Lusztig conjectures, old and new, I - Department of Mathematics

J. reine angew. Math. 455 (1994), 141-182 Journal fiir die reine undangewandte Mathematik0 Walter de GruyterBerlin New York 1994Lusztig conjectures,old and new, IBy Jie Du at Kensingtonand Leonard Scott at Charlottesviller)This paper grew from a first attempt to understand the relationship of a new conjectureof Lusztig [L2] for representations of quantized enveloping algebras to hisconjecture [Ll] for representations of algebraic groups in characteristic p. The mainquestion here is: Does the new conjecture imply the old? Using results of [APW], it is easyto see this question has an affirmative answer if appropriate irreducible representations of aquantum enveloping algebra at a pth root of unity remain irreducible upon ‘reductionmodp’. (Equivalently, the quantum and characteristic p irreducible modules involved,labelled by the same weight, must have the same dimension.) Indeed, Lusztig has conjecturedthis irreducibility in an equivalent context [L3], where the weights involved arerestricted, and p is sufficiently large. Another standard hypothesis (equivalent forp 2 2 h - 3) requires that p be at least as large as the Coxeter number h, and that the weightsin question belong to the Jantzen region, which is contained in the lowest p2-alcove.The main question above has been made even more interesting by an announcement ofKazhdan and Lusztig [KL] that they can prove the new conjecture, assuming resultsasserted by Casian, at least in the case of a simply-laced root system.We investigate here a cruder model of the question, a more general problem for quasihereditaryalgebras over regular local rings of Krull dimension two, where the ‘lowest p2-alcove’ condition is replaced by a hypothesis on localizations. We find that the more generalproblem has a positive answer under a multiplicity-free assumption on characteristic p Weylmodules, cf. 54, but is false in general. A counterexample by Alfred Wiedemann is given inan appendix. Of course, this does not mean that the main question itself has a negativeanswer, only that it is a difficult problem. Indeed, one can see with hindsight that suchcounterexamples had to exist, owing to differences between the ‘lowest p2-alcove’, which ourmodel captures, and the precise Jantzen region, which the model is too crude to recognize.‘) The authors would like to thank NSF for support under the Group Project Grant DMS-890-2661, andthe Universities of Virginia and Oklahoma for their cooperation. We would also like to thank MSRI at Berkeley forits hospitality. Most of the research in this paper was completed during the program on representations of finitegroups and related topics at MSRI in Fall 1990. The first author thanks ARC for support under the Large GrantL38.24210.10 Journal ftir Mathematlk Band 455

142 Du and Scott, Lusztig conjectures IThe main thread of the investigation is as follows: 5 1 contains many results of generalinterest regarding representations of algebras over commutative rings. $2 discusses thenotion of a highest weight category in this context, and shows how to construct quasihereditaryalgebras from them. This theory is applied to the quantum case in $3. Themultiplicity-free results mentioned above are then obtained in 94, after constructing suitablequasihereditary algebras. Equivalently, one may view the above results as giving a richsupply of ‘projective’ modules. The existence of (endomorphism ring idempotents associatedwith) these modules is already sufficient to prove, for a given multiplicity-free Weylmodule with high weight in the lowest p2-alcove, a diagonalization conjecture of Andersen-Polo-Wen [APW]. This in turn gives the required equality of dimensions for thecorresponding quantum and characteristic p irreducible modules. Even for Weyl moduleswhich are not multiplicity-free, our results give information on the equality of dimensions forsome weight spaces of these irreducible modules.An interesting feature of our work is that, though we consider only weights in thelowest p2-alcove, we require no serious restriction on p itself, beyond standard exclusionsfor p = 2 or 3. Since this paper was first written, Soergel has announced on behalf ofAndersen, Jantzen and himself that they can answer the original main question affirmativelyfor any sufficiently large prime (size unknown) depending on the root system2). While this‘large prime’ result is certainly a step forward, the main question itself remains open, asdoes the original characteristic p Lusztig conjecture. Much of the interest in the Lusztigconjecture (as well as James’ conjecture below) grew from issues in finite group theory [S]involving groups of arbitrary characteristic, so the size of the prime is important.We hope to make further remarks regarding the main question elsewhere. For acompletely different approach to the Lusztig conjecture, see [CPS 51 and [CPS 61.This investigation has proved rich in byproducts. In addition to the above generalresults in $1 and 0 2, they include the new notion of generalized q-Schur algebra (9 3, 5 5),and a proof that all finite rank representations of a quantized enveloping algebra (over thenatural Krull dimension two local rings) are integrable (5 3). Since the q-Schur algebra itselfhas an integral quasihereditary version [CPS4], our results have consequences for thedecompositions of standard permutation modules of symmetric groups in arbitrary characteristic.A few applications in this direction are discussed in 44. In particular, we prove, inthe special case of multiplicity-free Weyl modules, a conjecture of Gordon James for q-Schur algebras [J].James’ conjecture is analogous to that of Lusztig [L 31, but makes sense for specializingq to an arbitrary nonzero element of a$nitejkZd, where q automatically becomes a primitivelth-root of unity for some positive integer 1. As per the Dipper-James theory [DJI], theresults here also have consequences for representations of the finite general linear groupsin their nondescribing characteristics. At the 1990 MSRI algebraic groups conference, theauthors of [CPS7] conjectured that representations of quantum groups over finite fields(equivalently, of generalized q-Schur algebras) would eventually allow the extension of theDipper-James theory to all finite groups of Lie type. In 6 5 we indicate how our results extendto these quantum group representations, working with the lowest Ip-alcove.‘) This work has now appeared as A&risque 220

Du and Scott, Lusztig conjectures I 143After this paper was first written, Z. Lin pointed out that a crude version of thequantum multiplicity-free results for I= p could be obtained from the Jan&en sum formula.Inspired by his remarks, we expand upon this theme in a second appendix. In the finalanalysis, the sum formula adds no essentially new ingredient to the above theory, thoughthe general viewpoint we present may be of computational interest.A third appendix treats an integrability result for quantum groups over fields, and$2 itself has an appendix, which formalizes a criterion used in [PS] to check the highestweight category axioms.We would like to thank D. Costa and W. van der Kallen for a number of references anddiscussions regarding the ring n [q, q- ‘1 and its variations.1. Finite and complete k-categoriesThroughout this paper k is a Noetherian commutative ring, which we will momentarilyassume also to be local. A k-category ‘% is an additive category in which the setHom,(M, N) of morphisms between any two objects M, N of %? is equipped with thestructure of a k-module, and multiplication of morphisms is k-bilinear.1.0. Convention. All k-categories considered in this paper are abelian, and equippedwith a fixed exact embedding into the category of k-modules.In particular, the objects of the category objects may be regarded as k-modules withsome additional structure (e.g., the category of modules for a k-algebra). The full subcategoryof k-finite objects in %? (objects whose underlying k-module is finitely generated)is denoted G$. A k-category % is jinite (we also say k-jinite) if $7 = ‘$$.Even without our convention above, any abelian k-category in which the objects forma set may, through standard embedding theory [F], be exactly embedded into the categoryof k-modules. However, our convention allows a crude way, adaquate for the aims of thispaper, of keeping track of finiteness and other k-module properties.We will also make use of the fact that Yoneda Ext groups can be defined in any abeliancategory, subject to some care regarding set-theoretic issues, and that the long exact sequencesfor Ext are available [M], XII. 5, XI. 1.Suppose now for the rest of this section that k is local, with maximal ideal m and residuefield k = k/m. If M is any k-module or object in a k-category %‘, the quotient M/Mm isdenoted M. Note that Mm makes sense in 97, independent of any k-module structure on M,as the sum of the images of the multiplication maps M + M associated to a finite set ofgenerators of m. The full subcategory of objects M in V on which m acts trivially is denoted59. It is naturally a k-category.There are obviously many examples of k-categories. Here is an especially importantone for the purposes of this paper:

144 Du and Scott, Lusztig conjectures I1.1. Example. Let k be the localization of the polynomial ring Z [q] at the maximalideal m = (p, q - 1) where p is a fixed odd prime. Let U be the quantum enveloping algebraover k associated to a fixed classical root system as in [APW], introduced by Lusztig in [L 21.If the root system contains a component of type G,, make the assumption of [APW] thatp > 3. Let V = g(U) be the category of k-finite U-modules M for which M is type 1, inthat the elements standardly denoted Ki act as 1 on M, cf. [APW], 9.5. We will showmomentarily in 1.3 that % is an abelian category, and thus is a finite k-category. We call theobjects in (63 the k-finite U-modules of type 1.Differences in notation. The ring k is denoted by d by Andersen-Polo-Wen [APW],and they use k to denote what we call k. Also, they use the symbol %? to denote the categoryof “integrable modules of type 1” for U, in their sense. In the k-finite case their integrabletype 1 modules are easily seen to be type 1 in our sense also. A nontrivial consequenceof the present paper is that all k-finite U-modules are indeed integrable, so that, ultimately,our category V consists precisely of the d-finite objects in theirs.It is proved in [APW], 9.5 that @ (in our notation) coincides naturally with thecategory of finite-dimensional modules for the hyperalgebra over k of the semisimple groupassociated with the given root system. It is also proved that each Weyl module of @ lifts to ak-free module in %‘.Eventually, we will use these few facts here (with some additional input from the tensorproduct structure of V) to exhibit a “highest weight category” structure on %? in the spirit of[CPS 21. At the same time, we define the notion of a generalized q-Schur algebra. The notionof a generalized Schur algebra (without the q) is due independently to [CPS l] and Donkin[Do 11; these algebras made a notion of Schur algebra (studied originally by Schur for typeA) available for all types, collectively covering all finite-dimensional rational representationsof semisimple algebraic groups. The new generalized q-Schur algebras of this paper similarlymake available a notion of q-Schur algebra (defined originally by Dipper and James [DJ 11,[DJ 21) for all types, and collectively cover the type 1 finite-dimensional representations ofthe above quantum enveloping algebras (associated to finite root systems). We give integralversions of these algebras, in the spirit of the CPS integral version of the q-Schur algebra[CPS4], and prove in most cases these new algebras are also k-quasihereditary in the integralsense of [CPS4]. (Other localizations besides k of Z [q, q- ‘1 are also studied in the lastsection of this paper.)With these aims in mind, we now return to the general k-categoryfirst the extent that vanishing of Extl for objects in %? reduces to @.%‘, and investigateFor later use we observe that, if P is a k-projective object in V, and T is in @ thenExt&(P, T) g Exti(P, T). If each k-projective object (such as P) in %? is the image of aprojective and k-projective object of %?, this isomorphism may be extended to the higher Extgroups by dimension-shifting. Even without this hypothesis, there is always a naturalinjection Ext$ (P, T) -+ Exti(P, T), as follows from a standard interpretation of Ext’ interms of obstructions [M]; XII, Lemma 5.3.If % is a k-category and M is a k-finite object (an object in %“), we will call anycomposition factor of the finite length object &f = M/Mm a “composition factor” of M.“Composition factors” are only well-defined up to isomorphism. In this sense there are only

Du and Scott, Lusztig conjectures I 145finitely many of them for each object M. The following proposition“composition factor” agrees with another plausible definition.shows this notion of1.2. Proposition. Suppose k is local, $? is a k-category, and M is a k:jinite object in V.Then any simple object of %? which is a quotient of a subobject qf M is a composition factor ofii? = M/Mm.Proof. Let N’ E N s M be inclusions in %? such that S = N/N’ is simple. Thus, Sis a composition factor of NjNm in the usual sense. However, for some large n we haveNnt 2 Mm” n N. So S is a composition factor of the finite length object N/(Mm” n N), andthus of M/Mm”. However, for each nonnegative integer Y, the object Mm’/Mm*+’ is thesum of the images in M/Mm’+’ of M/Mm under the maps induced by multiplicationfrom each of a finite set of generators for m’. The proposition follows easily. q1.3. Corollary. Let k and %? be as above, and letbe an exact sequence of k-finite objects in W. Then the set of composition factors ofii? = M/Mm is the union of the set of composition,factors of L and the set of compositionfactors of N. 0That is, the set of “composition factors” of M is the union of the sets of “compositionfactors” for L and N. In particular, it is nowclear that the category described in example1.1 is abelian.If k is local and complete, with maximal ideal m, we say that a k-category 9 is completeif it is finite, and if every object M is the inverse limit in %7 of its quotient objectsM/Mm”.1.4. Proposition. Suppose k is local and complete, and %? is a finite k-category. AssumeV has an exact and full k-linear embedding into the category of k-finite modules for a k-algebra U (not necessarily itself k-finite).Then M z I& M/Mm” in %? for each object M of $9.Proof. Let M be an object of %‘. As in the proof above, the k-submodules Mm”, forn a nonnegative integer, all make sense as subobjects of M in %?. To test an isomorphismM E I@ M/Mm” in %‘, we may test the analogous statement in the category of k-finite U-modules. Hence we may assume ‘8 is the category of k-finite-U-modules. (Such reductionsare often a problem with limits, but here we already have the candidate for the inverselimit, and just need to check that it satisfies the defining universal property.)Certainly M z @ M/Mm” in the category of finite k-modules: This is clear for M afinite direct sum of copies of k, and follows for M finitely presented from the Artin-Reestheory, cf. [AM], Prop.lO.12. However, if N is a U-module, and f is a k-linear mapf: N + M such that f induces a U-equivariant map N + M/Mm” for each n 2 0, then fis obviously itself U-equivariant. The proposition follows. q

