Algebraic Group Actions in the Cohomology Theory of Lie Algebras of Cartan Type

JOURNAL OF ALGEBRA ARTICLE NO.

179, 852]888 Ž1996.

0040

Algebraic Group Actions in the Cohomology Theory of Lie Algebras of Cartan Type Zongzhu Lin* Department of Mathematics, Kansas State Uni¨ ersity, Manhattan, Kansas 66506

and Daniel K. Nakano† Department of Mathematics and Statistics, Utah State Uni¨ ersity, Logan, Utah 84322 Communicated by Wilberd ¨ an der Kallen Received October 27, 1994 DEDICATED TO PROFESSOR GEORGE SELIGMAN ON THE OCCASION OF HIS

65TH BIRTHDAY

Given a Lie algebra g of Cartan type we construct an infinite-dimensional cocommutative Hopf algebra DŽ G . uŽ g . which is the analog of the distribution algebra of a connected reductive algebraic group. We show that the simple uŽ g .-modules lift to a DŽ G . uŽ g .-structure. This additional structure is used to formulate and prove relative projectivity theorems for Lie algebras of Cartan type. The support varieties of certain DŽ G . uŽ g .-modules are computed by using these algebraic group techniques. Q 1996 Academic Press, Inc.

1. INTRODUCTION Let G be a connected reductive algebraic group scheme with Lie algebra g over an algebraically closed field k of characteristic p ) 0. One of the important methods in studying the cohomology of g is to use the rational G-module structure on the cohomology groups if the coefficients are in a G-module. This leads to many interesting results like the calcula* E-mail address: [email protected]. Research of the first author was supported in part by NSF Grant DMS-9401389. † E-mail address: [email protected]. Research of the second author was supported in part by NSF Grant DMS-9206284. 852 0021-8693r96 $12.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

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tion of cohomology ring for the restricted enveloping algebra uŽ g . for p sufficiently large wAJ, FP4x. One can identify the Žrestricted. representation theory of g with the representation theory of the first Frobenius kernel G1 of G. The relationship of the group scheme G1 with G provides the basic techniques for the study of the cohomology groups of the Lie algebra. This paper is aimed at developing similar techniques for the cohomology theory of the restricted simple Lie algebras of Cartan type. Block and Wilson wBWx have shown that restricted simple Lie algebras are either classical Žones which arise as the Lie algebras of connected simple algebraic groups. or of Cartan type for primes larger than 7. The Lie algebras of Cartan type are typically realized as Lie subalgebras of vector fields and fall into four subcategories W, S, H, and K. The Lie algebras of type S, H, and K are constructed by considering the subalgebra of the Lie algebra of divided power derivations W which stabilize a particular differential form. Unlike the classical restricted Lie algebras, the restricted representations of a simple Lie algebra of Cartan type have no characterization in terms of the representations of an infinitesimal subgroup of an algebraic group. However, one can still make use of algebraic group actions in both the representation and cohomology theory for restricted Lie algebras of Cartan type. The key idea involves creating a new ‘‘distribution algebra’’ by fusing the two representation theories, the representation theory of an algebraic group and that of the restricted Lie algebra, together in a coherent manner. In our setting the algebraic groups will be certain subgroups of the automorphism group G of the restricted Lie algebra. Our main approach is to use the representation theory of Hopf algebras studied in wLin1, Lin2x. Instead of dealing with the algebraic group directly, we translate its representation theory into the representation theory of its distribution algebra, which is a cocommutative Hopf algebra. Given this algebra we construct a larger Hopf algebra which contains the distribution algebra and the restricted enveloping algebra of g. One can then appeal to the results developed in wLin1, Lin2x to establish the Hopf algebra action on the cohomology ring and then translate it back to the action of the algebraic group on the cohomology ring and its associated cohomological varieties. The paper is organized as follows. In the second section, we provide a general construction of a Hopf algebra built out of two Hopf algebras that is analogous to taking the product of a subgroup with another which normalizes it in a larger group. In Section 3 we first define and construct the Lie algebras of Cartan type then review Wilson’s description wW2x of the automorphism group G of these algebras. With these constructions one can appeal to the results in the earlier section to create an infinite-dimensional cocommutative Hopf algebra whose modules are modules over

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the restricted enveloping algebra with a compatible action of an appropriate subgroup of G. A theorem, like the result of Curtis wCux which involves the lifting of G-structures on simple modules, is crucial in our construction. In the last two sections we provide applications of the algebraic group actions on the cohomology rings. In Section 4, we prove results about projective modules over Lie algebras of Cartan type similar to the results of Cline et al. wCPSx for G1T-modules. Later in Section 5 calculations of the support varieties and complexities of certain V Ž g .-modules are provided when p is large. The authors express their appreciation to Alexander Premet, Eric Friedlander, and George Seligman for useful discussions and comments. We are also grateful to the University of Washington for their hospitality during the completion of this work.

2. HOPF ALGEBRA ACTIONS ON THE COHOMOLOGY RINGS Throughout this section let k be a field. Unless otherwise indicated, all algebras are algebras over k and all tensor products will be taken over k. Let H be a Hopf algebra over k with D, g , and « being the comultiplication, antipode, and counit respectively. We will adopt the notations of wSwx on Hopf algebras. We call an Žassociative. algebra A over k an H-module algebra if A has an Žleft. H-module structure such that h Ž ab . s

Ý Ž hŽ1. a.Ž hŽ2. b .

and

h1 A s « Ž h . 1 A

for h g H and a, b g A with DŽ h. s ÝhŽ1. m hŽ2. . Here the first condition is equivalent to the fact that the multiplication is a homomorphism of H-modules. For a given H-module algebra A, the smash product algebra AaH, is the vector space A m H with multiplication defined by

Ž a m h . Ž b m g . s Ý a Ž hŽ1. b . m hŽ2. g for all a, b g A and h, g g H. EXAMPLE 2.1. Let A be a finite dimensional algebra and G s Aut alg Ž A. which is an algebraic group. Let H s DistŽ G . be the distribution algebra of G. Then A is an H-module algebra with the H-module structure on A given as the differential of the G-module structure on A. EXAMPLE 2.2. Let X be an affine variety on which a connected algebraic group G acts. Then the coordinate algebra k w X x is a DistŽ G .module algebra. DEFINITION 2.3. We say an A-module M has an H-structure if M is also an H-module such that hŽ am. s ÝŽ hŽ1. a.Ž hŽ2. m. for h g H, a g A,

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and m g M. The last condition is equivalent to the structure map A m M ª M being a homomorphism of H-modules. Here A m M is a H-module via comultiplication. One can easily verify that the natural embedding A ª AaH and H ª AaH defined by a ¬ a m 1 and h ¬ 1 m h respectively are homomorphisms of algebras. We can identify A and H with the subalgebras Aa1 and 1aH respectively. PROPOSITION 2.4. An A-module M has an H-structure if and only if the A-module structure on M extends to an AaH-module structure on M. Proof. If the A-module structure on M is the restriction of an AaHmodule structure, we take the H-module structure to be the restriction to H from AaH. The compatibility condition is the consequence of the commuting relation Ž1 m h.Ž a m 1. s ÝŽ hŽ1. a. m hŽ2. . On the other hand, if M has an H-structure, we can define the AaH-module structure by defining Ž a m h. m s aŽ hm.. We leave it to the readers to verify that this does define an AaH-module structure on M, which extends the original A-module structure. We further assume that A is a Hopf algebra such that the multiplication, comultiplication, unit, and counit are all H-module homomorphisms. In this case, we call A an H-module Hopf algebra. In general, one can construct a Hopf algebra structure on AaH provided that certain conditions are satisfied. See wRadx for a set of conditions on a bialgebra structure. In this paper H will be the distribution algebra of an algebraic group so it is enough to consider cocommutative Hopf algebras H. The bialgebra structure on AaH, with the tensor product coalgebra structure and tensor product counit, can be directly verified under the cocommutativity condition on H or one can use wRad, Theorem 1x with the trivial H-comodule structure on A. In the following, we construct the antipode on AaH. Note that C s Hom k Ž A, A. is an algebra with the convolution multiplication. We make C an H-module via the adjoint action defined by Ž hf .Ž a. s ÝhŽ1. f Žg H Ž hŽ2. . a. for all h g H, a g A and f g C. One can check directly that C is an H-module algebra with respect to this H-module structure using the cocommutativity of H. As a consequence, the antipode of A is a homomorphism of H-modules. In fact, ŽgA ) I .Ž a. s « AŽ a.1 for all a g A. Then we have, for each h g H, hŽgA ) I . s « H Ž h. « A1. This shows that gA ) I g C H s  c g C < hc s « N Ž h. c, for all h g H 4 . Here we call C H the set of H-fixed points in C. However, since I g C H , a direct calculation shows that hŽgA ) I . s hŽgA .) I and hgA s « H Ž h.gA . By Lemma 1.2 in wLin2x, we have C H s Hom H Ž A, A.. Thus, gA is a homomorphism of H-modules. Using this property, one can now directly verify that the

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map g : AaH ª AaH, defined by g Ž a m h. s Ýg H Ž hŽ2. .gAŽ a. m g H Ž hŽ1. . for all a g A and h g H, is an antipode of AaH. EXAMPLE 2.5. Let G be a finite group, H s kG the group algebra, and A the algebra of all k-valued functions on G. Then both A and kG are Hopf algebras in a natural way Ž A is a coalgebra because G is finite.. The cocommutative Hopf algebra kG acts on the Hopf algebra A via the conjugate action of G on G. Then it is straightforward to check that this action makes A into a kG-module Hopf algebra. The Hopf algebra AakG we constructed above is exactly the quantum double in the orbifold theory wDPRx. Let A be a Hopf algebra and let f l : A m A ª A and fr : A m A ª A be defined by

fl Ž x m y . s

Ý xŽ1. yg Ž xŽ2. .

and

fr Ž x m y . s

Ý g Ž xŽ1. . yxŽ2. .

f l Žor fr . makes A into a left Žor right. A-module, which we will call A the left Žor right. adjoint module. We say a Hopf subalgebra D of A is normal if f l Ž D m A. : D and fr Ž A m D . : D. For a given normal Hopf subalgebra, D, ADqs Dq A is a Hopf ideal of A and we denote the quotient Hopf algebra ArŽ ADq . by ArD Žby abusing the notation.. Here Dqs kerŽ « < D . is the augmentation ideal of D. LEMMA 2.6. If A is a Hopf algebra and H is cocommutati¨ e then A is a normal Hopf subalgebra of AaH. Proof. For a g A and h g H, we have

Ý g Ž 1 m hŽ1. . Ž a m 1. Ž1 m hŽ2. . s Ý Ž 1 m g Ž hŽ1. . . Ž a m hŽ2. . s

Ý Ž g Ž hŽ1.Ž2. . a m g Ž hŽ1.Ž1. . hŽ2. .

