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Representation Theory of the Witt Algebra
203, 447]469 Ž1998. JA977343
JOURNAL OF ALGEBRA ARTICLE NO.
Representation Theory of the Witt Algebra Jorg ¨ FeldvossU Mathematisches Seminar der Uni¨ ersitat, ¨ Bundesstrasse 55, D-20146, Hamburg, Germany
and Daniel K. Nakano† Department of Mathematics and Statistics, Utah State Uni¨ ersity, Logan, Utah 84322 Communicated by Georgia Benkart Received April 24, 1997
1. INTRODUCTION 1.1. Let g be a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic p ) 0. Every simple g-module is finite-dimensional and therefore admits a character x g gU Žsee wSF, Theorem 5.2.5x.. Conversely, for any such linear form x g gU , there exists a finite-dimensional algebra uŽ g, x . which is a quotient of the universal enveloping algebra UŽ g . and whose simple modules are exactly the simple modules for g with character x . Because of this fact, these algebras play an important role in studying the representations of g. In particular, the restricted g-representations coincide with the uŽ g, 0.-modules. Every finite-dimensional restricted simple Lie algebra over an algebraically closed field of characteristic p ) 7 is either classical or of Cartan type wBWx. For a classical Lie algebra g, the representation theory of uŽ g, x . was first studied by Kac and Weisfeiler wWD, KWx, and further * Research of the first author was supported by a Habilitandenstipendium of the DFG. E-mail: [email protected]. † Research of the second author was supported in part by NSF Grant DMS-9500715 and Utah State University Grant 84986. E-mail: [email protected]. 447 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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developed by Friedlander and Parshall wFP1, FP2x. Recently, Premet wP2x proved a conjecture of Kac and Weisfeilder on the dimension of the simple uŽ g, x .-modules by using support varieties. On the other hand, for restricted Lie algebras of Cartan type a classification of the simple g-modules is given for g s W Ž1, 1., H Ž2, 1., and W Ž2, 1. wC, Ko1, Ko2x. Unlike the situation for classical Lie algebras, it is not even known in general how to compute the number of simple uŽ g, x .-modules. In this paper we study the representation and cohomology theory of uŽ g, x . for the toral rank one Cartan type Lie algebra g s W Ž1, 1.. The dimensions of the projective indecomposable uŽ g, x .-modules are determined as well as the multiplicities of their composition factors. We also compute the orbits of characters under the action of the automorphism group. This provides a complete picture of the isomorphism classes of the reduced enveloping algebras. Moreover, support varieties for simple uŽ g, x .-modules are computed. We hope our results lend some insight into the general picture of the representation theory of uŽ g, x . for other restricted Lie algebras of Cartan type. 1.2. Notation. Let k be an algebraically closed field of characteristic p ) 3, and let W [ W Ž1, 1. be the p-dimensional Z-graded restricted simple Witt algebra with basis e i < y 1 F i F p y 24 . The Lie bracket and p-mapping are given by
w e i , e j x s Ž j y i . ? e iqj
; y 1 F i , j F p y 2,
where e iqj [ 0 if i q j f y1, . . . , p y 24 and e wi px [ d i0 ? e i for any y1 F i F p y 2. It will be useful to consider the p-subalgebras Wi [ ke i [ ??? [ ke py2
; y 1 F i F p y 2.
In this way W is a Z-filtered Lie algebra. The subalgebra W0 is strongly solvable with maximal torus ke0 and p-unipotent radical W1. Moreover, Wi is p-unipotent for every 1 F i F p y 2 because W1w px s 0. We will also need the p-subalgebra s [ key1 [ ke0 [ ke1 which is isomorphic to s l Ž2. with its Borel subalgebras b "[ ke0 [ ke " 1. Let x g W U . The centralizer of x in W is defined as W x s x g W < x Ž w x, W x . s 0 4 . It is easy to verify that W x is a p-subalgebra of W. In order to classify the simple uŽW, x .-modules, Strade wSx defines the height r Ž x . of x by r Ž x . [ min i < y 1 F i F p y 2 and x < W i s 0 4
REPRESENTATIONS OF THE WITT ALGEBRA
449
if x Ž e py 2 . s 0 and r Ž x . [ p y 1 if x Ž e py2 . / 0. The height of x is a very useful invariant for studying the representation and cohomology theory of uŽW, x .. 1.3. Simple W-Modules. We quickly review Chang’s description of the simple uŽW, x .-modules Žsee also wSx.. First consider x g W U with r Ž x . s y1, 0, 1. Observe that r Ž x . s y1 if and only if x s 0. In all these cases x vanishes on the p-unipotent radical W1 of W0 . In particular, W1 acts trivially on every simple uŽW0 , x .-module. Hence the isomorphism classes of simple uŽW0 , x .-modules are in one-to-one correspondence with the weights L Ž x . [ l g k < l p y l s x Ž e0 .
p
4.
If r Ž x . s y1, 0, then LŽ x . coincides with the prime field of k. The uŽW0 , x .-module corresponding to the weight l is given by kl [ k with action e0 ? 1 [ l ? 1,
e i ? 1 [ 0 ;1 F i F p y 2.