146 Du and Scott, Lusztig conjectures 1We now come to the main result of this section. The proof is inspired by Auslander-Goldman’s argument [AGI], [AG2] that separability of k-algebras, for k local andNoetherian, can be checked at the reductions modulo the maximal ideal of k. As before, ifk is local and M is an object in a k-category, we let M denote the object M/Mm, wherem is the maximal ideal of k.1.5. Theorem. Let k be a complete commutative Noetherian local ring, and V a completek-category. Suppose P and M are objects of % such that P is k-free, and Ext& (P, S) = 0(equivalently, Ext+(P, S) = 0) f or every composition factor S of it?.Then Ext&(P, M) = 0, and every morphism in %? from P to a quotient M’ of M ltfts toa morphism from P to M.Proof. The last assertion about lifting morphisms is a consequence of the Ext’ conclusion,applied with M replaced by the kernel of M + M’, since the “composition factors”of that kernel are among those for M.Thus, it is enough to prove Ext&(P, M) = 0. As is well-known,x E Extb(P, M), and E = E, is a corresponding short exact sequence:cf. [M], III, 1.7, ifO-+M-tE+P+O,then x maps to zero in Exti(P, E). Hence, it suffices to show that the functor Hom,(P, -)yields an exact sequence when applied to the above sequence.However,each of the sequences0 + M/Mm” + E/Em” + P/Pm” + 0,is exact, since P is k-free. Also, as noted in the first proposition of this section, all the compositionfactors of the (finite-length) object M/Mm” occur already as composition factorsof M. Thus Exti (P, M/Mm”) = 0, and so Horn, (P, -) applied to the above sequence givesagain a short exact sequence, for each nonnegative integer n.A similar argumentusing the exact sequences0 + Mm”/Mm”+’ -+ M/Mm”+’ + M/Mm” + 0shows that the transition maps Hom,(P, M/Mm”+’ ) + Horn, (P, M/Mm”) are all surjective.As is well-known, this implies that there is an exact sequence of inverse limits of k-modules0 -+ l@Hom,(P, M/Mm”) + @Horn, (P, E/Em”) -+ l$nHom,(P, P/Pm”) + 0The theorem now follows from the previous proposition. q1.6. Corollary. Suppose k is a commutative local Noetherian ring, and %? is a finitek-category. Assume % has an exact andfull k-linear embedding into the category of k-$nite

Du and Scott, Lusztig conjectures I 147modules for a k-algebra U (not necessarily itself k-finite). Also assume that the subcategory @is closed under extensions with respect to U-modules: for all objects T, S in @, any extensionof T by S as a U-module is isomorphic to a D-module in (the image of) $?.Suppose P and A4 are objects of %? such that P is k-free, and Ext&(P, S) = 0 for everycomposition factor of S of ii?. Then Ext&(P, M) = 0, and every morphism in ‘2 from P to aquotient of M lifts to a morphism from P to M.Proof As before, it is enough to show Ext,$(P, M) = 0. Let E be an extension of Pby A4 in % Let V denote the image of U in the k-endomorphism ring of E. Thus V acts onM and P, viewed as sections of E. The extension E is split as a V-module if and only if it issplit in U, and the latter occurs if and only if E is split in %?.Routine arguments from Noetherian ring theory show V is finitely generated as a k-module. In particular, it follows from standard arguments with resolutions thatExt;(X, Y) 0 k “= Ext;,ac(X@ & Y@ k)for any k-finite V-modules X and Y. Here k denotes the completionmaximal ideal.of k with respect to itsNow, any U-compositon factor of a is a V-composition factor, as well as a V@ k-composition factor. We also note that the reduction modulo m of I/@ k is just V, which isa homomorphic image of u. In particular Ext+(P, S) = Ext&(P, S) = Ext$(P, S) = 0 forall F-composition factors S of &?. (The second equation uses our hypothesis that G?? is closedunder extensions in the category of U-modules.) Thus Extb, i(P @ k, M@ k) = 0 by thetheorem, applied to the category of V@ k-modules. The displayed isomorphism now givesEx$(P, M) = 0, thus Ext$(P, M) = 0, and it follows from the full embedding of $9 thatExt:,(P, M) = 0. •I1.7. Corollary. If %? is a finite k-category which is complete or satis$es the assumptionsof the previous corollary, and P is an object of %’ such that P is projective in 0, then P isprojective in V. 02. k-finite highest weight categoriesTo begin, let k be a field; later we will consider Noetherian local rings, and even somenonlocal cases. Let /i be a locally finite poset (that is, the intervals of /1 are all finite). Theelements in /1 will be called weights. We introduce here a variation on the notion [CPS2],[CPS 31, of a highest weight category, which we shall call a k-finite highest weight categorywith weight poset /i.This is an abelian k-category%? in which every objects is k-finite, and in addition:(a) The nonisomorphicdistinct elements of /1.simple objects in %’ are indexed as L (2)) with 1 ranging over the

148 Du and Scott, Lusztig conjectures I(b) There are given objects V(n) and A (2) in V, called the Weyl modules and dual Weylmodules, respectively, for each i in A, such that(i) V(A) has a simple head L(n), and A (2) has a simple socle L(A). All othercomposition factors L(U) of V(n) or A (2) have weights p < ;1;(ii) Ext”,(V(1),A(p))= 0 for n = I,2 and each A, pin/i.These conditions are just translations of analogous conditions discovered in Chapter5 of [PS] in the study of perverse sheaves. Recall that an ideal in a poset is a subset containingany element of the poset smaller than some member of the subset; a coideal is asubset containing all elements of the poset larger than some member. Given a finitelygenerated ideal r of /1, the conditions imply, cf. 2 A. 3 below, the existence of projectivecovers in the category Q?[r] formed by objects whose composition factors L(p) all haveweights in r. These projective covers all have Weyl filtrations (filtrations whose successivequotients are Weyl modules). It follows that for /i finite, ‘% is a highest weight category inour previous sense [CPS2], and is the category of finite-dimensional modules for a quasihereditaryalgebra [CPS2], Thm. 3.6. The converse is clear, with (ii) even holding for alln > 0. The appendix to this section establishes a similar fact in general, cf. 2 A. 6.2.0. Remark. We remark that axiom (ii) is designed for easy verification, and may bechecked in any exact or abelian k-category @ in which V is fully and exactly embedded andclosed under extensions. These latter conditions imply Ext&(X, Y) g Ext&(X, Y), for anyobjects X, Y of V, and that the natural map Exti(X, Y) + Ext,$(X, Y) is an injection, cf.[M], XII, Lemma 5.3. Also, axiom (ii) behaves well with respect to the natural quotientmaps arising in the theory of quasihereditary algebras, cf. [PI, [CPS3].The above discussion of projective covers implies the following condition holds in ak-finite highest weight category, which we record for comparison with the local integraltheory which follows.(c) Let T be an object in %7. Then T is the epimorphichas a Weyl filtration.image of an object P in $F? whichIn fact, let r be the ideal in /i generated by all the weights h for which the simplemodule L(p) is a composition factor of T. By 2 A. 3 below, each L (,u) has a projective coverin %?[r], filtered by Weyl modules. Consequently T has such a projective cover. qWe shall now study the local integral case. For the rest of this section k is a commutativeNoetherian ring. We will also assume that k is local, unless explicitly allowed otherwise.As mentioned in 9 1, we are especially interested in the case where k is the ring Z [q],of [L4] and [APW], the localization of the polynomial ring Z[q] at the ideal generatedby a fixed prime p and q - 1. Our theory will also apply for the completion, which has anumber of advantages. Other important choices of k include localizations of Z [q], atvarious primes.At the end of 0 4 and in 9 5, in results oriented toward the Dipper-James theory, oneis interested in essentially all localizations of Z[q], or at least of Z [q, q- ‘1.

Du and Scott, Lusztig conjectures I 149In general we denote the maximal ideal of k as m and the residue field is k. Thereduction modulo m of any k-module A4 is denoted M.2.1. Definition. A k-Jinite highest weight category is a category @ of k-finitelygenerated objects, together with a locally finite poset /1, such that(i) the full subcategory 0 of objects M, with A4 in %‘, is a k-finite highest weightcategory with weight poset LI, as defined above;(ii) there are k-projective (thus free) objects V(J) and A(i) in @? whose reductionsV(L) and A (2) modulo m are, respectively, the Weyl modules and dual Weyl modules of @In addition,we will assume:(iii) Every object T in 5?? is the epimorphicfiltration;image of an object M in %? which has a Weyl(iv) if P, M are objects of %‘, with P k-free, andExt$(e L) = 0for all compositionfactors L of M, thenExt;(P, M) = 0We recall from 5 1 that this last axiom is satisfied if the k-category structure of Q? arisesfrom an exact and full embedding as subcategory, closed under extensions, of k-finitemodules for a k-algebra, the latter not necessarily itself k-finite.We also remark that the axiom also holds if the embeddings exist for enough abeliansubcategories to cover all finite sets of objects of %?, assuming the embeddings preserve k-freeness as prescribed by the k-category structure of ‘F, in the sense of our convention 1 .O. Inthe same spirit, one could probably weaken 1 .O to allow a family of embeddings to describethe needed k-category structure. No claim is made that the above axioms are optimal, butthey do seem adequate to relate the integral quantum enveloping algebra theory of [APW]to the integral quasihereditary theory of [CPS4], as we will shortly demonstrate. Noteespecially that2.1.1. Proposition. The category of k-jinite modules for a k-quasihereditary algebra isa k-finite highest weight category. qThe proof is obtained easily by imitating the argument for the corresponding resultover fields [CPS2]. In the nonlocal case we can say that the above proposition holds withrespect to some global objects V(A), A (A) provided the k-quasihereditary algebra is k-split,cf. the discussion at the end of [CPS4]. (As usual, the A’s may be constructed from the V’sfor the opposite algebra.) Explicitly, we have2.1.2. Proposition. Suppose k is a commutative Noetherian ring, not necessarily local,and Y is a split k-quasihereditary algebra. Then there exists aJinite poset A and k-projective

150 Du and Scott, Lusztig conjectures 1objects V(n) and A (3L) for 1 E A such that, for any prime ideal @, the category of k,-finite 5$modulesis a k,-finite highest weight category with weight poset A and with the localizationsof V(n) and A(1) as Weyl and dual Weyl modules, respectively, for 2 E A. q2.1.3. Remark and definition. This last proposition shows, in our view, that the categoryof k-finite modules for a split k-finite quasihereditary algebra deserves to be called ak-finite highest weight category even in the nonlocal case. This ‘definition’ for the split casecould easily be extended to (interval-finite) weight posets bounded below by finitely manyminimal elements: A (split) k-jinite highest weight category would be, simply, the compatibleunion of the categories of all k-finite modules for split k-quasihereditary algebras associatedto finite ideals of the weight poset. This definition is completely adequate for all thecases considered in this paper, even in the nonlocal case. The more abstract development ofthis section is, however, necessary to demonstrate the existence of the required algebras.For the rest of this section %? denotes a k-finite highest weight category, and the notationof 1 .I is in force, unless otherwise noted. We will also freely use the “composition factor”theory and notation of the previous section (and, henceforth, drop the quotation marks). Ifr is an ideal in the weight poset A, then %?[r] denotes the category of objects in %? all ofwhose composition factors are indexed by weights in r. It is easy to see from the axioms that%?[r] is also a k-finite highest weight category.2.2. Lemma. Let A E A. Then V(1) is projective in the full subcategory of objects X of$7 which have no composition factor L(v) satisfying A < v. In particular Ext& (V(n), V(v)) = 0unless 1 < v. (The same conclusion for higher Ext’s follows after 2.8 below.)Proof. By 2 A. 2 we have Ext$( V(J), R) = 0, and so Extk( V(n), X) = 0 by axiom(iv). The lemma follows. q2.3. Proposition. Let G?? be a k-finite highest weight category, let I be any finitelygenerated ideal in the weight poset A, and let T be the projective cover in @[I] of a simpleobject L(A) with 2 in I (cf 2A.3 below).Then there is a k-free object P in @[I], filtered by Weyl modules, with P g T. Such anobject P is projective, the projective cover of L(1) in %(I], and is uniquely determined up toisomorphism.Proof. If P exists, it is clearly projective by axiom (iv), and L(A) is its unique simplequotient. That is, P is a projective cover of L(A) in %[r]. Since k is Noetherian, and P isk-finite, every surjective endomorphism of P is an automorphism, and the uniqueness of Pfollows by a standard argument.We now address the existence. Let M be an object in V filtered by Weyl modules whichhas T as an epimorphic image. Applying the above lemma, we may assume all the Weylmodules appearing in M are indexed by members of r. We claim M contains a subobjectP filtered by Weyl modules with P z T, and we will prove this claim by induction on thenumber of weights in r which are 1 A. Write r = r’ u {v}, where v 2 A is maximal in r. Byinduction, we may assume that v appears as a weight of a composition factor of T.

