A geometric Schur-Weyl duality for quotients of affine Hecke algebras
aa r X i v : . [ m a t h . R T ] M a r A GEOMETRIC SCHUR-WEYL DUALITY FORQUOTIENTS OF AFFINE HECKE ALGEBRASGUILLAUME POUCHINAbstra t. After establishing a geometri S hur-Weyl duality in a gen-eral setting, we re all this duality in type A in the (cid:28)nite and a(cid:30)ne ase.We extend the duality in the a(cid:30)ne ase to positive parts of the a(cid:30)nealgebras. The positive parts have ni e ideals oming from geometry, al-lowing duality for quotients. Some of the quotients of the positive a(cid:30)neHe ke algebra are then identi(cid:28)ed to some y lotomi He ke algebras andthe geometri setting allows the onstru tion of anoni al bases.1. Introdu tionThe so- alled S hur-Weyl duality is a bi ommutant theorem whi h lassi- ally holds between GL d ( C ) and the symmetri group S n . The (cid:28)rst groupa ts naturally on V = C d and diagonally on V ⊗ n . The symmetri groupa ts on V ⊗ n by permuting the tensors. The theorem says that the algebrasof these groups are the ommutant of ea h other inside End C ( V ⊗ n ) when d ≥ n . More pre isely we have anoni al morphisms: φ : C [ GL d ( C )] → End C ( V ⊗ n ) , ψ : C [ S n ] → End C ( V ⊗ n ) su h that φ ( C [ GL d ( C )]) = End S n ( V ⊗ n ) and ψ ( C [ S n ]) = End GL d ( C ) ( V ⊗ n ) . The image of the (cid:28)rst map is alled the S hur algebra, and the se ondmap is inje tive when d ≥ n .The ase of a base (cid:28)eld of arbitrary hara teristi has been studied bynumerous autors ([CL℄, [Gr℄).This theorem has many other versions, for a(cid:30)ne algebras and for quantumones (see [J℄).The aim of this arti le is to establish geometri ally a S hur-Weyl dualitybetween some quotients of a (cid:16)half (cid:16) of a(cid:30)ne q -S hur algebras (whi h arethemselves quotients of the a(cid:30)ne quantum enveloping algebra of GL d ) andsome quotients of a (cid:16)half(cid:17) of the a(cid:30)ne He ke algebra b H n (of type A), when d ≥ n .The S hur-Weyl duality between a(cid:30)ne q -S hur algebras and a(cid:30)ne He kealgebras is already known for some time and an be expressed ni ely by onsidering onvolution algebras on some (cid:29)ag varieties (see [VV℄). We are1 GUILLAUME POUCHIN onsidering quotients of subalgebras whi h arise naturally in this geometri interpretation: the ideals whi h we use to de(cid:28)ne our quotients are just thefun tions whose support lies in some losed subvariety.In fa t some of the quotients of He ke algebras de(cid:28)ned this way are par-ti ular ases of y lotomi He ke algebras (where all parameters are equal tozero). One interesting out ome of our onstru tion is the existen e of anon-i al bases for su h algebras. These are simply de(cid:28)ned as the restri tion of ertain simple perverse sheaves to the open subvarieties we are onsider-ing. The S hur-Weyl duality also provides a strong link with a(cid:30)ne q -S huralgebras.The arti le is organized as follows: the (cid:28)rst part is the geometri settingneeded to have a S hur-Weyl duality. The se ond part is the appli ation ofthe (cid:28)rst part in type A for the (cid:28)nite and a(cid:30)ne ase. In the third part we showthat the geometri S hur-Weyl duality remains when we restri t ourselves tosome subalgebras verifying some onditions. We apply this in the next partto the a(cid:30)ne ase by taking positive parts of our a(cid:30)ne algebras. Thesepositive algebras have interesting two-sided ideals oming from geometry,so the (cid:28)fth part is dedi ated to quotients by su h ideals, in parti ular weestablish the S hur-Weyl duality for our quotient algebras. The quotients arealso identi(cid:28)ed. The sixth part deals with the onstru tion of anoni al basesof our quotients, using their onstru tion in terms of interse tion omplexes.The last part of the arti le is the study of the ase d < n , where only onehalf of the bi ommutant holds. This answers a question of Green ([G℄) inthe a(cid:30)ne ase.I deeply thank my supervisor O.S hi(cid:27)mann for his useful help, ommentsand his availability.2. S hur-Weyl duality in a general settingLet G be a group a ting on two sets X and Y . Let us assume that wehave the following data: • a de omposition Y = F i ∈ I Y i , where I is a (cid:28)nite set, • for ea h i ∈ I , a surje tive G -equivariant map, φ i : X → Y i , whi hhas (cid:28)nite (cid:28)bers of onstant ardinal m i , • An element ω ∈ I for whi h the map φ ω is bije tive.We equip the produ ts X × Y , X × X and Y × Y with the diagonal G -a tion.Let A = C G ( Y × Y ) be the set of G -invariant fun tions whi h take non-zero values on a (cid:28)nite number of G -orbits, and de(cid:28)ne B = C G ( X × X ) inthe same way. These are equipped with the onvolution produ t f ∗ g ( L, L ′′ ) = X L ′ f ( L, L ′ ) g ( L ′ , L ′′ ) . The spa e C = C G ( Y × X ) is endowed with a natural a tion by onvolutionof A (resp. B ) on the left (resp. on the right).EOMETRIC SCHUR-WEYL DUALITY 3Theorem 2.1 (Bi ommutant Theorem). We have: End B ( C ) = A, End A ( C ) = B. Proof. Let's prove the (cid:28)rst assertion. Let P ∈ End B ( C ) .From the de omposition Y = F i ∈ I Y i , we an split C as a dire t sum ofve tor spa es: C = M i ∈ I C G ( Y i × X ) and hen e End( C ) as:(1) End( C ) = M i,j Hom( C G ( Y i × X ) , C G ( Y j × X )) The ( i, j ) - omponent P ′ ( i,j ) = P ′ of P with respe t to (1) is the morphismde(cid:28)ned by P ′ ( f ) = 1 O ∆( j ) ∗ P (1 O ∆( i ) ∗ f ) , where ∆( l ) is the diagonal of Y l × Y l .The G -equivariant surje tive map φ i : X → Y i gives rise to the G -equivariant maps: Id × φ i : Y j × X → Y j × Y i φ i × Id : X × X → Y i × X From these surje tive maps we anoni ally build the inje tions: ψ i : C G ( Y j × Y i ) ֒ → C G ( Y j × X ) χ j : C G ( Y j × X ) ֒ → C G ( X × X ) For every f in C G ( Y i × X ) we have: P ′ ( f ) = P ′ ( m − i ∆( Y i × X ) ∗ χ i ( f )) = m − i P ′ (1 ∆( Y i × X ) ) ∗ χ i ( f ) where P ′ (1 ∆( Y i × X ) ) ∈ C G ( Y j × X ) .We will now prove that P ′ (1 ∆( Y i × X ) ) belongs to the image of C G ( Y j × Y i ) under ψ i .By de(cid:28)nition the image of ψ i in C G ( Y j × X ) is the set of fun tions takingthe same values on two orbits of Y j × X whi h have the same image in Y j × Y i .We introdu e: Z i = { ( L, L ′ ) ∈ X × X, φ i ( L ) = φ i ( L ′ ) } . Observe that P ′ (1 ∆( Y i × X ) ) ∗ Z i = P ′ (1 ∆( Y i × X ) ∗ Z i ) = m i P ′ (1 ∆( Y i × X ) ) .The following lemma implies that P ′ (1 ∆( Y i × X ) ) belongs to Im( ψ i ) Lemma 2.1.