As in the case of Lie algebras of classical type Žsee wFP1, Proposition 1.5x., every simple uŽW, x .-module is a homomorphic image of a x-reduced Verma module , x. V Ž l , x . [ Ind uŽW uŽW 0 , x . kl s u Ž W , x . muŽW 0 , x . kl
for some l g LŽ x .. Except for the restricted case all reduced Verma modules are simple, and the Žunique. maximal submodules of the restricted Verma modules can be determined by a straightforward computation Žcf. wFe1, III.3.2x.. With the exception of V Ž0, x . ( V Ž p y 1, x . for r Ž x . s 0 Žsee wC, Hilfssatz 7x., the reduced Verma modules are isomorphic if and only if the corresponding weights coincide. Hence the isomorphism classes of simple uŽW, x .-modules are in one-to-one correspondence with LŽ x . unless r Ž x . s 0. THEOREM A wC, Hauptsatz 2X x.
Let x g W U with r Ž x . s y1, 0, 1.
Ži. If r Ž x . s y1, then there are p isomorphism classes of simple Ž u W, x .-modules. These modules are represented by the one-dimensional tri¨ ial W-module, the Ž p y 1.-dimensional module V Ž p y 1, x .r Soc uŽW , x .V Ž p y 1, x ., and the p-dimensional modules V Ž l, x . for l s 1, 2, . . . , p y 2. Žii. If r Ž x . s 0, then there are p y 1 isomorphism classes of simple uŽW, x .-modules each of dimension p and represented by V Ž l, x . for l s 0, 1, . . . , p y 2.
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Žiii. If r Ž x . s 1, then there are p isomorphism classes of simple uŽW, x .-modules each of dimension p and represented by V Ž l, x . for l g LŽ x .. The reader should be made aware that the statement that there are p isomorphism classes of uŽW, x .-modules when r Ž x . s 0 given in wS, p. 601x is not correct. The simple uŽW, x .-modules in the case of r Ž x . s y1, 0, 1 will be denoted by S Ž l. if l is the eigenvalue of e0 on SŽ l.W1 . For the other cases when 1 - r Ž x . - p y 1, the simple uŽW, x .-modules are given by the following result. THEOREM B wC, Hauptsatz 1; Sx. Let x g W U with 1 - r Ž x . - p y 1. Then up to isomorphism there is a unique simple uŽW, x .-module S represented by u Ž W , x . muŽW s , x . kx , where s [ w r Ž x .r2x and kx is the one-dimensional simple uŽWx , x .-module. In particular, dim k S s p sq1. Finally, in the case r Ž x . s p y 1, every simple uŽW, x .-module is also a simple uŽW0 , x .-module and thus has dimension p py 1r2 wC, Satz 3, 5; S, pp. 601]602x. Moreover, Chang determined the number of isomorphism classes of simple uŽW, x .-modules in this case wC, Hauptsatz 3X x Žsee also wS, pp. 602]604x.. THEOREM C. sional.
Let x g W U with r Ž x . s p y 1. Then W x is one-dimen-
Ži. If W x is a torus, then there are p isomorphism classes of simple uŽW, x .-modules each of dimension p py 1r2 . Žii. If W x is p-unipotent, then there are p y 1 isomorphism classes of simple uŽW, x .-modules each of dimension p py 1r2 . It follows from the proof of the corollary on p. 604 in wSx that dim kW x s 1. Since W x is a p-subalgebra of W, it is either a torus or p-unipotent. A direct computation shows that W x s ke with e f W0 Žsee wG, Lemma 4.14Ž3.x.. One can now apply wP1, Lemma 4x with another straightforward computation Žsee the proof of wG, Lemma 4.15x. to prove that W x is p-unipotent if and only if there exists an automorphism h of W such that x Žh Ž e i .. s 0 for y1 F i F p y 3 and x Žh Ž e py2 .. / 0. In light of wC, Hauptsatz 3X x, the latter statement holds if and only if there are p y 1 isomorphism classes of simple uŽW, x .-modules.