Du and Scott, Lusztig conjectures I 151Therefore ,5(v) must appear as a composition factor of M, and, thus, of one of themembers of its Weyl filtration, V. We have I/ 2: V(v) by maximality of v. Shuffling terms ofthe Weyl filtration of M, using 2.2, we may assume all of the terms isomorphic to V(v)appear at the bottom, in a direct sum A4, of copies of V(v). Now the k-Weyl filtration ofT has a similar form, giving rise to an exact sequenceO+T,+T+T’+O,where TV is a direct sum of copies of V(v), and T’ is the projective cover of L(A) in @[PI].Obviously M,, maps into K, and even onto T,, since all composition factors of M/M,belong to $? [P’]. Now the radical quotient of A4, is a direct sum of copies of L(v), correspondingto the decomposition of M, as a sum of copies of V(v). Clearly, some subset ofthis set of copies of L(v) has sum which maps isomorphically onto the radical quotient ofTV. It follows that we may writewith each factor a direct sum of V(v)‘s, and q isomorphic to TV under the map M, -+ TV.The kernel X/m M,, of M,, + TV is isomorphic to M,“. Now V(v) is projective in the cateogryof objects in %? whose composition factors are indexed by weights 5 v, by 2.2. Thus, there is amap A!,” -+ X which maps surjectively onto X/tnM,. The resulting sum mapis also surjective, and thus an isomorphism. The conclusion is that we may choose M,” inthe displayed isomorphism so that it is in the kernel of M + T.’ The module M/M, is filtered by Weyl modules indexed by weights in P’, and mapssurjectively onto T’. By induction, it contains a submodule PI/M, filtered by Weyl moduleswith P’/M, z T’. Thus P//My g T. In particular, P’/M: is projective in the category ofobjects of ‘G9 whose composition factors are indexed by weights in P, so P’ contains a subobjectP complementary to M,“. Clearly P g T, and this completes the proof. q2.4. Definition. If r is a finitely generated ideal in the poset A, the object P constructedabove will be called the PIM associated to 3, in ‘%[P], and will be denotedP(1) = P,(A).2.5. Corollary. Suppose A is finite, and each End, (L (2)) is a separable k-algebra.Then V is equivalent to the cateogory of k-finite modules for a quasihereditary k-algebra Y ofseparable type, in the sense of [CPS 41, Also, a defining sequence of ideals Ji of Y may be chosenfor any linear order 1, >= 2, 2 . . . of the$nite set A, compatible with itsposet structure, so thatthe following property holds: For ri 5 A defined as the subset consisting of all weights 5 li inthe linear order, V[P,] is equivalent, under the given equivalence for V, to the categoryof k-finite modules for 9’1 Ji.Proof. Let P be the direct sum of the projective covers P(2) = P,(k), 1 E A, and putY = End,(P). Fix a linear order A, 1 AZ 2 ... as above, put Pi = @ P(lj), for each subjsiscript i, and defineJi = Horn,, (P, Pi) Horn, (Pi, P) s Horn&P, P) = Y.

152 Du and Scott, Lusztig conjectures IThen Ji is obviously an ideal in 9 Let Ji denote the image of Ji in 9 (This notation apparentlydiffers from our standard convention, though it will turn out to be the same.)Since P is projective, we have Hom,(P, Pi) g Hom,(P, Pi). SoJi = Horn, (P, Pi) Horn&P,, P)in 2 identifying--the latter with Hom,(P, P). Now $? is a highest weight category in thesense of [CPS2]. The proof of [CPS2], Theorem 3.6 shows that 9 is quasihereditary withdefining sequenceand it is clear that 9 is of separable type, since all the endomorphism algebra of its simplemodules are separable. Thus Y is quasihereditary as a k-algebra, and of separable type, by[CPS 41, Thm. 3.3 (b). q2.6. Remarks. (a) The separability hypotheses are just needed to restrict attention tomaximal ideals. If k is a DVR (e.g., the localization of Z [q, q- ‘1 at one of its height 1 primeideals), the separability hypothesis may be dropped if it is known, say, that the localizationof %? over the quotient field of k is semsimple, with the Weyl and dual Weyl modules becomingsimple. More generally, it would be sufficient to assume, for any k, that the localization atany prime ideal gave rise to a highest weight category, with the localizations of the Weyl anddual Weyl modules serving in the same role in the iocalizations.In fact the separability hypothesis is harmless in this paper, where in all examples theendomorphisms of simple modules are the underlying field. (That is, the split condition of[CPS4] holds in the examples.)(b) One can construct an algebra Morita equivalent to that in the above proof bytaking the endomorphism ring of a different projective generator. Another natural choicefor Y is obtained by taking P to bethe direct sum of dim Endy (L(A)) copies of each P(l)and putting Y = End,(P). This has the following advantage:Suppose it happens that there is a k-algebra U such that 5% consists of all k-finiteright U-modules with composition factors (L(A)},,,, in the sense of the previous section.Then Horn, (U, L(A)) g L (2) as a k-module, and it follows that the right U-module Y is ahomomorphic image of the right U-module U. Indeed it is easy to check that Y is themaximal such image in g, that the kernel of the map to Y is an ideal, and that the k-algebra structure of Y is inherited from U.That is, Y may be simply described as the unique maximal k-algebra homomorphicimage of U with composition factors of its right U-module structure among the L(A), i E A.Of course, it is not obvious a priori that any such k-algebra exists, not to mention thefact that it is quasihereditary (and, in particular, k-free). If U is k-free (or even torsion

Du and Scott, Lusztig conjectures I 153free), then there is an obvious candidate for Y: Let K be the quotient field of k. Then YKis a finite-dimensional K-algebra, and it is not hard, using lattices, to argue that it is themaximal homomorphic image of U with a given set of composition factors (indexed by /i).If ZK denotes the ideal of UK defining YK, then the ideal Z = ZK A U defines 9 Though it isnot easy to prove anything from this description, it at least provides a common startingpoint in the nonlocal case, where U has arisen by localization from an algebra over a commutativering localizing to k. See 9 5.We again let n be an arbitrary locally-finite poset, and give some further corollaries ofthe existence of the projective covers P,(k).2.7. Corollary. Let Z be a finitely-generated ideal of the poset A, and let A, p E Z.Then the multiplicity of V(p) in a Weyl$ltration of P,(A) is equal to the multiplicity of L(A)in A(p).The proof is the same as in [CPS 21. In standard examples where $? [Z] has a “duality”[CPS3], the multiplicity L(I) of in A(p) is equal to that in V(u). q2.8. Corollary. Let Z be a$nitely-generated ideal of the poset A. Then the inclusion%?[Z] 2 % of k-finite highest weight categories induces a full embedding of the correspondingbounded derived categories. Zn particular Ext,&,, (X, Y) E Ext”, (X, Y) for each X, Y in V [r]and each integer n 2 0.The proof is the same in spirit as that of the similar result 2 A. 5 below, and furtherdetails are omitted. For emphasis, we repeat our earlier remark that V[Z] is itself a k-finite highest weight category, as can be proved directly from the axioms. q2.9. Corollary. Let X, Y be objects in %?.(a) Zf X is k-free, thenfor all n 2 0.Ext$(X, 7) g Ext;(X, Y)(b) If Y is k-f ree, there is an E, spectral sequenceT$(Ext$(X, Y), E) =S Ext$-P(X, P) ,convergent with respect to ajiltration decreasing in p.Proof. It is enough to replace 9 with a suitable V[Z], where projective resolutions ofX are available, and the results above are obtainable from standard arguments with resolutionsand double complexes. (Note, if X is projective, thenHome(X, YO,F) E Hom,(X, Y) @JkFfor any free k-module F.) q

154 Du and Scott, Lusztig conjectures I2.10. Corollary. Let A, p E A, and n a nonnegative integer. If 2 =!= p or n + 0, thenAlso,Ext;( V-(n), A (p)) = Ext;( V(A), A (p)) = 0 .Horn, (V(n), A (1)) 2 Horn, (V(n), A (A)) z End, (L (1)) .Proof. By 2 A. 6 below, we have the analogous result for @. Suppose A+ p. Then thelimit of the spectral sequence above is zero. If any term E2ps4 is nonzero, choose q minimalwith that property, and observe E$q = ExtG( V(n), A(i)) @ kmust be nonzero. But thenthe minimality of q gives E$q z Et,q + 0, a contradiction. This gives the required vanishingfor II * p.For the remainder, we may assume 1 = p. Now, however, V(A) is projective in asubcategory ‘Z[r] containing A(1) and all of its composition factors. The rest of thecorollary follows easily. 02.11. Remark. In standard examples k is a domain, End,(L(I)) z k, and V(n)becomes irreducible over the quotient field of k. Thus 2.10 shows Horn,,, V(n), A (A)) is atorsion-free k-module generated by only one element, and so must be isomorphic to k.There is thus a canonical embedding of V(n) in A (A).Appendix. In this appendix we review some results largely implicit in Chapter 5 of[PSI. The hypothesis below will be in effect throughout the appendix. The convention 1 .Oof this paper is not required.2A.0. Hypothesis. Let k be a field, and V an abelian k-category which satisfiesconditions (a) and (b) of the beginning of this section with respect to a locally finite poset IIand objects V(n), A(2), and L(2).Assume also each object A4 in %? has a finite compositionis finite-dimensional over k, for each A E /i.series, and that End, (L(i))2 A. 1. Theorem. Suppose P is an object in @filtered by V(2)‘s and satisfyingExt&(P, V(p)) = 0for each p E A. Then P is projective in W.Proof. It suffices to show that Exti(P, X) = 0 for each object X in ‘3. Obviously, wemay assume X is irreducible, and thus equal to L(p) for some p c II. Let r = r(p) denotethe set of elements w in /i with A 5 w 5 p for some i for which V(2) occurs in the givenfiltration of P. Since A is locally finite, the set r is finite.We claim now that both that Ext& (P, L(p)) = 0 and Ext; (P, L (,u)) = 0. We will provethis claim, from which the theorem follows, by induction on the cardinality of the set r(p).First, we need a lemma:

Du and Scott, Lusztig conjectures I 1552 A. 2. Lemma. We have Ext& (V(n), L(v)) = 0 f or i = 1,2, unless 2 < v. (After 2A.6,the same conclusion may be proved for all i > 0.)Proof. By hypothesis Extk( V(A), A (v)) = 0 f or i = 1,2. Apply the long exactsequence of cohomology for Ext,* (V(n), -) to the short exact sequence0 + L(v) --, A(v) + Y --f 0,where Y = A (v)/ L(v). The conclusion of the lemma for i = 1 follows from the fact that allcomposition factors L(p) of Y satisfy p < v, which implies Horn, (V(A), Y) = 0 unless ,? < v.Applying the lemma for i = 1 in a similar way, we obtain the desired conclusion for i = 2.After 2 A. 6 below, the same argument works for all i > 0. c1We now return to the proof of the theorem, in particular, of the claim. If r is empty,the claim follows from the lemma. Hence we may assume the claim for p is true for allweights v for which T(v) has smaller cardinality; this includes all weights smaller than ,u. Wenow obtain that Ext$ (P, L (~1)) = 0 by arguing as in the proof of the above lemma, using ashort exact sequence0 + L(p) --) A(p) --f Y --) 0,induction, and the resulting long exact sequence for Extg (P, -). Similarly, we obtain thatby using a short exact sequenceExt&(P, L(p)) = 00 + z 4 V(p) + L(p) --t 0)together with induction and the hypothesis of the theorem. This proves the claim andcompletes the proof of the theorem. q2 A. 3. Theorem. Assume that the poset A is jinitely-generated. (There are finitelymany maximal elements, and all elements of /1 are bounded above by one of them.)Then each object T of Gf? has a projective cover P in Gf? and an injective hull Q. The projectiveP is filtered by V(I)‘s, and the injective Q is filtered by A (3L)‘s.Proof. We just apply the recursive procedure of [PSI, Chpt. 5, giving full details forcompleteness :First of all, it obviously is sufficient to take T simple, T = L(v) for some v E /1. Also,it is enough to treat the case of projective covers, since the existence of injective hulls, withthe desired filtrations, follows dually.Let D = Q(v, /i) denote the set of weights u) in /1 with o 2 v. Since /1 is both locallyfinite and finitely generated, the set 52 is finite. We will prove the existence of a projectivecover of L(v), with the desired filtration, by induction on the cardinality of G?. If !3 has onlyone element (equivalently, v is maximal), then Lemma 2A. 2 shows V(v) is a projective

