Schur-Weyl duality for higher levels

Transcription

1 Schur-Weyl uality for higher levels Jonathan Brunan an Alexaner Kleshchev Abstract. We exten Schur-Weyl uality to an arbitrary level l 1, level one recovering the classical uality between the symmetric an general linear groups. In general, the symmetric group is replace by the egenerate cyclotomic Hecke algebra over C parametrize by a ominant weight of level l for the root system of type A. As an application, we prove that the egenerate analogue of the quasi-hereitary cover of the cyclotomic Hecke algebra constructe by Dipper, James an Mathas is Morita equivalent to certain blocks of parabolic category O for the general linear Lie algebra. Mathematics Subject Classification (2000). 17B20. Keywors. Schur-Weyl uality, parabolic category O, finite W -algebras, cyclotomic Hecke algebras. 1. Introuction We will work over the groun fiel C (any algebraically close fiel of characteristic zero is fine too). Recall the efinition of the egenerate affine Hecke algebra H from [D]. This is the associative algebra equal as a vector space to the tensor prouct C[x 1,..., x ] CS of a polynomial algebra an the group algebra of the symmetric group S. We write s i for the basic transposition (i i+1) S, an ientify C[x 1,..., x ] an CS with the subspaces C[x 1,..., x ] 1 an 1 CS of H, respectively. Multiplication is then efine so that C[x 1,..., x ] an CS are subalgebras of H, s i x j = x j s i if j i, i + 1, an s i x i+1 = x i s i + 1. As well as being a subalgebra of H, the group algebra CS is also a quotient, via the homomorphism H CS mapping x 1 0 an s i s i for each i. The egenerate affine Hecke algebra arises when stuying the representation theory of the general linear Lie algebra g = gl N (C), as follows. Let V be the natural g-moule of column vectors, let M be any g-moule, an consier the Research supporte in part by NSF grant no. DMS

2 2 Jonathan Brunan an Alexaner Kleshchev tensor prouct M V V ( copies of V ). Let e i,j g enote the ij-matrix unit an Ω = N i,j=1 e i,j e j,i. Then, by an observation of Arakawa an Suzuki [AS, 2.2], there is a well-efine right action of H on M V, commuting with the left action of g, efine so that x 1 acts as the enomorphism Ω 1 1 an each s i acts as 1 i Ω 1 ( i 1). The goal of this article is to exten the classical Schur-Weyl uality to some remarkable finite imensional quotients of H : the egenerate cyclotomic Hecke algebras. Let Λ = i Z m iλ i be a ominant weight of level l for the root system of type A, so each m i is a non-negative integer an i Z m i = l. The egenerate cyclotomic Hecke algebra parametrize by Λ is the quotient H (Λ) of H by the two-sie ieal generate by the polynomial i Z (x 1 i) mi. If Λ = Λ 0 (or any other weight of level l = 1) then H (Λ) = CS as above. Fix now a partition λ = (p 1 p n ) of N, let (q 1 q l ) be the transpose partition, an take the ominant weight Λ from the previous paragraph to be the weight Λ = l i=1 Λ q i n. Up to change of origin, any Λ arises in this way for suitable λ. We ientify λ with its Young iagram, numbering rows by 1, 2,..., n from top to bottom in orer of increasing length an columns by 1, 2,..., l from left to right in orer of ecreasing length, so that there are p i boxes in the ith row an q j boxes in the jth column. Since these are not the stanar conventions, we give an example. Suppose λ = (p 1, p 2, p 3 ) = (2, 3, 4), so (q 1, q 2, q 3, q 4 ) = (3, 3, 2, 1), Λ = 2Λ 0 + Λ 1 + Λ 2, N = 9, n = 3, an the level l = 4. The Young iagram is We always number the boxes of the iagram 1, 2,..., N own columns starting from the first column, an write row(i) an col(i) for the row an column numbers of the ith box. This ientifies the boxes with the stanar basis v 1,..., v N of the natural g-moule V. Define e g to be the nilpotent matrix of Joran type λ which maps the basis vector corresponing the ith box to the one immeiately to its left, or to zero if there is no such box; in our example, e = e 1,4 +e 2,5 +e 5,7 +e 3,6 +e 6,8 + e 8,9. Finally, efine a Z-graing g = r Z g r on g by eclaring that e i,j is of egree (col(j) col(i)) for each i, j = 1,..., N, an set p = r 0 g r, h = g 0, m = r<0 g r. The element e g 1 is a Richarson element in the parabolic subalgebra p, so the centralizer g e of e in g is a grae subalgebra of p. Hence its universal enveloping algebra U(g e ) is a grae subalgebra of U(p). Our first main result is best unerstoo as a filtere eformation of the following extension of the classical Schur-Weyl uality to the centralizer g e. Let C l [x 1,..., x ] be the level l truncate polynomial algebra, that is, the quotient of the polynomial algebra C[x 1,..., x ] by the relations x l 1 = = x l = 0. Exten the usual right action of S on V to an action of the twiste tensor prouct C l [x 1,..., x ] CS by efining the action of x i to be as the enomorphism 1 (i 1) e 1 ( i). This commutes with the natural action of g e, making V

3 Schur-Weyl uality 3 into a (U(g e ), C l [x 1,..., x ] CS )-bimoule. We have efine a homomorphism φ an an antihomomorphism ψ : U(g e ) φ En C (V ) ψ C l [x 1,..., x ] CS. Now a result of Vust [KP, 6] asserts that these maps satisfy the ouble centralizer property, i.e. the image of φ is the centralizer of the image of ψ an vice versa. This is a surprising consequence of the normality of the closure of the conjugacy class containing e. In our filtere eformation of this picture, we replace C l [x 1,..., x ] CS with the egenerate cyclotomic Hecke algebra H (Λ) an U(g e ) with the finite W -algebra W (λ) associate to the partition λ. Let us recall the efinition of the latter algebra following [BK3]; see also [P1, P2, GG, BGK]. Let η : U(p) U(p) be the algebra automorphism efine by η(e i,j ) = e i,j + δ i,j (n q col(j) q col(j)+1 q l ) for each e i,j p. Let I χ be the kernel of the homomorphism χ : U(m) C efine by x (x, e) for all x m, where (.,.) is the trace form on g. Then, by the efinition followe here, W (λ) is the following subalgebra of U(p): W (λ) = {u U(p) [x, η(u)] U(g)I χ for all x m}. The graing on U(p) inuces a filtration on W (λ) so that the associate grae algebra gr W (λ) is naturally ientifie with a grae subalgebra of U(p); this is the loop or goo filtration of the finite W -algebra, not the more familiar Kazhan filtration. A key point is that, by a result of Premet [P2, Proposition 2.1], the associate grae algebra gr W (λ) is equal to U(g e ) as a grae subalgebra of U(p), hence W (λ) is inee a filtere eformation of U(g e ). We still nee to introuce a (W (λ), H (Λ))-bimoule V which is a filtere eformation of the (U(g e ), C l [x 1,..., x ] CS )-bimoule V from above. The left action of W (λ) on V is simply the restriction of the natural action of U(p). To efine the right action of H (Λ), let S act on the right by place permutation as usual. Let x 1 act as the enomorphism N N e + (q col(j) n)e j,j 1 ( 1) e i,j 1 (k 2) e j,i 1 ( k). j=1 k=2 i,j=1 col(i)<col(j) This extens uniquely to make V into a (W (λ), H (Λ))-bimoule. To explain why, let Q χ enote the g-moule U(g)/U(g)I χ. It is actually a (U(g), W (λ))- bimoule, the action of u W (λ) arising by right multiplication by η(u) (which leaves U(g)I χ invariant by the above efinition of W (λ)). Let C(λ) be the category of generalize Whittaker moules, that is, the g-moules on which (x χ(x)) acts locally nilpotently for all x m. By Skryabin s theorem [Sk], the functor Q χ W (λ)? : W (λ)-mo C(λ) is an equivalence of categories. The W (λ)- moule V correspons uner this equivalence to the g-moule M V, where M = Q χ W (λ) C an C is the restriction of the trivial U(p)-moule. The above

