Lusztig conjectures, old and new, Ipi.math.virginia.edu/~lls2l/lustig_conjectures_i.pdfLusztig conjectures, old and new, I ... for any sufficiently large prime (size unknown) ... conjecture

J. reine angew. Math. 455 (1994), 141-182 Journal fiir die reine und angewandte Mathematik 0 Walter de Gruyter Berlin New York 1994

Lusztig conjectures, old and new, I

By Jie Du at Kensington and Leonard Scott at Charlottesviller)

This paper grew from a first attempt to understand the relationship of a new conjecture of Lusztig [L2] for representations of quantized enveloping algebras to his conjecture [Ll] for representations of algebraic groups in characteristic p. The main question here is: Does the new conjecture imply the old? Using results of [APW], it is easy to see this question has an affirmative answer if appropriate irreducible representations of a quantum enveloping algebra at a pth root of unity remain irreducible upon ‘reduction modp’. (Equivalently, the quantum and characteristic p irreducible modules involved, labelled by the same weight, must have the same dimension.) Indeed, Lusztig has conjectured this irreducibility in an equivalent context [L3], where the weights involved are restricted, and p is sufficiently large. Another standard hypothesis (equivalent for p 2 2 h - 3) requires that p be at least as large as the Coxeter number h, and that the weights in 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 of Kazhdan and Lusztig [KL] that they can prove the new conjecture, assuming results asserted 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 quasihereditary algebras 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 general problem has a positive answer under a multiplicity-free assumption on characteristic p Weyl modules, cf. 5 4, but is false in general. A counterexample by Alfred Wiedemann is given in an appendix. Of course, this does not mean that the main question itself has a negative answer, only that it is a difficult problem. Indeed, one can see with hindsight that such counterexamples had to exist, owing to differences between the ‘lowest p2-alcove’, which our model 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, and the Universities of Virginia and Oklahoma for their cooperation. We would also like to thank MSRI at Berkeley for its hospitality. Most of the research in this paper was completed during the program on representations of finite groups and related topics at MSRI in Fall 1990. The first author thanks ARC for support under the Large Grant L38.24210.

10 Journal ftir Mathematlk Band 455

142 Du and Scott, Lusztig conjectures I

The main thread of the investigation is as follows: 5 1 contains many results of general interest regarding representations of algebras over commutative rings. $2 discusses the notion of a highest weight category in this context, and shows how to construct quasihereditary algebras from them. This theory is applied to the quantum case in $3. The multiplicity-free results mentioned above are then obtained in 9 4, after constructing suitable quasihereditary algebras. Equivalently, one may view the above results as giving a rich supply of ‘projective’ modules. The existence of (endomorphism ring idempotents associated with) these modules is already sufficient to prove, for a given multiplicity-free Weyl module 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 the corresponding quantum and characteristic p irreducible modules. Even for Weyl modules which are not multiplicity-free, our results give information on the equality of dimensions for some weight spaces of these irreducible modules.

An interesting feature of our work is that, though we consider only weights in the lowest p2-alcove, we require no serious restriction on p itself, beyond standard exclusions for p = 2 or 3. Since this paper was first written, Soergel has announced on behalf of Andersen, Jantzen and himself that they can answer the original main question affirmatively for 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, as does the original characteristic p Lusztig conjecture. Much of the interest in the Lusztig conjecture (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 a completely different approach to the Lusztig conjecture, see [CPS 51 and [CPS 61.

This investigation has proved rich in byproducts. In addition to the above general results 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 the natural Krull dimension two local rings) are integrable (5 3). Since the q-Schur algebra itself has an integral quasihereditary version [CPS4], our results have consequences for the decompositions of standard permutation modules of symmetric groups in arbitrary characteristic. A few applications in this direction are discussed in 4 4. In particular, we prove, in the 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 specializing q to an arbitrary nonzero element of a$nitejkZd, where q automatically becomes a primitive lth-root of unity for some positive integer 1. As per the Dipper-James theory [DJI], the results here also have consequences for representations of the finite general linear groups in their nondescribing characteristics. At the 1990 MSRI algebraic groups conference, the authors of [CPS7] conjectured that representations of quantum groups over finite fields (equivalently, of generalized q-Schur algebras) would eventually allow the extension of the Dipper-James theory to all finite groups of Lie type. In 6 5 we indicate how our results extend to 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 143

After this paper was first written, Z. Lin pointed out that a crude version of the quantum 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 final analysis, the sum formula adds no essentially new ingredient to the above theory, though the 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 highest weight category axioms.

We would like to thank D. Costa and W. van der Kallen for a number of references and discussions regarding the ring n [q, q- ‘1 and its variations.

1. Finite and complete k-categories

Throughout this paper k is a Noetherian commutative ring, which we will momentarily assume also to be local. A k-category ‘% is an additive category in which the set Hom,(M, N) of morphisms between any two objects M, N of %? is equipped with the structure of a k-module, and multiplication of morphisms is k-bilinear.

1.0. Convention. All k-categories considered in this paper are abelian, and equipped with a fixed exact embedding into the category of k-modules.

In particular, the objects of the category objects may be regarded as k-modules with some additional structure (e.g., the category of modules for a k-algebra). The full subcategory of 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 form a set may, through standard embedding theory [F], be exactly embedded into the category of k-modules. However, our convention allows a crude way, adaquate for the aims of this paper, 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 abelian category, subject to some care regarding set-theoretic issues, and that the long exact sequences for 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 residue field k = k/m. If M is any k-module or object in a k-category %‘, the quotient M/Mm is denoted 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 of generators of m. The full subcategory of objects M in V on which m acts trivially is denoted 59. It is naturally a k-category.

There are obviously many examples of k-categories. Here is an especially important one for the purposes of this paper:


1.1. Example. Let k be the localization of the polynomial ring Z [q] at the maximal ideal m = (p, q - 1) where p is a fixed odd prime. Let U be the quantum enveloping algebra over 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] that p > 3. Let V = g(U) be the category of k-finite U-modules M for which M is type 1, in that the elements standardly denoted Ki act as 1 on M, cf. [APW], 9.5. We will show momentarily in 1.3 that % is an abelian category, and thus is a finite k-category. We call the objects 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 category of “integrable modules of type 1” for U, in their sense. In the k-finite case their integrable type 1 modules are easily seen to be type 1 in our sense also. A nontrivial consequence of 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 the category of finite-dimensional modules for the hyperalgebra over k of the semisimple group associated with the given root system. It is also proved that each Weyl module of @ lifts to a k-free module in %‘.

Eventually, we will use these few facts here (with some additional input from the tensor product 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 notion of 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 type A) available for all types, collectively covering all finite-dimensional rational representations of semisimple algebraic groups. The new generalized q-Schur algebras of this paper similarly make 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 of the above quantum enveloping algebras (associated to finite root systems). We give integral versions 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 integral sense of [CPS4]. (Other localizations besides k of Z [q, q- ‘1 are also studied in the last section of this paper.)

With these aims in mind, we now return to the general k-category %‘, and investigate first the extent that vanishing of Extl for objects in %? reduces to @.

For later use we observe that, if P is a k-projective object in V, and T is in @ then Ext&(P, T) g Exti(P, T). If each k-projective object (such as P) in %? is the image of a projective and k-projective object of %?, this isomorphism may be extended to the higher Ext groups by dimension-shifting. Even without this hypothesis, there is always a natural injection Ext$ (P, T) -+ Exti(P, T), as follows from a standard interpretation of Ext’ in terms 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 any composition 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


finitely many of them for each object M. The following proposition shows this notion of “composition factor” agrees with another plausible definition.

1.2. 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 of ii? = M/Mm.

Proof. Let N’ E N s M be inclusions in %? such that S = N/N’ is simple. Thus, S is a composition factor of NjNm in the usual sense. However, for some large n we have Nnt 2 Mm” n N. So S is a composition factor of the finite length object N/(Mm” n N), and thus of M/Mm”. However, for each nonnegative integer Y, the object Mm’/Mm*+’ is the sum of the images in M/Mm’+’ of M/Mm under the maps induced by multiplication from each of a finite set of generators for m’. The proposition follows easily. q

1.3. Corollary. Let k and %? be as above, and let

be an exact sequence of k-finite objects in W. Then the set of composition factors of ii? = M/Mm is the union of the set of composition,factors of L and the set of composition factors of N. 0

That is, the set of “composition factors” of M is the union of the sets of “composition factors” for L and N. In particular, it is nowclear that the category described in example 1.1 is abelian.