REPRESENTATIONS OF THE WITT ALGEBRA
451
2. PROJECTIVE MODULES AND CARTAN INVARIANTS This section will be devoted to determining the projective indecomposable modules and to calculating their composition factors for the algebras uŽW, x .. 2.1. The projective indecomposable modules of a finite-dimensional associative k-algebra are in one-to-one correspondence with the simple modules via the map P Ž S . ª P Ž S .rRad P Ž S . ( S. The map P Ž S . ª P Ž S .rRad P Ž S . ( S is essential, so P Ž S . is the projective cover of S. If S is a simple module, then w M : S x will denote the number of times S appears as a composition factor of M. Moreover, if L is a set of representatives for the simple modules we will let w M x s ÝS g L w M : S xw S x denote the formal sum of composition factors of M in the Grothendieck group. In particular, w P Ž S . : SX x for S, SX g L are called Cartan in¨ ariants. In the case r Ž x . s y1, 0, 1 we will denote by P Ž l. the projective cover of the simple uŽW, x .-module SŽ l. for any l g LŽ x .. 2.2. w r Ž x . s y1x. For g a finite-dimensional restricted simple Lie algebra of Cartan type the projective indecomposable modules and Cartan invariants for uŽ g, 0. can be determined up to knowing information given by the Lusztig conjecture Žsee wN1, HN1, HN2x.. In particular, it was shown that uŽ g, 0. has one block. Since r Ž x . s y1 is precisely the case when x s 0 we have by wN1, Proposition 3.3.1x. THEOREM A. The dimensions of the projecti¨ e indecomposable uŽW, 0.modules and their composition factors are gi¨ en as follows: Ži. dim k P Ž l. s 2 p py 2 for l s 0, p y 1, and dim k P Ž l. s p py 2 for l / 0, p y 1. py4 Ž .x. Žii. w P Ž l.x s 4 p py4 w S Ž0.x q ŽÝmpy2 S m q 4 p py4 w S Ž p y s1 2 p 1.x for l s 0, p y 1. 2 py4 w Ž .x. Žiii. w P Ž l.x s 2 p py 4 w SŽ0.x q ŽÝmpy S m q 2 p py 4 w SŽ p y 1.x s1 p for l / 0, p y 1. In particular, uŽW, 0. has one block. The simple uŽW, 0.-modules SŽ l. have simple socles upon restriction to uŽ s, 0.. Let Ps Ž l. be the projective cover of the socle of S Ž l.< s . The next result provides an explicit realization of the projective indecomposable uŽW, 0.-modules. THEOREM B. The projecti¨ e indecomposable uŽW, 0.-modules can be realized as follows: Ži. Žii.
, 0. uŽW , 0. P Ž l. ( Coind uŽW uŽ b y , 0. kl ( Ind uŽ b y , 0. kl q1 for l / 0, p y 1. uŽW , 0. P Ž l. ( Ind uŽ s , 0. Ps Ž l. for l s 0, p y 1.
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FELDVOSS AND NAKANO
Proof. The first isomorphism in Ži. follows by wN1, Theorem 1.3.6x, because V Ž l, 0. is a simple uŽW, 0.-module for l / 0, p y 1. Moreover, the second isomorphism is an immediate consequence of wFe2, Corollary 1.2x. Now let l s 0, p y 1. According to Theorem 2.2A, dim k P Ž l. s , 0. Ž . Ž . 2 p py 2 . Furthermore, Ind uŽW uŽ s , 0. Ps l is a projective u W, 0 -module of py 2 dimension 2 p because Ps Ž l. is a projective uŽ s, 0.-module of dimen, 0. Ž . sion 2 p. The result will follow if Ind uŽW uŽ s , 0. Ps l has simple head isomorphic to SŽ l.. By Frobenius reciprocity and the knowledge of the composition factors of SŽ l. as a uŽ s, 0.-module, we have for l s 0, p y 1, , 0. Hom uŽW , 0. Ž Ind uŽW uŽ s , 0. Ps Ž l . , S Ž m . . ( Hom uŽ s , 0. Ž Ps Ž l . , S Ž m . .
(
½
k, 0,
msl m / l.
, 0. Ž . Ž . Ž . Hence, Ind uŽW uŽ s , 0. Ps l has simple head isomorphic to S l , and part ii follows.
2.3. w r Ž x . s 0x. We will now show that the representation theory of uŽW, x . for r Ž x . s 0 is somewhat similar to that of the case when r Ž x . s y1. THEOREM. If r Ž x . s 0, then the dimensions of the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. dim k P Ž0. s 2 p py 2 and dim k P Ž l. s p py 2 for all l / 0. 2 py4 w Ž .x Žb. w P Ž0.x s 4 p py 4 w SŽ0.x q Ýmpy S m . s1 2 p py 4 py 2 py4 Žc. w P Ž l.x s 2 p w SŽ0.x q Ýms1 p w SŽ m .x for l / 0. In particular, uŽW, x . has one block. Proof. Let l g 0, 1, 2, . . . , p y 24 and let Ps Ž l. be the projective cover of SŽ l.< s Žwhich still remains simple over s .. Note that x Ž ey1 . / 0 and x Ž e0 . s 0 s x Ž e1 ., so that x < s is a regular nilpotent character for s. This means that for any m g 0, 1, . . . , p y 14 , SŽ m .< s is simple and S Ž m .< s ( SŽ mX .< s where mX s p y 1 y m wFP1, Proposition 2.3x. We remark that the isomorphism between s and s l 2 is given by F ¬ ey1 , H ¬ 2 e0 , and E ¬ ey1 . , x. Ž . Ž . Consider the module QŽ l. [ Ind uŽW uŽ s , x . Ps l . Since Ps l is a projective uŽ s, x .-module, it follows that QŽ l. is a projective uŽW, x .-module. Furthermore, , x. Hom uŽW , x . Ž Ind uŽW uŽ s , x . Ps Ž l . , S Ž m . . ( Hom uŽ s , x . Ž Ps Ž l . , S Ž m . .