Ý Ž g Ž hŽ1. . a m g Ž hŽ2. . hŽ3. . s Ý g Ž hŽ1. . a m e Ž hŽ2. . s g Ž h . a m 1. s

Note that from the second to the third line above we are using the cocommutativity of H. This calculation implies that fr ŽŽ AaH . m A. : A. Similarly, one can verify f l Ž A g Ž AaH .. : A. From now on, we assume that H is cocommutative and A is an H-module Hopf algebra. By identifying A and H with the Hopf subalgebras Aa1 and 1aH of AaH respectively, the H-module structure on A can be realized from the Hopf algebra structure of AaH as follows: For each a g A and h g H, h Ž a. s

Ý Ž 1 m hŽ1. . Ž a m 1. Ž 1 m g Ž hŽ2. . . ,

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which is the restriction of the adjoint action of AaH on A to H. We call an ideal Žor left ideal of A. H-in¨ ariant if it is an H-submodule of A. LEMMA 2.7. For an A-module M with an H-structure, the annihilator Ann AŽ M . is H-in¨ ariant. Proof. Let a g Ann AŽ M . and h g H. By Proposition 2.4, M extends to an AaH-module. Thus we have, for any m g M, h Ž a. m s

Ý Ž 1 m hŽ1. . Ž a m 1. Ž 1 m g Ž hŽ2. . . m s 0.

This shows that hŽ a. g Ann AŽ M .. A full subcategory CA of the category of all left A-modules is called admissible if it consists of locally finite A-modules including the trivial module and it is closed under taking submodules, quotients, direct limits, tensor products of modules in CA , and the duals of finite dimensional modules in CA Žsee wLin1, Lin2x for a more precise definition.. Let LA be the set of ideals Ann AŽ M . with M finite-dimensional and in CA . The set LA consists of cofinite ideals and determines the category CA completely as follows: An A-module M is in CA if and only if M s  m g M
for each I g I , there is J g I such that J : gy1 Ž I ., there is an I g I such that I : kerŽ « ., for any I1 , I2 g I , there is J g I such that J : I1 l I2 , for any I1 , I2 g I , there is a J g I such that J : I1 n I2 . Here D

I1 n I2 is the kernel of the composition map H ª H m H ª HrI1 m HrI2 . It has been verified in wLin1x that the set LA we defined above is an admissible set of ideals in A. We call LA an admissible set of ideals defining CA . Note that the admissible set of ideals defining CA is not unique. LEMMA 2.8. Let CA be an admissible category and LA be an admissible set of ideals defining CA . If each finite dimensional module N in CA is a quotient of a finite dimensional module in CA with an H-structure, then the set LAH of all H-in¨ ariant ideals in LA is also an admissible set of ideals defining CA . Proof. It is straightforward to check that LAH is an admissible set of ideals using the conditions such as those in wLin1x. It is enough to show that for each J g LA , there exists I1 in LAH such that I1 : J. For each J g LA , ArJ is a finite-dimensional A-module in CA . By the assumption,

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there exists a finite dimensional module M in CA with an H-structure such that M ª ArJ ª 0 is exact. Thus I1 s Ann AŽ M . is H-invariant and I1 : J. The fact that LAH defines the category CA follows by a direct verification. Throughout the rest of this section, we will assume that all ideals in LA are H-invariant. Note that I m H q A m J is a left ideal of AaH if I is an H-invariant ideal of A and J is an ideal of H. In fact, for x g I, y g J, a, b g A, and h, g g H, we have that

Ž a m h. Ž x m g q b m y . s

Ý Ž ahŽ1. x m hŽ2. g q ahŽ1. b m hŽ2. y .

is still an element of I m H q A m J since I is H-invariant. For an admissible category CH of H-modules with a defining admissible set of ideals, LH of H, we can now define an admissible category of AaH-modules as follows. Let LAaH be the set of all left ideals I m H q A m J with I g LA and J g LH . One can easily verify that LAaH is an admissible set of left ideals and defines an admissible category CAaH of AaH-modules as follows: An AaH-module M is in CAaH if and only if for any m g M, Ann AaH Ž m. = J for some J g LAaH . LEMMA 2.9. The natural Hopf algebra embeddings A ª AaH and H ª AaH are compatible with the admissible categories, i.e., each module in CAaH is in CA and CH when restricted to A and H respecti¨ ely. Proof. We note that I m 1 and 1 m J are included in I m H q A m J. Observe that A is a normal Hopf subalgebra of AaH. It is easily verified that the map AaH ª H defined by a m h ¬ « Ž a. h is a Hopf algebra homomorphism and induces a Hopf algebra isomorphism Ž AaH .rA ( H such that the composition of the maps H ª AaH ª H is the identity map and all maps are compatible with the admissible categories. By wLin2, Lemma 1.4x, the induction functor Ind HAaH is exact. Furthermore, the conditions of wLin2, Prop. 1.9x for left ideals Žcf. wLin2, 1.10x. are satisfied. Therefore, we have A Res HAaH R i Ind AAaH M ( R i Ind H k Res k M

for all A-modules in CA since Ind kA is exact. Furthermore, the exactness of AaH Ind H . k in the above isomorphism implies the exactness of Ind A More generally, let D be a normal Hopf subalgebra of H. Then D is a left H-module with respect to the left adjoint action of H on D given by h ? d s ÝhŽ1. dg Ž hŽ2. . for all h g H and d g D. We further assume that there is an embedding f : D ª A of Hopf algebras such that f is also a

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homomorphism of H-modules and the action of d g D on A is given by the left adjoint action of f Ž d . on A, namely, d Ž a. s

Ý f Ž dŽ1. . ag Ž f Ž dŽ2. . .

for all a g A.

THEOREM 2.10. Let A, H, D be as abo¨ e. If A and H are free o¨ er D Ž as right and left D-modules respecti¨ ely . then there is a Hopf algebra P and embeddings r : A ª P and s : H ª P such that P s r Ž A. s Ž H . and s Ž H . l r Ž A. s s Ž D . s r Ž f Ž D ... Proof. We first consider the Hopf algebra AaH. Let I be the ideal of AaH generated by  f Ž d . m 1 y 1 m d < d g D4 . First of all, I is a Hopf ideal since DŽ f Ž d. m 1 y 1 m d. s

Ý Ž f Ž dŽ1. . m 1 y 1 m dŽ1. . m Ž f Ž dŽ2. . m 1 . q Ž 1 m dŽ1. . m Ž f Ž dŽ2. . m 1 y 1 m dŽ2. .

and

g Ž f Ž d . m 1 y 1 m d . s f ŽgH Ž d . . m 1 y 1 m gH Ž d . for all d g D because f is a Hopf algebra homomorphism. Set P s AaHrI, then P is a Hopf algebra. Set r to be the composition of maps A ª AaH ª P and s the composition of maps H ª AaH ª P. By the definition, we clearly have P s r Ž A. s Ž H . and s Ž D . : r Ž A. l s Ž H .. Note that A is a right D-module and H is a let D-module. We can consider the vector space A mD H. There is a natural surjective linear map F : AaH ª A mD H. We will show that kerŽ F . s I, then we will have P ( A mD H as vector spaces. First of all, kerŽ F . : I. In fact, the elements of kerŽ F . are linear combinations of elements of the form af Ž d . m h y a m dh with a g A, d g D, and h g H. We have af Ž d . m h y a m dh s Ž a m 1.Ž f Ž d . m 1 y 1 m d .Ž1 m h. g I. On the other hand, kerŽ F . contains all generators of the ideal I. We only need to show that kerŽ F . is a two-sided ideal. Using the identity af Ž d . m h y a m dh s Ž a m 1 . Ž f Ž d . m 1 y 1 m d . Ž 1 m h . and the fact that Ž A m 1.kerŽ F .Ž1 m H . : kerŽ F ., that kerŽ F . is a twosided ideal follows from the following two identities:

Ž a m g . Ž f Ž d. m 1 y 1 m d. s

Ý Ž agŽ1. f Ž d . m gŽ2. y a m Ž gŽ1. ? d . gŽ2. . g ker Ž F . Ž 2.1.

Ž f Ž d. m 1 y 1 m d. Ž a m g . s

Ý Ž dŽ1. Ž a. f Ž dŽ2. . m g y dŽ1. Ž a. m dŽ2. g . g ker Ž F . Ž 2.2.

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for all a g A, g g H, and d g D. Here we have used the formula gd s ÝŽ gŽ1. ? d . gŽ2. for d g D and g g H and its analog in A Žsee wLin1, Sect. 4x for the proof.. Since A and H are free D-modules, and therefore flat over D, we have the injective maps A ª A mD H and H ª A mD H. By choosing bases of A and H over D with the identity elements 1 in them, one can conclude that every element of r Ž A. l s Ž H . is in s Ž D .. EXAMPLE 2.11. Let A be any finite dimensional commutative algebra and G a connected closed subgroup of the algebraic group Aut alg Ž A.. We will denote by H the distribution algebra of G. Then A is an H-module algebra and the Lie algebra DerŽ A. of derivations of A over k is both an A-module and H-module Žderived from the conjugation action of G on DerŽ A.., which gives DerŽ A. an AaH-module structure. Observe that any Lie subalgebra g of DerŽ A. such that g is G- and A-invariant is an AaH-submodule of DerŽ A.. The natural H-module Hom AŽ g, A. carries a g-module structure defined by Ž yf .Ž x . s y Ž f Ž x .. y f Žw y, x x. for all x, y g g and f g Hom AŽ g, A., such that the g-module structure g m Hom AŽ g, A. ª Hom AŽ g, A. is a homomorphism of Hmodules. It is easily verified that the evaluation map d : A ª V s Hom AŽ g, A., defined by dŽ a.Ž D . s DŽ a. for all a g A and D g g, is a homomorphism of both H-modules and g-modules. Since A is commutative, we can consider the A-module Hom AŽ g, A. mA Hom AŽ g, A. which is both a G-module and a g-module in a consistent way. Indeed, we have for a g A, a , b g Hom AŽ g, A., and D g g, DŽ a a m b . s DŽ a.Ž a m b . q aŽ DŽ a m b .. s DŽ a m a b .. Consequently, we have natural G-module and g-module structures on mAr V and n rAV respectively. The Lie algebra LieŽ G . of G acts on A as derivations. Thus, there is a natural homomorphism of Lie algebras LieŽ G . ª DerŽ A., which is always injective and a homomorphism of G-modules. We will still use LieŽ G . to denote its image in DerŽ A.. Then there are two LieŽ G .-module structures on DerŽ A.. One is the differential of the G-module structure on DerŽ A., and the other is the left adjoint module structure since LieŽ G . is a Lie subalgebra of DerŽ A.. However, a direct verification shows that they coincide. For g and G as above with g being restricted, then g G s g l LieŽ G . is an ideal of LieŽ G . and a restricted Lie subalgebra of DerŽ A.. Thus the restricted enveloping algebra uŽ g G ., as a Hopf subalgebra of DistŽ G ., is invariant under the left adjoint action, and is thus a normal Hopf subalgebra. It is also a Hopf subalgebra of the restricted enveloping algebra uŽ g . of g. Using the PBW theorem for the restricted enveloping algebras one can see that both DistŽ G . and uŽ g . are free as uŽ g G .-modules Žboth right