Im( ψ i ) = { h ∈ C G ( Y j × X ) , h ∗ Z i = m i h } GUILLAUME POUCHINProof. For the in lusion of the left-hand side in the right-hand side, we anwrite for h ∈ Im( ψ i ) and ( L, L ′ ) ∈ Y j × X : h ∗ Z i ( L, L ′ ) = X L ′′ h ( L, L ′′ )1 Z i ( L ′′ , L ′ ) = X φ i ( L ′′ )= φ i ( L ′ ) h ( L, L ′′ ) = m i h ( L, L ′ ) For the other in lusion, let h ∈ C G ( Y j × X ) be su h that h ∗ Z i = m i h .Take ( L, M ) and ( L, N ) in Y i × X su h that φ j ( M ) = φ j ( N ) . Then we have Z i ( L ′ , M ) = 1 Z i ( L ′ , N ) for every L ′ in X . Then: m i h ( L, M ) = h ∗ Z i ( L, M ) = X L ′ h ( L, L ′ )1 Z i ( L ′ , M ) = X L ′ h ( L, L ′ )1 Z i ( L ′ , N ) = h ∗ Z i ( L, N ) = m i h ( L, N ) and so h ∈ Im( ψ i ) . ♦ Let g := ψ − i ( P ′ (1 ∆( Y i × X ) )) ∈ C G ( Y j × Y i ) So we have for ( L, M ) ∈ Y j × X , g ( L, φ i ( M )) = P ′ (1 ∆( Y i × X ) )( L, M ) . We an now prove that: ∀ f ∈ C ( Y i × X ) , P ′ ( f ) = g ∗ f Indeed we have seen that P ′ ( f ) = m − i P ′ (1 ∆( Y i × X ) ) ∗ χ i ( f ) . But we have: m − i P ′ (1 ∆( Y i × X ) ) ∗ χ ( f )( L, M ) = m − i X N ∈ X P ′ (1 ∆( Y i × X ) )( L, N ) χ i ( f )( N, M )= m − i X N ∈ X g ( L, φ i ( N )) f ( φ i ( N ) , M )= X N ′ ∈ Y i g ( L, N ′ ) f ( N ′ , M )= g ∗ f ( L, M ) So we have the result for P ′ .To have it for P , it su(cid:30) es to sum on the orthogonal idempotents. Forevery ( i, j ) ∈ I we have built g ( i,j ) ∈ C G ( Y j × Y i ) su h that ∀ f ∈ C G ( Y i × X ) , P ′ ( i,j ) ( f ) = g ( i,j ) ∗ f . Let g = P i,j g ( i,j ) .Then for f = ⊕ i f i ∈ C G ( Y × X ) = L i C G ( Y i × X ) , we have: P ( f ) = X i,j ∈ I O ∆( j ) ∗ P (1 O ∆( i ) ∗ f ) = X i,j ∈ I P ′ ( i,j ) ( f i ) = X i,j ∈ I g ( i,j ) ∗ f i = g ∗ f. We now turn to the se ond assertion.Take P ∈ End A ( C ) .The proje tor on C G ( Y i × X ) parallel to the rest of the sum is the onvo-lution on the left by the fun tion O ∆ i , where ∆ i is the diagonal of Y i × Y i .But P ommutes with the a tion of A , so these subspa es are stable.The next lemma will allow us to fo us on one su h subspa e.EOMETRIC SCHUR-WEYL DUALITY 5Lemma 2.2. The A -module C is generated by C G ( Y ω × X ) .Proof. It is su(cid:30) ient to verify that for every f in C G ( Y i × X ) , we have thefollowing formula: f = ψ − ω ( f ) ∗ ∆( Y ω × X ) where ψ ω is the isomorphism dedu ed from φ ω : ψ ω : C G ( Y i × Y ω ) → C G ( Y i × X ) ♦ As C is generated as a A -module by C G ( Y ω × X ) , the endomorphism P is entirely determined by its restri tion P ′ to C G ( Y ω × X ) . Then we an onsider P ′ ∈ End C G ( Y ω × Y ω ) ( C G ( Y ω × X )) .But the anoni al isomorphism φ ω : Y ω → X allows us to identify B = C G ( X × X ) with C G ( Y ω × X ) and C G ( Y ω × Y ω ) . This way we an see P ′ asan element of End B ( B ) = B . ♦
3. Appli ations3.1. The linear group. Let q be a power of a prime number p and F q the (cid:28)nite (cid:28)eld with q elements. We note G = GL n ( F q ) . In the followingeverything takes pla e in a ve tor spa e V on F q of dimension n . We (cid:28)x aninteger d ≥ n .The omplete (cid:29)ag manifold X is: X = { ( L i ) ≤ i ≤ n | L ⊆ L ⊆ · · · ⊆ L n = V , dim L i = i } The partial (cid:29)ag manifold Y of length d is: Y = { ( L i ) ≤ i ≤ d | L ⊆ L ⊆ · · · ⊆ L d ⊆ V } The group G a ts anoni ally on the varieties X and Y .A omposition of n of length d is a sequen e of integers d = ( d , · · · , d d ) whi h have a sum equal to n . Let Λ( n, d ) be the set of ompositions of n oflength d .For ea h omposition d of n we have a onne ted omponent Y d of Y de(cid:28)ned by Y d = { ( L • ) ∈ Y, ∀ i dim( L i +1 /L i ) = d i } and the de omposition: Y = G d ∈ Λ( n,d ) Y d Also we have a anoni al surje tive G -equivariant map φ d : X → Y d .As d ≥ n , the element ω = (1 , · · · , | {z } n , , · · · , belongs to Λ( n, d ) . The anoni al morphism φ ω is then an isomorphism.The hypotheses are veri(cid:28)ed so we an apply the theorem in se tion 1. Inthis ase the algebras onstru ted are well known:Proposition 3.1. The onvolution algebra C G ( X × X ) is isomorphi to theHe ke algebra H n with parameter q = v − . GUILLAUME POUCHINProposition 3.2. The onvolution algebra C G ( Y × Y ) is isomorphi to the q -S hur algebra S q ( n, d ) .Thus we obtain the standard S hur-Weyl duality.3.2. The a(cid:30)ne ase. Let us write M for the set of F q [[ z ]] -submodules of ( F q (( z ))) n whi h are free of rank n .The omplete a(cid:30)ne (cid:29)ag variety ˆ X is: ˆ X = { ( L i ) i ∈ Z ∈ M Z | ∀ i L i ⊆ L i +1 , L i + n = z − L i , dim F q L i /L i − = 1 } The a(cid:30)ne partial (cid:29)ag variety ˆ Y of length d is: ˆ Y = { ( L i ) i ∈ Z ∈ M Z | ∀ i L i ⊆ L i +1 , L i + d = z − L i } Let G be GL n ( F q (( z ))) . The varieties de(cid:28)ned above are equipped with a anoni al a tion of G .We still have a de omposition of ˆ Y : ˆ Y = G d ∈ Λ( n,d ) ˆ Y d where ˆ Y d is the subvariety of Y de(cid:28)ned by: ˆ Y d = { ( L • ) ∈ ˆ Y , ∀ i ∈ { , · · · , d } dim F q ( L i /L i − ) = d i } For ea h element d ∈ Λ( n, d ) , we have a G -equivariant surje tive map: φ d : ˆ X → ˆ Y d de(cid:28)ned by φ d ( L • ) = L and φ d ( L • ) i = L P ik =1 d k for i ∈ { , · · · , d } .We an identify these algebras as we did in the previous paragraph:Proposition 3.3 ([IM℄). The algebra C G ( ˆ X × ˆ X ) is isomorphi to the a(cid:30)neHe ke algebra b H