(
½
k, 0,
m s l , lX otherwise.
453
REPRESENTATIONS OF THE WITT ALGEBRA
The preceding calculation shows that
¡P Ž 0. ,
Q Ž l. (
~P
ž
py1 2
/
ls0 py1 ls 2 py1 l / 0, . 2
,
¢P Ž l. [ P Ž l . , X
Ž 2.3.1.
For l / Ž p y 1.r2, dim k Ps Ž l. s 2 p and dim k Ps Ž p y 1.r2 s p wFP1, Proposition 2.3x. Therefore, we have verified part Ža. for l s 0, Ž p y 1.r2. Now observe that x < W1 s 0, so uŽW1 , x . is a local algebra. This implies that p dim kW1 s p py 2 divides the dimension of every projective uŽW, x .module. Combining this fact with Ž2.3.1. shows that dim k P Ž l. s p py 2 for l / 0, Ž p y 1.r2. If l s 0, Ž p y 1.r2, then P Ž l. ( QŽ l.. By using transitivity and exactness of induction along with the fact that w uŽW0 , x . muŽ bq , x . k py1r2 x s Ým g LŽ x . p py 4 w km x we have py1 py1 , x. P s Ind uŽW uŽ s , x . Ps 2 2
ž
/
ž
/
, x. s Ind uŽW k uŽ b q , x . py 1r2
s u Ž W , x . muŽW 0 , x . Ž u Ž W0 , x . muŽ bq , x . k py1r2 . s
Ý
m gL Ž x .
p py 4 u Ž W , x . muŽW 0 , x . km py2
s 2 p py 4 S Ž 0 . q
Ý
p py4 S Ž m . .
ms1 uŽW , x . , x. Ž . w x wFP1, Proposition Moreover, w P Ž0.x s wInd uŽW uŽ s , x . Ps 0 s 2 Ind uŽ b q , x . k 0 x 2.3 . The argument above can now be used to show that
py2
P Ž 0 . s 4 p py 4 S Ž 0 . q
Ý
2 p py4 S Ž m . .
ms1
This completes the computation of w P Ž0.x and w P Ž p y 1.r2x. Now suppose that l / 0, p y 1r2. Since uŽW, x . is a symmetric algebra, we have for m s 0, p y 1r2, P Ž l. : S Ž m . s P Ž m . : S Ž l.
¡2 p ~ s ¢p
py 4
py 4
,
,
ms0 py1 ms . 2
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FELDVOSS AND NAKANO
Therefore, in order to finish the proof, it suffices to show that w P Ž l. : S Ž m .x s p py 4 for l, m / 0, p y 1r2. For l g LŽ0. with l / 0, p y 1, Let S Ž l. [ V Ž l, 0. be the simple uŽW, 0.-module corresponding to the weight l. Furthermore, let P Ž l. be the projective cover of S Ž l.. Suppose that P Ž l. m P Ž m .U ( P Ž l. m P Ž m .U as uŽW, 0.-modules for l, m / 0, p y 1. Then by Theorem 2.2A, p py 4 s P Ž l . : S Ž m . s dim k Hom uŽW , 0. Ž P Ž m . , P Ž l . . s dim k Hom uŽW , 0. Ž k, P Ž l . m P Ž m . s dim k Hom uŽW , 0. Ž k, P Ž l . m P Ž m .
U
U
.
.
s P Ž l. : S Ž m . . This shows that our problem reduces to proving that P Ž l. m P Ž m .U ( P Ž l. m P Ž m .U as uŽW, 0.-modules for all l, m / 0, p y 1. , 0. Ž Ž .U . by Theorem Observe that P Ž l. m P Ž m .U ( Ind uŽW uŽ b y , 0. kl q1 m P m y 2.2BŽi.. Let i be the injective uŽ b , 0. homomorphism from klq1 m P Ž m .U , 0. Ž Ž .U .. We will construct a uŽ by, 0. homomorphism ¨ Ind uŽW uŽ b y , 0. kl q1 m P m U s from klq1 m P Ž m . to P Ž l. m P Ž m .U in the following way. Since Soc uŽ by , 0. S Ž l. ( klq1 , there exists an injective uŽ by, 0. homomorphism r : klq1 ¨ S Ž l.. Let s 1 be the uŽ by, 0. homomorphism defined by r m id from klq1 m P Ž m .U to S Ž l. m P Ž m .U . Next we have S Ž l.< W 0 ( SŽ l.< W 0 and P Ž m .< W 0 ( P Ž m .< W 0 . This implies that for any simple uŽW, 0.-module L, U
U
Hom uŽW , 0. Ž S Ž l . m P Ž m . , L . ( Hom uŽW 0 , 0. Ž l m P Ž m . , L . U
( Hom uŽW 0 , 0. Ž l m P Ž m . , L . U
( Hom uŽW , 0. Ž S Ž l . m P Ž m . , L . . Therefore, S Ž l. m P Ž m .U ( SŽ l. m P Ž m .U because both modules are projective with the same projective summands appearing Žusing the computation above.. Let s 2 denote the uŽW, 0. isomorphism between these modules. Let a be the uŽW, x . homomorphism from P Ž l. ¸ P Ž l.rRadŽ P Ž l.. ( S Ž l.. Since S Ž l. is a projective uŽ by, x .-module, there exists a uŽ by, x . monomorphism b from S Ž l. into P Ž l. such that a ( b s id. Set s 3 [ b m id. Finally, let s [ s 3 ( s 2 ( s 1. The universal property of induction shows , 0. Ž that there exists a uŽW, 0. homomorphism C from Ind uŽW uŽ b y , 0. kl q1 m U U P Ž m . . to P Ž l. m P Ž m . . The dimensions of the two modules are the same. Therefore, to show that they are isomorphic, it suffices to prove that C is surjective.