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and left.. The restriction to uŽ g G . of the DistŽ G .-module structure on uŽ g . is actually the adjoint action of uŽ g G . on uŽ g . as Hopf algebras. Using the construction in Theorem 2.10 for H s DistŽ G ., A s uŽ g ., and D s uŽ g G ., we get a Hopf algebra P s DistŽ G . uŽ g . which contains both DistŽ G . and uŽ g . as Hopf subalgebras with uŽ g . normal and DistŽ G . l uŽ m . s uŽ g G .. Let A, H, D, and P be as in Theorem 2.10, and let CP be the category of P-modules M such that M is in CAaH when regarded as a AaH-module via the quotient map. The category CP can be defined by the admissible set of left ideals of the form IH q AJ with I g LA Žwhich is H-invariant. and J g LH . The category CP can be identified with the full subcategory of CAaH consisting of all modules on which the two D-actions given via restrictions from A and H coincide. By the definition of the admissible categories, the embedding maps A ª P and H ª P are compatible with the admissible categories Ži.e., each module M in CP is in CA Žor CH . when restricted to A Žor H ... We can now consider the induction functors Ind PA and Ind PH . THEOREM 2.12. If the induction functors Ind DA and Ind H D are exact then Ind PA and Ind PH are exact. Proof. The argument preceding the theorem shows that the condition of wLin2, Prop. 1.9x is satisfied by left ideals. Thus we have Res PH Ind PA M ( A Ind H D Res D M for each A-module M in CA . Applying this to short exact A sequences of A-modules and using the exactness of Ind H D and Res D , one P P can get the exactness of Ind A . The exactness of Ind H follows by a similar argument using the exactness of Ind DA . Note that A is a normal Hopf subalgebra of P. We want to describe the quotient Hopf algebra PrA [ PrPAq where Aq is the augmentation ideal of A. LEMMA 2.13. There is a Hopf algebra isomorphism PrA ( HrD, which induces an isomorphism between CP r A and CH r D of admissible categories. Proof. One clearly has HDq in the kernel of the composition map H ª AaH ª P ª PrA, which induces a homomorphism of Hopf algebras HrD ª PrA. Since P s H q Aq H, the above induced map is surjective. On the other hand, the kernel of the composition map AaH ª H ª HrD contains the ideal I Žin the proof of Theorem 2.10., and therefore induces a surjective map P ª HrD. Since Aq is sent to 0 under this map we have a map PrA ª HrD. By keeping track of each map involved, one can easily verify that the above two maps are inverses of each other, and thus we have an isomorphism of the Hopf algebras. In order to describe the isomorphism between admissible categories, we observe that the maps

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P ª HrD and H ª PrA are compatible with admissible categories. Each HrD-module in CH r D is in CP , on which A acts trivially Ži.e., via the counit., and can be thought of as a module in CP r A . In a similar manner, one can verify that each module in CP r A also defines a module in CH r D . From now on we can identify the algebras HrD and PrA together with their admissible categories. For any P-modules E, V in CP , Hom AŽ E, V . is a P-module with action defined via Ž yf .Ž x . s Ý yŽ1. f Žg Ž yŽ2. . x . for all y g P, x g E, and f g Hom AŽ E, V .. Hom AŽ E, V . is in CP if E is finite dimensional. Its restriction to A is trivial and it is, therefore, a module for PrA s HrD in CH r D if E is still finite dimensional. The following is the first of our main results of this section. THEOREM 2.14. Suppose Ind H D is exact. For a fixed finite dimensional P-module E in CP , the assignment V ¬ Hom AŽ E, V . defines a functor F : CP ª CH r D . The right deri¨ ed functor R i F can be identified with V ¬ Ext Ai Ž E, V .. Furthermore, there is a spectral sequence with the E2-term E2i , j s Ext iH r D Ž M, Ext Aj Ž E, V . . « Ext iqj P Ž M m E, V . for each M in CH r D and V in CP . Proof. This is a consequence of wLin2, Prop. 1.5x. Here the exactness of Ind PA has been proved in the Proof of Theorem 2.12 and the algebra PrA has been identified with HrD using Lemma 2.13. Using the above theorem, each Ext iAŽ E, V . for E, V in CP , with E finite dimensional, is an HrD-module in CH r D . Next we will show that the HrD-action on the above extension group preserves the Yoneda product. We remark that, since all the algebras are Hopf algebras, the cup product and Yoneda product are the same Žsee wCa2x for a proof.. In the rest of this section, we fix a Hopf algebra H Žnot necessarily cocommutative. and a normal Hopf subalgebra D such that Ind H D is exact with respect to fixed admissible categories CD and CH . This implies that the injective H-modules in CH remain injective when restricted to D Žsee wLin1, 3.7x.. For given H-modules E and M, the extension groups Ext iD Ž M, E . can be defined by an injective resolution of E in CH when it is restricted to D, which remains an injective resolution and actually gives the HrD-module structure on Ext iD Ž M, E .. The Yoneda product can be formulated using injective resolutions as follows: for H-modules N, M, and E, let us fix injective resolutions 0 ª E ª J 0 ª J1 ª ??? ª Ji ª ??? , 0 ª N ª I0 ª I1 ª ??? ª Ii ª ???

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in the category CH . Let Zi : Ji and K i : Ii be the kernels of the corresponding differentials. They are H-submodules since all the differentials are homomorphisms of H-modules. Therefore, we have surjective homomorphisms of HrD-modules Hom D Ž N, Zm . ª Ext Dm Ž N, E .

and Hom D Ž M, K n . ª Ext Dn Ž M, N . .

Each j g Ext Dm Ž N, E . can be represented by a D-map g : N ª Zm , which induces a homomorphism of complexes 0 ª N ª I0 ª I1 ª ??? ª In ª ??? g1

6

g0

gn

6

6

g

6

0 ªZm ª Jm ªJmq1 ª ??? ªJmqn ª ??? . By applying the functor Hom D Ž M, y. to the above diagram, g induces a sequence of linear maps Ext Dn Ž M, N . ª Ext Dmq n Ž M, E . for all n. These maps are uniquely determined by j and independent of the choice of g. In this way we get a linear map Ext Dn Ž M, N . m Ext Dm Ž N, E . ª Ext Dmq n Ž M, E ., which is the Yoneda product sending h m j to w f ( g n x. Here f : M ª K n represents h g Ext Dn Ž M, N . and w f ( g n x is the image of f ( g n : M ª Zmqn in Ext Dmq n Ž M, E .. It is a standard homological algebra argument to show that the class w f ( g n x is independent of the choices of f and g n . THEOREM 2.15.

For H-modules E, M, and N, the Yoneda product

Ext Dn Ž M, N . m Ext Dm Ž N, E . ª Ext Dmq n Ž M, E . is a homomorphism of HrD-modules. Proof. First of all, we have hŽ f ( g n . s ÝhŽ1. f ( hŽ2. g n for all h g H. In fact, for any x g M,

Ý Ž hŽ1. f ( hŽ2. g n . Ž x . s Ý hŽ1.Ž1. f ž g Ž hŽ1.Ž2. . hŽ2.Ž1. g n Ž g Ž hŽ2.Ž2. x . . / s

Ý hŽ1. f ž « Ž hŽ2. . g n Ž g Ž hŽ3. . x . /

s

Ý hŽ1. Ž f ( g n . Ž g Ž hŽ2. . x . .

To show the theorem, we only need to show that hŽ1.h m hŽ2. j is sent to w hŽ1. f ( hŽ2. g n x. Note that hŽ1.h is represented by hŽ1. f and hŽ2. j is represented by hŽ2. g. By the following lemma, we can take Ž hŽ2. g . n s hŽ2. g n and the theorem follows.

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Let E, F, M, and N be H-modules. If the diagram d1

g1

g2

d2

F

6

6

N

M

6

E

6

LEMMA 2.16.

commutes with d1 , d 2 being H-maps and g 1 , g 2 being D-maps, then for any h g H the diagram d1

6

hg 1

hg 2

d2

F

6

6

N

M

6

E

commutes. Proof. We have hg 2 Ž d1 Ž x . . s s

Ý hŽ1. g 2 Ž g Ž hŽ2. . d1Ž x . . s Ý hŽ1. g 2 ž d1 Ž g Ž hŽ2. . Ž x . . / Ý hŽ1. d2 ž g 1 Ž g Ž hŽ2. . Ž x . . / s d2 Ý hŽ1. ž g 1 Ž g Ž hŽ2. . Ž x . . /

s d 2 Ž hg 1 . Ž x . . for any x g E. Let ExtUD Ž M, E . denote the graded vector space [iG 0 Ext Di Ž M, E . for D-modules M and E. COROLLARY 2.17.

Let E, M, N be H-modules in CH .

ExtUD Ž E,

Ža. E . is a Ž graded. HrD-module algebra with the Yoneda product as multiplication. Žb. The Yoneda product ExtUD Ž E, E . m ExtUD Ž E, M . ª ExtUD Ž E, M . defines a left ExtUD Ž E, E .-module structure on ExtUD Ž E, M ., which is a graded HrD-module homomorphism. Žc. The annihilator Ann ExtU Ž E, E .ŽExtUD Ž E, M .. is an HrD-in¨ ariant D ideal of ExtUD Ž E, E .. Before we end this section, let us state several well known facts which are needed in order to apply the theorems of this section to Lie algebras of Cartan type. Let G be a connected affine algebraic group defined over the field k with DistŽ G . being its distribution algebra, and X an affine variety with coordinate algebra k w X x. Ži. The algebraic group G acts on X if and only if k w X x is a right rational G-module such that the action of each element of G on k w X x is

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an algebra automorphism, or equivalently, k w X x is a k w G x-comodule algebra, i.e., the multiplication map k w X x m k w X x ª k w X x is a homomorphism of comodules and 1 ¬ 1 m 1. Žii. Using the distribution algebra language we have that G acts on X if and only if k w X x is a DistŽ G .-module algebra such that the DistŽ G .module structure on k w X x comes from a rational G-module structure via differentiation. Žiii. If G acts on X, then any DistŽ G .-submodule of k w X x is a G-submodule. Živ. If I is a DistŽ G .-invariant ideal of k w X x, then I is also G-invariant. Furthermore, the closed subvariety V Ž I . in X defined by I is also G-invariant. The statement Ži. is the direct consequence of the definition of an algebraic group action on an affine variety, Žii. is the consequence of Ži., Žiii. follows from wJan3, I, 7.15x and Živ. is a consequence of Žiii..

3. CONSTRUCTION OF ALGEBRAS AND AUTOMORPHISMS We now fix an algebraically closed field k of characteristic p ) 5. For a given positive integer m and n s Ž n1 , . . . , n m . with n i being positive integers, let LŽ m, n. s  b s Ž b 1 , . . . , bm . g Z m < 0 F bi - p n i , i s 1, . . . , m4 . Let AŽ m, n. be the commutative associative algebra on linearly independent generators X a where a s Ž a 1 , a 2 , . . . , a m . g LŽ m, n.. The multiplication in AŽ m, n. is given by

XaX b s

ža a b/ q

X aq b

where

ža a b/ q

m

s

Ł

is1

ž

a i q bi . ai

/

The Lie algebra, W Ž m, n. : DerŽ AŽ m, n.., is the set

 u1 D1 q u 2 D 2 q ??? qu m Dm < u i g AŽ m, n . , i s 1, 2, . . . , m4 where the derivation Di is the divided power derivation which takes X a to X ay e i , i s 1, 2, . . . , m. The vector e i is the m-tuple with a 1 in the ith position and zeros everywhere else. Let G W be the automorphism group of g s W Ž m, n., which was first calculated by Jacobson for n s Ž1, . . . , 1. and by Wilson for n arbitrary.