If k is local and complete, with maximal ideal m, we say that a k-category 9 is complete if it is finite, and if every object M is the inverse limit in %7 of its quotient objects M/Mm”.

1.4. Proposition. Suppose k is local and complete, and %? is a finite k-category. Assume V 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”, for n a nonnegative integer, all make sense as subobjects of M in %?. To test an isomorphism M 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 reductions are often a problem with limits, but here we already have the candidate for the inverse limit, 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 a

finite direct sum of copies of k, and follows for M finitely presented from the Artin-Rees theory, cf. [AM], Prop.lO.12. However, if N is a U-module, and f is a k-linear map f: N + M such that f induces a U-equivariant map N + M/Mm” for each n 2 0, then f is obviously itself U-equivariant. The proposition follows. q

146 Du and Scott, Lusztig conjectures 1

We 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 and Noetherian, can be checked at the reductions modulo the maximal ideal of k. As before, if k is local and M is an object in a k-category, we let M denote the object M/Mm, where m is the maximal ideal of k.

1.5. Theorem. Let k be a complete commutative Noetherian local ring, and V a complete k-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 to a 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, cf. [M], III, 1.7, if x E Extb(P, M), and E = E, is a corresponding short exact sequence:

O-+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 sequences

0 + 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 composition factors of the (finite-length) object M/Mm” occur already as composition factors of M. Thus Exti (P, M/Mm”) = 0, and so Horn, (P, -) applied to the above sequence gives again a short exact sequence, for each nonnegative integer n.

A similar argument using the exact sequences

0 + Mm”/Mm”+’ -+ M/Mm”+’ + M/Mm” + 0

shows 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- modules

0 -+ l@Hom,(P, M/Mm”) + @Horn, (P, E/Em”) -+ l$nHom,(P, P/Pm”) + 0

The theorem now follows from the previous proposition. q

1.6. Corollary. Suppose k is a commutative local Noetherian ring, and %? is a finite k-category. Assume % has an exact andfull k-linear embedding into the category of k-$nite


modules 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 extension of 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 every composition factor of S of ii?. Then Ext&(P, M) = 0, and every morphism in ‘2 from P to a quotient 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 P by A4 in % Let V denote the image of U in the k-endomorphism ring of E. Thus V acts on M and P, viewed as sections of E. The extension E is split as a V-module if and only if it is split 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 that

Ext;(X, Y) 0 k “= Ext;,ac(X@ & Y@ k)

for any k-finite V-modules X and Y. Here k denotes the completion of k with respect to its maximal ideal.

Now, 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 is a homomorphic image of u. In particular Ext+(P, S) = Ext&(P, S) = Ext$(P, S) = 0 for all F-composition factors S of &?. (The second equation uses our hypothesis that G?? is closed under extensions in the category of U-modules.) Thus Extb, i(P @ k, M@ k) = 0 by the theorem, applied to the category of V@ k-modules. The displayed isomorphism now gives Ex$(P, M) = 0, thus Ext$(P, M) = 0, and it follows from the full embedding of $9 that Ext:,(P, M) = 0. •I

1.7. Corollary. If %? is a finite k-category which is complete or satis$es the assumptions of the previous corollary, and P is an object of %’ such that P is projective in 0, then P is projective in V. 0

2. k-finite highest weight categories

To begin, let k be a field; later we will consider Noetherian local rings, and even some nonlocal cases. Let /i be a locally finite poset (that is, the intervals of /1 are all finite). The elements 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 category with weight poset /i.

This is an abelian k-category %? in which every objects is k-finite, and in addition:

(a) The nonisomorphic simple objects in %’ are indexed as L (2)) with 1 ranging over the distinct elements of /1.


(b) There are given objects V(n) and A (2) in V, called the Weyl modules and dual Weyl modules, 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 other composition 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 Chapter 5 of [PS] in the study of perverse sheaves. Recall that an ideal in a poset is a subset containing any element of the poset smaller than some member of the subset; a coideal is a subset containing all elements of the poset larger than some member. Given a finitely generated ideal r of /1, the conditions imply, cf. 2 A. 3 below, the existence of projective covers in the category Q?[r] formed by objects whose composition factors L(p) all have weights in r. These projective covers all have Weyl filtrations (filtrations whose successive quotients are Weyl modules). It follows that for /i finite, ‘% is a highest weight category in our previous sense [CPS2], and is the category of finite-dimensional modules for a quasihereditary algebra [CPS2], Thm. 3.6. The converse is clear, with (ii) even holding for all n > 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 be checked in any exact or abelian k-category @ in which V is fully and exactly embedded and closed under extensions. These latter conditions imply Ext&(X, Y) g Ext&(X, Y), for any objects 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 quotient maps arising in the theory of quasihereditary algebras, cf. [PI, [CPS3].

The above discussion of projective covers implies the following condition holds in a k-finite highest weight category, which we record for comparison with the local integral theory which follows.

(c) Let T be an object in %7. Then T is the epimorphic image of an object P in $F? which has a Weyl filtration.

In fact, let r be the ideal in /i generated by all the weights h for which the simple module L(p) is a composition factor of T. By 2 A. 3 below, each L (,u) has a projective cover in %?[r], filtered by Weyl modules. Consequently T has such a projective cover. q

We shall now study the local integral case. For the rest of this section k is a commutative Noetherian 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 generated by a fixed prime p and q - 1. Our theory will also apply for the completion, which has a number of advantages. Other important choices of k include localizations of Z [q], at various primes.

At the end of 0 4 and in 9 5, in results oriented toward the Dipper-James theory, one is interested in essentially all localizations of Z[q], or at least of Z [q, q- ‘1.


In general we denote the maximal ideal of k as m and the residue field is k. The reduction modulo m of any k-module A4 is denoted M.

2.1. Definition. A k-Jinite highest weight category is a category @ of k-finitely generated 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 weight category with weight poset LI, as defined above;

(ii) there are k-projective (thus free) objects V(J) and A(i) in @? whose reductions V(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 epimorphic image of an object M in %? which has a Weyl filtration;

(iv) if P, M are objects of %‘, with P k-free, and

Ext$(e L) = 0

for all composition factors L of M, then

Ext;(P, M) = 0

We recall from 5 1 that this last axiom is satisfied if the k-category structure of Q? arises from an exact and full embedding as subcategory, closed under extensions, of k-finite modules 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 abelian subcategories 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. In the same spirit, one could probably weaken 1 .O to allow a family of embeddings to describe the needed k-category structure. No claim is made that the above axioms are optimal, but they 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. Note especially that

2.1.1. Proposition. The category of k-jinite modules for a k-quasihereditary algebra is a k-finite highest weight category. q

The proof is obtained easily by imitating the argument for the corresponding result over fields [CPS2]. In the nonlocal case we can say that the above proposition holds with respect 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’s for the opposite algebra.) Explicitly, we have

2.1.2. 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 1

objects V(n) and A (3L) for 1 E A such that, for any prime ideal @, the category of k,-finite 5$- modules is a k,-finite highest weight category with weight poset A and with the localizations of V(n) and A(1) as Weyl and dual Weyl modules, respectively, for 2 E A. q

2.1.3. Remark and definition. This last proposition shows, in our view, that the category of k-finite modules for a split k-finite quasihereditary algebra deserves to be called a k-finite highest weight category even in the nonlocal case. This ‘definition’ for the split case could easily be extended to (interval-finite) weight posets bounded below by finitely many minimal elements: A (split) k-jinite highest weight category would be, simply, the compatible union of the categories of all k-finite modules for split k-quasihereditary algebras associated to finite ideals of the weight poset. This definition is completely adequate for all the cases considered in this paper, even in the nonlocal case. The more abstract development of this 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 notation of 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). If r is an ideal in the weight poset A, then %? [r] denotes the category of objects in %? all of whose 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)) = 0 unless 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. q

2.3. Proposition. Let G?? be a k-finite highest weight category, let I be any finitely generated ideal in the weight poset A, and let T be the projective cover in @[I] of a simple object 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 an object P is projective, the projective cover of L(1) in %(I], and is uniquely determined up to isomorphism.

Proof. If P exists, it is clearly projective by axiom (iv), and L(A) is its unique simple quotient. That is, P is a projective cover of L(A) in %[r]. Since k is Noetherian, and P is k-finite, every surjective endomorphism of P is an automorphism, and the uniqueness of P follows by a standard argument.