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REPRESENTATIONS OF THE WITT ALGEBRA
First we claim that Im s 3 generates P Ž l. m P Ž m .U as a uŽW, 0.-module. This will follow by showing that Im b generates P Ž l. as a uŽW, x .module. Consider b ( r : klq1 ¨ P Ž l. as a uŽ t, 0.-module homomorphisms. Let iX be the injective uŽ t, 0. homomorphism from klq1 to 0 , 0. Ž . Ind uŽW uŽ t , 0. kl q1 . The universal property of induction yields a u W0 , 0 -map d 0 , 0. k Ž . Ž . Ž l. as a from Ind uŽW to P l . Since P l is the projective cover of S uŽ t , 0. l q1 uŽW0 , 0.-module with S Ž l.rRad SŽ l. ( klq1 , it follows that a : P Ž l. ª SŽ l. is an essential epimorphism. Now observe that b ( r s d ( iX , thus a ( b ( r s a ( d ( iX . But, a ( b s id, so r s a ( d ( iX . Suppose that Im b does not generate P Ž l. as a uŽW0 , 0.-module. Therefore, d cannot be surjective which implies that Im a ( d ( iX : Rad SŽ l.. This is a contradiction because Im r is not contained in Rad S Ž l.. Hence, Im b generates P Ž l. as a uŽW0 , 0.-module, and also as a uŽW, x .-module. To complete the proof observe that S Ž l. is generated by Im r as a Ž u W, 0.-module. Therefore, S Ž l. m P Ž m .U is generated by Im s 1 as a uŽW, 0.-module. Since s 2 is an isomorphism and Im s 3 generates P Ž l. m P Ž m .U as a uŽW, 0.-module, it follows that Im s generates P Ž l. m P Ž m .U as a uŽW, 0.-module. Hence, C is surjective. 2.4. w r Ž x . s 1x. We will show that for r Ž x . s 1 all the projective indecomposable uŽW, x .-modules have the same dimension and the same composition factors. THEOREM. If r Ž x . s 1, then the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. sp
py 2
.
, x. Ž . Ž . P Ž l. ( Ind uŽW uŽ b q , x . kl for e¨ ery l g L x . In particular, dim k P l
Žb. w P Ž l.x s Ým g LŽ x . p py 4 w SŽ m .x for e¨ ery l g LŽ x .. In particular, uŽW, x . has one block. Proof. Observe that x Ž e0 . / 0 and x Ž e1 . s 0 so that uŽ s, x . is semisimple. This means that there are p pairwise non-isomorphic simple uŽ s, x .-modules given by SŽ l.< s , for l g LŽ x . wFP1, Corollary 2.2x. By transitivity of induction and Frobenius reciprocity we have , x. Hom uŽW , x . Ind uŽW uŽ b q , x . kl , S Ž m .
ž
/
, x. uŽ s , x . ( Hom uŽW , x . Ind uŽW uŽ s , x . Ind uŽ b q , x . kl , S Ž m .
ž
ž
( Hom uŽ s , x . Ž S Ž l . , S Ž m . . (
½
k, 0,
msl m / l.
/
/
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, x. uŽW , x . Ž . Therefore, Ind uŽW uŽ b q , x . kl has a simple head S l . Moreover, Ind uŽ b q , x . kl is uŽ s , x . projective as a uŽW, x .-module because Ind uŽ bq , x . kl is projective as a uŽ s, x .-module and induction takes projective modules to projective mod, x. ules. Hence, P Ž l. ( Ind uŽW uŽ b q , x . kl. In order to prove the second assertion, we can use transitivity and exactness of induction to see that
P Ž l . s u Ž W , x . muŽ bq , x . kl s u Ž W , x . muŽW 0 , x . Ž u Ž W0 , x . muŽ bq , x . kl . s
Ý
p py 4 u Ž W , x . muŽW 0 , x . km
Ý
p py 4 S Ž m . .
m gL Ž x .
s
m gL Ž x .