n with parameter q = v − .Proposition 3.4 ([VV℄). The algebra C G ( ˆ Y × ˆ Y ) is isomorphi to the a(cid:30)ne q -S hur algebra ˆ S q ( n, d ) .4. Subalgebras dedu ed from subvarietiesWe will now see that the bi ommutant theorem remains true, under someadditionnal hypothesis, for a subspa e of C G ( Y × X ) and subalgebras of C G ( Y × Y ) and C G ( X × X ) .Let X, Y, Y i be as in se tion 1.Suppose we are given G -subvarieties Z ⊆ Y × X , X ⊆ X × X , Y ⊆ Y × Y satisfying the following onditions: • for every i, j ∈ I , when we write Z i = Z ∩ ( Y i × X ) and Y i,j = Y ∩ ( Y i × Y j ) , then ( φ i × Id X )( X ) = Z i and ( φ i × φ j )( X ) = Y ( i,j ) EOMETRIC SCHUR-WEYL DUALITY 7 • ∆ X ⊆ X and C G ( X ) is a subalgebra of C G ( X × X ) .From the above assumptions it follows that: • C G ( Z ) is stable for the a tion of C G ( X ) on C G ( Y × X ) . • C G ( Y ) is a subalgebra of C G ( Y × Y ) , and C G ( Z ) is stable for thea tion of C G ( Y × X ) . • the spa es C G ( X ) , C G ( Y ) and C G ( Z ) ontain the hara teri fun -tions of the diagonals O ∆( X × X ) , O ∆( Yi × X ) and O ∆( Yi × Yj ) (for every i , j in I ). • the subspa e C G ( Z ω ) generates C G ( Z ) as a C G ( Y ) -module. • the diagonal fun tion O ∆( Yi × X ) generates C G ( Z i ) as a C G ( X ) -module. • we have isomorphisms dedu ed from ψ and χ : C G ( Y ( ω,ω ) ) ≃ C G ( Z ω ) ≃ C G ( X ) Theorem 4.1. Under the previous onditions, the following bi ommutanttheorem holds:
End C G ( X ) ( C G ( Z )) = C G ( Y )End C G ( Y ) ( C G ( Z )) = C G ( X ) The proofs are the same as in the ase of the whole spa e.5. The positive part of the affine He ke algebraWe get ba k to the setting of se tion 2.2. Thus ˆ X and ˆ Y are resp. the omplete a(cid:30)ne (cid:29)ag variety and the partial a(cid:30)ne (cid:29)ag variety. We re all thatwe take d ≥ n , where n is the rank of the free modules and d is the periodi ityin the partial a(cid:30)ne (cid:29)ag variety. Consider the subvarieties: • X = ( ˆ X × ˆ X ) + = { ( L • , L ′• ) ∈ ˆ X × ˆ X, L ′ ⊆ L }• Y = ( ˆ Y × ˆ Y ) + = { ( L • , L ′• ) ∈ ˆ Y × ˆ Y , L ′ ⊆ L }• Z = ( ˆ Y × ˆ X ) + = { ( L • , L ′• ) ∈ ˆ Y × ˆ X, L ′ ⊆ L } whi h give rise to the onvolution algebras • A + = C G (( ˆ Y × ˆ Y ) + ) • B + = C G (( ˆ X × ˆ X ) + ) .The subspa e C + = C G (( ˆ Y × ˆ X ) + ) is a ( A + , B + ) -bimodule. It is easy to he k that the hypothesis of theorem3.1 are veri(cid:28)ed, so that the bi ommutant theorem still holds:Proposition 5.1. We have: End A + ( C + ) = B + End B + ( C + ) = A + Proof. It is a dire t appli ation of theorem 4.1. ♦ Our immediate aim is to identify pre isely the algebra B + . For this, weneed to re all in more details the stru ture of a(cid:30)ne Weyl group in type A. GUILLAUME POUCHIN5.1. The extended a(cid:30)ne Weyl group in type A. Let us (cid:28)rst re all(cid:28)rst the de(cid:28)nition of the extended a(cid:30)ne Weyl group in the general ase of a onne ted redu tive group G over C . We write T for a maximal torus of G , W = N G ( T ) /T is the Weyl group of G . The group W a ts on the hara tergroup X = Hom( T, C ∗ ) , whi h allows us to onsider the semi-dire t produ t W = W ⋊ X , whi h is alled the extended a(cid:30)ne Weyl group of G . Theroot system R of G generates a sublatti e of X , noted Y . The semi-dire tprodu t W ′ = W ⋊ Y is alled the a(cid:30)ne Weyl group of G . It's a Coxetergroup, unlike the extended a(cid:30)ne Weyl group. It is also a normal subgroupof W .There is an abelian subgroup Ω of W su h that ω − Sω = S for every ω ∈ Ω and W = Ω ⋊ W ′ .In the ase of G = GL n ( C ) , the Weyl group W is isomorphi to thesymmetri group S n . We write S = { s , · · · , s n − } for its simple re(cid:29)e tions.The group W ′ is still a Coxeter group, whi h is generated by the simplere(cid:29)e tions of W and an additional elementary re(cid:29)e tion s . The group Ω isisomorphi to Z , and is generated by an element ρ whi h veri(cid:28)es ρ − s i ρ = s i +1 for every i = 1 , · · · , n − , where we write s n for s .The group W ′ is then the group generated by the elements s , · · · , s n − ,with the following relations:(1) s i = 1 for every i = 1 , · · · n − (2) s i s i +1 s i = s i +1 s i s i +1 where the indi es are taken modulo n .We have W = Ω ⋊ W ′ , where the group Ω is isomorphi to Z , generatedby an element ρ whi h veri(cid:28)es: ρ − s i ρ = s i − . The hara ter group of a torus of GL n ( C ) is naturally isomorphi to Z n .Thus we have W = S n ⋉Z n , where the group S n a ts on Z n by permutation.The group W = S n ⋉Z n an also be onsidered as a subgroup of the groupof the automorphisms of Z in the following way: to ea h ( σ, ( λ i )) ∈ S n ⋉ Z n ,we asso iate the element ˜ σ ∈ Aut( Z ) de(cid:28)ned by: ˜ σ ( i ) = σ ( r ) + kn + λ r n where i = kn + r is the Eu lidian division of i by n , taking the rest between and n .In fa t if we write τ for the element of Aut( Z ) de(cid:28)ned by: τ : i i + n and set Aut n ( Z ) = { σ ∈ Aut( Z ) , στ = τ σ } . Then we obtain the followingisomorphism:Lemma 5.1. The previous map provides an isomorphism of groups: S n ⋉ Z n ≃ Aut n ( Z ) EOMETRIC SCHUR-WEYL DUALITY 9Under this isomorphism, the element s i ( ≤ i ≤ n − ) is mapped to ˜ s i de(cid:28)ned by: ˜ s i ( j ) = j if j = i, i + 1 mod ( n ) , ˜ s i ( j ) = j + 1 if j = i mod ( n ) , ˜ s i ( j ) = j − j = i + 1 mod ( n ) . The element ρ is mapped to ˜ ρ de(cid:28)ned by ˜ ρ ( i ) = i + 1 .The orbits of the a tion of G on ˆ X × ˆ X are parametrised by the elementsof the extended a(cid:30)ne Weyl group b S n . Then we an write O w for an orbit,with w in b S n . This an be done expli itly in the following way. A oupleof (cid:29)ags L • and L ′• are in the orbit w if there is a base e , · · · , e n of the F q (( z )) -module F q (( z )) n su h that L i = Y w ( j ) ≤ i F q e j and L ′ i = Y j ≤ i F q e j where we de(cid:28)ne e i for all i ∈ Z by the ondition e i + kn = z − k e i for all k ∈ Z .Theorem 5.1 ([IM℄). The algebra C G ( ˆ X × ˆ X ) is isomorphi to the a(cid:30)neHe ke algebra b H n spe ialized at v − = q and the isomorphism is given by: φ : 1 O w T w for every w ∈ b S n .To identify the positive part of the a(cid:30)ne He ke algebra, it is ne essary tore all its di(cid:27)erent presentations.5.2. The a(cid:30)ne He ke algebra.De(cid:28)nition 1 (The a(cid:30)ne He ke algebra b H n ). The a(cid:30)ne He ke algebra b H n is a C [ v, v − ] -algebra whi h may be de(cid:28)ned by generators and relations ineither of the following ways:(1) The generators are the T w , for w ∈ b S n = S n ⋉ Z n . The relations are:(1) T w T w ′ = T ww ′ if l ( ww ′ ) = l ( w ) + l ( w ′ ) ,(2) ( T s i + 1)( T s i − v − ) = 0 for s i = ( i, i + 1) .(2) The generators are T ± i , i = 1 . . . n − and X ± j , j = 1 . . . n . Therelations are:(1) T i T j = T j T i if | i − j | > ,(2) T i T i +1 T i = T i +1 T i T i +1 ,(3) T i T − i = T − i T i = 1 ,(4) ( T i + 1)( T i − v − ) = 0 ,(5) X i X − i = X − i X i = 1 ,(6) X i T j = T j X i if i = j, j + 1 ,(7) T i X i T i = v − X i +1 .0 GUILLAUME POUCHINThe isomorphism ψ between these two presentations is uniquely de(cid:28)nedby the following onditions: ψ ( T s i ) = T i ψ ( ˜ T − λ , ··· ,λ n ) ) = X λ · · · X λ n n if λ is dominant, whi h means λ ≥ λ ≥ · · · ≥ λ n , and if we write ˜ T w = v − l ( w ) T w , where l ( w ) is the length of w .One he ks that: T ρ v − n X − T · · · T n − . The multipli ation map de(cid:28)nes an isomorphism of C -ve tor spa es b H n ≃ C [ S n ] ⊗ C C [ v, v − ][ X ± , . . . , X ± n ] B + of G -invariant fun tions on the positive part X of theprodu t variety ˆ X × ˆ X .Theorem 5.2. The algebra C G ( X ) is isomorphi to the subalgebra b H + n of b H n generated by H n and the elements X i . This means that as a ve tor spa e,we have: C G ( X ) ≃ C [ S n ] ⊗ C C [ v, v − ][ X , · · · , X n ] Proof.The (cid:28)rst observation is that every element T i is in C G ( X ) .The element X is in C G ( X ) be ause, by the isomorphism ψ introdu edse tion 5.2, we have X = v − n T n − · · · T T − ρ . But the element T − ρ is the hara teristi fun tion of an orbit in X . As X is in C G ( X ) , the relations(7) prove that the X i s are in C G ( X ) as well.We will now prove that the algebra C G ( X ) is generated by the elements T − ρ , T , · · · , T n − . For this purpose we (cid:28)rst get ba k to the groups.Lemma 5.2. The subsemigroup S n ⋉ Z n − of S n ⋉ Z n is generated by theelements ρ − , s , · · · , s n − .Proof. First it is lear that the elements s , · · · , s n are in S n ⋉ Z n − , be- ause they are in S n . The element ρ − an be written in S n ⋉ Z n as (( n · · · , (0 , · · · , , − (to see it, use the bije tion S n ⋉ Z n ≃ Aut n ( Z ) ).This element belongs to S n ⋉ Z n − .We now prove that every element w of S n ⋉Z n − an be written as a produ tof elements among ρ − , s , · · · , s n − .We de(cid:28)ne the degree of an element w = ( σ, ( λ i )) ∈ S n ⋉ Z n by d = P ni =1 λ i . Let's prove the result by indu tion on the degree d of w ∈ S n ⋉ Z n − .If d = 0 , the result is true be ause w is an element in S n .For d < , let's onsider w ′ = ρw . The degree of w ′ is d + 1 , so we havethe result by re urren e if w ′ ∈ S n ⋉ Z n − . Only the ase where w ′ / ∈ S n ⋉ Z n − remains.EOMETRIC SCHUR-WEYL DUALITY 11Under the isomorphism S n ⋉Z n ≃ Aut n ( Z ) , the subset S n ⋉Z n − is mappedto { ˜ s ∈ Aut n ( Z ) , ∀ i = 1 , · · · , n, ˜ s ( i ) ≤ n } . The onditions w ∈ S n ⋉Z n − and w ′ = ρw / ∈ S n ⋉ Z n − give in Aut n ( Z ) : for every i = 1 , · · · , n , ˜ w ( i ) ≤ n and ∃ j , ≤ j ≤ n , ˜ w ′ ( j ) = ˜ w ( j ) + 1 ≥ n + 1 . For this j we have: ˜ w ( j ) + 1 = n + 1 hen e ˜ w ( j ) = n , whi h is equivalent to σ ( j ) = n and λ j = 0 .Besides, we have d < , so there is a k su h that λ k < . We write t ∈ S n for the transposition ( kj ) and we onsider w ′′ = wt . Then w ′′ = ( σt, ( λ i ) n ) and so for every i = 1 , · · · , n , ˜ w ′′ ( i ) < n . We just saw that this is equivalentto ρw ′′ ∈ S n ⋊ Z n − . So we an apply our re urren e to x = ρw ′′ , whi his of degree d + 1 , to obtain that x is in the subsemigroup generated by ρ − , s , · · · , s n − . As w = ρ − xt , this is also true for w . ♦ To lift this result to the He ke algebra, we need a little more: we haveto prove that every element of S n ⋉ Z n − has a redu ed de omposition as aprodu t of s , · · · , s n − , ρ − .Lemma 5.3. Every element of S n ⋉ Z n − has a redu ed de omposition whi hinvolves only the elements s , · · · , s n − , ρ − .Proof. For w ∈ S n ⋉ Z n − , we write k for its length and d for its degree( d ≤ ). We now pro eed by indu tion on k − d .If k = 0 , the element w is of length so it is a power of ρ , whi h is negativebe ause w ∈ S n ⋉ Z n − . We are done.Si d = 0 , the element w is of degree in S n ⋉ Z n − so it is an element of S n .Then we are done be ause an element of S n has a minimal de ompositionwhi h uses only s , · · · , s n − .The last ase is when d < and k > . We know that w has a minimalde omposition of the form: w = ρ l s i · · · s i k where ≤ i r ≤ n − .We now split the proof in two ases:If s i k = s , then we an apply the re urren e to ws i k , whi h has the samedegree as w and whose length is l ( w ) − . We dedu e from this a minimalwriting of ws i k whi h involves only the elements s , · · · s n − , ρ − , then aminimal writing of w using only these elements, by multiplying by s i k .The remaining ase is when s i k = s . In this ase we have l ( ws ) < l ( w ) .We know at this point by using the isomorphism S n ⋉ Z n − ≃ Aut n ( Z ) ( f [S℄,Cor 4.2.3 or [G℄, Cor 1.3.3)) that l ( ws ) < l ( w ) implies that ˜ w (0) > ˜ w (1) .Take the element w ′ = ws ρ , and let's show that it belongs to S n ⋉ Z n − . Weneed to show that for every i su h that ≤ i ≤ n , we have ˜ w ′ ( i ) ≤ n .By de(cid:28)nition w ′ ( i ) = ws ( i + 1) . So if ≤ i ≤ n − , we have that ˜ w ′ ( i ) = ˜ w ( i + 1) ≤ n be ause w ∈ S n ⋉ Z n − . We have ˜ w ′ ( n ) = ˜ w ( s ( n + 1)) =˜ w ( n ) ≤ n too be ause w ∈ S n ⋉ Z n − . We an dedu e that ˜ w ′ ( n −
1) =˜ w ( n + 1) = n + ˜ w (1) < n + ˜ w (0) = ˜ w ( n ) ≤ n , from whi h it follows that w ′ ∈ S n ⋉ Z n − .2 GUILLAUME POUCHINAs w ′ has a length equal to k − and degree d + 1 and is in the semigroup S n ⋉ Z n − , we an apply the re urren e hypothesis: w ′ has a minimal writingwhi h involves only s , · · · , s n − , ρ − . But as w = w ′ ρ − s = w ′ s ρ − , wehave a minimal writing of w using the s , · · · , s n − , ρ − . ♦ Observe that by onstru tion, we have: C G ( X ) = M w ∈ S n ⋉Z n − C [ v, v − ]1 O w = M w ∈ S n ⋉Z n − C [ v, v − ] T w By lemma 5.3, any T w may be written as a produ t of elements T − ρ , T , · · · , T n − . We easily he k that the algebra generated by these ele-ments is pre isely C [ S n ] ⊗ C C [ v, v − ][ X , · · · , X n ] . ♦
6. QuotientsNow that we have at our disposal a bi ommutant theorem for the positiveparts of the He ke algebra and the S hur algebra, we an try to (cid:28)nd subvari-eties whose orresponding subalgebras are two-sided ideals of these algebras,whi h allows us to take quotients and hope to still have a bi ommutanttheorem.Let λ = ( λ i ) ni =1 ∈ N n be a dominant partition (i.e. λ ≥ · · · ≥ λ n ).For every ( L • , L ′• ) ∈ X , as L ′ ⊆ L are two free F q [[ z ]] -modules of rank n ,the quotient L /L ′ is a torsion F q [[ z ]] -module of rank at most n . We knowthat the isomorphism lasses of torsion F q [[ z ]] -modules of rank at most n areparametrized by the dominant n -weights µ , · · · , µ n , with µ ≥ · · · ≥ µ n .We de(cid:28)ne: X λ = { ( L • , L ′• ) ∈ X , L /L ′ of type µ , with ∀ i = 1 , · · · , d, λ i ≤ µ i } Lemma 6.1. The set C G ( X λ ) is a two-sided ideal of C G ( X ) .Proof. We have to he k that if f and g are in C G ( X ) with f supported on X λ , then f ∗ g and g ∗ f are supported on X λ .But if ( L • , L ′• ) ∈ X and ( L ′• , L ′′• ) ∈ X λ , we have that L ′′ ⊆ L ′ ⊆ L and L /L ′ is of type µ , with ∀ i = 1 , · · · , n , λ i ≤ µ i . From the in lusion L ′ /L ′′ ⊆ L /L ′′ we dedu e that L /L ′′ is of type ν with ν i ≥ µ i ∀ i . So ν i ≥ λ i and ( L . , L ′′ . ) belongs to X λ . Finally f ∗ g is supported on X λ .If ( L • , L ′• ) ∈ X λ and ( L ′• , L ′′• ) ∈ X then we have L ′′ ⊆ L ′ ⊆ L and L /L ′′ is of type µ with µ i ≥ λ i ∀ i = 1 · · · n . As L /L ′ is a quotient of L /L ′′ , thetype ν of L /L ′′ veri(cid:28)es ν i ≥ µ i . So ν i ≥ λ i and ( L • , L ′′• ) ∈ X λ , whi h givesthat g ∗ f is supported on X λ . ♦ We now de(cid:28)ne for i, j in I : Z λ,i = ( φ i × Id )( X λ ) ⊆ Z i Y λ,i,j = ( φ i × φ j )( X λ ) ⊆ Y i,j EOMETRIC SCHUR-WEYL DUALITY 13 Z λ = G i ∈ I Z λ,i Y λ = G i,j ∈ I Y λ,i,j In the same way C G ( X λ ) is a two-sided ideal of C G ( X ) , we have thefollowing statement:Lemma 6.2. The set C G ( Y λ ) is a two-sided ideal of C G ( Y ) .The a tions of C G ( X λ ) and C G ( Y λ ) map the spa e C G ( Z ) to C G ( Z λ ) . Nowwrite A λ = C G ( Y ) / C G ( Y λ ) , B λ = C G ( X ) / C G ( X λ ) and C λ = C G ( Z ) / C G ( Z λ ) .The quotient spa e C λ is then a ( A λ , B λ ) -bimodule. We will write b H n,λ for B λ in the next part.We an now state:Theorem 6.1 (Bi ommutant of the quotient). End A λ ( C λ ) = B λ End B λ ( C λ ) = A λ Proof. We prove the (cid:28)rst assertion.Let P ∈ End A λ ( C λ ) . As in the theorem 2.1, the fa t that the endomor-phism P ommutes with the a tion of A λ implies that it ommutes withthe a tion of the proje tors on C λ,i where C λ,i = C [ Z i ] / C G [ Z λ,i ] , so thesubspa es C λ,i are stable by P .We also know that as an A λ -module, C λ is generated by C λ,ω . So weonly have to study the restri tion of P to C λ,ω , where P ommutes with thea tion of A λ,ω,ω .But there are anoni al isomorphisms A λ,ω,ω ≃ C λ,ω ≃ B λ . Then we an onsider that P belongs to End B λ ( B λ ) whi h is equal to B λ . This provesthe result.Now we prove the se ond point of the theorem.Let P be in End B λ ( C λ ) , and let P i be its