4 4 Jonathan Brunan an Alexaner Kleshchev formula for the action of x 1 on V arises by transporting its action on M V from [AS, 2.2] through Skryabin s equivalence. We have now efine a homomorphism Φ an an antihomomorphism Ψ W (λ) Φ En C (V ) Ψ H (Λ). Let W (λ) enote the image of Φ. This finite imensional algebra is a natural analogue of the classical Schur algebra for higher levels. Let H (λ) enote the image of the homomorphism Ψ : H (Λ) En C (V ) op, so that V is also a (W (λ), H (λ))-bimoule. Actually, if at least parts of λ are equal to l, then the map Ψ is injective so H (λ) = H (Λ). In general H (λ) is a sum of certain blocks of the finite imensional algebra H (Λ). In view of the fact that W (λ) an H (λ) are usually not semisimple algebras, the following result was to us quite surprising. Theorem A. The maps Φ an Ψ satisfy the ouble centralizer property, i.e. Moreover, the functor is an equivalence of categories. W (λ) = En H (λ)(v ), En W (λ)(v ) op = H (λ). Hom W (λ)(v,?) : W (λ)-mo H (λ)-mo Our secon main result is concerne with the parabolic analogue of the BGG category O for the Lie algebra g relative to the subalgebra p. Let P enote the moule U(g) U(p) C ρ inuce from the one imensional p-moule C ρ on which each e i,j p acts as δ i,j (q 1 + q q col(j) n). This is an irreucible projective moule in parabolic category O. Let O (λ) enote the Serre subcategory of parabolic category O generate by the moule P V. We note that O (λ) is a sum of certain integral blocks of parabolic category O, an every integral block is equivalent to a block of O (λ) for sufficiently large. Moreover, the moule P V is a self-ual projective moule in O (λ), an every self-ual projective inecomposable moule in O (λ) is a summan of P V. In equivalent language, P V is a prinjective generator for O (λ) where by a prinjective moule we mean a moule that is both projective an injective. Applying the construction from [AS, 2.2] once more, we can view P V as a (g, H )-bimoule. It turns out that the right action of H on P V factors through the quotient H (λ) of H to make P V into a faithful right H (λ)-moule, i.e. H (λ) En C (P V ) op. Theorem B. En g (P V ) op = H (λ). The link between Theorems A an B is provie by the Whittaker functor V : O (λ) W (λ)-mo introuce originally by Kostant an Lynch [Ly] an stuie recently in [BK3, 8.5]: the W (λ)-moule V from above is isomorphic to V(P V ). We actually

5 Schur-Weyl uality 5 show that Hom g (P V,?) = Hom W (λ)(v,?) V, i.e. the following iagram of functors commutes up to isomorphism: O (λ) V W (λ)-mo H (λ)-mo. Hom W (λ)(v,?) Hom g(p V,?) The categories W (λ)-mo an H (λ)-mo thus give two ifferent realizations of a natural quotient of the category O (λ) in the general sense of [Gab, III.1], the respective quotient functors being the Whittaker functor V an the functor Hom g (P V,?). In many circumstances, the Whittaker functor turns out to be easier to work with than the functor Hom g (P V,?), so this point of view facilitates various other important computations regaring the relationship between O (λ) an H (λ)-mo. For example, we use it to ientify the images of arbitrary projective inecomposable moules in O (λ) with the inecomposable summans of the egenerate analogues of the permutation moules introuce by Dipper, James an Mathas [DJM]. Also, we ientify the images of parabolic Verma moules with Specht moules, thus recovering formulae for the latter s composition multiplicities irectly from the Kazhan-Lusztig conjecture for g. The egenerate analogue of Ariki s categorification theorem from [A] follows as an easy consequence of these results, as we will explain in more etail in a subsequent article [BK4]. Theorem B shoul be compare with Soergel s Enomorphismensatz from [S]; the connection with Whittaker moules in that case is ue to Backelin [Ba]. As observe originally by Stroppel [S1, Theorem 10.1] (as an application of [KSX, Theorem 2.10]), there is also a version of Soergel s Struktursatz for our parabolic setup: the functor Hom g (P V,?) is fully faithful on projective objects. From this, we obtain the thir main result of the article. Theorem C. If at least parts of λ are equal to l, there is an equivalence of categories between O (λ) an the category of finite imensional moules over a egenerate analogue of the cyclotomic q-schur algebra of Dipper, James an Mathas. The proof of Theorem C explains how the category O (λ) itself can be reconstructe from the algebra H (λ). Unfortunately, this approach oes not give a irect construction of the iniviual blocks of O (λ), rather, we obtain all the blocks simultaneously. For this reason, it is important to fin some more explicit information about the iniviual blocks of the egenerate cyclotomic Hecke algebras, especially their centers; see [Kh, S2] for further motivation for such questions. As an application of the Schur-Weyl uality for higher levels evelope here, the first steps in this irection have recently been mae in [B2, B3, S3]: the centers of the blocks of egenerate cyclotomic Hecke algebras (an of the parabolic category

6 6 Jonathan Brunan an Alexaner Kleshchev O (λ)) are now known explicitly an are isomorphic to cohomology algebras of Spaltenstein varieties. Let us now explain the organization of the remainer of the article. In section 2, we introuce some notation for elements of the centralizer g e an then review Vust s ouble centralizer property. In section 3, we efine the algebras W (λ) an H (λ) an prove the filtere version of the ouble centralizer property, ultimately as a consequence of Vust s result at the level of associate grae algebras. In section 4, we review some known results about parabolic category O an the subcategory O (λ). In particular, we reformulate Irving s classification [I] of self-ual projective inecomposable moules in terms of stanar tableaux, an we introuce some natural projective moules in O (λ) built from ivie powers. In section 5, we prove Theorems A an B, moulo one technical fact the proof of which is eferre to section 6. We also formulate a characterization of the Whittaker functor V which is of inepenent interest, an we explain how to recover the egenerate analogue of a theorem of Dipper an Mathas [DM] in our framework. In section 6, we use certain weight iempotents in W (λ) to relate the projective moules in O (λ) built from ivie powers to the permutation moules of Dipper, James an Mathas, completing the proof of Theorem C. In similar fashion, we relate parabolic Verma moules to Specht moules. Finally there is an appenix giving a short proof of the fact that the egenerate cyclotomic Hecke algebras are symmetric algebras. Acknowlegements. We thank Steve Donkin for rawing our attention to [KP] an Catharina Stroppel, Anrew Mathas, Monica Vazirani an Weiqiang Wang for some helpful iscussions. 2. Vust s ouble centralizer property In this section we formulate Vust s ouble centralizer theorem for the centralizer of the nilpotent matrix e. This is the key technical ingreient neee to prove the main ouble centralizer property formulate in Theorem A in the introuction The centralizer g e We continue with the notation from the introuction. So, g = gl N (C) with natural moule V, an λ = (0 p 1 p n ) is a partition of N with transpose partition (q 1 q l 0). In [BK3], we worke in slightly greater generality, replacing the Young iagram λ with an arbitrary pyrami. There seems to be little to be gaine from that more general setup (see [MS, Theorem 5.4] for a more precise statement), an many things later on become harer to prove, so we will stick here to the left-justifie case. To help in translation, we efine the shift