Hence, the second assertion holds and the above formula shows that uŽW, x . has one block. 2.5. w1 - r Ž x . - p y 1x. Let S be the unique simple uŽW, x .-module given in Theorem 1.3B, and let P be its projective cover. THEOREM. If 1 - r Ž x . - p y 1, then the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. dim k P s p py sy1 and w P x s p py 2 sy2 w S x where s [ w r Ž x .r2x. Žb. If r s 2, 3, then P ( uŽW, x . muŽ s , x . St where St is a p-dimensional projecti¨ e simple uŽ s, x .-module. Proof. Since p p s dim k uŽW, x . s Ždim k S . ? Ždim k P ., the first assertion follows immediately from Theorem 1.3B. Furthermore, uŽW, x . muŽ s , x . St is a projective uŽW, x .-module because St is a projective uŽ s, x .-module. Also, if r s 2, 3, then this module has the same dimension as P because St is p-dimensional. Note that for all x < s there always exists such a p-dimensional projective simple module wFP1, Sect. 2x. It would be nice to have an explicit realization of the projective indecomposable uŽW, x .-modules for 3 - r Ž x . - p y 1. 2.6. w r Ž x . s p y 1x. The following theorem is a re-interpretation of an argument given in wFa, p. 83x by using Theorem 1.3C. THEOREM. If r Ž x . s p y 1, then the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. If W x is a torus, then e¨ ery simple uŽW, x .-module is projecti¨ e. In particular, uŽW, x . is semisimple.
REPRESENTATIONS OF THE WITT ALGEBRA
457
Žb. If W x is p-unipotent, then e¨ ery simple uŽW, x .-module with one exception is projecti¨ e. The remaining simple module L has a uniserial projecti¨ e co¨ er of dimension 2 p py 1r2 with two composition factors both isomorphic to L. In particular, uŽW, x . has p y 1 blocks.
3. ORBITS OF CHARACTERS 3.1. Conjugation Results In this section we will compute the orbits of characters under the action of the automorphism group AutŽW .. For classical Lie algebras one can identify the Lie algebra with its linear dual via the Killing form. This identification allows one to obtain a Jordan]Chevalley decomposition for characters. For the nilpotent characters, there are finitely many orbits under the action of the ambient algebraic group provided the characteristic of the field is good. We will see that this is not the case. The number of coadjoint orbits under AutŽW . for the Witt algebra will depend primarily on the parity of the height of the character. We should also remark that if x and x X are characters in the same AutŽW .-orbit, then the algebras uŽW, x . and uŽW, x X . are isomorphic. Therefore, knowledge of conjugacy classes provides information about the module theory of the algebras. Consider the following family of characters x a of height r where a g k. If r is odd and r / 1, let s s Ž r y 1.r2 and define 0, i / r y 1, s x a Ž e i . s 1, Ž 3.1.1. isry1 a, i s s.
½
If r is even or r s 1, then set
x 0 Ž ei . s THEOREM.
x0.
Ža.
½
0, 1,
i/ry1 i s r y 1.
Ž 3.1.2.
Let x g W U , r [ r Ž x ., and G [ AutŽW ..
If r is e¨ en with 0 F r F p y 3 or r s 1, then x is G-conjugate to
Žb. If r is odd with 3 F r F p y 2, then x is G-conjugate to x a for some a g k. Moreo¨ er, x a < a g k 4 forms a complete set of inequi¨ alent representati¨ es for the G-orbits of characters of height r. Proof. The automorphism group G s T h U where T is a one-dimensional torus and U is a unipotent algebraic group wWilx. Recall that G acts on W U by Ž g ? x .Ž w . s x Ž gy1 ? w . where g g G, x g W U , and w g W. Let x g W U , r s r Ž x ., and 0 F r - p y 1. By using the action of the
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FELDVOSS AND NAKANO
torus T we may assume that x Ž e ry1 . s 1, thus reducing the problem to finding u g U such that u ? x s x a for some a g k. This is equivalent to showing that for some a g k there exists u g U such that x aŽ u ? w . s x Ž w . for all w g W. An automorphism u g U arises from an automorphism su g AutŽ k w x xr Ž x p .. where suŽ1. s 1 and suŽ x . s x q b 2 x 2 q b 3 x 3 q ??? qbpy1 x py1 , bj g k for j s 2, 3, . . . , p y 1 Žset b1 s 1.. For any w g W, u ? w s suy1 wsu . One can express py2
u ? ej s ej q
a kj e k
Ý ksjq1
j for j s y1, 0, 1, . . . , r y 2. Moreover, observe that x aŽ u ? e j . s a ry1 qa? j a s . On the other hand,
Ž suy1 e j su . Ž x . s Ž u ? e j . Ž x . py2
ž
s ej q
Ý
/
a kj e k Ž x .
ksjq1 py2
sx
jq1
q
Ý
a kj x kq1 .
ksjq1
Therefore, x aŽ u ? e j . can be found by taking the coefficient of the x r-term in suy1 e j suŽ x . and adding it to a times the coefficient of the x sq 1-term. One can prove by induction that
Ž suy1 e j su . Ž x . s Ž suy1 Ž x . .
jq1
= 1 q 2 b 2 suy1 Ž x . q ??? q Ž p y 1 . bpy1 Ž suy1 Ž x . .