156 Du and Scott, Lusztig conjectures Icover of L(v), and it trivially has the desired filtration. Thus we may assume v is notmaximal.Let /i’ denote the poset obtained by deleting a maximal element p > v from /1, and letV’ = V [A’] be the full subcategory of objects in 5% whose composition factors L(A) all have;1 E /1’. As remarked at the beginning of this section, the category V’, which obviously is fullyembedded in V and closed under extensions, inherits the required Ext conditions. ObviouslyQ(v, /i’) has smaller cardinality than Q, and so we may apply induction. Thus, wemay assume L(v) has a projective cover P’ in V’, filtered by V(n)‘s.Observe Exth(P’, 2) = 0 for 2 in V’, since %?’ is closed under extensions. Taking 2 tobe the kernel of V(p) -+ L(p), we find that Extb(P’, V(p)) embeds naturally inExt; (P’, L(P)) .The latter is finite-dimensionalover k, as may be seen by forming the short exact sequence0 -+ L(p) + A(p) + Y+ 0,and applying the long exact sequence for Ext,*(P’, -).Note from 2A.2 that End,( I/@)) g End,,(L(p)) = D, a division algebra over k.Now choose arbitrarily a D-basis yi, . . . , yd of Ext&(P’, V(U)), and form the correspondingextension P of P’ by a direct sum Vd of d copies of F’(U). The corresponding extension Eof P’ by a direct sum Ld of d copies of L (p) remains nonsplit under any nonzero projectionLd + L(v), since such a projection may be viewed as just taking a specific D-linear combinationof the factors, and yl, . . . , yd have been chosen D-independent. Since P’ has asimple head, it follows that E has a simple head, which implies in turn that P has a simplehead. So, to prove P is a projective cover of L(v), it suffices to prove that P is projective.By 2 A. 1, it is enough to show Extb(P, X) = 0 for X any V(iz). Note, however, thatI’(p) is projective by 2A. 2, and so Ext&(P, X) is naturally a homomorphic image ofExt&(P’, X). The latter is zero for X = V(n) with i + ,u, as argued above from the projectivityof P’ in W. So only the case X= V(p) remains. However, the yi, . . . , yd spanExt:, (P: VP>), so that every element z may be viewed as the image in Exti(P’, V(p)) ofthe class x = (yi, . . . , yd) defining P in Extk(P’, V”) under the map induced by a morphismVd + V(p).Since x defines the exact sequenceits image is 0 in Extk(P, V”) under the map induced by P + P’, as is well-known. Consequently,the image of z in Ext&(P, V(,u)) is 0 under the map induced by P -+ P’. It followsthat Ext&(P, V(p)) = 0, and the proof is complete. •I2 A.4 Corollary. Let r z A be a finitely generated ideal. Then the full subcategory+?[T] of objects in %7 whose composition factors L(A) all satisfy A E r is an artinian highest

Du and Scott, Lusztig conjectures I 157weight category in the sense of [CPS2], as is the dual category %[I], with respect to theevident choices qf dual Weyl modules.The proof is immediate.qThere are many further corollariestwo that especially clarify matters.from the highest weight theory. We just mention2 A. 5. Corollary. Let I be as above. Then the inclusion %?[I] s w induces a fullembedding of bounded derived categories. In particular Extit,r (X, Y) g Ext”, (X, Y) for eachX, Y in %?[I] and each integer n 2 0.Proof. If r 5 r’ are finitely generated ideals of/i, then the inclusion %?[r] 5 Q? [r’]induces a full embedding of bounded derived categories by [CPS2], 3.9a. Clearly, anymorphism in Db(%?) is realized in some Db (V [r’]). The equality of two morphisms can bedetected in the homotopy category Kb (V) by a process involving only finitely many objectsof G??, consequently demonstrating equality in some Db(V [r’]). The corollary follows. q2 A. 6. Corollary. Let V be as above. Then Exti (M, N) isjinite-dimensional over k forall objects M, N in ‘F, and, for 2, ,LL E A,unless n = 0 and A = u.qExt#W, A (~4) = 03. Generalized q-Schur algebras: The natural localization caseIn this section we let k be the localization of the Laurent polynomial ringd = Z [q, q- ‘1 at the maximal ideal m = (q - 1, p), where p > 0 is a prime with the propertyas mentioned in the introduction. (We will consider other localizations and G? itself at theend of 5 4 and in 5 5.) As in the previous sections, we denote the residue field of k by k,with a similar notation for k-modules. The quotient field of k is denoted K.We let U’ be a quantized enveloping algebra over K in the sense of Lusztig [L2],associated with some finite root system C and with generators E,, Fi, Ki, Ki- ‘, i = 1,2,. . . , n.Let U, denote the integral subalgebra of U’ over ZZZ, following [AW], as based on Lusztig[L4]. The extension UK of U, to K is isomorphic to U’, and the localization of U, at m isdenoted by U. The latter notation agrees with [APW], though our commutative algebranotations and conventions do not agree - e.g., k and JZZ have different meanings.Let X be the set of weights, and X+ the set of dominant weights. Clearly, X+ is alocally finite poset. (This means that all intervals are finite. In fact, all finitely generated idealsof X+ are finite.) For each A E X+, let L(1) and L,(A) denote the irreducible modules forU and UF, respectively, with highest weight A, where F is a field of characteristic 0 in whichq is a primitive pth root of unity.haveLet %? be the category of k-finite U-modules of type 1, as defined in Example 1.1. We11 Journal fiir Mathematik. Band 455

158 Du and Scott, Lusztig conjectures I3.1. Lemma. The full subcategory G? of objects $3, with M in 59, is a k-$nite highestweight category.Proof. Since the image of q in k is 1, the action of Ki on each k-module M with Min %? is trivial. So, &? is actually a U/Z-module where Z is the ideal generated by Ki - 1(1 5 i 5 n). However, u/Z is isomorphic to the hyperalgebra QI corresponding to the sameroot system C ([L4]). Therefore, the category G? is isomorphic to the category of k-finitedimensional 2I-modules, which is a highest weight category (see [PSI, Example in Chapter6). qIf Z is a finitely generated ideal in X+, then the subcategory @[Z] of objects withcomposition factors all of whose highest weights lie in Z is a k-finite highest weight category([CPSl], [CPS2]). w e now can prove the following integral version.3.2. Theorem. The category %Y is a k-jinite highest weight category.Proof. The first two axioms follow from Lemma 3.1 and [APW], $9 1 - 2. Axiom (iv)is automatically true (1.6). We just have to check axiom (iii) for a k-finite highest weightcategory (over k or fi). Because of the Anderson-Polo-Wen observation [APW], (5.14),based on the work of Mathieu, that the tensor product of Weyl modules has a Weyl filtration,and our completion theory, which says projectivity relative to a finite set of weights canbe checked modulo m, it is sufficient to prove the following:Let Z be a finite saturated set of dominant weights, and 2 in Z. Then there exists amultiple tensor product X of k-Weyl modules (which thus lifts to a tensor product of k-Weyl modules) such that(a) the simple object L(J) is a homomorphic image of X,(b) Extb(X, L(p)) = 0 for every ZL in Z.Notice (b) is just a propertyfor the usual hyperalgebra.To determine such an X, one can just take the tensor product of two generalizedSteinberg modules (over k) with very large weights, together with a tensor product of the k-Weyl module associated with 1. By [CPSK], (3.7) and the finiteness of Z, we see that such amodule X exists.Once one has a module X = X(n) for each 2, and T is any k-finite U-module in @the map onto the radical quotient of T from a sum of X(A)‘s may be lifted to T by (b),assuming its composition factors are indexed by elements of Z.Finally, by [APW], (5.14), we get a k-free module filtered by Weyl modules which hasthe sum of X(A)‘s as its reduction modulo m. It is only necessary to use the same tensorproducts that went into the constructions of the X(n)‘s over k. qFor the complete definition of the term “integrable” used below, we refer the readerto [APW]. The main requirement on an integrable module is that it is a direct sum of‘weight’ spaces, where the standard generators Ki, Ki-’ and their variations act by scalarmultiplication.

Du and Scott, Lusztig conjectures I 1593.3. Corollary. Every k-jinite representation in %? is integrable.Proof. Let M be a k-finite U-module. By 3.2, there is a U-module P in $7, filtered byWeyl modules, such that i@ is the epimorphic image of P, and Extb(P, L) = 0 for everycomposition factor L of I%?. Applying 1.6 we obtain that Exth (P, M) = 0, and the epimorphismP + &? can be lifted to a map P + M, which is surjective by Nakayama’s lemma.Now, the fact that P is integrable implies that A4 is also integrable. qNote that 3.3 improves [APW], 9.12(a) from the field to the integral case.If Z is a finitely generated ideal in Xf (thus a finite ideal), the category V[Z] of allobjects in ‘% with composition factors all lying in G? [Z] is a k-finite highest weight categoryby the definition and 1.2; cf. also 2 A. 4. We have that End, (L (A)) = k is separable over kfor all ;1. So, by 2.5, %?[Z] is equivalent to the category of k-finite modules for a quasihereditaryalgebra. We denote this algebra by Y[Cr].3.4. Definition. The k-quasi-hereditary algebra Y [Z] is called the generalized q-Schur algebra over k associated with the poset Z.The name is justified because the q-Schur algebra over k can be recovered by a similarprocedure from a quantum enveloping algebra, but we omit the details. In $5 we willsimilarly define a generalized q-Schur algebra for k nonlocal.We now give an alternative description of Y [Z] like that of [Do 11, using quotients ofuniversal enveloping algebras. Our approach so far is like that of [CPS I], [CPS2] for theclassical case of generalized Schur algebras over a field.3.5. Proposition. Y[Cr] is a free k-module and is a quotient algebra of U.Proof. The freeness is automatic, since Y[Cr] is k-projective and k is local.For each v E Z, the U-module structure on L(v) gives rise to a U-epimorphismU + End, (L (v)) .However, as a U-module, we have End,(L(v)) g 1, L(v) (1, copies) where 1, = dim, L(v).So, we obtain a U-epimorphismrc’: U+ “2 End,(L(v)) g @&L(v).VETSince U is projective as a left U-module, the map rc’ can be lifted to a U-epimorphism7-c: u+@l,P,(v).vsrLet Z = ker n.Then Z is a left ideal. We claim that Z is a two-sidedthat IX + Z, for some x E U. Then we have a U-epimorphismideal. Indeed, supposeI--+ Ix + (Zx+Z)/Z*O.