7 Schur-Weyl uality 7 matrix σ = (s i,j ) 1 i,j n from [BK3, 3.1] by s i,j = p j p i if i j, s i,j = 0 if i j. Numbering the boxes of λ as in the introuction, let L(i) resp. R(i) be the number of the box immeiately to the left resp. right of the ith box in the iagram λ, or if there is no such box. Let I = {1, 2,..., N}, J = {(i, j) I I col(i) col(j), R(j) = }, K = {(i, j, r) 1 i, j n, s i,j r < p j }. The map J K, (i, j) (row(i), row(j), col(j) col(i)) is a bijection. The nilpotent matrix e g maps v i v L(i), interpreting v as 0. Let g e be the centralizer of e in g an let G e be the corresponing algebraic group, i.e. the centralizer of e in the group G = GL N (C). It is well known, see e.g. [BK2, Lemma 7.3], that the Lie algebra g e has basis {e i,j;r (i, j, r) K} where e i,j;r = e h,k. (2.1) h,k I row(h)=i,row(k)=j col(k) col(h)=r For example, if λ is the partition whose Young iagram is isplaye in the introuction an n = 3, then e 3,2;0 = e 3,2 + e 6,5 + e 8,7, e 2,3;1 = e 2,6 + e 5,8 + e 7,9, e 1,3;3 = e 1,9. Note accoring to this efinition that the notation e i,j;r still makes sense even if r p j, but then it is simply equal to zero. We will often parametrize this basis instea by the set J: for (i, j) J, let ξ i,j = e i,j + e L(i),L(j) + + e L k (i),l k (j) (2.2) where k = col(i) 1. This is the same as the element e row(i),row(j),col(j) col(i) from (2.1), so in this alternate notation our basis for g e is the set {ξ i,j (i, j) J} The grae Schur algebra associate to e Consier the associative algebra M e of all N N matrices over C that commute with e. Of course, M e is equal to g e as a vector space, inee, g e is the Lie algebra equal to M e as a vector space with Lie bracket being the commutator. Multiplication in M e satisfies e i,j;r e h,k;s = δ h,j e i,k;r+s, for 1 i, j, h, k n, r s i,j an s s h,k. Note also that the algebraic group G e is the principal open subset of the affine variety M e efine by the non-vanishing of eterminant. The coorinate algebra C[M e ] of M e is the polynomial algebra C[x i,j;r 1 i, j n, s i,j r < p j ], where x i,j;r is the coorinate function picking out the e i,j;r -coefficient of an element of M e when expane in terms of the above basis. The natural monoi structure on M e inuces a bialgebra structure on C[M e ],

8 8 Jonathan Brunan an Alexaner Kleshchev with comultiplication : C[M e ] C[M e ] C[M e ] an counit ε : C[M e ] C given on generators by n (x i,j;r ) = x i,k;s x k,j;t, k=1 s+t=r s s i,k t s k,j ε(x i,j;r ) = δ i,j δ r,0 for 1 i, j n an s i,j r < p j. The localization of C[M e ] at eterminant is the coorinate algebra C[G e ] of the algebraic group G e, so it is a Hopf algebra. We make C[M e ] into a grae algebra, C[M e ] = 0 C[M e], by efining the egree of each polynomial generator x i,j;r to be 1. Each C[M e ] is then a finite imensional subcoalgebra of the bialgebra C[M e ]. Hence the ual vector space C[M e ] has the natural structure of a finite imensional algebra. We pause to introuce some multi-inex notation. Let I enote the set of all tuples i = (i 1,..., i ) with each i k I. Let J enote the set of all pairs (i, j) of tuples i = (i 1,..., i ) an j = (j 1,..., j ) with each (i k, j k ) J. Let K enote the set of all triples (i, j, r) of tuples i = (i 1,..., i ), j = (j 1,..., j ) an r = (r 1,..., r ) with each (i k, j k, r k ) K. The symmetric group S acts on the right on the set I ; we write i j if the tuples i an j lie in the same S -orbit. Similarly, S acts iagonally on the right on the sets J an K. Choose sets of orbit representatives J /S an K /S. For i I, let row(i) = (row(i 1 ),..., row(i )) an efine col(i) similarly. Then the map J K, (i, j) (row(i), row(j), col(j) col(i)) (2.3) is an S -equivariant bijection. Finally, given i I an 1 j, let { (i1,..., i L j (i) = j 1, L(i j ), i j+1,..., i ) if L(i j ), otherwise. (2.4) Define R j (i) similarly. Now we can write own an explicit basis for the algebra C[M e ]. The monomials x i,j;r = x i1,j 1;r 1 x i,j ;r for (i, j, r) K /S clearly form a basis for C[M e ]. Let {ξ i,j;r (i, j, r) K /S } enote the ual basis for C[M e ]. Multiplication in the algebra C[M e ] then satisfies ξ i,j;r ξ h,k;s = a i,j,h,k,p,q;r,s,t ξ p,q;t, (2.5) (p,q,t) K /S where a i,j,h,k,p,q;r,s,t = # (m, u, v) (p, m, u) (i, j, r) (m, q, v) (h, k, s) u + v = t All this is a straightforwar generalization of [G, ch. 2]. Next we want to bring the universal enveloping algebra U(g e ) into the picture. The augmentation ieal of the bialgebra C[M e ] is the kernel of ε : C[M e ] C..