ž
py 2
/
Ž 3.1.3. for j s y1, 0, 1, . . . , r y 2. Let a j [ x Ž e j . for j s y1, 0, 1, . . . , r y 2 and suy1 Ž x . s x q c 2 x 2 q ??? qc py1 x py1 Žset c1 s 1.. Now assume that r s 1. Equation Ž3.13. shows that ay1 s 2 b 2 . Therefore, x is G-conjugate to x 0 . To handle both the even and odd cases simultaneously we abide by the following conventions Žfor r / 1.. If r Ž x . is even, then set s [ r y 2 and a [ 0. On the other hand, if r Ž x . is odd, then set s [ Ž r y 1.r2. Then by using Ž3.13. there exists f j Ž a, b 2 , . . . , bryŽ jq1. , c 2 , . . . , c ryŽ jq1. . g k for j s y1, 0, . . . , s and f j Ž b 2 , . . . , bryŽ jq1. ,
REPRESENTATIONS OF THE WITT ALGEBRA
459
c 2 , . . . , c ryŽ jq1. . g k for j s s q 1, . . . , r y 2 without constant term such that a j s Ž r y j . bryj q Ž j q 1 . c ryj q f j Ž a, b 2 , . . . , bryŽ jq1. , c 2 , . . . , c ryŽ jq1. . ;
Ž 3.1.4. for j s y1, 0, . . . , s. a j s Ž r y j . bryj q Ž j q 1 . c ryj q f j Ž b 2 , . . . , bryŽ jq1. , c 2 , . . . , c ryŽ jq1. .
Ž 3.1.5. that j s s q 1, . . . , r y 2. Furthermore, by using the fact that suŽ suy1 Ž x .. s x, it follows that there exist g j Ž b 2 , . . . , bjy1 , c 2 , . . . , c jy1 . g k without constant term such that bj s yc j q g n Ž b 2 , . . . , bjy1 , c 2 , . . . , C jy1 . .
Ž 3.1.6.
for j s 2, 3, . . . , p y 1. Suppose that r is even. We can recursively find b 2 , c2 , b 3 , c 3 , . . . , bpy1 , c py 1 satisfying the equations above as long as r y j / j q 1, or j / r y 1r2. This condition is always satisfied because r is even, thus x is G-conjugate to x 0 . On the other hand, suppose that r is odd and r / 1. Then s s Ž r y2 1 . . In particular, by similar reasoning as above we can find b 2 , . . . , bsy1 , c 2 , . . . , c sy 1 satisfying Eq. Ž3.1.5. for a sq1 , a sq2 , . . . , a ry2 . The equation involving a s given in Ž3.1.4. is a s s Ž r y s . brys q Ž s q 1 . c rys q a q h Ž b 2 , . . . , brysy1 , c 2 , . . . , c rysy1 . .
Ž 3.1.7. With an appropriate choice of a g k, this equation can be satisfied by some brys and c rys . One can then recursively find b 2 , . . . , bpy1 , c 2 , . . . , c py1 which satisfy Ž3.1.4.. Hence, x is G-conjugate to x a . Finally suppose that x a is G-conjugate to x aX . One can set x s x aX and use Eqs. Ž3.1.5. and Ž3.1.6.. Since a ry2 s a ry3 s ??? s a sq1 s 0, it follows that bj s c j s 0 for j s 2, 3, . . . , r y s y 1. This will imply that brys s yc rys , and by Ž3.1.7., x aX Ž e s . s a s s a s x aŽ e s .. Hence, x a s x aX and the proof of the theorem is complete. 3.2. Centralizers. Let x g W U . Observe that the definition of r Ž x . implies that Wr Ž x .q1 : W x. The next proposition calculates the centralizers for the canonical orbit representatives of characters as listed in Theorem 3.1.
460
FELDVOSS AND NAKANO
PROPOSITION. Ža. Žb. Žc.
Let 0 / x g W U .
If r Ž x . is e¨ en and 0 F r Ž x . F p y 3, then W x s Wr Ž x .q1. Let x [ x 0 with r s 1. Then W x s ke0 [ W2 . Let a g k, 3 F 4 [ r Ž x a . F p y 2 odd, and s [ r y 1r2. Then W
xa
s
½
k Ž e1 q 12 a ? e2 q 38 a2 ? e 3 . [ W4 ,
ss1
k Ž e2 y a ? e ry1 . [ Wrq1 ,
s ) 1.
1 2
Proof. Ža. If r Ž x . is even, the argument in wS, p. 604x shows that W x s Wr Ž x .q1. For Žb. and Žc., according to the proof of wG, Lemma 4.14.Ž2.x, W x arWrq1 is one-dimensional with basis x s Ý riss j i e i Ž j i s 0 for i - s or i ) r . which satisfies the equations 0 s x a Ž w x, e j x . s
r
Ý j i Ž j y i . xa Ž eiqj . iss
for all y1 F j F s y 1. From the definition of x 0 , it immediately follows that j 1 s 0 if s s 0. Furthermore, by the definition of x a ,
Ž 2 j y s . a ? j sy j q Ž 2 j y r q 1 . ? j ry1yj s 0 for all y1 F j F s y 1. In particular, the first summand is non-zero only if j F 0. Hence,
Ž s q 2 . a ? j sq 1 q Ž r q 1 . ? j r s 0, sa ? j s q Ž r y 1 . ? j ry1 s 0,
Ž 2 j y r q 1 . ? j ry1yj s 0 for all 1 F j F s y 1. If s s 1, the assertion follows from the first two equations. If s G 2, then one can conclude from 3 F r F p y 2 that 2 j y r q 1 k 0 Žmod p . for every 1 F j F s y 1. Then we obtain from the third equation that j j s 0 for all s q 1 F j F r y 2. Since r F p y 2, the first equation yields j r s 0. Finally, the result follows by using the second equation and s s Ž r y 1.r2.