160 Du and Scott, Lusztig conjectures ILet I,, be the kernel of this map. Thus, Z/Z0 g (Ix + Z)/Z is a submodule of U/I, hence,belongs to %?[Z]. Therefore, the cyclic U-module U/Z,, is a member in %?[Z], which hasU/Z as its image. So, U/Z0 g U/Z 0 N for some k-module N =+ 0, since U/Z is k-free. Onthe other hand, the radical quotient M (in %?[Z] !) of U/Z0 is still U-cyclic. So, A4 is a cyclicmodule for a finite dimensional k-algebra. It follows that A4 is an epimorphic image of@ 1, L(v). This implies U/Z, is an epimorphic image of @ 1, P(v), hence, an image of U/Z.VEI-YEI-So, U/Z0 is k-finitely generated, and so is N. Consequently, m = k’“’ for some m. However,U/Z g U/Z,,, so N = 0. Now N must be 0 by Nakayama’s lemma, a contradiction. Therefore,we must have Ix = Z, and so Z is an ideal. Thus, U/Z is an algebra and k-free as ak-module. Finally, we haveYCrl = End, @Z,&(v) g End,(U/Z) g U/Z. qL >4. Multiplicity-free resultsIn this section, our base ring k is a regular local ring of Krull dimension at least 2, withmaximal ideal m, and cp is a fixed height 1 prime ideal of k. As before, K denotes the quotientfield of k and k = k/m, the residue field. For any prime ideal @ of k and k-module M,we writeUP) = k&3, &f = MImM, M&J) = M&k(p).Let 9 be a k-quasihereditary algebra. According to $2, the category of k-finite Y-modules is a k-finite heighest weight category with a finite poset /i (see 2.1.1); in particularwe have Weyl modules V(A) and dual Weyl modules A (A), for each ,4 E A. These modulesreduce modulo m to their counterparts over k, and any nonzero map V’(n) -+ A (A) lifts toa map V(A) + A (A). This map is unique up to a multiple by a unit of k in the split case, andwe call it the canonical map. If YK is (split) semisimple, then the canonical maps are alwaysembeddings (see 2.11).We have the following general result.4.1. Theorem. Suppose 9 is a k-quasihereditary algebra such that(1) YK is split semisimple;(2) Y(g) is semisimple for each height 1 prime ideal @ + cp.Let V = V(I) be a Weyl module for Y for some 1 E A with L = L(A) the associatedirreducible module, and L’ the irreducible module for V(q). Zf V is multiplicity-free, thendim L = dim L’.Moreover, if A = A(1) is the corresponding “dual” of V then the canonical embeddingV + A is diagonalizable.Proof. Replacing Y by an appropriate quotient, we may assume A E /1 is maximal.Let .Z be a defining ideal such that 9/J g Y [A \{A>]. Obviously, we have

Du and Scott, Lusztig conjectures I 161Horny (Y/J, A) = 0by definition, and Ext& (Y/J, A) = 0 since A is a “relative” injective Y-module. (All k-split embeddings of k are split.) Thus, the exact sequenceinduces an isomorphismof k-modules(4.1a) A/v 2 Horn,,, (Y/J, A / V) E Horn, (Y/J, A/ V) E Ext$ (Y/J, V) .For each v E /1, let e, be a primitive idempotent in Y such that Ye, has simple headL(v). Thus, Y E @ 1, Ye, where Z, = dim L(v). Therefore, we have a k-space isomorphismVCAA/V2@ I,e,(A/V).veilSince V is multiplicity-free,we see that the k-modulek@ e,(A/ V) = e,(A/ V) E’ Horn, (ge,, A/ V)has dimension1 or 0. So, e,(A/ V) is cyclic, and, hence,e,(A/ V) E k/afor some ideal a c k. We now prove that a = cp” for some n, depending on v. Since k isregular (hence a UFD) and cp has height 1, it is enough to show that A/ V is cp-coprimary,that is, the set Ass(A/ V) of associated primes is (cp>. (The localization k, is a DVR, andthe formal powers k n cp”k, are actual powers here.) Suppose q E Ass(A/ V) has heighth > 1. Then q contains a height 1 prime ideal p + cp. Let @ be generated as an ideal by anelement p E k. Consider the exact sequenceWe have an exact sequencePo-+v- v+ v/pv+ 0.. . . + Horn, (sP/ J, V/p V) + Ext$(Y/ J, V) P Ext:,(Y/J, V) + . . . .Clearly, Horn, (Y’/ J, V/p V) g Horn, ((91 J)/p(Y/ J), V/p V) = 0, since the localizationsat @ of (Y/ J)/p(Y/ J), V/p V have no common composition factors (4.1 a). Thus, pannihilates no nonzero element of A/V. On the other hand, (A/V)” = 0 since 9” issemisimple, so all members of Ass(A/ V) have height exactly 1. However, (A/ V)@ = 0 foreach height 1 prime ideal @ =I= cp since Y(p) is semisimple by our hypothesis. So, (A/ V)@ + 0implies p = cp and therefore, Ass(A/ V) = (cp}.From the argumentabove we have(4.1 b)

162 Du and Scott, Lusztig conjectures Iwhere cp = (4). This implies thatdimA-dimL=dimA/V=dimA/V(cp)=dimA(cp)-dimL’.Therefore.dim L = dim L’.We are now going to show that the matrix of the canonical embedding V + A isdiagonalizable.Let a,, . . . , a, be the generators of all cyclic modules in A / V. Pick elements vi E A suchthat zli + V = ai for 1 5 i s r. In particular, the set {Ui} is part of a basis of A; indeed theimage in A/V of the set is the basis (a,} of the latter module. Consequently, {vi} is a k-linearly independent set in A, since k is a local ring. Put Y = i k$“‘vi. We claim thati=lV/Y is free. If so, vi, . . . , v, together with a basis of submodule of V which splits V + V/Ydiagonalizes the embedding V + A. The theorem would be proved.To show that V/Y is a free k-module, we first take the localization at the prime idealcp. We see that Y, is a submodule of I$ also, a submodule of A,, andWe show that this module is torsion-free. Pick x E V, such that the image x’ of x in V, /Gis a torsion element. So, x’ E (A,/Y,),, the torsion submodule of A,/Y,. Consider thenatural homomorphismn:A,/Y,+ AJVI.Since rc(ui + Y,) = ai, 1 5 i 5 r, we have n(Aq/ YJ, = Ao/ V,. Hence, the restriction of 7-c to(A,/ Y,), is an isomorphism by the comparison of the number of elements. Now, rc (x’) = 0implies x’ = 0. Therefore, V,/ Y, is a k-free module with rank, say m. So, m = dim V/Y.Choose xi, . . . , x, E V/Y such that Xi, . . . , X, form a basis of V/Y. Thus, xi, . . . , x,generate V/Y by Nakayama’s lemma. So we have a surjective.map rc’ : k’“’ + V/Y. Clearly,the images of xi, . . . , x, in I$/ Y, form a basis, too. So, the natural map V/Y -+ V,/Yp isinjective. Now, in the following commutative diagram/p 2l+v/yI 4k’“’ H;rp (Vi YIP,the vertical maps are injective and 7~1, is an isomorphism. It follows that rc’ is injective.Therefore, E’ is an isomorphism, and hence, V/Y is free. q4.1.1. An improvement. Suppose 9’ satisfies the hypotheses of 4.1 above, and e E 9’ isan idempotent. Let V, L, L’ be as in 4.1, but do not assume V is multiplicity-free. Then theargument for 4.1 yields the conclusiondimeL= dimeL’

Du and Scott, Lusztig conjectures I 163under the weaker hypothesis that eP is filtered by terms e L (v) with L(v) appearing in V withmultiplicity , and let Y = Y [/iI be the corresponding generalizedq-Schur algebra over k. Let 4, denote the nth cyclotomic polynomial and cp = (4,).By Theorem 4.1, we only need to prove that(4.1 c) Y(p) is semisimple for each height I prime ideal @ + cpIt is known from [L4], [RI-21 that the specialization of Y is semisimple at any fieldwhere q is not mapped to a root of unity. So if Y(p) is not semisimple, then q is sent to aroot of 1 under the map k + k(p). Since @ is of height 1, we may assume M = (4,). Since@ c m, we have n = pe for some e > 0. However, if e 2 2 then 9’(p) is still semisimple bythe strong linkage principle [APW], 8.1, since il is in the lowest p2-alcove. So, (4.1 c) isproved. Now our conclusion is a consequence of Theorem 4.1. q4.3. Remarks. (a) 0 ne can get the equality of dimensions here from a moretraditional character theory argument. See Appendix 2. However, the latter argument wasonly discovered after the more abstract version given here, and it does not give any diagonalizationresult. Similarly, the equality of dimensions in 4.6 can be obtained from charactertheory, as well as the equalities in 4.1,4.5, though one must impose the additional hypothesisthat k has Krull dimension two.(b) Let k be the localization of B [q] at a maximal ideal m such that the residue fieldk of k is the finite field s[c]. Thus, 5 is a root of unity of order prime to p. If 5 + 1, welet 1 denote the order of 5, but if 5 = 1, it is convenient to set 1 = p. We remark that onecan generalize 4.2 from the natural localization case to such a general localization case, theso-called “mixed case” in [PW] and [AW]; see 5 5. An especially attractive situation likethis is that of the original q-Schur algebras of Dipper and James, which play a role both inthe modular representation theory of the symmetric group and in the representation theoryof finite general linear groups in nondescribing characteristics. As mentioned in the introduction,James has independently made conjectures here parallel to Lusztig [L3]. We nowapply Theorem 4.1 to these algebras.

164 Du and Scott, Lusztig conjectures ILet Y = 3 (n, r) be the q-Schur algebra of degree (n, r) over & (we adopt the versiondiscussed in [DJ2] or [Du]), and letq((n, r; k) = 0 then YV(@) is still semisimple by the hypothesis r < lp(see [DJO], (4.2)). So %(@a> is semisimple unless M = cp. (This is again a Hecke algebraresult, and holds in the general case.) Now our conclusion follows again from Theorem4.1. 0We will return to q-Schur algebras later, but we now continue the general discussionof multiplicity-free results. Let k and Y be as described at the beginning of this section. Wedenote by 9? = Q?(Y) the category of k-finite Y-modules. By 2.1 .l, %? is a k-finite hightestweight category with the poset A. Similarly, we write @ = V(9) and %?’ = %?(Y(q)).For each 1 E A, we defineQ2, = {v E A I [V(A) : L(p)] s 1 for all p 2 v} .Obviously, Sz, is a coideal of ,4, that is, if v’ >= v with v E fi2, then V’E Q2,, and II E Q,since [V(A) : L (h-1>] =l= 0 implies A 1 p. We now can prove the following result.4.5. Theorem. Let Y satisjiy the hypotheses (1) and (2) in 4.1 with k being (a regularlocal ring) of Krull dimension 2 and assume cp $ m2. Let V = V(A) be a Weyl module in $7

Du and Scott, Lusztig conjectures I 165for some A E A with L(A) the associated irreducible module, and L(A)’ the irreducible modulefor V(A)’ = V(q). Suppose[L@)I = c a,, C JWI and C-W)1 = c 4, C v(v)‘1VEAVEAin the Grothendieck groups of S?? and V’ respectively.Then a,, = a;, for all v E Sz,.Proof. Let 0 = fi2, and r = /1 \s2. Then r is an ideal and the full subcategory %?[r]consisting of objects in %? with composition factors L(i), A E r, is a k-finite highest weightcategory (5 2).Let P be the direct sum of the projective covers in %? of the simple objects L(p) forp E Q. Then the algebra Y = End,(P) is a quasihereditary k-algebra. We denote by %‘(a)the corresponding k-finite highest weight category. We claim that %?(a) is the quotient category%?/%‘[Cr]. (This is a general property of quasihereditary k-algebras, for k a commutativelocal Noetherian ring.)Indeed, if 0 = JO 2 J1 E . . . & J, = 9 is the defining sequence of Y and Ai denotesthe set indexing the irreducible left Y-modules in the radical quotient of Ji /Ji _ i, thenA = u Ai. By 2.1.1 and 2.3 projective covers of simple objects exist. (In particular Y issemipkrfect.) It follows easily that Ji /Ji _ 1 is the projective cover of its radical quotient, asan Y/J,- ,-module. The summands corresponding to isotypic components of the radicalquotient are easily seen to be ideals, and it follows we may assume that each Ai containsa single element. Thus r = u Ai for some r, and the category 5~? [r] is isomorphic to thei>*category sP/ J, -f mod of finitely generated Y/J,-modules.Now, Y is semiperfect, as remarked above. So, we can find idempotents e,, . . . , e, = 1in Y such that e, ej = ej e, = e, for i sj and Ji = Yei9 It follows thatW(Q) g e, Ye, -f mod .This implies the claim (see [P], $2). (The reader uncomfortable with quotient categoriescan just use e,.Ye, -fmod in the remainder of the argument.)We conclude from the claim that g(Q) g V/V[r] and V’(Q) E %7/V [r], and thefunctorsF: @ + g(Q) and F’: %?’ + %“(a)are induced by the functor %? + %‘(a) given by sending M to eM where e = e,. By [CPS 33,1.4, F and F’ take irreducible modules and Weyl modules to irreducible modules and Weylmodules, respectively (excluding only the cases where the images are zero). In particular,the images of Weyl modules associated to weights in Q are again Weyl modules, and thusgive rise to linearly independent elements at the Grothendieck group level. Therefore, wehave[eL(A)] = C anv [eI/o] and [eL(I)‘] = C a;, [eV(v)‘]veRveR