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According to Wilson wW1x, the natural identification gives the following isomorphism of algebraic groups: G W (  g g Aut Ž A Ž m, n . . : g ? D ? gy1 g W Ž m, n . , ;D g W Ž m, n . 4 , in which the latter is denoted by AutŽ AŽ m, n., W Ž m, n... Applying our general argument given in Example 2.11 for G s G W and g s W Ž m, n., we have the evaluation map d : AŽ m, n. ª Hom AŽ m, n.ŽW Ž m, n., AŽ m, n.., which is a homomorphism of G-modules. It follows that Hom AŽ m, n.ŽW Ž m, n., AŽ m, n.. is a free AŽ m, n.-module with base  dx 1 , dx 2 , . . . , dx m 4 dual to  D 1 , . . . , Dn4 and m

da s

Ý Ž Di a. dx i . is1

Let V Ž m. be the exterior algebra of Hom AŽ m, n.ŽW Ž m, n., AŽ m, n.. over AŽ m, n.. The action of W Ž m, n. on Hom AŽ m, n.ŽW Ž m, n., AŽ m, n.. is given by D ? f s D( f y f ( Ž ad D . for D g W Ž m, n. and f g Hom AŽ m, n.ŽW Ž m, n., AŽ m, n... In particular, d is a homomorphism of W Ž m, n.-modules. By the argument in Example 2.11, the action of W Ž m, n. can be extended to V Ž m. by letting D ? Ž a n b . s D ? a n b q a n D ? b. Similarly, the action of G W can be extended to V Ž m. by letting g? Žanb. sg?ang?b

Ž 3.1.

with g g G, D g W Ž m, n., and a , b g V Ž m.. We can define the other Lie algebras of Cartan type by considering the action of W Ž m, n. on the differential forms:

v S s dx 1 n dx 2 n ??? n dx m

Ž 3.2.

r

vH s

Ý dx i n dx iqr

for m s 2 r ,

Ž 3.3.

is1

and r

v K s dx 2 rq1 q

Ý Ž x i dx iqr y x iqr dx i . is1

for m s 2 r q 1. Ž 3.4.

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COHOMOLOGY OF LIE ALGEBRAS

The Lie algebras of type S, H, and K are subalgebras of a Lie algebra of type W defined by S Ž m, n . s  D g W Ž m, n . : D ? v S s 0 4

Ž 3.5.

CS Ž m, n . s  D g W Ž m, n . : D ? v S g k ? v S 4 ,

Ž 3.6.

H Ž m, n . s  D g W Ž m, n . : D ? v H s 0 4 ,

Ž 3.7.

CH Ž m, n . s  D g W Ž m, n . : D ? v H g k ? v H 4 ,

Ž 3.8.

K Ž m, n . s  D g W Ž m, n . : D ? v K g A Ž m, n . ? v K 4 .

Ž 3.9.

Let X Ž m, n. generically denote the Lie algebras W Ž m, n., SŽ m, n., H Ž m, n., or K Ž m, n.. We will also denote by CX the corresponding algebras in the cases of the types S and H. Since p ) 5 the second derived algebra X Ž m, n.w2x is simple. The Lie algebras, X Ž m, n.w2x and X Ž m, n., have the structure of a graded Lie algebra with the grading inherited from grading on AŽ m, n.. More precisely, if g s X Ž m, n.w2x then g s gy1 [ g 0 [ g 1 [ ??? [ g s for X of type W, S, or H; and g s gy2 [ gy1 [ g 0 [ g 1 [ ??? [ g s for X if type K Žfor more details, see Section 4.. For i G 0 set Ui s Aut i Ž X Ž m, n .

w2 x

.

s g g Aut Ž X Ž m, n .

½

w2 x

.:

Ž g y id. ? g j ; g iqj [ g iqjq1 [ ??? [ g s for j ) 0 .

5

Wilson wW2, Lemma 1x has shown that for i G 1 any automorphism in Ui can be lifted to an automorphism of Aut i ŽW Ž m, n... Thus, for each i G 1, we can naturally embed Ui as a subgroup of Aut i ŽW Ž m, n... Let G 0, X s AutU Ž X Ž m, n.w2x . s  g g AutŽ X Ž m, n.w2x . : g ? g i ; g i for all i4 . The group G 0, X acts on k m ( ² X e 1 , X e 2 , . . . , X e m :. Let V Ž m, n.1 s ² X e j : n j s max n k : 1 - k F m4: and for i ) 1 inductively set V Ž m, n. i s ² X e j : n j ) max n k : 1 - k F m, X e k f V Ž m, n. iy14:. Moreover, let V Ž m, n. be the flag: V Ž m, n . : V Ž m, n . m = V Ž m, n . my1 = ??? = V Ž m, n . 1 =  0 4 . From wW2, Theorem 2x, the automorphism group GX s AutŽ X Ž m, n.w2x . is isomorphic to G 0, X h U1 where G 0, X (

½

StabG L m Ž k . V Ž m, n .

X of type W or S,

StabCS P m Ž k . V Ž m, n .

X of type H or K .

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Let gqs [i) 0 g i , gys [i- 0 g i and b "s g 0 [ g ". From Kreknin wKrx it follows that GX s U0 and GX ? bq; bq. Moreover, from wW2, Theorem 1x, GX 2 U1 2 U2 ??? 2 U2 2  1 4 is a normal series for GX and there exists an isomorphism of varieties Žand commutative algebraic groups. such that UrU i iq1 ( g i for i G 1; thus G X is irreducible. Let c bq Xs

½

bq bq W l CX Ž m, n .

X of type W or K X of type S or H.

For the rest of this section we will only consider the case when g s X Ž m, 1.w2x Ži.e., when g is a restricted simple Lie algebra.. The following fundamental lemma will be used throughout the rest of the paper. LEMMA 3.1. Let T s Ž kU . n be a torus and V a rational T-module with the decomposition V s Ý lVl . Here l runs through all characters l: T ª kU and Vl s  ¨ g V < t¨ s lŽ t . ¨ 4 . Suppose X is a T-in¨ ariant closed conical sub¨ ariety of V and x s Ý l xl g X. For a fixed l0 with xl 0 / 0, if there is a cocharactwr a : kU ª T such that ² a , l0 : is strictly smaller than any other ² a , l: with xl / 0. Then xl g X. Here, ² a , l: is defined by lŽ a Ž a.. s 0 a² a , l: for all a g kU . Proof. Consider the map f : kU ª X defined by f Ž a. s ay² a , l 0 :a Ž a. x for all a g kU . The image of f is in X because X is T-invariant and conical. Therefore, f Ž a. s Ý l a² a , l :y ² a , l 0 : xl . Now f Ž x . can be easily extended to a morphism f : k ª V, and ImŽ f . : Im Ž f . : X since X is closed. However, by the assumption, xl0 s f Ž0. g X. The next proposition will allow us to identify a normal Hopf subalgebra of DistŽ G ., so we can use the results in the previous section. PROPOSITION 3.2. If g s X Ž m, 1.w2x where X is of type W, S, H, or K and Ž . Ž . GX s AutŽ g ., then LieŽ GX . s c bq X : W m, 1 . In particular, Lie G X l bqs bq. Proof. First let g s W Ž m, 1.. By a standard resultwChex, LieŽ G W . : DerŽ g .. Let g s gy1 [ g 0 [ g 1 [ ??? [ g s denote the grading of W Ž m, 1.. Wilson wW1x has shown that DerŽ g . ( g; hence LieŽ G W . : g. Now suppose that x s xyq y g LieŽ G W . with 0 / xyg gy1 and y g bq. The Lie algebra g will be viewed as a rational T-module with T being the maximal torus of G W . Let a : kU ª T be defined by a Ž a.Ž X e i . s aX e i for all a g kU . Since LieŽ G W . is a closed, T-invariant, conical subvariety of g it

COHOMOLOGY OF LIE ALGEBRAS

869

follows that xyg LieŽ G W . by Lemma 3.1. Furthermore, G 0, W acts transitively on the set of nonzero elements in gy1 , thus gy1 : LieŽ G W .. Consider the subgroup Us in G W with s ) 0. It is easy to see that LieŽUs . s g s . Since g s and gy1 generate g as a Lie algebra we have g : LieŽ G W .. This is a contradiction because dim k LieŽ G W . - dim k g. Therefore, LieŽ G W . : bq. By counting dimensions we have LieŽ G W . s c bq. Now we consider the case when g s SŽ m, 1., H Ž2 r, 1., and K Ž2 r q 1, 1.. The action of GX on differential forms is given by Ž3.1.. By wW2, Lemma 1x, one can verify that g g GX if and only if g g G W and g ? vX g

½

k ? vX

X of type S or H

A Ž m, 1 . v K

X of type K .

Thus GX is a closed subgroup of G W and LieŽ GX . is a subalgebra of LieŽ G W .. Hence, LieŽ GX . : bq W . By considering the gradations and the dimensions, LieŽ GX . s c bq coincides with other types except the type K, X whose gradation is not quite compatible with the gradation of type W. However, a close inspection shows that the vectors in gyl bq W are of the r form fD 2 rq1 with f s Ý2is1 c i X e i . If there is an element y g LieŽ GK . with the form y s fD 2 rq1 q x such that x has a higher degree with respect to the gradation defined on W Ž m, 1. with X e m being of degree two Žsee wW2x., by choosing a suitable cocharacter of the torus of G 0, K we can show that fD 2 rq1 g LieŽ GK .. However, by a direct calculation, the only possible r f s Ý2is1 c i X e i with fD 2 rq1 v K g AŽ m, 1. v K is zero. Now by comparing the dimensions we have LieŽ GX . s c bq X. By wW2x, GX is a subgroup of G W and GX will act on V Ž m.. Note that GX s G 0, X h U1 with g G s LieŽ GX . l g : bq and g 0 : LieŽ G 0, X .. Here g G is a restricted ideal of LieŽ GX .. The group G 0, X is reductive and LieŽw G 0, X , G 0, X x. is contained in g 0 . Therefore, g 0 is the Lie algebra of a reductive subgroup of G 0, X . Note that every Žfinite dimensional. g 0-module can be extended to a LieŽ G 0, W .-module by letting certain central elements act trivially. Throughout the rest of this paper we will use the standard terminology for the representation theory of reductive groups given in wJan3x. Let X 1ŽT . denote the set of restricted weights for uŽ g 0 . Žwith fixed positive roots.. For each l g X 1ŽT . let LŽ l. be the simple uŽ g 0 .-module, qŽ . which extends to a bq-module by letting gq act as zero. Set Virr l s uŽ g . muŽ bq . LŽ l., which is a Žleft. uŽ g .-module by the left multiplication on qŽ . uŽ g .. The module Virr l is simple for g unless l is an exceptional weight wSh, N1, Hol1, Hol2x. Moreover, if L is the set of exceptional weights then qŽ . V Ž m. ( [lg LVirr l as both uŽ g .-module and GX -module.