9 Schur-Weyl uality 9 The algebra of istributions Dist(M e ) = {u C[M e ] u((ker ε) m ) = 0 for m 0} has a natural bialgebra structure ual to that on C[M e ] itself. It is even a Hopf algebra because there is an isomorphism U(g e ) Dist(M e ) inuce by the Lie algebra homomorphism g e Dist(M e ), e i,j;r ε x i,j;r ; see [J, I.7.10(1)]. We will always from now on ientify U(g e ) with Dist(M e ) in this way, i.e. elements of U(g e ) are functions on C[M e ]. Define an algebra homomorphism π : U(g e ) C[M e ] by letting π (u)(x) = u(x) for all u U(g e ) an x C[M e ]. Lemma 2.1. π is surjective. Proof. Suppose not. Then we can fin 0 x C[M e ] such that π (u)(x) = 0 for all u U(g e ). Hence, for each m 1, we have that u(x) = 0 for all u C[M e ] with u((ker ε) m ) = 0, i.e. x (ker ε) m. But m 1 (ker ε)m = The grae ouble centralizer property The natural g-moule V can be viewe as a right C[M e ]-comoule with structure map V V C[M e ], v j v i x row(i),row(j);col(j) col(i) (2.6) i I where the sum is over all i satisfying s i,j col(j) col(i) < p j. Since C[M e ] is a bialgebra we get from this a right C[M e ]-comoule structure on the tensor space V. The image of the structure map of this comoule lies in V C[M e ], so V is actually a right C[M e ] -comoule, hence a left C[M e ] -moule. Let ω : C[M e ] En C (V ) be the associate representation. For i, j I, let v i = v i1 v i V an e i,j = e i1,j 1 e i,j En C (V ). For (i, j) J, let ξ i,j = ξ h1,k 1 ξ h,k, (2.7) (h,k) (i,j) recalling (2.2). Using (2.6), one checks for (i, j) J that ξ i,j = ω (ξ row(i),row(j);col(j) col(i) ). Hence, recalling (2.3), ω maps the basis {ξ i,j;r (i, j, r) K /S } of the algebra C[M e ] to the set {ξ i,j (i, j) J /S } in En C (V ). Lemma 2.2. ω is injective. Proof. The vectors {ξ i,j (i, j) J /S } are linearly inepenent. Introuce the twiste tensor prouct algebra C l [x 1,..., x ] CS acting on the right on the space V as in the introuction.

10 10 Jonathan Brunan an Alexaner Kleshchev Lemma 2.3. Suppose we are given scalars a i,j C such that the enomorphism i,j I a i,je i,j of V commutes with the action of C l [x 1,..., x ] CS. Then a i,j = a i w,j w for all w S, a i,j = 0 if col(i k ) > col(j k ) for some k, an a i,j = a Rk (i),r k (j) for all k with R k (j) (interpreting the right han sie as 0 in case R k (i) = ). Proof. The image of v j uner the given enomorphism is i I a i,jv i. Applying w S, we get i I a i,jv i w. This must equal the image of v j w uner our enomorphism, i.e. i I a i w,j wv i w. Equating coefficients gives that a i,j = a i w,j w. A similar argument using x k in place of w gives that a Rk (i),j = a i,lk (j), interpreting the left resp. right han sie as zero if R k (i) = resp. L k (j) =. The remaining statements follow easily from this. Finally let φ : U(g e ) En C (V ) be the representation inuce by the natural action of the Lie algebra g e on the tensor space. By the efinitions, φ is precisely the composite map ω π, so Lemmas 2.1 an 2.2 show that its image is isomorphic to C[M e ]. Also let ψ : C l [x 1,..., x ] CS En C (V ) op be the homomorphism arising from the right action of C l [x 1,..., x ] CS on V. Thus, we have efine maps U(g e ) φ En C (V ) ψ C l [x 1,..., x ] CS. (2.8) The Z-graing on g from the introuction extens to a graing U(g) = r Z U(g) r on its universal enveloping algebra, an U(g e ) is a grae subalgebra. Make V into a grae moule by eclaring that each v i is of egree (l col(i)). There are inuce graings on V an its enomorphism algebra En C (V ), so that the map φ is then a homomorphism of grae algebras. Also efine a graing on C l [x 1,..., x ] CS by eclaring that each x i is of egree 1 an each w S is of egree 0. The map ψ is then a homomorphism of grae algebras too. Theorem 2.4. The maps φ an ψ satisfy the ouble centralizer property, i.e. φ (U(g e )) = En Cl [x 1,...,x ] CS (V ), En ge (V ) op = ψ (C l [x 1,..., x ] CS ). Moreover, if at least parts of λ are equal to l, then the map ψ is injective. Proof. The secon equality follows by a theorem of Vust; see [KP, 6]. For the first equality, note by Lemma 2.3 that any element of En Cl [x 1,...,x ] CS (V ) is a linear combination of the elements ξ i,j from (2.7). These belong to the image of φ by Lemma 2.1. Finally, assume that at least parts of λ are equal to l. Take any i I such that i 1,..., i are all istinct an col(i 1 ) = = col(i ) = l. Then, for 0 r 1,..., r < l an w S, we have that v i x r1 1 xr w = v j where j = L r1 1 Lr (i) w. These vectors are obviously linearly inepenent, hence the vector v i V generates a copy of the regular C l [x 1,..., x ] CS moule. This implies that ψ is injective.

11 Schur-Weyl uality 11 Remark 2.5. Theorem 2.4 is true over an arbitrary commutative groun ring R (instea of just over the fiel C) proviing one replaces the universal enveloping algebra U(g e ) everywhere with the algebra of istributions of the group scheme G e over R. To obtain this generalization, one first treats the case of algebraically close fiels of positive characteristic, using the arguments above together with a result of Donkin [Do, Theorem 2.3c] in place of [KP, 6]. From there, the result can be lifte to Z, hence to any other commutative groun ring. 3. Higher level Schur algebras In this section we evelop the filtere eformation of Vust s ouble centralizer property. This leas naturally to the efinition of certain finite imensional quotients W (λ) of finite W -algebras, which we call higher level Schur algebras The finite W -algebra as a eformation of U(g e ) Recall that m = r<0 g r, h = g 0, p = r 0 g r. Define the finite W -algebra W (λ) associate to the nilpotent matrix e as in the introuction; see also [BK3, 3.2]. As an algebra, W (λ) is generate by certain elements { T (r+1) i,j } (i, j, r) K of U(p) escribe explicitly in [BK3, 3.7]. The graing on p extens to give a graing U(p) = r 0 U(p) r on its universal enveloping algebra. In general, W (λ) is not a grae subalgebra, but the graing on U(p) oes at least inuce a filtration F 0 W (λ) F 1 W (λ) on W (λ) with r F r W (λ) = W (λ) U(p) s. The associate grae algebra gr W (λ) is canonically ientifie with a grae subalgebra of U(p). The first statement of the following lemma is a special case of general result about finite W -algebras ue to Premet [P2, Proposition 2.1]; see also [BGK, Theorem 3.8]. Lemma 3.1. We have that gr W (λ) = U(g e ). In fact, for (i, j, r) K, we have that T (r+1) i,j F r W (λ) an gr r T (r+1) i,j = ( 1)e i,j;r, where e i,j;r is the element of U(g e ) from (2.1). Proof. The fact that gr r T (r+1) i,j = ( 1) r e i,j;r follows by a irect calculation using [BK3, Lemma 3.4]. By [BK3, Lemma 3.6], orere monomials in the elements (i, j, r) K} form a basis for W (λ). Combining this with the PBW theorem for U(g e ), we euce that gr W (λ) = U(g e ). {T (r+1) i,j s=0