We now address the existence. Let M be an object in V filtered by Weyl modules which has T as an epimorphic image. Applying the above lemma, we may assume all the Weyl modules appearing in M are indexed by members of r. We claim M contains a subobject P filtered by Weyl modules with P z T, and we will prove this claim by induction on the number of weights in r which are 1 A. Write r = r’ u {v}, where v 2 A is maximal in r. By induction, we may assume that v appears as a weight of a composition factor of T.


Therefore ,5(v) must appear as a composition factor of M, and, thus, of one of the members of its Weyl filtration, V. We have I/ 2: V(v) by maximality of v. Shuffling terms of the 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 of T has a similar form, giving rise to an exact sequence

O+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), corresponding to the decomposition of M, as a sum of copies of V(v). Clearly, some subset of this set of copies of L(v) has sum which maps isomorphically onto the radical quotient of TV. It follows that we may write

with 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 cateogry of objects in %? whose composition factors are indexed by weights 5 v, by 2.2. Thus, there is a map A!,” -+ X which maps surjectively onto X/tnM,. The resulting sum map

is also surjective, and thus an isomorphism. The conclusion is that we may choose M,” in the 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 maps surjectively onto T’. By induction, it contains a submodule PI/M, filtered by Weyl modules with P’/M, z T’. Thus P//My g T. In particular, P’/M: is projective in the category of objects of ‘G9 whose composition factors are indexed by weights in P, so P’ contains a subobject P complementary to M,“. Clearly P g T, and this completes the proof. q

2.4. Definition. If r is a finitely generated ideal in the poset A, the object P constructed above will be called the PIM associated to 3, in ‘%[P], and will be denoted P(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 of separable type, in the sense of [CPS 41, Also, a defining sequence of ideals Ji of Y may be chosen for any linear order 1, >= 2, 2 . . . of the$nite set A, compatible with itsposet structure, so that the following property holds: For ri 5 A defined as the subset consisting of all weights 5 li in the linear order, V[P,] is equivalent, under the given equivalence for V, to the category of 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 put Y = End,(P). Fix a linear order A, 1 AZ 2 ... as above, put Pi = @ P(lj), for each sub-

script i, and define jsi

Ji = Horn,, (P, Pi) Horn, (Pi, P) s Horn&P, P) = Y.


Then Ji is obviously an ideal in 9 Let Ji denote the image of Ji in 9 (This notation appa- rently differs 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). So

Ji = Horn, (P, Pi) Horn&P,, P)

-- in 2 identifying the latter with Hom,(P, P). Now $? is a highest weight category in the sense of [CPS2]. The proof of [CPS2], Theorem 3.6 shows that 9 is quasihereditary with defining sequence

and it is clear that 9 is of separable type, since all the endomorphism algebra of its simple modules are separable. Thus Y is quasihereditary as a k-algebra, and of separable type, by [CPS 41, Thm. 3.3 (b). q

2.6. Remarks. (a) The separability hypotheses are just needed to restrict attention to maximal ideals. If k is a DVR (e.g., the localization of Z [q, q- ‘1 at one of its height 1 prime ideals), the separability hypothesis may be dropped if it is known, say, that the localization of %? over the quotient field of k is semsimple, with the Weyl and dual Weyl modules becoming simple. More generally, it would be sufficient to assume, for any k, that the localization at any prime ideal gave rise to a highest weight category, with the localizations of the Weyl and dual Weyl modules serving in the same role in the iocalizations.

In fact the separability hypothesis is harmless in this paper, where in all examples the endomorphisms 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 by taking the endomorphism ring of a different projective generator. Another natural choice for Y is obtained by taking P to be

the 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-finite right 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 a homomorphic image of the right U-module U. Indeed it is easy to check that Y is the maximal 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 homomorphic image 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 the fact that it is quasihereditary (and, in particular, k-free). If U is k-free (or even torsion


free), then there is an obvious candidate for Y: Let K be the quotient field of k. Then YK is a finite-dimensional K-algebra, and it is not hard, using lattices, to argue that it is the maximal 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 is not easy to prove anything from this description, it at least provides a common starting point in the nonlocal case, where U has arisen by localization from an algebra over a commutative ring localizing to k. See 9 5.

We again let n be an arbitrary locally-finite poset, and give some further corollaries of the 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). q

2.8. 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 corresponding bounded 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 further details 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. q

2.9. Corollary. Let X, Y be objects in %?.

(a) Zf X is k-free, then

for all n 2 0.

Ext$(X, 7) g Ext;(X, Y)

(b) If Y is k-f ree, there is an E, spectral sequence

T$(Ext$(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 of X are available, and the results above are obtainable from standard arguments with resolutions and double complexes. (Note, if X is projective, then

Home(X, YO,F) E Hom,(X, Y) @JkF

for any free k-module F.) q


2.10. Corollary. Let A, p E A, and n a nonnegative integer. If 2 =!= p or n + 0, then

Ext;( V-(n), A (p)) = Ext;( V(A), A (p)) = 0 .

Also,

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 the limit of the spectral sequence above is zero. If any term E2ps4 is nonzero, choose q minimal with that property, and observe E$q = ExtG( V(n), A(i)) @ kmust be nonzero. But then the minimality of q gives E$q z Et,q + 0, a contradiction. This gives the required vanishing for II * p.

For the remainder, we may assume 1 = p. Now, however, V(A) is projective in a subcategory ‘Z[r] containing A(1) and all of its composition factors. The rest of the corollary follows easily. 0

2.11. 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 a torsion-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 .O of this paper is not required.

2A.0. Hypothesis. Let k be a field, and V an abelian k-category which satisfies conditions (a) and (b) of the beginning of this section with respect to a locally finite poset II and objects V(n), A(2), and L(2).

Assume also each object A4 in %? has a finite composition series, and that End, (L(i)) is finite-dimensional over k, for each A E /i.

2 A. 1. Theorem. Suppose P is an object in @filtered by V(2)‘s and satisfying

Ext&(P, V(p)) = 0

for 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, we may assume X is irreducible, and thus equal to L(p) for some p c II. Let r = r(p) denote the set of elements w in /i with A 5 w 5 p for some i for which V(2) occurs in the given filtration 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 prove this claim, from which the theorem follows, by induction on the cardinality of the set r(p). First, we need a lemma:


2 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 exact sequence of cohomology for Ext,* (V(n), -) to the short exact sequence

0 + 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 all composition 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. c1

We 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 all weights v for which T(v) has smaller cardinality; this includes all weights smaller than ,u. We now obtain that Ext$ (P, L (~1)) = 0 by arguing as in the proof of the above lemma, using a short exact sequence

0 + L(p) --) A(p) --f Y --) 0,

induction, and the resulting long exact sequence for Extg (P, -). Similarly, we obtain that

Ext&(P, L(p)) = 0

by using a short exact sequence

0 + z 4 V(p) + L(p) --t 0)

together with induction and the hypothesis of the theorem. This proves the claim and completes the proof of the theorem. q

2 A. 3. Theorem. Assume that the poset A is jinitely-generated. (There are finitely many 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 projective P 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 for completeness :

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, with the 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 locally finite and finitely generated, the set 52 is finite. We will prove the existence of a projective cover of L(v), with the desired filtration, by induction on the cardinality of G?. If !3 has only one element (equivalently, v is maximal), then Lemma 2A. 2 shows V(v) is a projective


cover of L(v), and it trivially has the desired filtration. Thus we may assume v is not maximal.

Let /i’ denote the poset obtained by deleting a maximal element p > v from /1, and let V’ = 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 fully embedded in V and closed under extensions, inherits the required Ext conditions. Ob- viously Q(v, /i’) has smaller cardinality than Q, and so we may apply induction. Thus, we may 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 to be the kernel of V(p) -+ L(p), we find that Extb(P’, V(p)) embeds naturally in

Ext; (P’, L(P)) .