restri tion to C λ,i , so that P i ∈ Hom B λ ( C λ,i , C λ ) .The anoni al morphisms given in theorem 2.1 go through to the positiveparts to give for every i ∈ I an inje tion: χ i : C + i ֒ → B + whose left inverse is given by the left multipli ation by m i ∆ i , where ∆ i isthe diagonal in ( ˆ Y i × ˆ X ) + .We write α for the map from C λ to C + whi h asso iates to a fun tion in C λ its unique representative in C G ( Z ) whose restri tion to Z λ is zero.Now de(cid:28)ne P ′ i ∈ Hom B + ( C + ,i , C + ) by the formula: P ′ i ( f ) = α ( P i ( 1 m i ∆ i )) ∗ χ i ( f ) And P ′ = L i ∈ I P i .4 GUILLAUME POUCHINWe easily he k that P ′ ommutes with the a tion of B λ and se ondlythat through the anoni al map End B + ( C + ) ֒ → End B λ ( C λ ) the morphism P ′ maps to P .We have lifted P and got P ′ ∈ End B + ( C + ) . The bi ommutant theoremfor the positive parts gives us the fa t that P ′ ∈ A + , hen e P ∈ A λ asdesired. We are done. ♦ We an now identify the two-sided ideal in question.Proposition 6.1. The two-sided ideal C G ( X λ ) is generated by the element X λ ′ = Q ni =1 X λ n − i i .Proof. From the de(cid:28)nition of C G ( X λ ) , it is obvious that as a ve tor spa ewe an write: C G ( X λ ) = M σ ∈ S n dom( µ ) ≥ λ C T ( σ, − µ ) where dom( µ ) is the partition dedu ed from µ by reordering, and where thepartial order between ompositions is given by λ ≥ µ ⇔ ∀ i = 1 , · · · , n, λ i ≥ µ i . In parti ular the element Q ni =1 X λ n − i i = v − l ( λ ) T ( Id, − λ ′ ) belongs to C G ( X λ ) ,and thus b H n X λ ′ b H n ⊆ C G ( X λ ) .The in lusion of the right-hand side in the left-hand side is done.For the other in lusion, prove (cid:28)rst:Lemma 6.3. For every dominant omposition ν the following holds: H n T ( Id,ν ) H n = M σ,σ ′ ∈ S n C T ( σ,ν σ ′ ) Proof. As the element T ( Id,ν ) belongs to the sum on the right-hand side andthis spa e is stable by the a tion of H n on the right and on the left, thein lusion of the left-hand side in the right-hand side is lear.We show by indu tion on the length of σ ′ that every element in the right-hand side belong to H n T ( Id,ν ) H n .For σ ′ = Id : as we have l ( σ, ν ) = l ( σ,
0) + l ( Id, ν ) be ause ν is domi-nant, the equation T ( Id,ν ) .T ( σ, = T ( σ,ν ) holds, whi h implies that T ( σ,ν ) ∈H n T ( Id,ν ) H n .For l ( σ ′ ) > : we write σ ′ = ts , where s ∈ S and l ( t ) = l ( σ ′ ) − . Byindu tion we know that for every u ∈ S n , we have T ( u,ν t ) ∈ H n T ( Id,σ ) H n .The following holds: T ( u,ν t ) .T ( s, = ( T ( us,ν σ ′ ) if l ( us, ν σ ′ ) = l ( u, ν t ) + 1(1 − q ) T ( u,ν t ) + qT ( us,ν σ ′ ) if l ( us, ν σ ′ ) = l ( u, ν t ) − As the left-hand side term and T ( u,ν t ) belong to H n T ( Id,ν ) H n for every u ∈ S n , we have also T ( σ,ν σ ′ ) ∈ H n T ( Id,ν ) H n for every σ ∈ S n . ♦ EOMETRIC SCHUR-WEYL DUALITY 15To prove the proposition, we (cid:28)rst remark that we have the equality T ( Id, − dom( µ ) ′ ) = T ( Id, − λ ′ ) .T ( Id, − dom( µ ) ′ + λ ′ ) for ea h omposition µ su h that dom( µ ) ≥ λ , whi h implies that T ( Id, − dom( µ ) ′ ) ∈ C G ( X λ ) . By applying thelemma to − dom( µ ) ′ for ea h µ su h that dom( µ ) ≥ λ we obtain the in lusion: C G ( X λ ) ⊆ M dom( µ ) ≥ λ H n X dom( µ ) ′ H n = b H + n X λ ′ b H + n whi h gives the equality. ♦ So far we have de(cid:28)ned for ea h partition λ a losed subset X λ and a twosided ideal I λ = C G ( X λ ) generated by X λ ′ . We an ask if every two sidedideal omes this way.The (cid:28)rst remark to make is that we an asso iate to every (cid:28)nite set ofpartition λ = ( λ ( i ) ) i the losed subset S i X λ ( i ) . The orresponding two-sidedideal is the sum I λ = P i I λ ( i ) . It is easy to see that the bi ommutant theoremholds in this ase too (the proofs are the same).Theorem 6.2. Every G -stable losed subset F of X su h that C G ( F ) isa two-sided ideal of C G ( X ) is of the form X λ , for a (cid:28)nite set of partition λ = ( λ ( i ) ) i .Proof. As C -ve tor spa es, we have(2) C G ( F ) = M O w ⊆F C T w . The next lemma is a re(cid:28)nement of lemma 6.3.Lemma 6.4. For every w ∈ b S n we have: H n T w H n = M σ,σ ′ ∈ S n C T σwσ ′ We use the lemma to rewrite the sum (2) as:(3) C G ( F ) = X O ( Id,λ ) ⊆F H n T ( Id,λ ) H n where the sum is over the dominant partitions λ . Indeed, every w ∈ b S n belongs to a lass S n ( Id, λ ) S n for a dominant λ .We have the usual partial order on the partitions λ , and the set of minimalpartitions ( λ ( i ) ) = λ is (cid:28)nite. Using that C G ( F ) is in fa t a ( b H + n , b H + n )-bimodule, the equality (3) gives: C G ( F ) = X i b H + n T ( Id,λ ( i ) ) b H + n = C G ( X λ ) Then F = X λ . ♦ q -S hur algebras ( f [SS℄, [A℄),but only in the semi-simple ase. Our quotients are y lotomi He ke alge-bras (with all parameters equal to zero) when we take the partition ( d, , · · · , and the semi-simpli ity is obviously not veri(cid:28)ed in this ase.7. Canoni al basis of b H n,λ The geometri onstru tion of our algebras allows us to onstru t anoni- al bases for them. Su h bases, whi h are also alled Kazhdan-Lusztig basesfor He ke algebras, were introdu ed for quantum enveloping algebras byKashiwara and Lusztig (see [L1℄). These bases have several important prop-erties, whi h in lude positivity of the stru ture onstants and ompatibilitywith bases