12 12 Jonathan Brunan an Alexaner Kleshchev 3.2. Filtere eformation of tensor space For the remainer of the article, c = (c 1,..., c l ) C l will enote a fixe choice of origin; in the introuction we iscusse only the most interesting situation c = 0, i.e. when c 1 = = c l = 0. Let η c : U(p) U(p) enote the automorphism mapping e i,j e i,j + δ i,j c col(i) for each e i,j p. Let Vc enote the grae U(p)-moule equal as a grae vector space to V, but with action obtaine by twisting the natural action by the automorphism η c, i.e. u v = η c (u)v. Also let C c = C1 c enote the one imensional p-moule on which each e i,j p acts by multiplication by δ i,j c col(i) ; in particular, C 0 is the trivial moule C. Obviously, we can ientify C c V = Vc so that 1 c v = v for every v V. By restriction, we can view Vc also as a grae g e -moule. The resulting representation φ,c : U(g e ) En C (V c ) (3.1) is just the composition φ η c where φ is the map from (2.8). Because the automorphism η c of U(p) leaves the subalgebra U(g e ) invariant, the image of φ,c is the same as the image of φ itself. So the statement of Theorem 2.4 remains true if we replace φ with φ,c. Of course, we can also consier C c an Vc as W (λ)-moules by restriction. Let Φ,c : W (λ) En C (Vc ) (3.2) be the associate representation. View the grae algebra En C (Vc ) as a filtere algebra with egree r filtere piece efine from F r En C (Vc ) = En C (Vc ) s. s r The homomorphism Φ,c is then a homomorphism of filtere algebras. Moreover, ientifying the associate grae algebra gr En C (Vc ) simply with En C (Vc ) in the obvious way, Lemma 3.1 shows that the associate grae map gr Φ,c : gr W (λ) En C (V c ) coincies with the map φ,c. We stress that the automorphism η c of U(p) oes not in general leave the subalgebra W (λ) invariant (unless c 1 = = c l ), so unlike φ,c the image of the map Φ,c efinitely oes epen on the choice of c Skryabin s equivalence of categories Recall the (U(g), W (λ))-bimoule Q χ = U(g)/U(g)I χ an the automorphism η from the introuction. We write 1 χ for the coset of 1 in Q χ. Let C(λ) enote the category of all g-moules on which (x χ(x)) acts locally nilpotently for all x m. Then, by Skryabin s theorem [Sk], the functor Q χ W (λ)? : W (λ)-mo C(λ)

13 Schur-Weyl uality 13 is an equivalence of categories; see also [BK3, 8.1]. Conversely, given M C(λ), the subspace Wh(M) = {v M xv = χ(x)v for all x m} (3.3) is naturally a W (λ)-moule with action u v = η(u)v for u W (λ), v Wh(M). This efines a functor Wh : C(λ) W (λ)-mo which gives an inverse equivalence to Q χ W (λ)?. Given any M C(λ) an any finite imensional g-moule X, the tensor prouct M X belongs to C(λ). This efines an exact functor? X : C(λ) C(λ). Transporting through Skryabin s equivalence of categories, we obtain a corresponing exact functor? X : W (λ)-mo W (λ)-mo, M Wh((Q χ W (λ) M) X); see [BK3, 8.2]. Given another finite imensional g-moule Y an any W (λ)-moule M, there is a natural associativity isomorphism [BK3, (8.8)]: a M,X,Y : (M X) Y M (X Y ). (3.4) Suppose that M is actually a p-moule, an view M an M X as W (λ)- moules by restricting from U(p) to W (λ). Consier the map ˆµ M,X : (Q χ W (λ) M) X M X (3.5) such that (η(u)1 χ m) x um x for u U(p), m M an x X. The restriction of this map to the subspace Wh((Q χ W (λ) M) X) obviously efines a homomorphism of W (λ)-moules µ M,X : M X M X. (3.6) Much less obviously, this homomorphism is actually an isomorphism; see [BK3, Corollary 8.2]. We are going to apply these things to the th power (? V ) of the functor? V ; we will often enote this simply by (? V ). To start with, take the moule M to be the one imensional p-moule C c. Composing isomorphisms built from (3.4) an (3.6) in any orer that makes sense, we obtain an isomorphism µ : C c V Cc V = V c (3.7) of W (λ)-moules. The associativity pentagons from [BK3, (8.9) (8.10)] show that this efinition is inepenent of the precise orer chosen. For example, µ 0 : C c C c is the ientity map an µ is the composite (C c V ( 1) ) V µ 1 i V Vc ( 1) V for 1. Alternatively, µ is the composite C c V a Cc V µ V ( 1) c,v V c µ Cc,V V c (3.8) (3.9)

14 14 Jonathan Brunan an Alexaner Kleshchev where a 0 : C c C c is the ientity map an a is the composite (C c V ( 1) ) V a 1 i V (Cc V ( 1) ) V a Cc,V ( 1),V C c V for 1. The following technical lemma will be neee to compute the isomorphism µ appearing in (3.9) explicitly later on. Recall Cc,V I enotes the set of multi-inexes i = (i 1,..., i ) I an v i = v i1 v i. Lemma 3.2. For all i, j I, there exist elements x i,j U(p) such that (i) [e i,j, η(x i,j )] + k η(x k,j) U(g)I χ for each e i,j m, where the sum is over all k I obtaine from i by replacing an entry equal to i by j; (ii) x i,j acts on the moule C c as the scalar δ i,j. The inverse image of v j Vc uner the isomorphism µ Cc,V is equal to (η(x i,j )1 χ 1 c ) v i C c V (Q χ W (λ) C c ) V i I for any such choice of elements x i,j. Proof. The existence of such elements follows from [BK3, Theorem 8.1]; one nees to choose the projection p there in a similar way to [BK3, (8.4)] so that all u U(p) with p(u 1 χ ) = 0 act as zero on C c. The secon statement then follows from the explicit escription of the map µ Cc,V given in [BK3, Corollary 8.2] Action of the egenerate affine Hecke algebra Now we bring the egenerate affine Hecke algebra H into the picture following the approach of Chuang an Rouquier [CR, 7.4]. Consier the functor? V on g-moules. Define an enomorphism x :? V? V by letting x M : M V M V (3.10) enote the enomorphism arising from left multiplication by Ω = N i,j=1 e i,j e j,i, for each g-moule M. Define an enomorphism s : (? V ) V (? V ) V by letting s M : M V V M V V (3.11) enote the enomorphism arising from left multiplication by 1 Ω, i.e. m v v m v v. More generally, for any 1, there are enomorphisms x 1,..., x, s 1,..., s 1 of the functor (? V ) = (? V ) efine by x i = 1 i x1 i 1, s j = 1 j 1 s1 j 1. (3.12) By [CR, Lemma 7.21], these enomorphisms inuce a well-efine right action of the egenerate affine Hecke algebra H on M V for any g-moule M. This is precisely the action from [AS, 2.2] escribe at the beginning of the introuction. Transporting x an s through Skryabin s equivalence of categories, we get enomorphisms also enote x an s of the functors? V an (? V ) V ; see [BK3, (8.13)-(8.14)]. Once again, we set x i = 1 i x1 i 1, s j = 1 j 1 s1 j 1, (3.13)