The latter is finite-dimensional over k, as may be seen by forming the short exact sequence

0 -+ 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 corresponding extension P of P’ by a direct sum Vd of d copies of F’(U). The corresponding extension E of P’ by a direct sum Ld of d copies of L (p) remains nonsplit under any nonzero projection Ld + L(v), since such a projection may be viewed as just taking a specific D-linear com- bination of the factors, and yl, . . . , yd have been chosen D-independent. Since P’ has a simple head, it follows that E has a simple head, which implies in turn that P has a simple head. 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, that I’(p) is projective by 2A. 2, and so Ext&(P, X) is naturally a homomorphic image of Ext&(P’, X). The latter is zero for X = V(n) with i + ,u, as argued above from the projectivity of P’ in W. So only the case X= V(p) remains. However, the yi, . . . , yd span

Ext:, (P: VP>), so that every element z may be viewed as the image in Exti(P’, V(p)) of the class x = (yi, . . . , yd) defining P in Extk(P’, V”) under the map induced by a morphism Vd + V(p).

Since x defines the exact sequence

its image is 0 in Extk(P, V”) under the map induced by P + P’, as is well-known. Conse- quently, the image of z in Ext&(P, V(,u)) is 0 under the map induced by P -+ P’. It follows that Ext&(P, V(p)) = 0, and the proof is complete. •I

2 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


weight category in the sense of [CPS2], as is the dual category %[I], with respect to the evident choices qf dual Weyl modules.

The proof is immediate. q

There are many further corollaries from the highest weight theory. We just mention two that especially clarify matters.

2 A. 5. Corollary. Let I be as above. Then the inclusion %?[I] s w induces a full

embedding of bounded derived categories. In particular Extit,r (X, Y) g Ext”, (X, Y) for each X, 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, any morphism in Db(%?) is realized in some Db (V [r’]). The equality of two morphisms can be detected in the homotopy category Kb (V) by a process involving only finitely many objects of G??, consequently demonstrating equality in some Db(V [r’]). The corollary follows. q

2 A. 6. Corollary. Let V be as above. Then Exti (M, N) isjinite-dimensional over k for

all objects M, N in ‘F, and, for 2, ,LL E A,

Ext#W, A (~4) = 0

unless n = 0 and A = u. q

3. Generalized q-Schur algebras: The natural localization case

In this section we let k be the localization of the Laurent polynomial ring d = Z [q, q- ‘1 at the maximal ideal m = (q - 1, p), where p > 0 is a prime with the property as mentioned in the introduction. (We will consider other localizations and G? itself at the end 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 is denoted by U. The latter notation agrees with [APW], though our commutative algebra notations 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 a locally finite poset. (This means that all intervals are finite. In fact, all finitely generated ideals of X+ are finite.) For each A E X+, let L(1) and L,(A) denote the irreducible modules for U and UF, respectively, with highest weight A, where F is a field of characteristic 0 in which q is a primitive pth root of unity.

Let %? be the category of k-finite U-modules of type 1, as defined in Example 1.1. We have

11 Journal fiir Mathematik. Band 455


3.1. Lemma. The full subcategory G? of objects $3, with M in 59, is a k-$nite highest weight category.

Proof. Since the image of q in k is 1, the action of Ki on each k-module M with M in %? 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 same root system C ([L4]). Therefore, the category G? is isomorphic to the category of k-finite dimensional 2I-modules, which is a highest weight category (see [PSI, Example in Chapter 6). q

If Z is a finitely generated ideal in X+, then the subcategory @[Z] of objects with composition 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 weight category (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 can be 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 a multiple 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 property for the usual hyperalgebra.

To determine such an X, one can just take the tensor product of two generalized Steinberg 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 a module 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 has the sum of X(A)‘s as its reduction modulo m. It is only necessary to use the same tensor products that went into the constructions of the X(n)‘s over k. q

For the complete definition of the term “integrable” used below, we refer the reader to [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 scalar multiplication.


3.3. 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 by Weyl modules, such that i@ is the epimorphic image of P, and Extb(P, L) = 0 for every composition factor L of I%?. Applying 1.6 we obtain that Exth (P, M) = 0, and the epimorphism P + &? 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. q

Note 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 all objects in ‘% with composition factors all lying in G? [Z] is a k-finite highest weight category by the definition and 1.2; cf. also 2 A. 4. We have that End, (L (A)) = k is separable over k for all ;1. So, by 2.5, %?[Z] is equivalent to the category of k-finite modules for a quasihereditary algebra. 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 similar procedure from a quantum enveloping algebra, but we omit the details. In $5 we will similarly 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 of universal enveloping algebras. Our approach so far is like that of [CPS I], [CPS2] for the classical 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-epimorphism

U + 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-epimorphism

rc’: U+ “2 End,(L(v)) g @&L(v). VET

Since U is projective as a left U-module, the map rc’ can be lifted to a U-epimorphism

7-c: u+ @l,P,(v). vsr

Let Z = ker n.Then Z is a left ideal. We claim that Z is a two-sided ideal. Indeed, suppose that IX + Z, for some x E U. Then we have a U-epimorphism

I--+ Ix + (Zx+Z)/Z*O.


Let 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 has U/Z as its image. So, U/Z0 g U/Z 0 N for some k-module N =+ 0, since U/Z is k-free. On the other hand, the radical quotient M (in %? [Z] !) of U/Z0 is still U-cyclic. So, A4 is a cyclic module 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. There- fore, we must have Ix = Z, and so Z is an ideal. Thus, U/Z is an algebra and k-free as a k-module. Finally, we have

YCrl = End, @Z,&(v) g End,(U/Z) g U/Z. q L >

4. Multiplicity-free results

In this section, our base ring k is a regular local ring of Krull dimension at least 2, with maximal ideal m, and cp is a fixed height 1 prime ideal of k. As before, K denotes the quotient field of k and k = k/m, the residue field. For any prime ideal @ of k and k-module M, we write

UP) = 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 particular we have Weyl modules V(A) and dual Weyl modules A (A), for each ,4 E A. These modules reduce modulo m to their counterparts over k, and any nonzero map V’(n) -+ A (A) lifts to a map V(A) + A (A). This map is unique up to a multiple by a unit of k in the split case, and we call it the canonical map. If YK is (split) semisimple, then the canonical maps are always embeddings (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 associated irreducible module, and L’ the irreducible module for V(q). Zf V is multiplicity-free, then dim L = dim L’.

Moreover, if A = A(1) is the corresponding “dual” of V then the canonical embedding V + 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


Horny (Y/J, A) = 0

by 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 sequence

induces an isomorphism of 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 head L(v). Thus, Y E @ 1, Ye, where Z, = dim L(v). Therefore, we have a k-space isomorphism

VCA

A/V2 @ I,e,(A/V). veil

Since V is multiplicity-free, we see that the k-module

k@ e,(A/ V) = e,(A/ V) E’ Horn, (ge,, A/ V)

has dimension 1 or 0. So, e,(A/ V) is cyclic, and, hence,

e,(A/ V) E k/a

for some ideal a c k. We now prove that a = cp” for some n, depending on v. Since k is regular (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, and the formal powers k n cp”k, are actual powers here.) Suppose q E Ass(A/ V) has height h > 1. Then q contains a height 1 prime ideal p + cp. Let @ be generated as an ideal by an element p E k. Consider the exact sequence

P o-+v- v+ v/pv+ 0.

We have an exact sequence

. . . + 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 localizations at @ of (Y/ J)/p(Y/ J), V/p V have no common composition factors (4.1 a). Thus, p annihilates no nonzero element of A/V. On the other hand, (A/V)” = 0 since 9” is semisimple, so all members of Ass(A/ V) have height exactly 1. However, (A/ V)@ = 0 for each height 1 prime ideal @ =I= cp since Y(p) is semisimple by our hypothesis. So, (A/ V)@ + 0 implies p = cp and therefore, Ass(A/ V) = (cp}.

From the argument above we have

(4.1 b)


where cp = (4). This implies that

dimA-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 is diagonalizable.

Let a,, . . . , a, be the generators of all cyclic modules in A / V. Pick elements vi E A such that zli + V = ai for 1 5 i s r. In particular, the set {Ui} is part of a basis of A; indeed the image 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 that i=l

V/Y is free. If so, vi, . . . , v, together with a basis of submodule of V which splits V + V/Y diagonalizes 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 ideal cp. We see that Y, is a submodule of I$ also, a submodule of A,, and

We show that this module is torsion-free. Pick x E V, such that the image x’ of x in V, /G is a torsion element. So, x’ E (A,/Y,),, the torsion submodule of A,/Y,. Consider the natural homomorphism

n: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’) = 0 implies 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 is injective. Now, in the following commutative diagram

/p 2l+ v/y

I 4 k’“’ 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. q

4.1.1. An improvement. Suppose 9’ satisfies the hypotheses of 4.1 above, and e E 9’ is an idempotent. Let V, L, L’ be as in 4.1, but do not assume V is multiplicity-free. Then the argument for 4.1 yields the conclusion

dimeL = dimeL’


under the weaker hypothesis that eP is filtered by terms e L (v) with L(v) appearing in V with multiplicity <= 1. (Since the irreducible eye-modules are precisely the nonzero modules eL(v), we can state this weaker hypothesis by saying eV is multiplicity-free.) The generalization appears useful, especially for e a projection onto a ‘weight space’, and will be discussed further later.