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PROPOSITION 3.3. If g s X Ž m, 1.w2x where X is of type W, S, H, or K qŽ . then Virr l has a GX -module structure such that the two induced g G -structures on it coincide for each l g X 1ŽT .. Proof. As we discussed above, the module LŽ l. can be extended to a G 0, X -module wJan3x and can be further lifted to a GX -module by letting U1 act trivially. Note that the G 0, X -module structure on LŽ l. is compatible with the g 0-module structure, i.e., g Ž Dx . s g Ž D .Ž gx . for all y g G 0, X , D g g 0 , and x g LŽ l.. Moreover, since bq is a GX -submodule of g, we can show that the bq-module structure on LŽ l. is compatible with the GX -module structure, i.e., g Ž Dx . s g Ž D .Ž gx . for all g g GX , D g bq, and x g LŽ l.. Since bqs g 0 [ gq we have for each g 0 g G 0, X , g 1 g U1 , D 0 g g 0 , and D 1 g gq, g 0 g 1 Ž Ž D 0 q D 1 . x . s g 0 Ž D 0 x . s g 0 Ž D 0 . g 0 x s g 0 g 1 Ž D 0 q D 1 . g 0 g 1 x. Here we have used the fact that g 1Ž D 0 . y D 0 g gq for all D 0 g g 0 . Now let us define a GX -module structure on uŽ g . muŽ bq . LŽ l. as follows. First, uŽ g . mk LŽ l. is a GX -module as a tensor product module. The subspace generated by  ub m x y u m bx < x g LŽ l., u g uŽ g ., b g uŽ bq.4 is GX -invariant. In fact, g Ž ub . m Ž gx . s y Ž gu . m g Ž bx . Ž gu . Ž gb . m gx y Ž gu . m Ž gb . Ž gx . . Thus the GX -module structure on uŽ g . muŽ bq . LŽ l. is given as the quotient GX -module of uŽ g . mk LŽ l.. By the definition, one can see easily that the GX -module structure defined above is compatible with the uŽ g .-module structure, since GX acts on uŽ g . as automorphisms of the algebra. In order to finish off the proof of the proposition, we still need to show that the two g G -module structures on uŽ g . muŽ bq . LŽ l. coincide. Let D g g G . The action of D on uŽ g . muŽ bq . LŽ l. via differentiation of GX module structure is given by D Ž u muŽ bq . x . s D ? u muŽ bq . x q u muŽ bq . Dx. Here D ? u s Du y uD is the action of LieŽ GX . on uŽ g . induced by the GX -action on g and Du, uD should be understood as product in uŽ g .. Since D is in bq and the two actions of D on LŽ l. coincide, we have D Ž u muŽ bq . x . s Ž Du y uD . muŽ bq . x q uD muŽ bq . x s Du muŽ bq . x, which is the same as the action of D via left multiplication as an element of uŽ g ..

COHOMOLOGY OF LIE ALGEBRAS

871

qŽ . For each l g X 1ŽT . the module Virr l has a unique maximal submodqŽ . ule RVirr l , where R s Rad uŽ g . wN1, Prop. 1.2.3x, such that qŽ . qŽ . Virr l rRVirr l is a simple uŽ g .-module, which we denote by L Ž l.. All simple uŽ g .-modules are obtained in this way. Since elements of GX permute the maximal submodules, the uniqueness implies that the maximal submodule is GX -stable. Hence, we can state the following result.

COROLLARY 3.4. If l g X 1ŽT . then L Ž l. has a GX -module structure such that the two induced g G -module structures coincide. y Ž . Let P Ž l. be the uŽ g 0 .-projective cover of LŽ l. and set Vproj l s uŽ g . y muŽ by . P Ž l.. The projective module P Ž l. is lifted to a b -module in an analogous manner as for LŽ l.. If P Ž l. is the projective indecomposable module with the simple head L Ž l. then for each l f L we have P Ž l. s y Ž . Vproj l for X of type W, S, H, and K. Let h be the Coxeter number associated to the reductive group G 0, X . There are compatible module structures for the algebraic group G 0, X on these projective modules for large primes.

COROLLARY 3.5. If X is of type W, S, H, or K with l f L and p ) 2 h y 2 then P Ž l. has a G 0, X -module structure. Proof. For each l g X 1ŽT . the modules P Ž l. have a G 0, X -structure if p ) 2 h y 2 where h is the Coxeter number of the reductive group G 0, X wJan3x. As in the case for simple modules, the G 0, X -module and by-module structures on P Ž l. are compatible. The rest of the proof follows from the same argument as in the proof of Proposition 3.3. In fact, the proof of Proposition 3.3 works for any g 0-module M with a compatible G 0, X -module structure. From now on, we fix a connected closed subgroup G : GX . Let DistŽ G . be the distribution algebra of G. Then X Ž m, 1. and g s X Ž m, 1.w2x are DistŽ G .-modules and the two actions of g G s LieŽ G . l g on X Ž m, 1.w2x Žby the restriction from DistŽ G . and by the adjoint action. coincide. Since g and g G are G-invariant restricted subalgebras of the restricted Lie algebra X Ž m, 1., the restricted enveloping algebras uŽ X Ž m, 1.., uŽ g ., and uŽ g G . are rational G-modules and DistŽ G .-module algebras. We can use Example 2.11 along with the construction of the Lie algebras of Cartan type, given at the beginning of the section, to show that there exists a Hopf algebra DistŽ G . uŽ g . which contains both DistŽ G . and uŽ g . as Hopf subalgebras such that the images of uŽ g G . in both DistŽ G . and uŽ g . are identified. We call a module M a DistŽ G . uŽ g .-module if M is a uŽ g .-module with a DistŽ G .-module structure which comes from a rational G-module structure on M and the two uŽ g G .-module structures on M coincide. We will consider the category CuŽ g . of all uŽ g .-modules and the category CDistŽG. of all Žlocally. rational G-modules. Since uŽ g . is a rational G-mod-

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ule it has a DistŽ G .-structure. Therefore, every finite dimensional uŽ g .module is a quotient of uŽ g . n which has a DistŽ G .-structure. Now one can verify that the category of all DistŽ G . uŽ g .-modules forms an admissible category which can be constructed from CuŽ g . and CDistŽG. as in Section 2 ŽLemma 2.8, Theorem 2.10.. The Hopf algebra DistŽ G . uŽ g . is the analog of the distribution algebra of a reductive algebraic group and the Hopf subalgebra uŽ g . is the analog of the distribution algebra of the first Frobenius kernel of the reduction group. We end this section with several remarks. Remark 3.6. Since the set  L Ž l.< g l g X 1ŽT .4 contains a full set of non-isomorphic irreducible uŽ g .-modules, the results in Corollary 3.4 resemble Curtis’s wCux results for classical semisimple Lie algebras. Remark 3.7. If G 0, X is replaced by TX , Corollary 3.5 is valid without the assumption on primes. Remark 3.8. For the prime p s 5 there is another class of restricted simple Lie algebras known as the Melikyan algebras wKuzx. These algebras are graded by using a G 2 root system. The afore-mentioned constructions can also be used to create an infinite dimensional cocommutative Hopf algebra for these Lie algebras. Remark 3.9. Let GX be considered now as an affine algebraic group scheme over k and let F : GX ª GX be the Frobenius morphism. Furthermore, if F r is the r th iteration of the Frobenius morphism let Ž GX . r be the scheme theoretic kernel of F r. The procedure in Section 2 ŽTheorem 2.12. can be used to produce a finite dimensional Hopf algebra DistŽŽ GX . r . uŽ g . Ža finite dimensional subalgebra of DistŽ GX . uŽ g .. along with the category of all DistŽŽ GX . r . uŽ g .-modules. This construction yields a ‘‘hyperalgebra’’ analog for Lie algebras of Cartan type Žsee wHux.. The simple DistŽ GX . uŽ g .and DistŽŽ GX . r . uŽ g .-modules can be classified as in the classical situation and described via twisted tensor products. We will pursue this investigation in a forthcoming paper wLNx. 4. PROJECTIVITY RESULTS FOR RESTRICTED LIE ALGEBRAS OF CARTAN TYPE Let g s X Ž m, 1.w2x be a Lie algebra of Cartan type and let GX s AutŽ g . ( G 0, X h U1. In this section we will assume that GX is defined over the prime field Fp . Let TX be a maximal torus for the reductive subgroup G 0, X . We will first show that the construction of the Ivanovskii spectral sequence is compatible with the action of GX . As a consequence the map on the cohomological support variety into affine space obtained from the edge homomorphism of this spectral sequence will be GX -equivariant.

COHOMOLOGY OF LIE ALGEBRAS

873

Using the concrete realizations of these varieties along with TX -stability Žresp. G 0, X -stability ., we will deduce projectivity theorems for DistŽTX . uŽ g . and DistŽ G 0, X . uŽ g . modules. Let M be a GX -module. Since GX is defined over Fp , we can define the Frobenius twist M Ž1. of M as follows: let M Ž1. s M as an abelian group with a new scalar structure such that a) m s a p m Ž a g k . and a new GX -action such that g ) m s FrŽ g . m Žfor g g GX . ŽSee wJan3, I, Chap. 9x for more details about Frobenius twisted modules.. As a DistŽ GX .-module, M Ž1. is trivial when restricted to uŽLieŽ GX .., which is a subalgebra of DistŽ GX .. Using the quotient map DistŽ GX . uŽ g . ª DistŽ GX .ruŽLieŽ GX .. Žsee Lemma 2.13., one can make M Ž1. a DistŽ GX . uŽ g .-module on which uŽ g . acts trivially. Conversely, by Lemma 2.13, for any DistŽ GX . uŽ g .-module N on which uŽ g . and uŽLieŽ GX .. act trivially there is a DistŽ GX .ruŽLieŽ GX ..-module structure on N. Furthermore, there is a GX -module M Ž M is rational since N is in the category of DistŽ GX . uŽ g .modules. such that N s M Ž1.. We will write M s N Žy1.. In particular, the GX -module structure on g gives rise to GX -module structures on g a and S s Ž g a .. In the following proposition we have a spectral sequence of DistŽ GX . uŽ g .-algebras on which uŽ g . acts trivially. PROPOSITION 4.1. For p G 3 and g s X Ž m, 1.w2x, there exists a spectral sequence of GX -algebras of the form E2s, t s S s Ž g a .

Ž1 .

m H t Ž U Ž g . , k . « H 2 sqt Ž u Ž g . , k . ,

where UŽ g . is the uni¨ ersal en¨ eloping algebra of g and H U ŽUŽ g ., k . is the ordinary Lie algebra cohomology. Proof. For a description of the construction of the spectral sequence we refer the reader to wFP1, FP2x. In order to check that the spectral sequence carries a GX -structure it is necessary to check that the cobar resolution is a GX -resolution and the filtration by powers of the augmentation ideal I of uŽ g .a is compatible with the action of GX . Note that GX acts on g as automorphisms of the restricted Lie algebra. One can extend the GX -action to uŽ g . and uŽ g .a as automorphisms of the Hopf algebras Ž uŽ g . is finite dimensional and uŽ g .a denotes the dual Hopf algebra.. Thus the cobar resolution is a GX -resolution. In order to show that GX stabilizes the filtration induced by powers of the augmentation ideal, it suffices to show that GX stabilizes powers of the augmentation ideal. For g g GX , x g uŽ g ., and f g uŽ g .a we have Ž g ? f .Ž x . s f Ž gy1 ? x .. Since I s  f g uŽ g .a : f Ž1. s 04 it follows that g ? I ; I for all g g GX . For x g g the comultiplication DX in uŽ g . is given by DX Ž x . s x m 1 q 1 m x and we have DX Ž g ? x . s g ? DX Ž x .. By using the Poincare]Birkhoff]Witt ´ theorem along with the fact that DX is an algebra homomorphism, it follows

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LIN AND NAKANO

that GX acts as automorphisms in uŽ g .a and g ? I n ; I n for all positive integers n. From this spectral sequence there is an edge homomorphism FU : S U Ž g a . ª H 2 U Ž u Ž g . , k .