15 Schur-Weyl uality 15 to get enomorphisms x 1,..., x, s 1,..., s 1 of the functor (? V ) = (? V ) for all 1. By [BK3, (8.18) (8.23)], these inuce a well-efine right action of the egenerate affine Hecke algebra H on M V, for any W (λ)-moule M. Returning to the special case M = C c once again, we have mae C c V into a (W (λ), H )-bimoule. Using the isomorphisms C c V = C c V = Vc from (3.9), we lift the H -moule structure on C c V first to C c V then to Vc, to make each of these spaces into (W (λ), H )-bimoules too. Let us escribe the resulting actions of the generators of H on C c V an on Vc more explicitly. For C c V, this is alreay explaine in [BK3, 8.3]: the conclusion is that x i an s j act on C c V (Q χ W (λ) C c ) V in the same way as the enomorphisms efine by multiplication by i h=1 Ω[h,i+1] an Ω [j+1,j+2], respectively. Here, Ω [r,s] enotes N i,j=1 1 (r 1) e i,j 1 (s r 1) e j,i 1 (+1 s), treating Q χ W (λ) C c as the first tensor position an the copies of V as positions 2, 3,..., ( + 1). To escribe the action of H on Vc, each s i acts by permuting the ith an (i + 1)th tensor positions, as follows from the naturality of µ : C Cc,V c V Vc. The action of each x j is more subtle, an is escribe by the following lemma; the formula for the action of x 1 on V from the introuction is a special case of this. Lemma 3.3. For i I an 1 j, the action of x j on Vc satisfies v i x j = v Lj(i) + (c col(ij) + q col(ij) n)v i + v i (k j) 1 k<j col(i k ) col(i j) j<k col(i k )<col(i j) (Recalling (2.4), the term v Lj(i) shoul be interprete as 0 in case L j (i) =.) v i (j k). Proof. Note that x j+1 = s j x j s j + s j. Using this an inuction on j, one reuces just to checking the given formula in the case j = 1. Let µ enote the map ˆµ Cc,V from (3.5). Choosing elements x i,j U(p) as in Lemma 3.2, we have that ) v i x 1 = µ (Ω [1,2] j I (η(x j,i )1 χ 1 c ) v j = µ((e j1,k 1 η(x j,i )1 χ 1 c ) v k ) j,k where the last sum is over j, k I with j 2 = k 2,, j = k. Suppose first that col(j 1 ) col(k 1 ). By Lemma 3.2(ii), µ((e j1,k 1 η(x j,i )1 χ 1 c ) v k ) = 0 unless i = j = k, in which case µ((e j1,k 1 η(x j,i )1 χ 1) v k ) = (c col(i1) + q col(i1) + + q l n)v i. Now consier the terms with col(j 1 ) > col(k 1 ). By Lemma 3.2(i), we have that e j1,k 1 η(x j,i )1 χ = η(x j,i )e j1,k 1 1 χ h η(x h,i)1 χ where the sum is over all h I obtaine from j by replacing an entry equal to j 1 by k 1. Since χ(e j1,k 1 ) is zero unless k 1 = L(j 1 ) when it is simply equal to 1, an using Lemma 3.2(ii), it follows that these terms contribute v L1(i) j,k v k where the sum is over all j, k I

16 16 Jonathan Brunan an Alexaner Kleshchev such that col(j 1 ) > col(k 1 ), j 2 = k 2,..., j = k, an i is obtaine from j by replacing an entry equal to j 1 by k 1. This simplifies further to give v L1(i) (q col(i1) q l )v i which ae to our earlier term give the conclusion Degenerate cyclotomic Hecke algebras 1<k col(i k )<col(i 1) v i (1 k) Now we pass from the egenerate affine Hecke algebra to its cyclotomic quotients. Lemma 3.4. For 1, the minimal polynomial of the enomorphism of V c efine by the action of x 1 is l i=1 (x (c i + q i n)). Proof. In the special case = 1, this is clear from Lemma 3.3. By the efinition of x 1, it follows that for any 1, the minimal polynomial certainly ivies l i=1 (x (c i+q i n)). Finally, by Lemma 3.3, the enomorphism of Vc efine by x 1 belongs to F 1 En C (Vc ), an the associate grae enomorphism of Vc is equal to e 1 ( 1). This clearly has minimal polynomial exactly equal to x l, since l is the size of the largest Joran block of e. This implies that the minimal polynomial of the filtere enomorphism cannot be of egree smaller than l, completing the proof. Let Λ = l i=1 Λ c i+q i n, an element of the free abelian group generate by the symbols {Λ a a C}. The corresponing egenerate cyclotomic Hecke algebra is the quotient H (Λ) of H by the two-sie ieal generate by l i=1 (x 1 (c i + q i n)). In view of Lemma 3.4, the right action of H on Vc factors through the quotient H (Λ), an we obtain a homomorphism Ψ : H (Λ) En C (V c ) op. (3.14) Define a filtration F 0 H (Λ) F 1 H (Λ) by eclaring that F r H (Λ) is the span of all x i1 1 xi w for i 1,..., i 0 an w S with i i r. Recalling the grae algebra C l [x 1,..., x ] CS from the previous section, the relations imply that there is a well-efine surjective homomorphism of grae algebras ζ : C l [x 1,..., x ] CS gr H (Λ) such that x i gr 1 x i for each i an s j gr 0 s j for each j. Moreover, by Lemma 3.3, the map Ψ from (3.14) is filtere an (gr Ψ ) ζ = ψ. (3.15) Lemma 3.5. The map ζ : C l [x 1,..., x ] CS gr H (Λ) is an isomorphism of grae algebras. If in aition at least parts of λ are equal to l then the map gr Ψ, hence also the map Ψ itself, is injective.

17 Schur-Weyl uality 17 Proof. We can a extra parts equal to l to the partition λ if necessary to assume that at least parts of λ are equal to l without affecting Λ or the map ζ. Uner this assumption, the last sentence of Theorem 2.4 asserts that the map ψ is injective. Hence by (3.15) the maps ζ an gr Ψ are injective too The filtere ouble centralizer property To prove the main result of the section we just nee one more general lemma. Lemma 3.6. Let Φ : B A an Ψ : C A be homomorphisms of filtere algebras such that Φ(B) Z A (Ψ(C)), where Z A (Ψ(C)) enotes the centralizer of Ψ(C) in A. View the subalgebras Φ(B), Ψ(C) an Z A (Ψ(C)) of A as filtere algebras with filtrations inuce by the one on A, so that the associate grae algebras are naturally subalgebras of gr A. Then (gr Φ)(gr B) gr Φ(B) gr Z A (Ψ(C)) Z gr A (gr Ψ(C)) Z gr A ((gr Ψ)(gr C)). Proof. Exercise. We are going to apply this to the maps from (3.2) an (3.14): W (λ) Φ,c En C (V c ) Ψ H (Λ). (3.16) As we have alreay sai, Lemma 3.1 means that the associate grae map gr Φ,c is ientifie with the map φ,c from (3.1). Also Lemma 3.5 an (3.15) ientify gr Ψ with ψ. Hence, taking A = En C (Vc ), Theorem 2.4 establishes that (gr Φ)(gr B) = Z gr A ((gr Ψ)(gr C)) if (B, C, Φ, Ψ) = (W (λ), H (Λ) op, Φ,c, Ψ ) or if (B, C, Φ, Ψ) = (H (Λ) op, W (λ), Ψ, Φ,c ). The following theorem now follows by Lemma 3.6. Theorem 3.7. The maps Φ,c an Ψ satisfy the ouble centralizer property, i.e. Φ,c (W (λ)) = En H (Λ)(V c ), En W (λ) (Vc ) op = Ψ (H (Λ)). Moreover, at the level of associate grae algebras, gr Φ,c (W (λ)) = φ,c (U(g e )) an gr Ψ (H (Λ)) = ψ (C l [x 1,..., x ] CS ) Basis theorem for higher level Schur algebras Define the higher level Schur algebra W (λ, c) to be the image of the homomorphism Φ,c : W (λ) Hom C (Vc ). Also let H (λ, c) enote the image of Ψ : H (Λ) Hom C (Vc ) op. In the most important case c = 0, we enote W (λ, c) an H (λ, c) simply by W (λ) an H (λ). Both W (λ, c) an H (λ, c) are filtere algebras with filtrations inuce by the filtration on En C (Vc ). The last part of Theorem 3.7 shows that gr W (λ, c) = φ,c (U(g e )) = φ (U(g e )), gr H (λ, c) = ψ (C l [x 1,..., x ] CS ).