166 Du and Scott, Lusztig conjectures Iin the Grothendieck groups of g(Q) and %“(a) respectively. However, [V(n) : L(v)] 5 1for all v E Q by definition. So, by 4.1 or 4.1.1, we have dimeL(A) = dimeL(1)‘.Now, by the hypothesis that k is regular and cp 3 nt2, the image 0 of k in k(cp) is aregular local ring of Krull dimension I, thus a DVR, and O/w E k/m where y is themaximal ideal of 0. So, the standard reduction modulo tp process from Q?‘(a) to 5?(O) isindependent of the choice of the O-lattices at the Grothendieck group level. Therefore, wehave a group homomorphism from the Grothendieck group of the category %7’(Q) to thatof the category g-(Q) which sends [eF’(v)‘] to [ek’(v)] and [eL(v)‘] to [eL(v)] by thedimension equality. Consequently, we have a,, = a;, for all v E s2. qAs before, we can apply 4.5 to the generalized q-Schur algebras in the natural localizationcase. Let chM denote the formal character as usual (see, for example, [APW] in thequantum case).4.6. Corollary. Let 3, be a dominant weight in the lowest p2-alcove. SupposechL(il) = c a,,ch V(v) and chL,(i) = c a;,ch v,(v)vvwhere V,(v) is the quantum Weyl module associated to weight v. Then a,, = a;, for all vsuch that [V(A) : L(p)] 5 1 for all ,u 2 v.Note that the validity of Lusztig’s conjecture in the quantum case (with q specializedto a pth root of unity) implies that the a;, are described by Kazhdan-Lusztig polynomials.The same is true (with exactly the same description) of the an,, for weights in the Jantzenregion, assuming the Lusztig characteristic p conjecture is true.Proof. Let A = {,u E X + 1 p in the lowest p2-alcove} and Y = Y [A] the generalizedq-Schur algebra associated to .4 (9 3). By the proof of 4.2, we see that Y satisfies thehypothesis of 4.1. Now the conclusion is the consequence of 4.5. qA similar consequence can be concluded for the q-Schur algebras $ (n, r; k) assumingY < lp (and for the generalized q-Schur algebras of 9 5 for weights in the ‘lowest /p-alcove’).We leave the details to the reader.We do think it worth mentioning that, through Yq(n, r; k), the result 4.1.1 has aninteresting consequence for standard permutation modules of the symmetric group 6, on rletters: We recall that the generic Hecke algebra X = Xq(r; k) is a k-algebra which is a q-analogue of the symmetric group, that it has representations which are q-analogs of eachpermutation module for the action of the symmetric group on partitions with a givennumber of parts, and that Yq(n, r; k) may be viewed as the endomorphism ring of a directsum T (called ‘tensor space’) of such q-analogs. Here the (q-analog) permutation modulefor X appears with nonzero multiplicity if and only if the number of parts in the partitionsinvolved in the associated symmetric group action is at most n. In particular, forY = q(n, r; k), there exist idempotents e(v) for each weight v in A+@, r) such that e(v) Ye(v)is the endomorphism ring of the 2 permutation module associated with the (placepermutation)action of the symmetric group on the sequences (iI, . . . , i,) (ii E (I, . . . , n>) ofshape v. The result 4.1 .I for e = e(v) now translates into a result stating the equality of the

Du and Scott, Lusztig conjectures I 167multiplicities of certain Young module components of this permutation module. Moreprecisely, let PA and iV’ (resp. Y” and M”) denote the Young modules and permutationmodules for SS, (resp. for the corresponding Hecke algebra Z(4,)) associated to partitionA, and let s”“’ be the dual of the Specht module S” for /$Gr. (The labelling here forYoung modules and Specht modules coincides with the labelling for PIMs and Weylmodules. One way to do this is to apply the functor T By - to any Y module to get itscorresponding A? module. The dual Specht module gA’ for 6, may be described as theimage of the Weyl module WA for 2 and the Young module r” may be described as theimage under T a9 - of the PIM P* for 9 which has WA as a homomorphic image. Themodule PA is the reduction modulo nt of a PIM PA for x and the ‘Young module’ Y”for X(4,) is just the image of P”(c$,). The functor TOY - is additive and right exact,takes PIM’s to Young modules over any specialization, takes Weyl modules to dual Spechtmodules, and preserves all the associated decomposition numbers. The functor even takesWeyl filtrations to dual Specht filtrations in an exact manner, as may be proved from [PW],10.4.1(2), 11.3 at specializations where q is not sent to a root of unity of even order.)We define for a partion v E /i’(n, Y)0, = {A E ,4+ (n, r) 1 [F” : s”“‘] 5 1 for all p with rp a direct summand of M”) .We have4.7. Corollary. If r < p2, then [I@” : P”] = CM’” : YIA] for all 2 E 0,.qThe proof is a direct application of 4.1.1 and is left to the reader. The left hand sideis the dimension of the v-weight space of the irreducible 9 module indexed by II, and the righthand side may be interpreted similarly for 9’($&.We note that the hypothesis 1 E 0, is always satisfied (by any Specht module S’,assuming r < p2) for partitions of 2 or fewer parts, and probably for partitions of 3 orfewer parts (checked for p s 5).We also remark that an analogue of 4.7 holds in the mixed case, if we think of theresult as expressing a characteristic p permutation module multiplicity in terms of somethingknown for the q-Schur algebra ~O(C#+). The permutation modules relevant to the mixedcase are standard permutation modules for the finite general linear group GL(r, q). Thesepermutation modules, associated to coset spaces of parabolic subgroups corresponding topartitions, are taken over a field ffp where q is a power of a prime different from p. We take1 to be the smallest power of q such that p divides q’ - 1, unless p divides q - 1, where weset I = p. The set 0, above can be defined precisely as above, or by analogy using GL (r, q)-modules. (The Dipper-James theory gives an equivalence of categories between GL(r, q)-modules finitely presented by standard Fp-permutation modules as above and the correspondingcategory of smodules.) One assumes r < lp, and the conclusion is that the lefthandside, expressed in terms of s-permutation modules, is computable, if A E O,, as thev-weight space of an irreducible Y($,)-module indexed by A. (In turn computable fromLusztig’s quantum group conjecture.) If Jame’s conjecture were true, one could dispensewith the hypothesis ;Z E 0,.We end this section with an application of 4.1 in a different direction.

168 Du and Scott, Lusztig conjectures I4.8. Corollary. Let 2 be a dominant weight in the lowest p2-alcove such that, if;I = 1, + p;l, with A0 restricted, 1, is regular and not in the lowest p-alcove. Then the Weylmodule V(A) associated with 2 is not multiplicity-free.ProojI Let U be as in 1.1. According to the tensor product theorem for U(q) ([L 2]),we haveL,(A) = L&J0 L&)[l’where L, (A,) is an irreducible module for the usual Lie algebra enveloping algebra % overk(cp), where cp = (4& and [l] means the pullback through the Frobenius morphismU(q) + 2I (see [AW]). Since A, is regular and not in the lowest p-alcove, it follows thatdim L,(A,) > dim L(A,). So, by the Steinberg tensor product theorem in the characteristicp case, we have dim L,(A) > dim L(n). Therefore, the previous theorem implies that V(A)is not multiplicity free. qWe remark that such weights II are not in the Jantzen region required by the characteristicp Lusztig conjecture. As the proof shows, the associated irreducible modules L,(1)and L(1) do not have the same dimension, in spite of the favorable commutative algebrasituation obtained from the weight lying in the lowest p2-alcove. The original example ofthis kind was obtained by Alfred Wiedemann by an explicit construction, and is given inan appendix.5. Generalized q-Schur algebras: General caseWe should say at the outset that the results of this section are tentative and not likelyto be best possible. The problem is that, outside of type A, where one has the coalgebraresults of Parshall-Wang [PW], the theory of quantum enveloping algebras over arbitraryfields is still in a formative stage. Even in [PW], when 9 is specialized to an Zth root ofunity, it is required that 1 be odd. For other types, a reasonable start toward a theory, butwith many further restrictions on I, has been given by Andersen and Wen in their preprint[AW]. We will take their point of view here, using a number of their results, as well asresults of L.Thams announced in [AW]. It is also necessary to sketch the proof of someresults similar in spirit to those of [AW], but which were not included in the presentversion of that work.As in 9 3, let U, (denote the quantized enveloping algebra of [L4] and [AW] for agiven finite root system over the ring d = Z [q, q- ‘1. The quotient field of d is called K, asbefore, and U’ denotes the quantized enveloping algebra over K. Let r be any saturated setof dominant weights, and let V’(y) denote the irreducible type 1 U’-module with highweight y, for y E l7 Put V’(r) = @ V’(y), and let 9, = Y, [r] be the image of U, underthe natural mapYETU, & U’ --+ End, (V’(r)) .We define Yf to be the generalized q-Schur algebra over & associated with r.It is, of course, difficult to prove anything directly from this definition. Nevertheless,the discussion in $3 shows that, if p > 0 is a prime number and k denotes the localization

Du and Scott, Lusztig conjectures I 169of G? at the maximal ideal (p, q - I), then the localization y&k = Y& @ k is the generalizedq-Schur algebra Y [r] over k described there. In particular Ydk is k-quasihereditary. (Asequence of defining ideals may be chosen by using any linear order compatible with theusual partial order on weights, all such orderings associated with the same Weyl and dualWeyl modules.)Now let &’ be any localization (not necessarily local) of d with respect to a multiplicativesystem, and put Y’ = Y& @ &I. (All tensor products here and below are over &) Wealso refer to Y ’ = 9; as a generalized q-Schur algebra (over die’). A number of convenientcriteria are available to determine if Y ’ is &I-quasihereditary. First of all, there are naturalcandidate choices for sequences of defining ideals Ji’ in Y ‘, described as follows: Take anylinear order on the dominant weights compatible with the usual partial order. Let Ji’ be thekernel in 9 ’ of the projection of Y K onto the sum of the matrix components of Y K associatedto modules V’(y) with y at least as big as the ith weight (from the bottom) in the linearorder. According to the theory of [CPS4], the algebra Y’ is d’-quasihereditary with {Ji’}as a sequence of defining ideals if and only if the corresponding statement for the localizationsat all height 1 prime ideals of &02’ is true. (It is also sufficient to check the ‘separablequasihereditary’ property at all maximal ideals, though we do not require this criterion.)To make use of the results of Andersen-Wen [AW], we shall assume that d1 containsthe reciprocals of all cyclotomic polynomials +I where 1 is either divisible by 2, 3, or is aninteger smaller than the Coxeter number h of the root system. (Improvements to the resultsof [AW] would produce corresponding improvements in our results here. For applicationsto finite groups of Lie type in nondescribing characteristics, one really does not want torequire inverses of any cyclotomic poloynomials to be in d’.) For each weight y E r, there is,by [AW], 5.4, an integrable type 1 U 0 &‘-module Ho (A), of finite rank and projective overd’, with a number of additional properties: (For definitions of terms involved, we refer thereader to [APW] and [AW]. The definition of ‘integrable’ is only implicit in the latter paper,cf. [AW], 4.2, but presumably agrees with [APW], 1.6. Using the analog of [APW], 9.1, amodule is ‘integrable’ in the finite rank case iff it is a direct sum of its weight spaces. Thiscondition can be checked locally. We remark that it is proved by Polo, see [AW], $4, thatH’(1) is free over -02’; we shall only use the fact that it is projective, which, of course, is alsoa local property.) First, the localization of Ho (A) with respect to any multiplicative systemis the corresponding module over the localization. Next, there is a version Hj (1) over anyfield Fwhich is a specialization of ,Oe’, and &? (1) satisfies ‘Kempf’s vanishing theorem’. Thelatter implies that H’(A) @ F g @(/2) by a spectral sequence argument. (In the case thatF is the residue field of a height 1 prime, which is the only case we require, the argument isa simple application of the long exact sequence of cohomology.)Put A (A) = H’(1) and let V(1) be the ‘dual’ of A (A). (Take the linear dual, change theaction from right to left using the antipode, and then twist through the automorphism ofUk = U, @ JZZ’ associated with the ‘opposition involution’ on the root system. As usual,this sequence of operations applied to any irreducible type 1 U,F-module with high weightin r just returns an isomorphic copy. Here the type of a U’-module is as described inAppendix 3.) We note that the theory developed in $3 l-2 allows us to assume that thespecialization F of d1 has positive characteristic, and the image q in F is not 1 (cf. 5 3).Standard arguments with weights and the vanishing theorem show, cf. [CPSK], thatExt”(V(A)F, A(l)F) = 0 for n > 0

170 Du and Scott, Luszfig conjectures Iin the category of integrable type 1 U,F-modules. Fortunately, we will prove in Appendix 3that, over such a field F of characteristic p > 2, all finite dimensional Uz-modules are integrable.In particular, one has the above vanishing for n = 1,2 in the cateogory qF [r] offinite-dimensional U,F-modules whose composition factors are all indexed by weights in l7(See Remark 2.0.) The following proposition follows easily:5.1. Proposition. The category wF [r], with the Weyl and dual Weyl modules above,is an F-finite highest weight category. qThis does not yet tell us what we want to know about Y r (or YF), but it is a start. Toprove that Y1 above is #-quasihereditary, we just need to check (using the indicatedsequence of defining ideals) the case where d1 is the localization at a height 1 prime ideal of&. That is, the localization &’ may be assumed to be a DVR with residue field F. Thus,the proposition above puts us in similar position for d’ and F as we found ourselvesfacing in 5 3 for k and k. (In some ways we are even in a better position, since d1 is aDVR.) We solved the problem in $3 by demonstrating that the category of k-finite U-modules formed a k-finite highest weight category, and the same program may be followedhere. We just need to show that the category of &“-finite I&?-modules is an &l-finitehighest weight category, with the expected Weyl and dual Weyl modules. This follows as in9 3 from the following proposition, communicated verbally to us by H. Andersen at theMSRI conference.5.2. Proposition. Zf 2, I’ are dominant weights, and d’ is a DVR as above, the tensorproductHO(i) @ HO(X)of &j-modules is filtered by modules H’(A”), with A” dominant.Discussion and sketch of the proof. For the field k the corresponding result is atheorem of Mathieu [Ma], completing previous investigations by Wang [W] and Donkin[DON]. (Donkin observed at the MSRI conference that Lusztig’s results [L5] regardingnew bases for Weyl modules implied Mathieu’s general result.) (‘Good filtrations’ can bechecked by Ext’ vanishing with Weyl module arguments in the first variable. This is equivalentto an H1 vanishing using cohomology for the unipotent radical of a Bore1 subgroup,cf. [Do 23. For modules that can be lifted, say to rational modules over the p-adic integers,one can just check that fixed points for the unipotent radical are the same p-adically ormodularly, and this can be seen directly from Lusztig’s results.) Andersen observed that thesame arguments also applied for F. Formal arguments, cf. [DON], show the result for d’is a consequence. Anticipating a more complete treatment by Andersen or his students inthe future, we omit further details. (It is not actually necessary to assume d’ is a DVR forthe argument to work, only that it is local, though we need below only the DVR case.)Putting the above arguments together, we now have the following main result. Thenotion of ‘composition factor’ used below is the natural one, meaning a simple quotient ofa submodule. (Obviously, all composition factors occur locally, where they fall under thetheory of $1.)

