n Y i =1 (1 − v X i )(1 − v − qX i ) − n Y i =1 (1 − X i )(1 − qX i ) (cid:27) S = X r,t ( − r r X u =0 (cid:18) ru (cid:19) ( v r − u ) − q u + t m (1 r t ) ( X , . . . , X n ) . (B.10)Let A denote the right-hand side of (B.9) multiplied by (1 − v n )( v − − B stand for the right-hand side of (B.10) multiplied by ( v n − v n − ).Equation (B.8) is simply that A = B . To show this, we check that the term q u + t m (1 r t ) ( X , . . . , X n ) appears in A and B with the same coefficient. For B it isclearly equal to ( − r (cid:18) ru (cid:19) ( v r − u ) − v n − v n − ) . As far as A is concerned, it is equal to (1 − v n )( v − − − r (cid:18) ru (cid:19) (cid:18) v r − u + t ) (cid:20) n − u + t (cid:21) + (cid:20) nu + t (cid:21) + − v r − u + t ) (cid:20) n − r − u + t (cid:21) + (cid:20) nr − u + t (cid:21) + (cid:19) = (1 − v n )( v − − − r (cid:18) ru (cid:19) v r − u + t ) (cid:18) v n − r + u − t ) − v n − u − t ) − v n (cid:19) = ( v − − v n (1 − v r − u ) )( − r (cid:18) ru (cid:19) = ( v n − v n − )( v r − u ) − − r (cid:18) ru (cid:19) as wanted. Equation (B.8) and Proposition 3.3 are (finally) proved ! ⊓⊔ Appendix C. Proof of Theorem 6.3 C.1. We begin with equation (6.3). Since E r ( z ) = E vecr ( z ) E ( ν r z ) and since[ E ( z ) , E ( ν r z )] = 0, the relation (6.3) is equivalent to(C.1) [ T (0 , , E vecr ( z )] = ν X ( F l ) ν − r − ν − − z − E vecr ( z ) . We prove (C.1) by showing that for any x ∈ X ( F l ),[ O x , vecr,d ] = ν − r − ν r ν − − vecr,d +1 , where O x is the structure sheaf at x . Indeed, we have vecr,d · O x = ν − r X F vec. bdle F =( r,d ) F⊕O x , whereas, since every nonzero map to O x is onto, O x · vecr,d = ν r X G vec. bdle G =( r,d +1) G , O x ) − ν − − G + X F vec. bdle F =( r,d ) F , O x ) F⊕O x . We conclude using dim Hom( G , O x ) = dim Hom( F , O x ) = r . C.2. We now turn to the proof of (6.4). We begin with a Lemma : Lemma C.1. For any torsion sheaf T , the series E r ( z ) is an eigenvector for theadjoint action of T .Proof. It is essentially the same as for the above case of T = O x (see C.1.). Itsuffices to notice that the number Surj( G , T ) of surjective maps from a vector bundle G of rank r to T is independent of the choice (and of the degree) of G . This laststatement is clear when T is stable, and may be proved in general by inductionusing the formula G , T ) = X T ′ ⊆T G , T ′ ) . ⊓⊔ ALL, CHEREDNIK, EISENSTEIN, MACDONALD 45 From the above Lemma and from the formula E ( z ) = exp (cid:16)P r T (0 ,r ) [ r ] z r (cid:17) wededuce that there exists a series E r ( z /z ) ∈ C [[ z /z ]] such that(C.2) E ( z ) E r ( z ) = E r (cid:18) z z (cid:19) E r ( z ) E ( z ) . Let us first determine E ( z /z ). Relation (C.2) for r = 1 is equivalent to(C.3) E ( z ) E vec ( z ) = E (cid:18) z z (cid:19) E vec ( z ) E ( z ) . Thus, in order to compute E ( z /z ) it is enough to consider the restriction of E ( z ) E ( z ) to line bundles of degree, say, zero. If L is such a line bundle thenfor any d > (0 ,d ) · (1 , − d ) ( L ) = ν d X L − d ∈ P ic − d ( X ) L − d , L ) − ν − − ν d X ( F l ) ν − d − ν − − E ( z ) E ( z )( L ) = 1 + X d> ν − d (cid:18) z z (cid:19) d (0 ,d ) (1 , − d ) ( L )= 1 + X ( F l ) ν − − X d> (cid:18) z z (cid:19) d ( ν − d − z z X ( F l )(1 − z z )(1 − ν − z z )= ζ (cid:18) z z (cid:19) . This shows that E ( z /z ) = ζ ( z /z ). Finally, to determine E r ( z /z ), observethat by the coproduct formulas in Proposition 6.2, E r (cid:18) z z (cid:19) ∆ ,..., ( E r ( z ) E ( z )) = ∆ ,..., ( E ( z ) E r ( z ))= E ( z ) E ( z ) ⊗ E ( z ) E ( ν z ) ⊗ · · · ⊗ E ( z ) E ( ν r − z )= r − Y i =0 ζ (cid:18) ν − i z z (cid:19) E ( z ) E ( z ) ⊗ · · · ⊗ E ( ν r − z ) E ( z )= r − Y i =0 ζ (cid:18) ν − i z z (cid:19) ∆ ,..., ( E r ( z ) E ( z )) . It follows that E r ( z /z ) = Q r − i =0 ζ ( ν − i z /z ) as desired. Theorem 6.3 is proved. ⊓⊔ Appendix D. Proof of Lemma 7.4. Let us denote by F ( α , . . . , α r ) the r.h.s. of (7.7). Each ζ (cid:16) α i α j (cid:17) is a rationalfunction in degree zero and leading coefficient one in both α i and α j . From thisand the expression (7.7) we see that F ( α , . . . , α r ) is a rational function of degreeone in each of the variables α , . . . , α r , and whose leading coefficient in any of these variables is also equal to one. Next, since ζ ( z ) has a simple pole at z = 1 and z = v , the function F ( α , . . . , α r ) has at most simple poles and these are locatedalong the hyperplanes α i = α j and α i = v α j . We claim that the residues oneach of these hyperplanes in fact vanish, so that F ( α , . . . , α r ) is a polynomial in α , . . . , α r .Indeed, the residues along hyperplanes α i = α j vanish because F ( α , . . . , α r ) issymmetric in α , . . . , α r ; as for the hyperplanes a i = v α j , we computeRes v α j − α i F ( α , . . . , α r )= 1 v − − (cid:20) Y l = il = j ζ (cid:18) α l α j (cid:19) · Res v α j − α i ζ (cid:18) α i α j (cid:19) · (cid:18) X l = il = j α l + v α j (cid:19) − Y l = il = j ζ (cid:18) v α j α l (cid:19) · Res v α j − α i ζ (cid:18) α i α j (cid:19) · (cid:18) X l = il = j α l + α j (cid:19)(cid:21) + Y l = il = j ζ (cid:18) v α j α l (cid:19) · Res v α j − α i ζ (cid:18) α i α j (cid:19) v α j . Using the relation ζ (cid:16) v α j α l (cid:17) = ζ (cid:16) α l α j (cid:17) we simplify this toRes v α j − α i F ( α , . . . , α r ) = Y l = il = j ζ (cid:18) α l α j (cid:19) · Res v α j − α i ζ (cid:18) α i α j (cid:19) · (cid:26) v α j − α j v − − − v α j (cid:27) = 0as wanted. Combining all the information we have on the function F ( α , . . . , α r )we see that necessarily F ( α , . . . , α r ) = α + . . . + α + r + u for some u ∈ K ′ . Itremains to observe that (for instance) we have F (1 , . . . , 1) = r . We are done. ⊓⊔ Acknowledgments. We would like to thank Iain Gordon, Mark Haiman and Fran¸cois Bergeron for in-teresting discussions. O.S. would like to thank Pavel Etingof for a crucial suggestionmade a few years ago concerning spherical DAHA’s. ALL, CHEREDNIK, EISENSTEIN, MACDONALD 47 References [A] M. Atiyah, Vector bundles over an elliptic curve , Proc. Lond. Math. Soc, III Ser. 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