18 18 Jonathan Brunan an Alexaner Kleshchev In particular, recalling Lemmas 2.1 an 2.2, the algebra W (λ, c) is a filtere eformation of the algebra C[M e ], so ( ) im ge + 1 im W (λ, c) =. (3.17) We want to construct an explicit basis for W (λ, c) lifting the basis {ξ i,j (i, j) J /S } for the associate grae algebra from (2.7). Lemma 3.8. Suppose we are given scalars a i,j C such that the enomorphism i,j I a i,je i,j of Vc commutes with the action of H (Λ). Then a i,j = a i w,j w for all w S, a i,j = 0 if col(i k ) > col(j k ) for some k, an a i,j = a Rk (i),r k (j) + (c col(ik ) + q col(ik ) c col(jk )+1 q col(jk )+1)a i,rk (j) + a i (h k),rk (j) h k col(i h ) col(i k ) col(j h ) col(j k ) h k col(i h )>col(i k ) col(j h )>col(j k ) a i (h k),rk (j) for all k with R k (j) (interpreting the first term on the right han sie as 0 in case R k (i) = ). Proof. The fact that a i,j = a i w,j w for all w S is prove as in Lemma 2.3. By Theorem 3.7 we know that all enomorphisms commuting with H (Λ) belong to W (λ, c). By the efinition of the latter algebra, they are therefore containe in the image of the algebra U(p) uner its representation on Vc. This proves that a i,j = 0 if col(i k ) > col(j k ) for some k. For the final formula, a calculation like in the proof of Lemma 2.3, using Lemma 3.3 to compute the action of x k, gives that a Rk (i),j + (c col(ik ) + q col(ik ) n)a i,j + a i (h k),j = a i,lk (j)+(c col(jk )+q col(jk ) n)a i,j + 1 h<k col(i h ) col(i k ) 1 h<k col(j h ) col(j k ) a i,j (h k) k<h col(i h )>col(i k ) k<h col(j h )<col(j k ) a i (h k),j a i,j (h k), interpreting the first terms on either sie as zero if R k (i) or L k (j) is. Now replace j by R k (j) an simplify. Theorem 3.9. For each (i, j) J, there exists a unique element Ξ i,j = a h,k e h,k W (λ, c) h,k I such that the coefficients a h,k satisfy a i,j = 1 an a h,k = 0 for all (h, k) J with (h, k) (i, j). Moreover: (i) The elements {Ξ i,j (i, j) J /S } form a basis for W (λ, c). (ii) Letting r = col(j 1 ) col(i 1 ) + + col(j ) col(i ), Ξ i,j belongs to F r W (λ) an gr r Ξ i,j = ξ i,j.

19 Schur-Weyl uality 19 Proof. Lemma 3.8 an Theorem 3.7 show that any element i,j I a i,je i,j of W (λ, c) is uniquely etermine just by the values of the coefficients a i,j for (i, j) J /S. Since im W (λ, c) = J /S, this implies that for each (i, j) J there exists a unique element Ξ i,j W (λ, c) with coefficients a h,k satisfying a i,j = 1 an a h,k = 0 for all (h, k) J with (h, k) (i, j). Repeating this argument with W (λ, c) replace by F r W (λ, c) implies that Ξ i,j actually lies in F r W (λ, c), where r = col(j 1 ) col(i 1 ) + + col(j ) col(i ). Hence the coefficients a h,k of Ξ i,j are zero unless col(k 1 ) col(h 1 ) + + col(k ) col(h ) r. Moreover, by Lemma 3.8, a h,k = 0 if col(h j ) > col(k j ) for some j, an a h,k = a h w,k w for all w S. Now the recurrence relation in Lemma 3.8 gives a recipe to compute all remaining coefficients from the known coefficients a h,k for (h, k) J, proceeing by ownwar inuction on col(k 1 )+ +col(k ). This establishes in particular that the coefficients of top egree r, i.e. the a h,k s with col(k 1 ) col(h 1 )+ +col(k ) col(h ) = r, satisfy the same recurrence as the coefficients of ξ i,j from Lemma 2.3. Hence gr r Ξ i,j = ξ i,j. Finally, the fact that the elements {Ξ i,j (i, j) J /S } form a basis for W (λ, c) follows because the elements {ξ i,j (i, j) J /S } form a basis for gr W (λ, c). Remark Assume throughout this remark that c = 0. In that case, it is clear from the proof of Theorem 3.9 that all the coefficients of Ξ i,j are actually integers. Using this an Remark 2.5, we can exten some aspects of Theorem 3.7 to an arbitrary commutative groun ring R. Let H (Λ) Z be the subring of H (Λ) generate by x 1,..., x an S. Set H (Λ) R = R Z H (Λ) Z. Let V Z be the Z- = R Z V Z. By Lemma 3.3, the action of H (Λ) restricts to a well-efine action of H (Λ) Z on V Z, hence we also get a right action of H (Λ) R on V R by extening scalars. Let H (λ) R enote the image of the resulting homomorphism H (Λ) R En R (V R )op. Let W (λ) R = En H (Λ) R (V ). It is then the case that submoule of V spanne by {v i i I } an V R R W (λ) R = R Z W (λ) Z, H (λ) R = R Z H (λ) Z. Moreover, the ouble centralizer property En W (λ) R (V R )op = H (λ) R still hols over R. These statements are prove by an argument similar to the above, but using Remark 2.5 in place of Theorem 2.4. The first step is to observe that the elements {Ξ i,j (i, j) J /S } from Theorem 3.9 make sense over any groun ring R, an give a basis for W (λ) R as a free R-moule. 4. Parabolic category O A basic theme from now on is that results about the representation theory of H (Λ) shoul be euce from known results about category O for the Lie algebra g relative to the parabolic subalgebra p. In this section we give a brief review of the latter theory, taking our notation from [BK3, ch. 4]. We will assume for the