Žy1 .

which is a finite map of GX -algebras. For a finite dimensional uŽ g .-module M, H U Ž uŽ g ., M a m M . Žy1. s ExtUuŽ g .Ž M, M . Žy1. is a graded H 2U Ž uŽ g ., k .Žy1.-module via the Yoneda product as stated in Section 2. The cohomological support ¨ ariety < g < M of M is the closed affine homogeneous subvariety of < g < k s Maxspec Ž H 2U Ž uŽ g ., k .Žy1. . obtained by considering the annihilator of H U Ž uŽ g ., M a m M .Žy1. in the commutative graded ring H 2U Ž uŽ g ., k .Žy1.. The map FU on rings induces a morphism of algebraic varieties F: < g < k ª Adim k g . Under this map one has dim < g < M s dim F Ž< g < M .. Moreover, the image of < g < M for any uŽ g .-module M can be characterized as wFP3x: F Ž < g < M . s x g g : x w px s 0 and M² x : p is not projective j  0 4 . Ž 4.1.

½

5

Furthermore, the following were proved in wFP5x. Ža. M is projective if and only if F Ž< g < M . s  04 . Žb. Let ??? ª Pn ª ??? ª P1 ª P0 ª M ª 0 be a minimal projective resolution. The complexity c g Ž M . of M is defined to be the smallest nonnegative integer c such that dim Pnrn c is bounded for all n Žand ` if c does not exist.. Then dim F Ž< g < M . s c g Ž M .. Note that the group GX acts on both < g < k and Adim k g since GX acts on their coordinate rings as an automorphism of algebras. In particular, the GX -action on Adim k g is the obvious action of G on g. Since FU is a GX -map of rings, the morphism F is GX -equivariant. Furthermore, if the uŽ g .-module M has a DistŽ G X . uŽ g .-module structure, then ExtUuŽ g .Ž M, M .Žy1. has a rational GX -module structure, which is compatible with the GX -action on the algebra H 2U Ž uŽ g ., k .Žy1. Žsee Corollary 2.17 and the following remarks.. Therefore, < g < M is a GX -invariant subvariety of < g < k and so is its image F Ž< g < M . in g. In order to understand the action of TX on the support varieties F Ž< g < k . it is necessary to compute the characters of TX on g. This will be accomplished by using the explicit basesrgenerators for the Lie algebras of Cartan type. We will start with the generalized Cartan-type Lie algebras

875

COHOMOLOGY OF LIE ALGEBRAS

and then consider the restricted cases later on. For the details and proofs we refer the reader to wSF, Chap. 4x. For each b g LŽ m, n., we set
½

b 1 q b 2 q ??? qbm b 1 q b 2 q ??? qbmy1 q 2 bm

for g of type W , S, or H for g of type K .

For W Ž m, n. Žs W Ž m, n.w2x . a basis is given by

 X b Dk : k s 1, 2, . . . , m, 0 F bi F p n y 1 for i s 1, 2, . . . , m4 . i

Set eb , k s X b D k and let TW be a maximal torus in GLmŽ k . given y1 4 by TW s  t a1 , a 2 , . . . , a m : a i g kU 4 with t a1 , a 2 , . . . , a m s diag a1y1 , ay1 2 , . . . , am . b Note that the action of t a1 , . . . , a m on AŽ m, n. is given by t a1 , . . . , a m X s a1b 1 ??? a mb m X b and the induced action on DerŽ AŽ m, n.. gives t a1 , . . . , a m D k s ay1 k D k . Thus we have t a1 , a 2 , . . . , a m ? eb , k s

a1b1 a2b 2 , . . . , a mb m ak

eb , k .

Ž 4.2.

For SŽ m, n.w2x, let Di, j : AŽ m, n. ª W Ž m, n. be the map defined by Di , j Ž f . s Dj Ž f . Di y Di Ž f . Dj for i, j s 1, 2, . . . , m. A set of generators for SŽ m, n.w2x can be given by

 Di , j Ž X b . : 1 F i - j F m, < b < / 0 4 . From Section 3 we have TS s TW , the maximal torus of GLmŽ k .. Then by applying Ž4.2. twice it follows that t a1 , a 2 , . . . , a m ? Di , j Ž X b . s

a1b1 a2b 2 ??? a mb m

? Di , j Ž X b . .

ai a j

Ž 4.3.

For H Ž m, n.w2x Ž m s 2 r ., let DH : AŽ m, n. ª W Ž m, n. be the map defined by 2r

Ý s Ž j . Dj Ž f . Dj

DH Ž f . s

X

js1

where

s Ž j. s

½

1 y1

1FjFr r q 1 F j F 2r

876

LIN AND NAKANO

and jX s

½

jqr jyr

1FjFr r q 1 F j F 2r.

A basis for H Ž m, n.w2x is given by

 DH Ž X b . : b g L Ž m, n . , 0 - < b < - p n q ??? qn 1

y m4 .

m

Let TH be the maximal torus in CSP2 r Ž k . ( SP2 r Ž k . h kU given by TH s  t a , a , . . . , a , b : a j , b g kU 4 with 1 2 r y1 t a1 , a 2 , . . . , a r , b s diag  a1y1 by1 , . . . , ay1 , a1 by1 , . . . , a r by1 4 . r b

The action of TH on H Ž m, n.w2x is t a1 , a 2 , . . . , a r , b ? DH Ž X b . s Ž a ab1yb rq 1 a2b 2yb rq 2 ??? a rb ryb m . = Ž b b1qb 2q ??? q b 2 ry2 . ? DH Ž X b . .

Ž 4.4.

For K Ž m, n.w2x Ž m s 2 r q 1., let DK : AŽ m, n. ª W Ž m, n. be the map defined by DK Ž f . s Ý mjs1 f j Dj , where f j s X e j Dm Ž f . q s Ž jX . DjX Ž f . ,

j s 1, 2, . . . , 2 r ,

Ž 4.5.

2r

fm s 2 f y

Ý s Ž j . X e fj . j

Ž 4.6.

X

js1

The map DK is a linear isomorphism onto its image, K Ž m, n.w2x, and a basis for the Lie algebra K Ž m, n.w2x can be given by

 DK Ž X b . : 0 F bi F p n y 1, i s 1, 2, . . . , m4 , i

where DK Ž X b . s DH Ž X b . q X b y e n

ž

2r

Ý js1

X e j Dj q 2 y

/ ž

2r

Ý bj js1

/

X b Dn .

The operator DH is defined for the indices 1, . . . , 2 r s m y 1. The Lie algebras of type K are graded differently from the other Lie algebras of Cartan type. Consequently, the action of the scalars is given as follows. If t b s diag by1 , . . . , by1 4 g kU then t b ? Dj s

½

by1 Dj y2

b

D 2 rq1

j s 1, 2, . . . , 2 r , j s 2 r q 1.

877

COHOMOLOGY OF LIE ALGEBRAS

It follows that t b ? DK Ž X b . s b b1qb 2q ??? q b 2 rq2 b 2 rq 1y2 DK Ž X b . . Let TK be a maximal torus in CSP2 r Ž k . given by TK s  t a1 , a 2 , . . . , a 2 r , b : a j , b y1 y1 g kU 4 where t a1 , a 2 , . . . , a 2 r , b s diag a1y1 by1 , ay1 , . . . , ay1 , a1 by1 , 2 b r b y1 y1 y1 4 a2 b , . . . , a r b , b . By applying Ž4.2. we obtain t a1 a 2 , . . . , a r , b ? DK Ž X b . s t b ? Ž t a1 , a 2 , . . . , a r , 1 ? DK Ž X b . . s t b Ž a1b1yb rq 1 a2b 2yb rq 2 ??? a rb ryb 2 r . DK Ž X b . s Ž a1b1yb rq 1 a2b 2yb rq 2 ??? a rb ryb 2 r . =b b1qb 2q ??? q b 2 rq2 b 2 rq 1y2 DK Ž X b . . For the rest of this section we will consider only the case g s X Ž m, 1.w2x Žwhen g is a restricted Lie algebra.. In this situation G 0, X s GLmŽ k . for X of type W or S and G 0, X s CSPmŽ k . for X of type H or K. We will first provide a description of the TX decomposition of these restricted Lie algebras. Let X Ž m, 1.w2x s W Ž m, 1. and t j s X e j Dj for j s 1, 2, . . . , m. Let Gj,yW s  b : b j s 04 and Gj,qW s  b : bi s p y 1 for i / j and 0 F b j - p y 14 for j s 1, 2, . . . , m. For each b g Gj,yW Žresp. Gj,qW . set Fby, W s ² X b Dj : Žresp. y q q Fbq, W s ² X b t j :.. Let GW"s D mjs1 Gj,"W and GW s GW DGW . For a f GW set Ea , W s ² X a t j : j s 1, 2, . . . , m: ; g < a < ; W Ž m, 1 . . One should note that dim k Fb", W s 1 for all b g GW" and 2 F dim k Ea , W F q m for all a f GW and Ea , W / 0. For type S let Gj,yS s Gj,yW and for each b g Gj,yS let Fby, S s Fby, W ; SŽ m, 1.. If i / j let Gi,qj, S s  b : b k s p y 1 for k / i, j and 0 F bi , b j - p y 14 . Again let GSqs D i, j Gi,qj, S, GSys D j Gj,yS, and GS s GSqD GSy. Moreover, for b g Gi,qj, S let Fbq, S s

m

½Ý

5

c k X b t k : c i Ž bi q 1 . q c j Ž b j q 1 . s 0 .

ks1

Note that dim k Fbq, S s 1. For a f GSq set

½ ž

Ea , S s X a

m

Ý c j X e Dj j

js1

m

/

:

Ý Ž a j q 1. c j s 0 js1

5

.

The last condition imposes the condition that Ea , S ; S Ž m, 1.w2x. er For g s X Ž m, 1.w2x where X is of type W or S, let s s Ý m rs1 X Dr and X b q set Eb , X s ² X s : ; Eb , X for b f GX and for X of type W. For X of

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type S we impose the condition that < b < q m ' 0 mod p to insure that EbX , X ; SŽ m, 1.w2x. For X of type H or K we can use the fact that the character of t a1 , a 2 , . . . , a r , 1 on X b is the same as the character on D X Ž X b .. For each j s 1, 2, . . . , 2 r define Gj,yX s  b : b j / 0, b l s 0 for all l / j4 and, for each j, k such that j / k, kX , set Gj,qk, X s  b : b j , b k / 0 and b l s 0 for all l / j,k 4 . Let GXys D j Gj,yX , GXqs D j, k Gj,qk, X , and GX s GXq DGXy. For each b g GX set Fb , X s ² DH Ž X b .:. Moreover, for each fixed a s Ž a 1 , a 2 , . . . , a r , N . such that there exists g f G with g i y g iqr s a i , i s 1, 2, . . . , r and < g < X y 2 s N, set Ea , X s ² D X Ž X g . : g i y g iqr s a i ,

i s 1, 2, . . . , r , < g < X y 2 s N : .

.w2x

For g s X Ž m, 1 , where X is type W, S, H, or K, we have the following decomposition of X Ž m, 1.w2x as a TX -module. X Ž m, 1 .

w2 x

s

[ Ea , X [ [

bgGXy

Fby, X [

[

b gGXq

Fbq, X .

Ž 4.7.

The following technical lemma which provides the computation of the action of GLmŽ k . on W Ž m, 1. and S Ž m, 1. will be useful in proving the main result of this section. For each i / j, let g i, j s e i, j q I g SL mŽ k . where e i, j denotes the matrix with 1 in the Ž i, j .-entry and zeros everywhere else. A direct computation using the actions given in Section 2 shows that g i , j ? Ž X e k D k . s X e k D k q d j, k X e i D k y d i , k X e k Dj .

Ž 4.8.

er If y s Ý m rs1 c r X Dr , then

g i , j ? Ž X b y . s g i , j Ž X b . y q Ž c j y c i . X e i Dj . .

Ž 4.9.