For now, we apply the basic version of Theorem 4.1 to the generalized q-Schur algebras introduced in 8 3. We will see that in the ‘multiplicity-free case’ Lusztig’s new conjecture on quantum enveloping algebras does imply his old conjecture on algebraic groups.

Keep the notation in $3. Let L,(A) and L(1) denote irreducible modules for U(q) and 0, respectively, with highest weight A, where q = (4,).

4.2. Corollary. Let k be the localization of d [q] at the maximal ideal m = (p, q - I). Thus k is a regular local ring of Krull dimension 2. Let 3, be a dominant weight in the lowest p2-alcove with the property that V(A) is multiplicity-free. Then the APW diagonalization conjecture [APW], 10.15 is true for A. In particular, dim L(A) = dim L,(A).

ProoJ Let n = {p E X+ 1 p 5 d>, and let Y = Y [/iI be the corresponding generalized q-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 @ + cp

It is known from [L4], [RI-21 that the specialization of Y is semisimple at any field where q is not mapped to a root of unity. So if Y(p) is not semisimple, then q is sent to a root 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 by the strong linkage principle [APW], 8.1, since il is in the lowest p2-alcove. So, (4.1 c) is proved. Now our conclusion is a consequence of Theorem 4.1. q

4.3. Remarks. (a) 0 ne can get the equality of dimensions here from a more traditional character theory argument. See Appendix 2. However, the latter argument was only discovered after the more abstract version given here, and it does not give any diagonalization result. Similarly, the equality of dimensions in 4.6 can be obtained from character theory, as well as the equalities in 4.1,4.5, though one must impose the additional hypothesis that k has Krull dimension two.

(b) Let k be the localization of B [q] at a maximal ideal m such that the residue field k of k is the finite field s[c]. Thus, 5 is a root of unity of order prime to p. If 5 + 1, we let 1 denote the order of 5, but if 5 = 1, it is convenient to set 1 = p. We remark that one can generalize 4.2 from the natural localization case to such a general localization case, the so-called “mixed case” in [PW] and [AW]; see 5 5. An especially attractive situation like this is that of the original q-Schur algebras of Dipper and James, which play a role both in the modular representation theory of the symmetric group and in the representation theory of finite general linear groups in nondescribing characteristics. As mentioned in the introduction, James has independently made conjectures here parallel to Lusztig [L3]. We now apply Theorem 4.1 to these algebras.


Let Y = 3 (n, r) be the q-Schur algebra of degree (n, r) over & (we adopt the version discussed in [DJ2] or [Du]), and let

q((n, r; k) = <((n, r) Qdk and $’ = q(r, r; k)

with k as in 4.3. By [CPS4], (3.7.2) $( n, r; k) is a quasihereditary k-algebra, so the category of k-finite $(n, r; k)-modules is a k-finite highest weight category. The usual poset is the set A’(n, r) of partitions of r with at most n parts. We denote the Weyl modules for 3 (n, r; k) by W” for each A E A + (n, r). (Here our Weyl modules are as described in [DJ 21.)

1 Let F” (resp. FA) denote the simple head of W”(q) where cp = (&) (resp. of WA = W ). It is known from [DJ2] or [PW] that the set {F” 13, E A’@, r)} (resp. (F’I 1 E A+(n, r)}) is a complete set of nonisomorphic irreducible Y(q)-modules (resp. .9-modules). In [J], 9 4 Gordon James conjectured (for the case r = n) that, when r < lp, F” remains irreducible upon ‘reduction modulo p’. Our result 4.1 does support this conjecture in the multiplicity- free case.

4.4. Corollary. Maintain the above notation. If r < lp, and WA is multiplicity-free, then dim F” = dim F”. In particular, James’ conjecture is true for such a A.

Proof. We give the proof for r = n, and just indicate the changes required for the general case.

We only need to prove that ,u?,(@) is semisimple for all height 1 prime ideals M =l= cp. We know from [DJO] that Y,(p) is semisimple when q is not mapped to a root of unity under the map k + k(p). (The same is true in the general case, and follows from semisimplicity of the associated generic Hecke algebra.) Suppose q is sent to a root of 1 in k(p). Thus @ = (4,) for some n. If n = 1, then Yr(@) is the classical Schur algebra over Q, which is certainly semisimple. Otherwise, the element 5 generating the finite residue field k/m is a primitive lth-root of unity, and n = lp” for some e 2 0, since @ s m. (Recall that roots of unity of order prime to p always remain distinct when mapped into a finite field of characteristic p.) However, if e > 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 algebra result, and holds in the general case.) Now our conclusion follows again from Theorem 4.1. 0

We will return to q-Schur algebras later, but we now continue the general discussion of multiplicity-free results. Let k and Y be as described at the beginning of this section. We denote by 9? = Q?(Y) the category of k-finite Y-modules. By 2.1 .l, %? is a k-finite hightest weight category with the poset A. Similarly, we write @ = V(9) and %?’ = %?(Y(q)).

For each 1 E A, we define

Q2, = {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 regular local ring) of Krull dimension 2 and assume cp $ m2. Let V = V(A) be a Weyl module in $7


for some A E A with L(A) the associated irreducible module, and L(A)’ the irreducible module for V(A)’ = V(q). Suppose

[L @)I = c a,, C JWI and C-W)1 = c 4, C v(v)‘1 VEA VEA

in 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 weight category (5 2).

Let P be the direct sum of the projective covers in %? of the simple objects L(p) for p 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 commutative local Noetherian ring.)

Indeed, if 0 = JO 2 J1 E . . . & J, = 9 is the defining sequence of Y and Ai denotes the set indexing the irreducible left Y-modules in the radical quotient of Ji /Ji _ i, then A = u Ai. By 2.1.1 and 2.3 projective covers of simple objects exist. (In particular Y is

semipkrfect.) It follows easily that Ji /Ji _ 1 is the projective cover of its radical quotient, as an Y/J,- ,-module. The summands corresponding to isotypic components of the radical quotient are easily seen to be ideals, and it follows we may assume that each Ai contains a single element. Thus r = u Ai for some r, and the category 5~? [r] is isomorphic to the

i>*

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, = 1 in Y such that e, ej = ej e, = e, for i sj and Ji = Yei9 It follows that

W(Q) g e, Ye, -f mod .

This implies the claim (see [P], $2). (The reader uncomfortable with quotient categories can 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 the functors

F: @ + 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 Weyl modules, 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 thus give rise to linearly independent elements at the Grothendieck group level. Therefore, we have

[eL(A)] = C anv [eI/o] and [eL(I)‘] = C a;, [eV(v)‘] veR veR


in the Grothendieck groups of g(Q) and %“(a) respectively. However, [V(n) : L(v)] 5 1 for 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 a regular local ring of Krull dimension I, thus a DVR, and O/w E k/m where y is the maximal ideal of 0. So, the standard reduction modulo tp process from Q?‘(a) to 5?(O) is independent of the choice of the O-lattices at the Grothendieck group level. Therefore, we have a group homomorphism from the Grothendieck group of the category %7’(Q) to that of the category g-(Q) which sends [eF’(v)‘] to [ek’(v)] and [eL(v)‘] to [eL(v)] by the dimension equality. Consequently, we have a,, = a;, for all v E s2. q

As before, we can apply 4.5 to the generalized q-Schur algebras in the natural localization case. Let chM denote the formal character as usual (see, for example, [APW] in the quantum case).