20 20 Jonathan Brunan an Alexaner Kleshchev remainer of the article that the origin c = (c 1,..., c l ) C l satisfies the conition c i c j Z c i = c j. (4.1) There is no loss in generality in making this assumption, because for every choice of a 1,..., a l C it is possible to write Λ = Λ a1 + + Λ al as l i=1 Λ c i+q i n for suitable λ an c satisfying (4.1) Combinatorics of tableaux By a λ-tableau we mean a filling of the boxes of the iagram λ by complex numbers. We aopt the following notations. Tab(λ) enotes the set of all λ-tableaux. Tab (λ) is the subset of Tab(λ) consisting of the tableaux whose entries are non-negative integers summing to. γ(a) = (a 1,..., a N ) C N is the column reaing of a λ-tableau A, that is, the tuple efine so that a i is the entry in the ith box of A. θ(a) enotes the content of A, that is, the element N i=1 γ a i of the free abelian group on generators {γ a a C}. A 0 enotes the groun-state tableau, that is, the λ-tableau having all entries on its ith row equal to (1 i) (recall we number rows 1,..., n from top to bottom). A c enotes the tableau obtaine from A 0 by aing c i to all of the entries in the ith column for each i = 1,..., l. Col(λ) enotes the set of all column strict λ-tableaux, where a tableau is calle column strict if in every column its entries are strictly increasing from bottom to top, working always with the partial orer on C efine by a b if (b a) N. Col c (λ) enotes the set of all A Col(λ) with the property that every entry in the ith column of A belongs to c i + Z for each i = 1,..., l. St c (λ) enotes the set of all stanar tableaux in Col c (λ), where a column strict tableau A is stanar if in every row its entries are non-ecreasing from left to right; this means that if x < y are entries on the same row of A then x appears to the left of y. Col c(λ) enotes the set of all A Col c (λ) such that the entrywise ifference (A A c ) lies in Tab (λ). St c(λ) enotes St c (λ) Col c(λ). Finally let enote the Bruhat orering on the set Col(λ), efine as in [B1, 2] (where it is enote ). Roughly speaking, this is generate by the basic move of swapping an entry in some column with a smaller entry in some column further to the right, then reorering columns in increasing orer. For example, > > >

21 Schur-Weyl uality 21 We note by [B1, Lemma 1] that the restriction of this orering to the subset St c (λ) coincies with the restriction of the Bruhat orering on row stanar tableaux from [BK3, (4.1)]. The first combinatorial lemma is the reason the assumption (4.1) is necessary. Lemma 4.1. Suppose that A Col c(λ) an B Col c (λ) satisfy θ(a) = θ(b). Then B Col c(λ) too. Proof. Suppose that B / Col c(λ). Then, for some i an j, the entry in the ith row an jth column of B is < (c j + 1 i). Hence as B is column strict, the entry in the nth row an jth column of B is < (c j + 1 n). As θ(a) = θ(b), this implies that some entry of A is < (c j + 1 n) too. This contraicts the assumption that A Col c(λ) Parabolic category O Now let O(λ) enote the category of all finitely generate g-moules that are locally finite imensional over p an semisimple over h; recall here that h is the stanar Levi subalgebra gl q1 (C) gl ql (C) of g. Moules in O(λ) are automatically iagonalizable with respect to the stanar Cartan subalgebra consisting of iagonal matrices in g. We say that a vector v is of weight (a 1,..., a N ) C N if e i,i v = a i v for each i = 1,..., N. There is the usual ominance orering on the set C N of weights, positive roots coming from the stanar Borel subalgebra of upper triangular matrices in g. Also let? # enote the uality on O(λ) mapping a moule to the irect sum of the uals of its weight spaces with g-action efine via the antiautomorphism x x t (matrix transposition). To a column strict tableau A Col(λ) we associate the following moules: N(A) is the usual parabolic Verma moule in O(λ) of highest weight (a 1, a 2 + 1,..., a N + N 1) where γ(a) = (a 1,..., a N ) is the column reaing of A. K(A) is the unique irreucible quotient of N(A). P (A) is the projective cover of K(A) in O(λ). V (A) enotes the finite imensional irreucible h-moule of highest weight γ(a A 0 ), viewe as a p-moule via the natural projection p h. V (A 0 ) is the trivial p-moule C. V (A c ) is the one imensional p-moule C c on which e i,j acts as δ i,j c col(i) as in the previous section. We point out that N(A) = U(g) U(p) (C ρ V (A)) (4.2) where C ρ is the one imensional p-moule from the introuction; cf. [BK3, (4.38)]. Moreover, {K(A) A Col(λ)} is a full set of irreucible moules in O(λ). A stanar fact is that K(A) an K(B) have the same central character if an only if θ(a) = θ(b); inee, the composition multiplicity [N(A) : K(B)] is zero unless A B in the Bruhat orering. Recall also that BGG reciprocity in

22 22 Jonathan Brunan an Alexaner Kleshchev this setting asserts that P (A) has a parabolic Verma flag, an the multiplicity of N(B) in any parabolic Verma flag of P (A) is given by (P (A) : N(B)) = im Hom g (P (A), N(B) # ) = [N(B) : K(A)]. (4.3) One consequence of this is that Ext 1 O(λ)(N(A), N(B)) = 0 (4.4) for all A, B Col(λ) with A B. From now on, we restrict our attention to the Serre subcategory O c (λ) of O(λ) generate by the irreucible moules {K(A) A Col c (λ)}. This is a sum of certain blocks of O(λ), an conversely every block of O(λ) is a block of O c (λ) for some c chosen as in (4.1). For 0, let O c(λ) enote the Serre subcategory of O c (λ) generate by the irreucible moules {K(A) A Col c(λ)}. Letting c r1 enote the tuple (c 1 r,..., c l r) for each r N, it is obvious that Col c(λ) Col +rn c r1 (λ) an Col c (λ) = Col c r1 (λ). Hence, O c(λ) O +rn c r1 (λ), O c(λ) = O c r1 (λ). (4.5) This is useful because, by Lemma 4.1, O c(λ) is a sum of blocks of O c (λ), an conversely every block of O c (λ) is a block of O c r1(λ) for some r, 0. In this way, questions about O(λ) can usually be reuce to questions about the subcategories O c(λ). In particular, since Col 0 c(λ) = {A c }, we get that K(A c ) is the unique (up to isomorphism) irreucible moule in its block, hence P (A c ) = N(A c ) = K(A c ). This moule plays a special role; we enote it simply by P c. Lemma 4.2. Let A Col(λ). Then K(A) is a composition factor of P c V if an only if A Col c(λ). Proof. Recall that C c V = Vc. By the classical Schur-Weyl uality, Vc ecomposes as an h-moule as a irect sum of copies of the moules V (A) for A Col c(λ), an each such V (A) appears at least once. Hence, Vc has a filtration as a p-moule with sections of the form V (A) for A Col c(λ), an each such V (A) appears at least once. By the tensor ientity, P c V = U(g) U(p) (C ρ V c ). Recalling (4.2), these two statements imply that P c V has a parabolic Verma flag with sections of the form N(A) for A Col c(λ), an each such N(A) appears at least once. This shows that every K(A) for A Col c(λ) is a composition factor of P c V. Conversely, if K(A) is a composition factor of P c V for some A Col(λ), it is also a composition factor of N(B) for some A B Col c(λ). Hence by Lemma 4.1 we see that A Col c(λ) too.