By using the definition of the divided power algebra and the fact that g i, j acts as an automorphism, it follows that bj

g i , j X b s bj !

ž ž Ý

bi q b j y n

ns0

bi

/

/

X bqŽ ny b j . e jqŽ b jyn. e i .

Ž 4.10.

LEMMA 4.2. Let X Ž m, 1. s W Ž m, 1. or SŽ m, 1. and M be a er b Ž< < . DistŽ G 0, X . uŽ g .-module with y s Ý m rs1 c r X Dr and z s X y g F g M . Ža. If there exist i, j with c i / c j and bi q b j F p y 2 then X by b j e jqŽ b jq1 . e i Dj g F Ž< g < M .. Žb. If there exist i, j with i / j and bi q b j G p y 1 then X bqŽ b iyŽ py1.. e jqŽŽ py1.y b i . e i y y

Ž bi q 1 . Ž c j y c i . bi q b j q 2

? X e j Dj g F Ž < g < M . .

COHOMOLOGY OF LIE ALGEBRAS

879

Žc. If b j s p y 1 and bi - p y 1 with c j / c i and c i s c k for some k / i, j then X bqŽ b iyŽ py1.. e jqŽŽ py1.y b i . e i y y

Ž bi q 1 . Ž c j y c i . bi q b j q 2

? X e j Dj g F Ž < g < M . ,

with c j y Ž bi q 1.Ž c j y c i .rŽ bi q b j q 2. / c k . Proof. For part Ža. suppose that there exist i, j such that c i / c j and X y g F Ž< g < M .. Since F Ž< g < M . is G 0, X -stable, it follows that g i, j ? Ž X b y . g F Ž< g < M .. Now set a k s 1 for k / j and a j s a for t a1 , a 2 , . . . , a m g TW s TS . From Lemma 3.1 and the TX -stability of F Ž< g < M . it follows that the term in Ž4.10. with n s 0 multiplying Ž c j y c i . X e i Dj must be in F Ž< g < M .. Hence, X by b j e jqŽ b jq1 . e i Dj g F Ž< g < M .. The proof of part Žb. is similar to part Ža. and is left to the reader. Part Žc. follows immediately from part Žb.. Note that if c j y Ž bi q 1.Ž c j y c i .rŽ bi q b j q 2. s c k then Ž bi q 1.Ž c j y c i . s 0 which contradicts the assumptions. b

THEOREM 4.3.

Let g s X Ž m, 1.w2x where X is of type W, S, H, or K.

Ža. A finite dimensional DistŽTX . uŽ g .-module M is uŽ g .-projecti¨ e if and only if F Ž< g < M . l Ea , X s 0 for all a f GXq and M² Fb", X : p is projecti¨ e for all b g GX". Žb. Let g s SŽ m, 1.. A finite dimensional DistŽ G 0, S . uŽ g .-module M is uŽ g .-projecti¨ e if and only if M² EbX , S : p is projecti¨ e for all b f GSq and M² Fb", S : p is projecti¨ e for all b g GS". Žc. Let g s W Ž m, 1. and m G 2. A finite dimensional DistŽ G 0, W . uŽ g .-module M is uŽ g .-projecti¨ e if and only if M² EbX , W : p is projecti¨ e for all b f GSq, M² Fb", W : p is projecti¨ e for all b g GW", and M² Fbq, W : p is projecti¨ e for all b g GSq. Proof. First observe that Fb", X being one dimensional implies that F Ž< g < M . l Fb", X s 0 if and only if M² Fb , X : p is projective. Therefore, the first statement Ža. can be deduced immediately from the decomposition Ž4.7. and Lemma 3.1 with X s F Ž< g < M . and V s g. er . Ž< < . Let z s X a ŽÝ m rs1 c r X Dr g F g M l Ea , X with z / 0. In order to prove part Žb. it suffices to prove, using the G 0, X -stability, that one can conjugate z to a sum involving a term in some Fb", X. By using the TX -stability of F Ž< g < M . and appealing to Lemma 3.1 it will follow that Fb", X : F Ž< g < M . which contradicts the hypothesis. Moreover, according to the assumption involving EaX , X we can assume that c i / c j for some i, j with i / j. We will prove by induction on m Ž m G 2. that part Žb. holds. For SŽ2, 1. the results follow because the definition of Ea , S becomes vacuous in this

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case. Let m ) 2 and X b y g F Ž< g < M . l Eb , S . If b k s p y 1 for some k then Lemma 4.2Žc. insures that we may choose k such that there exist distinct i, j with c i / c j for i, j / k and thus we may appeal to the induction hypothesis. Next assume that b k - p y 1 for all k and there exist i / j with bi q b j G p y 1. According to Lemma 4.2Žb., X bqŽ b iyŽ py1.. e jqŽŽ py1.y b i . e i ? yX g F Ž < g < M . where yX s y y Ž bi q 1.Ž c j y c i .rŽ bi q b j q 2. ? X e j Dj . Note that we may assume that yX / 0. For if yX s 0 then c r s 0 for r / j and c j Ž b j q 1. s 0 so c j s 0 and y s 0. If b k q bi F p y 2 then use Lemma 4.2Ža. to get X by b j e jqŽ b jq1 . e k Dj g F Ž< g < M . which contradicts the hypothesis. Otherwise, if b k q bi G p y 1 then replace j with k and appeal to Lemma 4.2Žb, c. and the induction hypothesis. Finally, if bi q b j F p y 2 for all i, j we may choose i, j such that c i / c j . From the argument used above we can again reach a contradiction. Hence, F Ž< g < M . l Ea , S s 0 for all a f GSq. In order to prove part Žc. we will again use induction. Let X b y g F Ž< g < M . l Eb , W . For m s 2 if b 1 q b 2 F p y 2 then the argument used in part Žb. leads to a contradiction. If b 1 q b 2 G p y 1 and X b y g SŽ2, 1. then X b y g Fbq, S l F Ž< g < M .. On the other hand, if X b y f SŽ2, 1. then by X X b Lemma 4.2Žb. X y g FbqX , W l F Ž< g < M . where b X s b q Ž b 2 y Ž p y 1.. e 1 q Ž p y 1. y b 2 . e 2 . Now assume that the result holds for m y 1 variables where m ) 2. If X b y g SŽ m, 1. then the result holds by part Žb.. If b k s p y 1 then the same argument in part Žb. can be used to reduce to m y 1 variables. Now assume that b k - p y 1 for all k and there exists i / j such that bi q b j G p y 1. Once again we can apply Lemma 4.2Žb., but observe that yX s y y Ž bi q 1.Ž c j y c i .rŽ bi q b j q 2. ? X e j Dj / 0 since X b y f SŽ m, 1.; so again we can reduce it to m y 1 variables. The case when bi q b j F p y 2 for all i, j can be handled as in part Žb.. EXAMPLE 4.4. Let g s W Ž1, 1. s ² ey1 , e0 , e1 , . . . , e py2 : with Lie relations

w ei , e j x s

½

Ž i y j . e iqj

for y1 F i q j F p y 2,

0

else, e wj px

and pth power operations s 0 for j / 0 and e0p s e0 . Set T s TW Ž1, 1. . From Theorem 4.3Ža. we obtain a statement like the result for classical Lie algebras wCPSx: a DistŽT . uŽ g .-module M is projective as a uŽ g .-module if and only if M < ² e j : p is projective for j s y1, 1, 2, . . . , p y 2. Moreover, the same argument as that used in wN3x can be applied to obtain a bound on the complexity for DistŽT . uŽ g .-modules: cg Ž M . F

Ý c² e : Ž M . . j p

j/0

COHOMOLOGY OF LIE ALGEBRAS

881

We refer the reader to wFP5x for the definition and properties of complexity for restricted Lie algebras. EXAMPLE 4.5. Let g s H Ž2, 1. s SŽ2, 1.w2x. In this case since dim k Ea , S s 1 there is a similar projectivity result and complexity bound as in Example 4.4 for DistŽTS . uŽ g .-modules. EXAMPLE 4.6. For a Lie algebras of Cartan type with total rank larger than one, the root spaces need not be one dimensional. For example, consider g s W Ž2, 1. and ² X e 1qe 2 D 2 , X 2 e 1 D 1 : : g 1 . Let x s X e 1qe 2 D 2 q X 2 e 1 D 1. Then t a1 , a 2 ? x s a1 x. By using the same methods in wFP2x one can conclude that there exists an indecomposable DistŽTW . uŽ g .-module M whose support variety is ² x :. This example illustrates the necessity to use ‘‘m-planes’’ to provide sufficient criteria for projectivity. We now give a criterion for a DistŽ GX . uŽ g .-module to be projective over uŽ g .. PROPOSITION 4.7.

Let g s X Ž m, 1.w2x where X is of type W, S, H, or K.

Ža. If M is a DistŽTX . uŽ g .-module which is projecti¨ e upon restriction Ž to u bq. then F Ž< g < M . : F Ž< gy< M .. Žb. If M is a DistŽ GX . uŽ g .-module then M is projecti¨ e when restricted to uŽ g . if and only if M is projecti¨ e when restricted to uŽ bq. . Proof. Note that F Ž< g < M . is a TX -invariant closed conical subvariety of g. Consider the scalar homomorphism f : kU ª TX . Every element x g g can be written as x s Ý x i with x i g g i . Here we have f Ž t . x i s t iy1 x i . If 0 / x g F Ž< g < M . and r is the largest integer such that x r / 0, by Lemma 3.1 we have x r g F Ž< g < M .. Since M is projective over uŽ bq. we have x r g gyl F Ž< g < M . s F Ž< gy< M .. In order to prove part Žb., we only need to show that M is projective over uŽ g . if it is projective over uŽ bq. since all projective uŽ g .-modules are projective over uŽ bq. . Suppose that M is not projective over uŽ g .. Then F Ž< g < M . /  04 . By Ža., we have F Ž< g < M . : F Ž< gy < M .. The DistŽ GX . uŽ g .-structure on M implies that F Ž< g < M . is GX -invariant. Now one can use the description of GX in Section 3 to see that U1 gy ­ gy while G 0 acts on all nonzero elements in gy1 transitively. This shows that F Ž< g < M . cannot be contained in gy.