4.6. Corollary. Let 3, be a dominant weight in the lowest p2-alcove. Suppose

chL(il) = c a,,ch V(v) and chL,(i) = c a;,ch v,(v) v v

where V,(v) is the quantum Weyl module associated to weight v. Then a,, = a;, for all v such 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 specialized to 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 Jantzen region, 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 generalized q-Schur algebra associated to .4 (9 3). By the proof of 4.2, we see that Y satisfies the hypothesis of 4.1. Now the conclusion is the consequence of 4.5. q

A similar consequence can be concluded for the q-Schur algebras $ (n, r; k) assuming Y < 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 an interesting consequence for standard permutation modules of the symmetric group 6, on r letters: 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 each permutation module for the action of the symmetric group on partitions with a given number of parts, and that Yq(n, r; k) may be viewed as the endomorphism ring of a direct sum T (called ‘tensor space’) of such q-analogs. Here the (q-analog) permutation module for X appears with nonzero multiplicity if and only if the number of parts in the partitions involved in the associated symmetric group action is at most n. In particular, for Y = 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 (place- permutation) action of the symmetric group on the sequences (iI, . . . , i,) (ii E (I, . . . , n>) of shape v. The result 4.1 .I for e = e(v) now translates into a result stating the equality of the


multiplicities of certain Young module components of this permutation module. More precisely, let PA and iV’ (resp. Y” and M”) denote the Young modules and permutation modules for SS, (resp. for the corresponding Hecke algebra Z(4,)) associated to parti- tion A, and let s”“’ be the dual of the Specht module S” for /$Gr. (The labelling here for Young modules and Specht modules coincides with the labelling for PIMs and Weyl modules. One way to do this is to apply the functor T By - to any Y module to get its corresponding A? module. The dual Specht module gA’ for 6, may be described as the image of the Weyl module WA for 2 and the Young module r” may be described as the image under T a9 - of the PIM P* for 9 which has WA as a homomorphic image. The module PA is the reduction modulo nt of a PIM PA for x and the ‘Young module’ Y” for X(4,) is just the image of P”(c$,). The functor TOY - is additive and right exact, takes PIM’s to Young modules over any specialization, takes Weyl modules to dual Specht modules, and preserves all the associated decomposition numbers. The functor even takes Weyl 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 have

4.7. Corollary. If r < p2, then [I@” : P”] = CM’” : YIA] for all 2 E 0,. q

The proof is a direct application of 4.1.1 and is left to the reader. The left hand side is the dimension of the v-weight space of the irreducible 9 module indexed by II, and the right hand 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 or fewer parts (checked for p s 5).

We also remark that an analogue of 4.7 holds in the mixed case, if we think of the result as expressing a characteristic p permutation module multiplicity in terms of some- thing known for the q-Schur algebra ~O(C#+). The permutation modules relevant to the mixed case are standard permutation modules for the finite general linear group GL(r, q). These permutation modules, associated to coset spaces of parabolic subgroups corresponding to partitions, are taken over a field ffp where q is a power of a prime different from p. We take 1 to be the smallest power of q such that p divides q’ - 1, unless p divides q - 1, where we set 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 corresponding category of smodules.) One assumes r < lp, and the conclusion is that the left- hand side, expressed in terms of s-permutation modules, is computable, if A E O,, as the v-weight space of an irreducible Y($,)-module indexed by A. (In turn computable from Lusztig’s quantum group conjecture.) If Jame’s conjecture were true, one could dispense with the hypothesis ;Z E 0,.

We end this section with an application of 4.1 in a different direction.


4.8. 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 Weyl module 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 have

L,(A) = L&J 0 L&)[l’

where L, (A,) is an irreducible module for the usual Lie algebra enveloping algebra % over k(cp), where cp = (4& and [l] means the pullback through the Frobenius morphism U(q) + 2I (see [AW]). Since A, is regular and not in the lowest p-alcove, it follows that dim L,(A,) > dim L(A,). So, by the Steinberg tensor product theorem in the characteristic p case, we have dim L,(A) > dim L(n). Therefore, the previous theorem implies that V(A) is not multiplicity free. q

We remark that such weights II are not in the Jantzen region required by the characteristic p 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 algebra situation obtained from the weight lying in the lowest p2-alcove. The original example of this kind was obtained by Alfred Wiedemann by an explicit construction, and is given in an appendix.

5. Generalized q-Schur algebras: General case

We should say at the outset that the results of this section are tentative and not likely to be best possible. The problem is that, outside of type A, where one has the coalgebra results of Parshall-Wang [PW], the theory of quantum enveloping algebras over arbitrary fields is still in a formative stage. Even in [PW], when 9 is specialized to an Zth root of unity, it is required that 1 be odd. For other types, a reasonable start toward a theory, but with 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 as results of L.Thams announced in [AW]. It is also necessary to sketch the proof of some results similar in spirit to those of [AW], but which were not included in the present version of that work.

As in 9 3, let U, (denote the quantized enveloping algebra of [L4] and [AW] for a given finite root system over the ring d = Z [q, q- ‘1. The quotient field of d is called K, as before, and U’ denotes the quantized enveloping algebra over K. Let r be any saturated set of dominant weights, and let V’(y) denote the irreducible type 1 U’-module with high weight y, for y E l7 Put V’(r) = @ V’(y), and let 9, = Y, [r] be the image of U, under

the natural map YET

U, & 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


of G? at the maximal ideal (p, q - I), then the localization y&k = Y& @ k is the generalized q-Schur algebra Y [r] over k described there. In particular Ydk is k-quasihereditary. (A sequence of defining ideals may be chosen by using any linear order compatible with the usual partial order on weights, all such orderings associated with the same Weyl and dual Weyl modules.)

Now let &’ be any localization (not necessarily local) of d with respect to a multiplicative system, and put Y’ = Y& @ &I. (All tensor products here and below are over &) We also refer to Y ’ = 9; as a generalized q-Schur algebra (over die’). A number of convenient criteria are available to determine if Y ’ is &I-quasihereditary. First of all, there are natural candidate choices for sequences of defining ideals Ji’ in Y ‘, described as follows: Take any linear order on the dominant weights compatible with the usual partial order. Let Ji’ be the kernel in 9 ’ of the projection of Y K onto the sum of the matrix components of Y K associated to modules V’(y) with y at least as big as the ith weight (from the bottom) in the linear order. 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 localizations at all height 1 prime ideals of &02’ is true. (It is also sufficient to check the ‘separable quasihereditary’ 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 contains the reciprocals of all cyclotomic polynomials +I where 1 is either divisible by 2, 3, or is an integer smaller than the Coxeter number h of the root system. (Improvements to the results of [AW] would produce corresponding improvements in our results here. For applications to finite groups of Lie type in nondescribing characteristics, one really does not want to require 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 over d’, with a number of additional properties: (For definitions of terms involved, we refer the reader 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, a module is ‘integrable’ in the finite rank case iff it is a direct sum of its weight spaces. This condition can be checked locally. We remark that it is proved by Polo, see [AW], $4, that H’(1) is free over -02’; we shall only use the fact that it is projective, which, of course, is also a local property.) First, the localization of Ho (A) with respect to any multiplicative system is the corresponding module over the localization. Next, there is a version Hj (1) over any field Fwhich is a specialization of ,Oe’, and &? (1) satisfies ‘Kempf’s vanishing theorem’. The latter implies that H’(A) @ F g @(/2) by a spectral sequence argument. (In the case that F is the residue field of a height 1 prime, which is the only case we require, the argument is a 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 the action from right to left using the antipode, and then twist through the automorphism of Uk = 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 weight in r just returns an isomorphic copy. Here the type of a U’-module is as described in Appendix 3.) We note that the theory developed in $3 l-2 allows us to assume that the specialization 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], that

Ext”(V(A)F, A(l)F) = 0 for n > 0

170 Du and Scott, Luszfig conjectures I

in the category of integrable type 1 U,F-modules. Fortunately, we will prove in Appendix 3 that, 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] of finite-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. q

This does not yet tell us what we want to know about Y r (or YF), but it is a start. To prove that Y1 above is #-quasihereditary, we just need to check (using the indicated sequence 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 ourselves facing in 5 3 for k and k. (In some ways we are even in a better position, since d1 is a DVR.) 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 followed here. We just need to show that the category of &“-finite I&?-modules is an &l-finite highest weight category, with the expected Weyl and dual Weyl modules. This follows as in 9 3 from the following proposition, communicated verbally to us by H. Andersen at the MSRI conference.