Du and Scott, Lusztig conjectures I 1715.3. Theorem. Let di be the localization of d = Z [q, q- ‘1 with respect to a multiplicativesystem containing all cyclotomic polynomials 4, with 1 divisible by 2, 3, or 1 smallerthan the Coxeter number. (Thus, these cyclotomic polynomials are allowed as denominatorsin ,Pe’.) Let I be anyJinite saturated set of dominant weights. Then the generalized q-Schuralgebra 9” = Yt? defined above is dae’-quasihereditary and split. The category $9’ [I] ofd’-finite type 1 modules of UJ,l = U, @ d’ whose composition factors all have high weightsin I coincides precisely with the category of #-finite Yt?-modules. In particular, %?I[I] is and’-finite highest weight category. qThe last assertion of the theorem also holds, essentially by definition, for the categoryV1 of &l-finite type 1 modules of U,l; see the discussion of the nonlocal case in $2.As a further consequence of the theorem (or its proof), we have the followingcorollary.5.4. Corollary. Let ~2’ be as above. Then all &l-jinite U,‘-modules are integrable. qOf course, a main consequence of Theorem 5.3 is that we can apply the quasihereditarytheory of the previous section. Suppose F is a finite field which is a quotient field of& by a maximal ideal m, and suppose the image of q in F is a primitive 1 ‘h-root of unity. IfI+ 1, we will assume that 1 is at least as large as the Coxeter number h of the root system,and not divisible by 2 or 3. (These assumptions are required to quote [AW]. However,according to [AW], 0 5, these assumptions are not required for type A, where Parshall andWang have given a different approach [PW] without restrictions on 1 (though they doassume 1 is odd). Still another way to approach type A is to use integral q-Schur algebras, asin the previous section, where we obtained the theorem below for q-Schur algebras withoutrestriction, as Corollary 4.4.) Let p be a fixed prime, and let r = r (1, p) be the set of dominantweights y satisfying(y + e, a;> S lp if I* 1 ,(y + Q, al;) 5 p2 if 1 = 1 .Our result below for the second case (I= 1) has already been obtained in 9 3, so we havejust included this case for perspective. We may regard ouselves as replacing 1 by p in theI= 1 case, as in 4.3, so the reader might think of r as the ‘lowest lp-alcove’, including theclosure. (The replacement makes some sense in the finite general linear group theory [J],where q is itself a prime power, and one may choose 1 to be the smallest nonnegativeqi- 1integer for which p divides __ q _ I .) Beyond the weights in the boundary, a few other lp-singular dominant weights could also be allowed into r, as the proof of the theorem belowdemonstrate. (The additional weights are those minimal with respect to the strong linkageorders associated with the affine Weyl group I+$,.)5.5. Theorem. Fix a root system, a prime p, and a nonnegative integer 1 restricted asabove, and let I be the ‘lowest lp-alcove’, as de$nedprecisely above. Let 4 denote the cyclotomicpolynomial 4i if I+ 1, and put 4 = q5r if I= 1. Let M be a maximal ideal ofd = H [q, q- ‘1 containing p and 4, and such that q becomes a primitive lth-root of unity inthe residue field F = ~41 m. Suppose y E I is a weight such that the Weyl module V(y)’ forUi = U, 0 F is multiplicity-free. Let d(4) denote the residue$eld of the localization dc6,.

172 Du and Scott, Lusztig conjectures IThen the irreducible quotients of V(y)’ and of the Weyl module V(Y)&(~) for U$@)have the same dimension.Proof. Let k = H [q], and r, = {p E r 1 p s y}. We denote by Y the generalized q-Schur algebra Yr’, as in 5.4 with JZZ’ = k. By theorems 4.1 and 5.4 we only need to check thehypothesis for Y in 4.1. It is obvious that Y K is (split) semisimple for the quotient field Kof k. To show that Y(p) is semisimple for each height one prime ideal @ + ($), we onlyconsider the case where q is sent to an nth primitive root of 1 in k(p) (cf. the proof of 4.2).In fact, n has to be equal to lp’ for some e 2 0, by the proof of 4.4.Suppose e > 0. Then char k(p) = 0. We claim that the linkage principle anologous to[APW], 8.1 holds over k(p). We just sketch the proof of the claim. Indeed, the assumptionon 1 allows us to use the results in [AW], $5. In particular, we see that H;(1) satisfies the“Kempf vanishing theorem”. Consequently, we have a k-free ‘Weyl module’ V,(A) over kwhose character is given by Weyl’s formula (this is implied by the argument in [AW], 4.3,and they use the notation D(n) there), and a k-free ‘induced module’ Hf (A) for il E X+.Both are k-free modules. Further, one can compute for the rank one case the structure ofH’(1) for i 2 0 and II dominant, as done in [APW], 54. (Such an analysis is announced in[AW], 5.1 without explicit statement as part of the work of L.Thams.) In particular, wehave a similar result over k(p) as [APW], 4.7 (ii). With this and some spectral sequencesas given in [APW], Cor. 2.15, one obtains vanishing properties for H&) as stated in [A],Lemma 3 or [PW], (10.1.16) by arguing as in the proof of [APW], Cor.5.7. One alsoobtains some long exact sequences relating these cohomology groups, as in [A], Cor. 2 or[PW], (10.2.2). Also, one can prove Serre duality over k(g) as done in [APW], 5 7 (see also[PW], (10.3)), which implies thatwhere V = V,(A). It follows that there is a non-zeroU,(p)-homomorphismunique up to scalar multiple (see also 2.10). Now, a similar argumentof [A], Theorem 1 proves our claim.to that in the proofBy the claim, if e > 0 then Y(p) is still semisimple, since y is in the lowest Ip-alcove.Therefore, our result follows from 4.1. qAppendix 1 (due to A. Wiedemann, Stuttgart). Example of quasihereditary algebraY over a regular local ring k of Krull dimension 2 having two simple modules, two Weylmodules and being maximal as an order after localization at each but one height 1 prime idealcp. However the projective indecomposableWeyl module P occurs with multiplicity 1 as a directsummand of Y: whereas P, occurs as a direct summand of Yq with multiplicity 2.We think of k being the localization of H [q] at the maximal ideal m = (p, q - 1) fora prime number p. We put 4 = $p and y = q - 1. Then M = (4, rp). Beyond this equation,we do not need the explicit form of k or 4 and tp. LetK = the quotient field of k,

where Z3 is the identity matrix in M3(K).Du and Scott, Lusztig conjectures I 173A = Kx M3(K),co = (1,0), 61 = @,I,),In the set of columns of size 2 over k, we define a k-submoduleA4 as follows:and define a k-orderY in A bySince M is k-free, 9’ is also k-free. One has,y:=%“n(0)XM3(K)E (; ; ;)=($ ; ;)(i f +kkk)c3)as right Y-module. (We always work with right modules in this appendix.) HenceEnd, (J$) g M3 (k). Secondly, 3 is the kernel of the projection of Y onto the first component(= “multiplication with so”). Hence, Y/f g k. Moreover, 9 = 9’eY where000e=O,OOO .!( 0 0 111Thus 9 is an idempotent ideal, which is right projective over 9 Altogether, this showsthat Y is k-quasihereditary in the sense of [CPS4].Observethate’= (I,(: :)) E Y: and so # % The latter fact implies that the kernelof the projections 9’ + 9’e1 is contained in (Rad k) Ed. Now(e’%e’).zl z kl, + (MM)5 M,(k)is local, since (MM) s M, (Rad k). Thus End, (e’Y) z e’9’e’ is local. This shows that 9’has two indecomposable projective summands e’Y and P := eU: each occurring withmultiplicity 1.The base change providing the isomorphism M 2 k0 k is given by the matrix12 Journal fiir Mathematik. Band 455

174 Du and Scott, Lusztig conjectures Iwith determinant 4’. Since cp = (4) is the only height 1 prime ideal containing +2,for each height 1 prime @ + cp, showing that 9@ is maximal for each height 1 prime idealp + cp. (In fact, 9@ 2 k, x M3(kJ.)Considerpotentnow M,. It is easy to see thatE M,,,. Hence 5$ containsthe idem-err= (o,( i (:,. i))and e”,4”, g (k, k, kJ E P,. Therefore, Yq has Pv as a summand with multiplicity at least 2.Note that E,, $ Yq and so the multiplicity of P, as a summand of $$ must be exactly2. In particular, the irreducible quotient of P (or of P/mP) has dimension 1, but the irreduciblequotient of P, (or of Pv / cp P,) has dimension 2.Appendix 2. In this appendix, we use more traditional character theory for algebraicgroups, and the quantum analog of that theory, to give an alternative proof of the equalityof dimensions in 4.2. The idea that such a proof might exist began with Z. Lin, thoughthe argument below is due to us. (Lin’s approach required stronger assumptions.) In turn,Lin was working from earlier announcements of our work in this paper.We use the notation of 8 3 and 0 4. Thus, (8 is the category of k-finite U-modules oftype 1. If M is a free k-finite U-module in %, then M is integrable by 3.3. So, M is thedirect sum of weight spaces M,, ;1 E X. We setch A4 = c rank, (MA) e’ ,lEXcalled the formal character of M. If M = v(n), the Weyl module with highest weight1 E X+, we denote ch A4 simply by x(A). The q-analog for U-modules of Jantzen’s sumformula has been established in [APW], 9 10 over various DVR’s associated with k. For ourpurposes here, let R be the DVR k@ (cp = (&,)) or k/(q - 1) with a unique maximal idealdenoted by @,, and residue field R. Denote by u@ the valuation on the quotient field Fof R.By 2.11, we have a canonical embedding for 2 E X+,lp : v(n)” -+ A (A)” .Now, define a filtrationof V(A)” as followsv’ = {x E v(n)” 1 yx E @‘A (A)“)

Du and Scott, Lusztig conjectures I 175Let Vi be the image of Vj in V(A)” = V(A)” OR R. Then we obtain a filtrationWe denote by Si the j-th section Vi/ Vi+ I. Obviously,we have(1) chV,‘= c ch$.j>OA2.1. Theorem ([APW]). Let C+ be the positive root system of C, s, the reflectionwith respect to c( E C ‘, Then we havec chV,‘= c 1 v,([m])~(s;A+ma),jZ0 aSZ+ m=lwhere [m] = q” - q-m.q--4-lWhen R = k,, the image 5 of q in R is a primitive pth root of 1. ,In this case, wedenote M, = ME for M a UR-module, and we write u4 = vxJ and Si = Sk. We write s(1)for the Weyl module V(A), = V(A)” over U, = U(q), and note that L,(A) is the correspondingirreducible quotient in the notation of 4.2. If R = k/(q - l), then i? = E, and wewrite &? = M”, up = up%, and sj = Si. Clearly, i;;(l)R = V(A) in the notation of $4, theWeyl module for D with irreducible quotient L(1). Note also that S: g L,(A) andso z L(i).Let “e, be the category of k(cp)-finite U(q)-modules of type 1. Thus, there is acanonical map rc from the Grothendieck group of w* to that of @ given byrc [M] = [MO @ k], where MO is a Z, [Oj>O