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5. COMPLEXITY AND SUPPORT VARIETIES FOR DistŽ G . uŽ g .-MODULES Let g s X Ž m, 1.w2x where X is of type W, S, H, or K and let M be a DistŽ GX . uŽ g .-module. From the results in Section 4 the support variety, F Ž< g < M ., will be a GX -stable affine homogeneous variety. Since the mody Ž . qŽ . ules Virr l , Vproj l , L Ž l. have compatible algebraic group actions for all l g X 1ŽT ., we can use this fact to look at the support varieties for these modules. Recall that g s gyr [ ??? [ gy1 [ g 0 [ ??? [ g s is the grading of the Lie algebra and that gys [i- 0 g i , gqs [i- 0 g i , and b "s g 0 [ g ". Let h be the Coxeter number of the reductive group G 0, X . The first result shows that the complexity and support varieties are the same for all the simple modules corresponding to exceptional weights Ž l g L . wSh, N1, Hol1, Hol2x. PROPOSITION 5.1. If g s X Ž m, 1.w2x where X is of type W, S, H, or K, p ) h, and l g L, then c g Ž L Ž l . . s dim  x g g : x w px s 0 4 . Proof. In wSh, N1x all restricted simple modules for X Ž m, 1.w2x of types W, S, and H are described. Holmes wHol1, Hol2x has recently computed all the simple modules for X Ž m, 1.w2x s K Ž2 r q 1, 1.w2x. From this information the dimensions of the simple modules, L Ž l., for l g L are given by a binomial coefficient times a integer prime to p. For p ) h, p does not divide this binomial coefficient so p ¦ dim k L Ž l.. Hence, F Ž< g < L Ž l. . s F Ž< g < k . for all l g L. We should remark that Premet wPx has calculated the nilpotent variety n N s  x g g: x w px s 0 for some n4 and shown that for g s W Ž m, 1., dim N s dim G W . In fact, if one considers z s D 1 q X Ž py1. e 1 D 2 q ??? qX Ž py1. e 1q ??? qŽ py1. e ny 1 Dn then z g N and dim GX s dim GX ? z with GX ? z s N . Since z w px / 0 for g / W Ž1, 1. it follows that not every p-unipotent element has a trivial restriction. Note that this fact is independent of the characteristic of the field and differs considerably from the case of classical Lie algebras because for large primes the nullcone equals the variety of elements with trivial restriction. In general, since N is irreducible and F Ž< g < k . : N we will have for g / W Ž1, 1., c g Ž L Ž l.. - dim GX for l g L. We will now calculate the dimension of the GX -orbit in g of a nonzero elements in gy1 . Consider X Ž m, 1. s W Ž m, 1.. The group G 0, X acts on the set of nonzero elements in gy1 transitively so it suffices to simply take D 1 in gy1 and to consider the U1-orbit of D 1. Since GX s AutŽ AŽ m, 1.. each f g GX is uniquely determined by the values f Ž X e i . s Sf d Ž X e i . in

COHOMOLOGY OF LIE ALGEBRAS

883

AŽ m, 1.. Here f d Ž X e i . is the homogeneous part of degree d and f 1Ž X e i . defines an element in G 0, X ( GLm Ž k .. Observe that f g U1 if and only if f 1Ž X e i . s X e i . Let c s f y id. Then we have c Ž1. s 0 and f g StabU1Ž D 1 . if and only if D 1Ž c Ž X e i .. s 0 for all i. This is equivalent to the statement that X n e 1 does not appear in c Ž X e i . for any n ) 0, or c Ž X e i . g AŽ m y 1, 1. for all i. Therefore, dim StabU1Ž D 1 . s m Ž dim k A Ž m y 1, 1 . y m . s mp my 1 y m2 and dim U1 ? D 1 s dim Ž U1 . y mp my 1 y m2 s m Ž p m y m y 1 . y m Ž p my 1 y m . s m Ž p my 1 Ž p y 1 . y 1 . . Now by taking G 0 into consideration we get dim GX ? D 1 s m Ž p my 1 Ž p y 1 . . . Hence, it follows that c g Ž L Ž l.. G mŽ p my 1 Ž p y 1.. for g s W Ž m, 1. and l g L. The next proposition shows that one can use the properties of the G 0, X -action to compute the support varieties for the induced modules y Ž . Vprog l with l g X 1ŽT .. PROPOSITION 5.2. Let g s X Ž m, 1.w2x where X is of type W, S, or H and p ) 2 h y 2. For l g X 1ŽT . we ha¨ e y c g Ž Vproj Ž l. . s

F Ž < g < VyprojŽ l. . s

½

½

0 m

 04 y

g

for l f L for l g L for l f L for l g L .

Proof. From the Brauer]Humphreys reciprocity law wHolN1, N1x and qŽ . w y Ž . the decomposition of Virr l Sh, N1x, Vproj l is a projective uŽ g .-module if and only if l f L. Moreover, following an argument similar to that in the proof of Proposition 3.3 with bq replaced by by and using the fact

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y Ž . that G 0, X bys by, one can show that Vproj l is a DistŽ G 0, X . uŽ g .-module. y Ž . q Since Vproj l is uŽ b . -projective it follows from Proposition 4.7Ža. that

F Ž < g < VyprojŽ l. . s G 0, X ? F Ž < gy< VyprojŽ l . . s gy for l g L because F Ž< gy < VyprojŽ l . . / 0 and G 0, X acts transitively on gy. We say that a weight l g X 1ŽT . is a Steinberg weight if and only if LŽ l. is a projective uŽ g 0 .-module. Since g 0 is a reductiverclassical Lie algebra g 0 has a triangular decomposition g 0 s ny[ t [ nq where t is a maxi" " " mal torus for both g 0 and g. If we let n " and b " gsg [n gst[n y q then we have a triangular decomposition for g: g s n g [ t [ n g . There qŽ . exists an isomorphism of DistŽ G . uŽ g .-modules; Virr l s L Ž l. for l f L. qŽ . y Since Virr l is projective over uŽ g . it follows from an argument similar to that in the proof of Proposition 4.7Ža. that F Ž < g < Vqirr Ž l . . s F Ž < bq< Vqirr Ž l . . .

Ž 5.1.

This means that computing the support varieties for simple modules corresponding to nonexceptional weights involves looking at the restriction to bq. The next theorem provides some information on support varieties for these modules. In particular, there exists a large linear subvariety contained in the support variety for these modules independent of the given weight. THEOREM 5.3. Let g s X Ž m, 1.w2x where X is of type W, S, H, or K and p ) h. Moreo¨ er, let q s m if x / K Ž2 r q 1, 1. and q s m q 1 if x s K Ž2 r q 1, 1.. Ža.

If M is a DistŽ GX .-module then < F Ž < bq< M . s G 0, X ? F Ž < bq g M ..

Žb. Žc.

If l g X 1ŽT . then F Ž< g 0 < LŽ l. . ; F Ž< g 0 < Vqirr Ž l. .. If m G 2 and l g X 1ŽT . is not a Steinberg weight then g q [ g qq1 [ ??? [ g s ; F Ž < g < Vqirr Ž l. . ; F Ž < g 0 < Vqirr Ž l. . [ gq.

Žd.

If m G 2 and l g X 1ŽT . is a Steinberg weight then g q [ g qq1 [ ??? [ g s : F Ž < g < Vqirr Ž l. . ; gq.

Proof. For part Ža. let G s GX s AutŽ g . ( G 0, X h U and let T be a maximal torus of G 0 s G 0, X with Borel subgroups B0 and B0q coinciding

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with the triangular decomposition g 0 ( ny[ t [ nq. If Bqs B0qh U then w x Bq is a closed subgroup of G with GrBq( G 0rBq 0 . From Jan3, I, 6.12 and Proposition 3.2, if M is a DistŽ G .-module then there exists a first quadrant spectral sequence E2s, t Ž M . s H s GrBq, H t Ž u Ž bq g ., M .

ž

Žy1 .

« H sqt Ž u Ž bq . , M . .

By using this spectral sequence and the fact that GrBq( G 0rB0q we are in a position to apply the argument given in wFP3, Ž1.2.x to conclude that < . Ž< q< . is U-stable so it follows F Ž< bq< M . s G ? F Ž< bq b M . Moreover, F b g M q q that F Ž< b < M . s G 0, X ? F Ž< b g < M .. g. uŽ b y. Ž . qŽ . Ž . Ž . The module Virr l s coind uŽ uŽ b q . L l ( coind uŽ g 0 . L l as a u g 0 -module. Moreover, there is the following direct sum decomposition as uŽ g 0 . modules: q Virr Ž l . ( Ž 1 m L Ž l . . [ Ž nyu Ž ny . m L Ž l . . .

Ž 5.2.

The second part, Žb., follows by applying the fact that the support variety for the direct sum of the two modules is the union of the support varieties for each module wFP2x. From Lemma 3.1 and the fact that the torus TX can be used to describe the gradation on the Lie algebra Žsee Proposition 3.2. we have F Ž < g < Vqirr Ž l. . ; F Ž < g 0 < Vqirr Ž l. . [ gq qŽ . for any l g X 1ŽT .. Moreover, if l is a Steinberg weight then Virr l is a projective module over uŽ g 0 .. It follows from Lemma 3.1 again that F Ž< g < Vqirr Ž l. . ; gq for a Steinberg weight l. Let y g g with y w px s 0. If M is a uŽ² y :. projective module then there exist l s dim k Mrp linearly independent elements m1 , m 2 , . . . , m l g M with y py 1 m1 , y py 1 m 2 , . . . , y py 1 m l are linearly independent. Now observe qŽ . that M s Virr l is a graded module with respect to the grading on g. If y g g q [ g qq1 [ ??? [ g s where q s m for g / K Ž2 r q 1, 1. and q s m q 1 for g s K Ž2 r q 1, 1. then y py 1 M has dimension no greater than dim k LŽ l.. However, for m G 2, dim k Mrp s dim k LŽ l. p my 1 ) dim k LŽ l., thus parts Žc. and Žd. follow.

For the simple Witt algebra W Ž1, 1. we can provide a complete description of the complexity and support varieties for all simple uŽW Ž1, 1.. modules.

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THEOREM 5.4. Let g s W Ž1, 1.. The complexity and support ¨ arieties for the simple uŽ g . modules are gi¨ en by

cg Ž L Ž l. . s

¡p y 1

for l s 0, 1

~

for l / 0, 1,

py2

¢p y 3

for l s

2

py1 2

¡G ? e

for l s 0, 1

~ke

for l / 0, 1,

y1

F Ž < g < L Ž l. . s

py1

1

[ ke2 [ ??? [ ke py2

¢ke [ ke [ ??? [ ke 2

3

py2

for l s

py1 2

py1 2

Proof. If g s W Ž1, 1. then there are p nonisomorphic simple uŽ g . modules indexed by l s 0, 1, . . . , p y 1 wChax. By using the fact qŽ . ² ey1 , e0 , e1 : ( s l Ž2. it follows that Virr l is a projective uŽ² e1 :. module if Ž . Ž Ž and only if l s p y 1 r2 i.e., l s p y 1.r2 is a Steinberg weight.. qŽ . Moreover, if y g gq and l / Ž p y 1.r2 then y py 1 Virr l s 0. Since qŽ . qŽ . dim k Virr l s p, Virr l is not projective over uŽ² y :.. Therefore, gq: F Ž< g < Vqirr Ž l. . for l / Ž p y 1.r2. An argument similar to the one used in Theorem 5.3 shows that l s Ž p y 1.r2 implies that ke2 [ ??? [ ke py2 : Ž< g < Vqi r r Ž l . .. On the other hand, if z s D 1 then z g F Ž< g < k . : N and G ? zs N . Therefore, in the case when g s W Ž1, 1., F Ž< g < k . s N . The statement of the theorem now follows from Proposition 5.1 and the fact that l s 0, 1 are the exception weights. In wN1, HolN2x it is shown that the Cartan invariants of uŽ g . can be computed once one knows the Cartan invariants for uŽ g 0 .. For g s W Ž2, 1. Žresp., H Ž2, 1.. we have g 0 s g l Ž2. Žresp., g 0 s s l Ž2... In this situation, for all l g X 1ŽT ., F Ž< g 0 < LŽ l . . s F Ž< g 0 < Vqirr Ž l. .. One might ask if equality holds in general for all Lie algebra of Cartan type. Also, if g s H Ž2, 1. and qŽ . l s 0 one can explicitly calculate the support variety of Virr 0 by using the < . module action given in wN1x. This yields F Ž< g < Vqirr Ž0. . s G 0 ? F Ž< bq g k . These examples seem to suggest how the support varieties for simple modules of uŽ g . might depend on having to compute the support varieties for simple uŽ g 0 . modules.

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