5.2. Proposition. Zf 2, I’ are dominant weights, and d’ is a DVR as above, the tensor product

HO(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 a theorem of Mathieu [Ma], completing previous investigations by Wang [W] and Donkin [DON]. (Donkin observed at the MSRI conference that Lusztig’s results [L5] regarding new bases for Weyl modules implied Mathieu’s general result.) (‘Good filtrations’ can be checked by Ext’ vanishing with Weyl module arguments in the first variable. This is equivalent to 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 or modularly, and this can be seen directly from Lusztig’s results.) Andersen observed that the same 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 in the future, we omit further details. (It is not actually necessary to assume d’ is a DVR for the 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. The notion of ‘composition factor’ used below is the natural one, meaning a simple quotient of a submodule. (Obviously, all composition factors occur locally, where they fall under the theory of $1.)


5.3. Theorem. Let di be the localization of d = Z [q, q- ‘1 with respect to a multiplicative system containing all cyclotomic polynomials 4, with 1 divisible by 2, 3, or 1 smaller than the Coxeter number. (Thus, these cyclotomic polynomials are allowed as denominators in ,Pe’.) Let I be anyJinite saturated set of dominant weights. Then the generalized q-Schur algebra 9” = Yt? defined above is dae’-quasihereditary and split. The category $9’ [I] of d’-finite type 1 modules of UJ,l = U, @ d’ whose composition factors all have high weights in I coincides precisely with the category of #-finite Yt?-modules. In particular, %?I [I] is an d’-finite highest weight category. q

The last assertion of the theorem also holds, essentially by definition, for the category V1 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 following corollary.

5.4. Corollary. Let ~2’ be as above. Then all &l-jinite U,‘-modules are integrable. q

Of course, a main consequence of Theorem 5.3 is that we can apply the quasihereditary theory 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. If I+ 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 and Wang have given a different approach [PW] without restrictions on 1 (though they do assume 1 is odd). Still another way to approach type A is to use integral q-Schur algebras, as in the previous section, where we obtained the theorem below for q-Schur algebras without restriction, as Corollary 4.4.) Let p be a fixed prime, and let r = r (1, p) be the set of dominant weights 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 have just included this case for perspective. We may regard ouselves as replacing 1 by p in the I= 1 case, as in 4.3, so the reader might think of r as the ‘lowest lp-alcove’, including the closure. (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 nonnegative

qi- 1 integer 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 below demonstrate. (The additional weights are those minimal with respect to the strong linkage orders associated with the affine Weyl group I+$,.)

5.5. Theorem. Fix a root system, a prime p, and a nonnegative integer 1 restricted as

above, and let I be the ‘lowest lp-alcove’, as de$nedprecisely above. Let 4 denote the cyclotomic polynomial 4i if I+ 1, and put 4 = q5r if I= 1. Let M be a maximal ideal of d = H [q, q- ‘1 containing p and 4, and such that q becomes a primitive lth-root of unity in the residue field F = ~41 m. Suppose y E I is a weight such that the Weyl module V(y)’ for Ui = U, 0 F is multiplicity-free. Let d(4) denote the residue$eld of the localization dc6,.


Then 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 the hypothesis for Y in 4.1. It is obvious that Y K is (split) semisimple for the quotient field K of k. To show that Y(p) is semisimple for each height one prime ideal @ + ($), we only consider 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 assumption on 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 k whose 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 of H’(1) for i 2 0 and II dominant, as done in [APW], 5 4. (Such an analysis is announced in [AW], 5.1 without explicit statement as part of the work of L.Thams.) In particular, we have a similar result over k(p) as [APW], 4.7 (ii). With this and some spectral sequences as 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 also obtains 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 that

where V = V,(A). It follows that there is a non-zero U,(p)-homomorphism

unique up to scalar multiple (see also 2.10). Now, a similar argument to that in the proof of [A], Theorem 1 proves our claim.

By 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. q

Appendix 1 (due to A. Wiedemann, Stuttgart). Example of quasihereditary algebra Y over a regular local ring k of Krull dimension 2 having two simple modules, two Weyl modules and being maximal as an order after localization at each but one height 1 prime ideal cp. However the projective indecomposable Weyl module P occurs with multiplicity 1 as a direct summand 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) for a 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. Let

K = the quotient field of k,


A = Kx M3(K),

co = (1,0), 61 = @,I,),

where Z3 is the identity matrix in M3(K).

In the set of columns of size 2 over k, we define a k-submodule A4 as follows:

and define a k-order Y in A by

Since 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.) Hence End, (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 where

0 0 0 e=O,OOO .

!( 11 0 0 1

Thus 9 is an idempotent ideal, which is right projective over 9 Altogether, this shows that Y is k-quasihereditary in the sense of [CPS4].

Observethate’= (I,(: :)) E Y: and so # % The latter fact implies that the kernel

of 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 with multiplicity 1.

The base change providing the isomorphism M 2 k 0 k

is given by the matrix

12 Journal fiir Mathematik. Band 455


with 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 ideal p + cp. (In fact, 9@ 2 k, x M3(kJ.)

Consider now M,. It is easy to see that

potent

E M,,,. Hence 5$ contains the 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 exactly 2. In particular, the irreducible quotient of P (or of P/mP) has dimension 1, but the irreducible quotient of P, (or of Pv / cp P,) has dimension 2.

Appendix 2. In this appendix, we use more traditional character theory for algebraic groups, and the quantum analog of that theory, to give an alternative proof of the equality of dimensions in 4.2. The idea that such a proof might exist began with Z. Lin, though the 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 of type 1. If M is a free k-finite U-module in %, then M is integrable by 3.3. So, M is the direct sum of weight spaces M,, ;1 E X. We set

ch A4 = c rank, (MA) e’ , lEX

called the formal character of M. If M = v(n), the Weyl module with highest weight 1 E X+, we denote ch A4 simply by x(A). The q-analog for U-modules of Jantzen’s sum formula has been established in [APW], 9 10 over various DVR’s associated with k. For our purposes here, let R be the DVR k@ (cp = (&,)) or k/(q - 1) with a unique maximal ideal denoted by @,, and residue field R. Denote by u@ the valuation on the quotient field F of R.

By 2.11, we have a canonical embedding for 2 E X+,

lp : v(n)” -+ A (A)” .

Now, define a filtration of V(A)” as follows

v’ = {x E v(n)” 1 yx E @‘A (A)“)


Let Vi be the image of Vj in V(A)” = V(A)” OR R. Then we obtain a filtration

We denote by Si the j-th section Vi/ Vi+ I. Obviously, we have

(1) chV,‘= c ch$. j>O

A2.1. Theorem ([APW]). Let C+ be the positive root system of C, s, the reflection with respect to c( E C ‘, Then we have

c chV,‘= c 1 v,([m])~(s;A+ma), jZ0 aSZ+ m=l

where [m] = q” - q-m. q--4-l

When R = k,, the image 5 of q in R is a primitive pth root of 1. ,In this case, we denote 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 corresponding irreducible quotient in the notation of 4.2. If R = k/(q - l), then i? = E, and we write &? = M”, up = up%, and sj = Si. Clearly, i;;(l)R = V(A) in the notation of $4, the Weyl module for D with irreducible quotient L(1). Note also that S: g L,(A) and so z L(i).

Let “e, be the category of k(cp)-finite U(q)-modules of type 1. Thus, there is a canonical map rc from the Grothendieck group of w* to that of @ given by rc [M] = [MO @ k], where MO is a Z, [<]-lattice of M E ob Vq and [N] denotes the class of N in the corresponding Grothendieck group. It is well known that the map rc respects formal characters.

We can now prove from these considerations the part of 4.2 dealing with equality of dimensions.

A 2.2. Theorem. Let 1 be a dominant weight in the lowest p2-alcove with the property that V(A) is multiplicity-free. Then dim L(A) = dim L,(A).

Proof We first note that v,([m]) = v,(m) f or all m, 0 < m < (A + Q, a”), since iz is in the lowest p2-alcove. It follows from the sum formula that

Consequently,

(2) 1 j7c[S(g = c j[S’] . j>O j>O


Since v(n) is multiplicity-free, the reduction modulo p of each composition factor of <(A) is also multiplicity-free. Therefore, all sections Si are multiplicity-free on reduction modulo p. Also, the reductions of these sections have disjoint characters (no characters of irreducible modules in common), since their sum is the character of s(n)‘. So, (2) implies that rc[S,j] = [sj] for all j > 0. Consequently, by (1) we obtain dim c(n)’ = dim V(n)‘, which implies dim L,(A) = dim L(A). q

A2.3. 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. The same 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 some calculation 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 by commutative algebra arguments. (The argument for 4.1 shows the idempotent projections

e, (A (4 / W>) are cp-coprimary. Under the hypothesis of 4.5, we have m = cp + z, where z is a (principal) height 1 prime ideal (e.g. z = (q - 1) above). The length of the modules obtained by tensoring these projections with R = k, or k/z may be used to determine the Grothendieck group image, and the coprimary condition guarantees corresponding lengths are 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 the information available from the commutative algebra approach we took in 4.1 and 4.2, and indeed the more sophisticated point of view gave diagonalization information not available from the character formulas. At the same time, it is interesting to note the argument of the above paragraph makes a Jantzen sum formula over k available for computations in any k-quasihereditary algebra satisfying the hypotheses of 4.5 for which the characters of the irreducible modules over k(q) are known.