of representations (see [A2℄, [LLT℄,[VV℄).Write ζ : b H + n → b H n,λ for the quotient map.Let us all B the anoni al basis of the a(cid:30)ne He ke algebra b H n . Thisbasis B = ( b O ) is de(cid:28)ned by the formula: b O = X i, O ′ v − i + dim O dim H i O ′ ( IC O )1 O ′ where H i O ′ ( IC O ) is the (cid:28)ber at any point in O ′ of the ohomology sheaf ofthe interse tion omplex of O .As X is a losed subset of ˆ X × ˆ X , the subset B + of B de(cid:28)ned by B + = { b ∈ B, b ∈ C G ( X ) } is a basis of C G ( X ) .Theorem 7.1. The set of elements: B ′ = { ζ ( b ) , b ∈ B + | ζ ( b ) = 0 } form a basis of b H n,λ .Proof. It su(cid:30) es to see that: Ker( ζ ) = M φ ( b )=0 C b As Ker( ζ ) is the set of fun tions supported on the losed subset X λ , theelements b O , where O ⊆ X λ form a basis of Ker( ζ ) . The theorem follows. ♦
8. The ase d < n
The previous bi ommutant theorems are true only in the ase d ≥ n . Inthe ase d < n , one half of the result still holds.Theorem 8.1. If d < n the map: C G ( X × X ) → End C G ( Y × Y ) ( C G ( Y × X )) is surje tive, when X is the omplete (resp. a(cid:30)ne) (cid:29)ag variety and Y the(resp. a(cid:30)ne) (cid:29)ag variety of length d .EOMETRIC SCHUR-WEYL DUALITY 17Proof. Let d < n . We asso iate as before to ea h omposition d of n of length d with a onne ted omponent Y d of the a(cid:30)ne (cid:29)ag variety of length d . To a omposition of n of length d we asso iate a subset of S = { , · · · , n − } oforder at least n − d , in the way that the omposition of n give the sequen edimensions of su essive fa tors in the (cid:29)ag while the set I give whi h stepin a omplete (cid:29)ag are forgotten. We have a bije tion between these subsetsand isomorphism lasses of onne ted omponents in Y .We then write Y I for a onne ted omponent in the orresponding lass.Let W be the extended a(cid:30)ne Weyl group of GL n . We re all that it is thesemi-dire t produ t W ′ ⋊ Ω , where W ′ is the a(cid:30)ne Weyl group of GL n and Ω is isomorphi to Z , generated by ρ . The group W ′ is a Coxeter group,whi h is equipped with the length fun tion l . Ea h element w of W an beuniquely written w ′ ρ z , where w ′ is an element of W ′ and z ∈ Z . We de(cid:28)nethe length l ( w ) of w by l ( w ′ ) and its height h ( w ) = | z | .For ea h I ⊆ S there is a G -invariant surje tive map: φ I : X ։ Y I and for ea h I ⊆ J there is also a surje tive G -morphism: φ I,J : Y I ։ Y J From these we dedu e the maps: θ I : C G ( X × X ) → C G ( Y I × X ) θ I,J : C G ( Y I × X ) → C G ( Y J × X ) given by: θ I ( f )( L, L ′ ) = X L ′′ ∈ φ − I ( L ) f ( L ′′ , L ′ ) = 1 ∆( Y I × X ) ∗ f and θ I,J ( g )( L, L ′ ) = X L ′′ ∈ φ − I,J ( L ) g I ( L ′′ , L ) = 1 ∆( Y J × Y I ) ∗ g I They ommute with the right a tion of C G ( X × X ) by onvolution.By summing over the I ⊆ S of order greater or equal to n − d , we de(cid:28)nea map: θ : C G ( X × X ) → C G ( Y × X ) Lemma 8.1. The image of the map θ is the set: { f = X | I |≥ n − d f I ∈ C G ( Y × X ) | ∀ I ⊆ J, f J = θ I,J ( f I ) } Proof.The in lusion of the image of θ in this set omes from the equality θ I,J ◦ θ I = θ J .Let's prove the other in lusion. We must (cid:28)nd, given a family of fun tions ( f I ) | I |≥ n − d verifying for ea h I ⊆ J the equality θ I,J ( f I ) = f J , a fun tion f in C G ( X × X ) su h that f I = θ I ( f ) for ea h I .8 GUILLAUME POUCHINIn order to give a fun tion in C G ( X × X ) , we have to give a value for ea horbit O w , with w ∈ W . By abuse of notation, we will denote that value by f ( O w ) . We pro eed in two steps.Consider the set M of all w ∈ W su h that for ea h I of order at least n − d , the orbit O w is not open in the (cid:28)ber ( φ I × Id ) − ( O W I w ) , where O W I w is the image of O w in Y I × X . It is equivalent to say that for ea h I of orderat least n − d (and stri tly less than n ), the element w is not of maximallength in the lass W I w (seen as a subset of W ), where W I is the Youngsubgroup of W generated by the elements s i with i ∈ I (it is a (cid:28)nite groupbe ause | I | < n ).The fun tions f I have a ompa t support, so we an hoose k su h thatfor ea h w ∈ W of length or height greater or equal to k , f I vanishes on O W I w for every I .Fix an integer l > k . For ea h element w of M of length less or equal to l we assign an arbitrary value to f ( O w ) , and for the element of length orheight greater or equal to l we set f ( O w ) = 0 .Now we have to give a value to to the elements w ∈ W − M . By de(cid:28)nition,for su h an element w there exists a set I for whi h the orbit O w is dense inthe (cid:28)ber ( φ I × Id ) − ( O W I w ) .We know that w is of maximal length in W I w if and only if for ea h i ∈ I we have l ( s i w ) < l ( w ) . We an dedu e from this that there is a maximal set I ( w ) su h that w is of maximal length in W I ( w ) w .Now we give a value to the elements w ∈ M , pro eeding by indu tion onthe length of w . To that purpose we use the following equation, where wewrite simply I for I ( w ) : m I ( w ) f ( O w ) = f I ( O W I w ) − X w ′ ∈ W I ww ′ = w m I ( w ′ ) f ( O w ′ ) where m I ( w ′ ) = |{ L ′′• ∈ φ − I ( L • ) , ( L ′′• , L ′• ) ∈ O w ′ }| for any L • ∈ Y I su hthat ( L • , L ′• ) ∈ O W I w .This determines the values f ( O w ) be ause ea h element in the sum