176 Du and Scott, Lusztig conjectures ISince v(n) is multiplicity-free, the reduction modulo p of each composition factor of 0. Consequently, by (1) we obtain dim c(n)’ = dim V(n)‘,which implies dim L,(A) = dim L(A). qA2.3. Remark. We note that the proof above by using only the Jan&en sum formula,though initially surprising, does not add any real new ingredients to our theory. Thesame proof can be made for any k-quasihereditary algebra under the hypothesis of 4.5,without the use of formal characters. The Jantzen sum formula essentially just gives somecalculation for the image in a Grothendieck group of the quotient module A (A)“/ V(n)“.Under the hypothesis of 4.5 one can compare these images and prove they are the same bycommutative algebra arguments. (The argument for 4.1 shows the idempotent projectionse, (A (4 / W>) are cp-coprimary. Under the hypothesis of 4.5, we have m = cp + z, wherez is a (principal) height 1 prime ideal (e.g. z = (q - 1) above). The length of the modulesobtained by tensoring these projections with R = k, or k/z may be used to determine theGrothendieck group image, and the coprimary condition guarantees corresponding lengthsare the same for the two choices of R [N], Chpt. 7.)Thus, the character formula approach may be viewed as using just a part of theinformation available from the commutative algebra approach we took in 4.1 and 4.2, andindeed the more sophisticated point of view gave diagonalization information not availablefrom the character formulas. At the same time, it is interesting to note the argument ofthe above paragraph makes a Jantzen sum formula over k available for computations inany k-quasihereditary algebra satisfying the hypotheses of 4.5 for which the characters ofthe irreducible modules over k(q) are known.Appendix 3. In this appendix, we shall prove the result promised in 4 5 that a finitedimensionalU,F-module A4 is integrable, where F is a field of characteristic p > 2, and theimage of q in the specialization & + F is not 1. By abuse of notation we denote this imageby the same symbol q. When q is a primitive Ph-root of I, we assume that 2 is odd and primeto 3 whenever the corresponding root system has a component of type G,. (These requirements,and more stringent ones, are implicit in all the specializations of 5 5.)The proof splits into two cases. We first consider the case when q is a primitive Zth-rootof 1. The other case where q is not a root of 1 is easier. For completeness, we will sketch theargument given in [RI] at the end of this appendix.We now assume that q E F is a primitive lth -root of 1 (with I satisfying the aboveassumptions). A similar argument to that given in [L2], 4.4 shows that K: is central andK.2z=1 1 in UF d8’Let V be a U$-module.(cf. [L2], 4.6)For any sequence E = (si, . . . , E,) with si E {I, - I} we defineE = {VE VlK,‘v = ~ZI} .

Du and Scott, Lusztig conjectures I 177Then V, is a U,F-submoduleof I/ (since K/ is central), and I/ = @ V, (since (p, 21) = 1). Wesay that V has type E if V = V,. By convention, V has type 1 if” V = V, with E = (1, . . . , 1).As Lusztig observes [L2], 4.6, there is an automorphism U$ + U,F induced by definingEi + si Ei, Fi -+ si Fi, Ki + si Ki, which interchanges type E modules with those of type 1.Thus the category of U,F-modules of type E is equivalent to the category of U’-modules oftype 1. Therefore, it is enough to consider type 1 Uz-modules.That is, we may assume, without loss of generality, that the finite dimensional moduleM (that we wish to show integrable) is of type 1. Be finiteness, it suffices to show M is adirect sum of its integral weight spaces. Since any commuting set of semi-simple transformationson a finite dimensional vector space is diagonalizable, we just need to show thatthe action of the O-part of UF for each fixed i is diagonalizable with integral weights, whereUi is the subalgebra of U’ generated by Er), F/‘) and Ki, Ki- ‘. Obviously, Ui is isomorphicto the quantum enveloping algebra of type A r. So, we may assume that U = U, is of typeA,, and denote the generators by E, F, K.Let U” be the &-subalgebra of U generated byK,K-‘, K;c , tEN,[ 1 CEZ,S+l -K-lq-C+S-tK; cmwhereas defined in [L2], 4.1. Also, letq” - q-st[ 1[I nn 4 m-s+1 -m+s-1-4denote the Gaussian polynomial n Ed.s=lq” - q-sK; cWe shall denote the image of q, E d in Fand of K, t EU’in UFO= U”@F[ 1by the same letters. By [L2], 3.2 (or [PW], (7.1.3)), we have, for 12 2,(A 3.1)where m = m. + lm,, n = no + In, 2 0 with 0 5 m,, no < 1.Consider two subalgebras u. and Y of Vi such that u. (resp. 9’) is generated by K,K; 0K-’ (resp. t E N). It is easy to see ([L4], 8.10) that V is isomorphic to the hyper-[ tl 1 ’algebra hy (G,) of the multiplicative group &&, and that u. is isomorphic to the groupalgebra over F of the cyclic group of order 21 since the order of K is 21 ([L3], 5.8).A 3.2. Lemma.We have an isomorphism of algebras

178 Du and Scott, Lusztig conjectures IProof. By [L3], (g8) and (A3.1), for any t = to + t, 2 0 with 0 5 to < 1 and IIt,,we have[“;“I = [:o] [K;o]= 5 (- l)jqtl@O-.i)j=O[tl+;-qKj[tf?][y]We also see thatK; 0[ t0 1E u.since 0 s to < 1. Now, the elementssE{O,l},tE Iv/,form a basis for U,“, and similarly, the elementswith 0 5 t < 1(resp.K; 0tl , tEN)[ 1form a basis for u. (resp. for V) by [L3], (2.21) (5.4). Therefore, our result is clear. qFor each n > 0, let u, (resp. K) be the subspace of C.Ji spanned by [“,“],K[“;“],K; 00 5 t < Ipn (resp. tl , 0 5 t < p”). By A 3.2, we see that u, is a subalgebra, and,c 1u, z u. @ ^yr,. Thus, Uj has a filtration(A 3.3) u. c u1 c u2 c . . . and Ui = U u, .Let V be a u,-module (or a Ui-module). We say that I/ has type 1 ifV= {VE VIK’v = v}.Obviously, if V is a &module of type 1, then V has also type 1 as a u,-module for each n.For each integer c, we define an algebra homomorphismx, : u, + Fnt0such that x,(K) = qc and x, ([“;“I) = [:I.Thus, it is easy to see that F via xc is a1 -dimensional u,-module of type 1.A 3.4. Lemma. (1) Every u,-module is a direct sum of one-dimensional u,-modules.

Du and Scott, Lusztig conjectures I 179(2) The elements (x,1 0 5 c < 1~“) f orm the set of non-isomorphic irreducible u,-modules of type 1.Proof. The first statement is obvious when n = 0. If II > 0, then every %-module is adirect sum of one-dimensional K-modules. This implies the first statement.To prove (2), we first note that the elements give lp” distinct homomorphisms from u,to F. By A 3.1, one also sees easily that x, = xd if c - d E 1p”Z.Finally, let x : u, + F be an algebra homomorphism. Then x(K) = q’ for some c,0 I c < 1 since K2’ = 1 (and we consider type 1 modules), and-is also an algebra homomorphism. As is well-known in the classical (nonquantum) case,xI”Y;;mw [:f] to(y) for some fixed c’, 0 5 c’ < p”. Now, for any t = t, + t, 1 2 0with 0 5 t, < 1,([“,“1) =qrpl)([:;P]) = [fl(fl) = [c+tc’l]by A 3.1 again. Therefore,x = x, + C,l. qA 3.5. Corollary. AnyJinite dimensional &f-module is a direct sum of one-dimensionalUi-modules.Proof. This follows immediately from (A 3.3) and A 3.4. qLet X( U,“) be the set of the algebra homomorphisms x from Q? to F such that F via2 is a Ui-module of type 1. This is an abelian subgroup of the dual space (U,“)*. We candescribe easily X(Ui) as the ring of I-p-adic integers as follows:(A 3.6)We denote the latter by Z,,,.Indeed, suppose x E X( U,“). Let x, = x llln : u, + F. Then, x,, E Zjlp”.Zclearly, x, + 1 I”, = x,. Therefore, x = (QnciO, E Z,,,.by A 3.4, andNote that a “number”c in Z,,, has the formc~,+c,l+c,lp+c21p2+ . ..)

180 Du and Scott, Lusztig conjectures Iwhere 0 s q, < 1, 0 5 ci < p for all i 2 0. If t is a finite number such that 0 5 t < lp”, thenthe notationnotation,n-lc[I%I means the numbert[Iwe havefor any x E X(U,O).tin F where s, = c-r + 1 cilpi. With thisx([“;“]) = [:Ii=OA 3.7. Theorem. Every finite dimensional &-module is integrable.haveProof. Let M be a finite-dimensional &-module of type 1. By A 3.5 and (A 3.6), wewhere M, = (v E Mlav = x(a)v, Va E Ui}. We want to prove x E Z.Since E”‘M c M and F@)M G M -2r, it sufftces to consider a I&-moduleM, = 0 M, wh&e X”=’ f’ + 22 for &me i’ with Mx, + 0. So we may assume thatXEXM = M,. Define a partial order on X : x 2 6 if and only if x - 6 2 0, and let x E X be themaximal element such that M, + 0. Pick v E M, with v + 0. Then E”‘v = 0 for all r > 0.By the commutation formula [L2], (4.1) (a), we haveE”‘l;“‘v= [K;o]+]v.Suppose x is not a finite number. Then there are infinitely many numbers of r such that: + 0 in F. This implies F@‘v + 0 for infinitely many numbers of r. This is impossible[Isince M is finite-dimensional. This contradiction shows that x E Z. Therefore, all weightsof M are integral, i.e. M is integrable. qWe note that it is proved ([Awl, 1.6 (c)) that the integrable finite-dimensional simple&modules (of type 1) correspond bijectively to the dominant weights. We now canstrengthen this result as the following corollary of the theorem. When F has characteristic0, a similar result is proved in [L2], 6.4. The corollary, as well as the theorem, is valid forall types of finite root systems, not just type A,.A 3.8. Corollary. There is a one to one correspondence between the set of jinitedimensionalirreducible Q-modules of type 1 and the set of dominant weights. qWe now turn to the case when q E F is not a root of 1. As before, we can also assumethat U, is of type A, with generators E @), F”), K, K-‘. With this assumptions, the algebraU, = U: is generated just by the elements E, F, K, K- ‘.

Du and Scott, Lusztig conjectures I 181Let 17: be the subalgebra of U, generated by the K, K - ‘, and oj’ the abelian groupof all algebra homomorphisms x from Uj’ to F. Then the elementsdefined by X,-~(K) = eq’, form an abelian subgroup X of 0:.Let M be a Q-module. For any x E vi, writeM, = {ZIEM~KZI = x(K)u}.We are going to show that if M is finite-dimensional, then M = c M,. The proof belowis due to Rosso [Rl] (see also [APW], 9.3). XEXBy the assumptionsand the equalitiesKEK-‘= q2E and K-‘FK= q2F,one sees easily that 0 is the only possible eigenvalue of E and F. Therefore, E and F arenilpotent on M.Choose r such that F”‘M = 0, and let2t-1h, = n (Kq”-“‘- K-lq(S-r)), t 2 1 _s=lWe have, for 0 5 t’ 5 t, [K;v;l][K;:;t]E ht. I!$‘. Thus, using the commutationformula [L4], 6.5 (a2), also (a6), we prove by induction on t that h, F”- ‘) = 0. So, h, = 0,2r-1and hence n (K2 - q2(s-r)) = 0, on M. This implies that the eigenvalues of K2 are amongs=lthe elements q 2(s-rq1~s~22r-1), so, K2 is diagonalizable on M. Consequently, K isdiagonalizable on M with eigenvalues in the set { f q” 11s 15 r - l}. Therefore, M is adirect sum of the weight spaces M,(x E A’).This completes the proof of the integrabilityappendix, in all cases.of M, as asserted at the beginning of thisReferencesCAMI M. At&ah, I. MacDonald, Introduction to commutative algebra, AddisonWesley, Reading, Mass. 1969.CA1 H.H. Andersen, The strong linkage principle, J. reine angew. Math. 315 (1980), 53-59.[APW] H. Andersen, P. Polo, K. Wn, Representations of quantum algebras, Invent. Math. 104 (1991), l-59.CA’W H. Andersen, K. Wen, Representations of quantum algebras: The mixed case, J. reine angew. Math. 427(1992), 35-50.[AGl] M. Auslander, 0. Goldman, Maximal orders, Trans. Amer. Math. Sot. 97 (1960) l-24.[AG2] M. Auslander, 0. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Sot. 97 (1960),367-409.

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