Appendix 3. In this appendix, we shall prove the result promised in 4 5 that a finite- dimensional U,F-module A4 is integrable, where F is a field of characteristic p > 2, and the image of q in the specialization & + F is not 1. By abuse of notation we denote this image by the same symbol q. When q is a primitive Ph-root of I, we assume that 2 is odd and prime to 3 whenever the corresponding root system has a component of type G,. (These require- ments, 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-root of 1. The other case where q is not a root of 1 is easier. For completeness, we will sketch the argument 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 above

assumptions). A similar argument to that given in [L2], 4.4 shows that K: is central and K.2z=1 in UF 1 d8’

Let V be a U$-module. For any sequence E = (si, . . . , E,) with si E {I, - I} we define (cf. [L2], 4.6)

E = {VE VlK,‘v = ~ZI} .


Then V, is a U,F-submodule of I/ (since K/ is central), and I/ = @ V, (since (p, 21) = 1). We

say 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 defining Ei + 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 of type 1. Therefore, it is enough to consider type 1 Uz-modules.

That is, we may assume, without loss of generality, that the finite dimensional module M (that we wish to show integrable) is of type 1. Be finiteness, it suffices to show M is a direct sum of its integral weight spaces. Since any commuting set of semi-simple trans- formations on a finite dimensional vector space is diagonalizable, we just need to show that the action of the O-part of UF for each fixed i is diagonalizable with integral weights, where Ui is the subalgebra of U’ generated by Er), F/‘) and Ki, Ki- ‘. Obviously, Ui is isomorphic to the quantum enveloping algebra of type A r. So, we may assume that U = U, is of type A,, and denote the generators by E, F, K.

Let U” be the &-subalgebra of U generated by

K,K-‘, K;c , [ 1 tEN, CEZ, where

K; c [ 1 S+l -K-lq-C+S-t

as defined in [L2], 4.1. Also, let m

t q” - q-s [I n

n 4 m-s+1

-4 -m+s-1

denote the Gaussian polynomial n Ed. s=l

q” - q-s

We shall denote the image of q, E d in Fand of K, K; c [ 1 t EU’in UFO= U”@F

by 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-’ (resp. K; 0 [ 1 tl ’

t E N). It is easy to see ([L4], 8.10) that V is isomorphic to the hyper-

algebra hy (G,) of the multiplicative group &&, and that u. is isomorphic to the group algebra 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


Proof. 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)[email protected])

j=O

[tl+;-qKj[tf?] [y]

We also see that K; 0 [ 1 t0

E u. since 0 s to < 1. Now, the elements

sE{O,l}, tE Iv/, form a basis for U,“, and similarly, the elements

with 0 5 t < 1 (resp. K; 0 [ 1 tl , tEN)

form a basis for u. (resp. for V) by [L3], (2.21) (5.4). Therefore, our result is clear. q

For each n > 0, let u, (resp. K) be the subspace of C.Ji spanned by [“,“],K[“;“],

0 5 t < Ipn (resp. K; 0 c 1 tl , 0 5 t < p”). By A 3.2, we see that u, is a subalgebra, and,

u, z u. @ ^yr,. Thus, Uj has a filtration

(A 3.3) u. c u1 c u2 c . . . and Ui = U u, . nt0

Let V be a u,-module (or a Ui-module). We say that I/ has type 1 if

V= {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 homomorphism

x, : u, + F

such that x,(K) = qc and x, ([“;“I) = [:I.

Thus, it is easy to see that F via xc is a

1 -dimensional u,-module of type 1.

A 3.4. Lemma. (1) Every u,-module is a direct sum of one-dimensional u,-modules.


(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 a direct 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 0

with 0 5 t, < 1,

([“,“1) =qrpl)([:;P]) = [fl(fl) = [c+tc’l] by A 3.1 again. Therefore, x = x, + C,l. q

A 3.5. Corollary. AnyJinite dimensional &f-module is a direct sum of one-dimensional Ui-modules.

Proof. This follows immediately from (A 3.3) and A 3.4. q

Let X( U,“) be the set of the algebra homomorphisms x from Q? to F such that F via 2 is a Ui-module of type 1. This is an abelian subgroup of the dual space (U,“)*. We can describe 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”.Z by A 3.4, and

clearly, x, + 1 I”, = x,. Therefore, x = (QnciO, E Z,,,.

Note that a “number” c in Z,,, has the form

c~,+c,l+c,lp+c21p2+ . ..)


where 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”, then

the notation c [I means the number %I n-l

t [I t

in F where s, = c-r + 1 cilpi. With this i=O

notation, we have

x([“;“]) = [:I for any x E X(U,O).

A 3.7. Theorem. Every finite dimensional &-module is integrable.

Proof. Let M be a finite-dimensional &-module of type 1. By A 3.5 and (A 3.6), we have

where 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&-module M, = 0 M, wh&e X”=’ f’ + 22 for &me i’ with Mx, + 0. So we may assume that

XEX M = M,. Define a partial order on X : x 2 6 if and only if x - 6 2 0, and let x E X be the maximal 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 have

E”‘l;“‘v= [K;o]+]v.

Suppose x is not a finite number. Then there are infinitely many numbers of r such that

[I : + 0 in F. This implies F@‘v + 0 for infinitely many numbers of r. This is impossible

since M is finite-dimensional. This contradiction shows that x E Z. Therefore, all weights of M are integral, i.e. M is integrable. q

We 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 can strengthen this result as the following corollary of the theorem. When F has characteristic 0, a similar result is proved in [L2], 6.4. The corollary, as well as the theorem, is valid for all types of finite root systems, not just type A,.

A 3.8. Corollary. There is a one to one correspondence between the set of jinite- dimensional irreducible Q-modules of type 1 and the set of dominant weights. q

We now turn to the case when q E F is not a root of 1. As before, we can also assume that U, is of type A, with generators E @), F”), K, K-‘. With this assumptions, the algebra U, = U: is generated just by the elements E, F, K, K- ‘.


Let 17: be the subalgebra of U, generated by the K, K - ‘, and oj’ the abelian group of all algebra homomorphisms x from Uj’ to F. Then the elements

defined by X,-~(K) = eq’, form an abelian subgroup X of 0:.

Let M be a Q-module. For any x E vi, write

M, = {ZIEM~KZI = x(K)u}.

We are going to show that if M is finite-dimensional, then M = c M,. The proof below

is due to Rosso [Rl] (see also [APW], 9.3). XEX

By the assumptions and the equalities

KEK-‘= q2E and K-‘FK= q2F,

one sees easily that 0 is the only possible eigenvalue of E and F. Therefore, E and F are nilpotent on M.

Choose r such that F”‘M = 0, and let

2t-1

h, = n (Kq”-“‘- K-lq(S-r)), t 2 1 _ s=l

We have, for 0 5 t’ 5 t, [K;v;l][K;:;t]

E ht. I!$‘. Thus, using the commutation

formula [L4], 6.5 (a2), also (a6), we prove by induction on t that h, F”- ‘) = 0. So, h, = 0, 2r-1

and hence n (K2 - q2(s-r)) = 0, on M. This implies that the eigenvalues of K2 are among s=l

the elements q 2(s-rq1~s~22r-1), so, K2 is diagonalizable on M. Consequently, K is diagonalizable on M with eigenvalues in the set { f q” 1 1 s 1 5 r - l}. Therefore, M is a direct sum of the weight spaces M,(x E A’).

This completes the proof of the integrability of M, as asserted at the beginning of this appendix, in all cases.

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School of Mathematics, University of New South Wales, Kensington, NSW 2033, Australia

Department of Mathematics, University of Virginia, Charlottesville, VA 22903, USA

Eingegangen 10. August 1992, in revidierter Fassung 14. Dezember 1993

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