has alength stri tly smaller than l ( w ) , so that f ( O w ′ ) is already de(cid:28)ned.It remains to show that beyond a (cid:28)xed length or height the obtained values f ( O w ) are zero (for the fun tion to be ompa tly supported), and that thefun tion given by this method is a solution to our problem.We start with the (cid:28)rst point.We prove (cid:28)rst that for ea h element w of length greater or equal to l + n ( n − | I ( w ) | , f ( O w ) is zero.If w is not maximal in any of its lasses W I w , then we have taken f ( O w ) =0 . If not, for I = I ( w ) , the element w is maximal in its lass W I w , and f ( O w ) is given by: m I ( w ) f ( O w ) = f I ( O W I w ) − X w ′ ∈ W I ww ′ = w m I ( w ′ ) f ( O w ′ ) EOMETRIC SCHUR-WEYL DUALITY 19In the sum the elements w ′ have a length greater or equal to l + n ( n − | I ( w ) |− n ( n − . But as they are not maximal in W I w , if they are in M they satisfy | I ( w ′ ) | < | I ( w ) | . Then if w ′ is in M it has a length greater or equal to l + n ( n − | I ( w ′ ) | , and if w ′ is not in M we have f ( O w ′ ) = 0 . By indu tion on | I ( w ′ ) | , we easily see that ea h f ( O w ′ ) in the right sum is zero, and f I ( O W I w ) too, so f ( O w ) is zero. We also know that f ( O w ) is zero for ea h element w of length greater or equal to l . Therefore f is non-zero only on a (cid:28)nite set.It remains to he k that the fun tion just built is a solution to our problem.We have to show that for ea h I and for ea h w ∈ W , the following holds: f I ( O W I w ) = X w ′ ∈ W I w m I ( w ′ ) f ( O w ′ ) Obviously, it is su(cid:30) ient to he k this equation when w is of maximal lengthin its lass W I w . If I = I ( w ) =: J , then the equation is true by onstru tionof f ( O w ) . If not we have I ⊆ J . We pro eed by indu tion: suppose that theequality is true for ea h x su h that l ( x ) < l ( w ) .By onstru tion of f ( O w ) we have: f J ( O W J w ) = X w ′′ ∈ W J w m J ( w ′′ ) f ( O w ′′ ) If we de ompose the sum along the elements of the lass W I \ W J w we obtain: f J ( O W J w ) = X x ∈ W I \ W J w X z ∈ W I x m J ( z ) f ( O z ) f J ( O W J w ) = X x ∈ W I \ W J wx = W I w X z ∈ W I x m J ( z ) f ( O z ) + X y ∈ W I w m J ( y ) f ( O y ) For w ′ ∈ W we de(cid:28)ne m I,J ( W I w ′ ) = |{ L ′′• ∈ φ − I,J ( L • ) , ( L ′′• , L ′• ) ∈ O W I w ′ }| for any L • ∈ Y J su h that ( L • , L ′• ) ∈ O W J w . From the identity φ J = φ I,J φ I we dedu e m J ( w ′ ) = m I,J ( W I w ′ ) m I ( w ′ ) . So we an write: f J ( O W J w ) = X x ∈ W I \ W J wx = W I w m I,J ( W I x ) X z ∈ W I x m J ( z ) f ( O z )+ m I,J ( W I w ) X y ∈ W I w m I ( y ) f ( O y ) (where W I \ W J w is seen as a subset of the quotient W I \ W )But ea h z ∈ W I x for x = W I w is of length less than l ( w ) . Then we anuse X z ∈ W I x m I ( z ) f ( O z ) = f I ( O W I x ) So we have: f J ( O W J w ) = X x ∈ W I \ W J wx = W I w m I,J ( W I x ) f I ( O W I x ) + X y ∈ W I w m I ( y ) f ( O y ) f J = θ I,J ( f I ) the following holds: f J ( O W J w ) = X x ∈ W I \ W J wx = W I w m I,J ( W I x ) f I ( O W I x ) + m I,J ( W I w ) f I ( O W I w ) We (cid:28)nally obtain: f I ( O W I w ) = X x ∈ W I w m I ( x ) f ( O x ) ♦ Let P be an element of End C G ( Y × Y ) ( C G ( Y × X )) . As P ommmutes withthe a tion of the hara teristi fun tions of the diagonals of the omponents Y I × X , the subspa es C G ( Y I × X ) are stable for the endomorphism P . Sowe an write P = ⊕ I P I , where P I ∈ End C G ( Y I × Y I ) ( Y I × X ) .Write ψ I : C G ( Y I × Y I ) ֒ → C G ( Y I × X ) for the inje tion dedu ed from thesurje tion φ I : X → Y I . Write f I := P I (1 ∆( Y I × X ) ) . For ea h g ∈ C G ( Y I × X ) the following equalities hold: P I ( g ) = P I ( ψ I ( g ) ∗ ∆( Y I × X ) ) = ψ I ( g ) ∗ P I (1 ∆( Y I × X ) ) = ψ I ( g ) ∗ f I The next step is to apply the lemma 8.1 to lift the f I s to some f ∈ C G ( X × X ) .For this we have to he k that the fun tions f I = P I (1 ∆( Y I × X ) ) satisfythe hypothesis of the lemma.But for I ⊆ J , we have ∆( Y J × Y I ) ∗ ∆( Y I × X ) = 1 ∆( Y J × X ) , thus f J = θ I,J ( f I ) .We dedu e that there exists f ∈ C G ( X × X ) su h that for ea h I oforder greater or equal to n − d we have f I = θ I ( f ) . But by onstru tion for g ∈ C G ( Y I × X ) the produ t ψ I ( g ) ∗ f I is equal to g ∗ f . We have shownthat P ( g ) = g ∗ f , ∀ g ∈ C G ( Y × X ) . The theorem follows. ♦ Referen es[A℄ Ariki S., Cy lotomi q -S hur algebras as quotient of quantum algebras, J. reineangew. Math. 513, 53-69 (1999).[A2℄ Ariki S., On the de omposition numbers of the He ke algebra of G ( m, , n ) , J. Math.Kyoto Univ. 36, 789-808 (1996).[CL℄ Carter R.W., Lusztig G., On the modular representations of general linear andsymmetri groups, Math. Z. 136 193-242 (1974).[G℄ Green R.M., The a(cid:30)ne q -S hur algebra, J. Alg. 215, 379-411 (1999).[Gr℄ Green J.A.Polynomial Representations of GL n , Le t. Notes in Math. Vol. 830,Springer-Verlag, 1980.[IM℄ Iwahori N.,Matsumoto H., On some Bruhat de omposition and the stru ture of theHe ke rings of p -adi Chevalley groups, Inst. Hautes Etudes S i. Publ. Math. No25, 5-48 (1965).[J℄ Jimbo M., A q -analogue of U ( gl ( N + 1)) , He ke algebras and the Yang-Baxterequation, Lett. Math. Phys. 11, No 3, 247-252) (1986).[K℄ Kashiwara, M., On rystal bases of the qq