4. SUPPORT VARIETIES 4.1. Let g be a finite-dimensional restricted Lie algebra and x g gU . n Ž . For any two uŽ g, x .-modules M and N, let Ext uŽ g , x . M, N be the nth extension group of M by N. The commutative ring R [ Ext 2uŽ?g , 0.Ž k, k . is
REPRESENTATIONS OF THE WITT ALGEBRA
461
a finitely generated k-algebra. Furthermore, for any finite-dimension ? Ž . is a finitely generated R-module uŽ g, x .-module M, Ext uŽ g , x . M, M wFP1, Sect. 6x. Let < g < M [ SpecŽ RrJM . where JM is the annihilator of R ? Ž . on Ext uŽ g , x . M, M . There exists a natural finite morphism F : < g < k ª g of varieties. The image of < g < M under F is given by
x g g < x w px s 0 and
M < k x is not projective4 j 0 4
wFP1, Theorem 6.4x. The goal of this section will be to compute F Ž< W < M . for simple W-modules M. In particular, this enables us by using cW Ž M . s dim F Ž< W < M . wFP1, Remark 6.3x to determine the complexity cW Ž M . for a simple W-module M. Since many of the simple W-modules are induced, the following proposition will be used extensively. PROPOSITION. Let h be a p-subalgebra of a finite-dimensional restricted Lie algebra g and x g gU . Moreo¨ er, let M be a finite-dimensional uŽ h, x .ˆ [ Ind uŽuŽgh ,, xx .. M. Then module and M Ža. Žb.
F Ž< g < Mˆ . : F Ž< h < M .. F Ž< g < Mˆ . s F Ž< h < Mˆ ..
Proof. Consider the commutative diagram r
? Ž ˆ ˆ. Ext uŽ g , x . M, M
6
i
? ˆ. Ž Ext uŽ h , x . M, M
6
6
S ?Ž h U .
6
S ?Ž g U .
where the right-hand vertical arrow is the isomorphism given by using Frobenius reciprocity. Here, i is the map induced by the inclusion h ¨ g ? Ž ˆ ˆ. and r is composition S ?Ž gU . ª H 2 ?Ž uŽ g ., k . ª Ext uŽ ˆ g , x . M, M . Let IM be the kernel of r and V Ž IMˆ . be the associated variety. Then F Ž< g < Mˆ . s ?Ž U . ? ˆ. Ž V Ž IMˆ .. Now let K M , Mˆ be the annihilator of Ext uŽ h , x . M, M in S h . The commutativity of the diagram shows that iy1 Ž K M , Mˆ . : IMˆ . Therefore, F Ž< g < Mˆ . : V Ž iy1 Ž K M , Mˆ ... Furthermore, by an argument given in wBen, p. 181x, it follows that V Ž iy1 Ž K M , Mˆ . . : F Ž < h < M . l F Ž < h < Mˆ . . Hence, the proof of part Ža. is complete. The inclusions above also show that F Ž< g < Mˆ . : F Ž< h < Mˆ .. By wFP1, Proposition 7.1Ža.x, F Ž< h < Mˆ . : F Ž< g < Mˆ ., thus F Ž< g < Mˆ . s F Ž< h < Mˆ .. A more general statement of part Žb. of the above proposition can be found in wFa, Proposition 3.4x which in view of wFP1, Proposition 7.1Ža., Žb.x
462
FELDVOSS AND NAKANO
is an immediate consequence of our result. Part Ža. is a stronger version of wFa, Corollary 3.5x because there are no additional hypotheses on the Lie algebras g and h Žsee also wFP1, Remark 7.5x.. 4.2. w r Ž x . s y1x. The computation of the support varieties for all simple restricted modules of the Witt algebra was first done in wLN, Theorem 5.4x. This result is given below. THEOREM A. The support ¨ arieties and the complexities of the simple uŽW, 0.-modules are gi¨ en by
¡AutŽ W . ? e
y1
F Ž < W < SŽ l. . s
~W
1
¢W ¡p y 1 2
cW Ž S Ž l . . s
for l s 0, p y 1 py1 for l / 0, , py1 2 py1 for l s 2 for l s 0, p y 1 py1 for l / 0, , py1 2 py1 for l s . 2
~p y 2
¢p y 3
The computation of the support varieties for all restricted Verma modules V Ž l, 0. is provided in the following theorem. This result will be used in the proof of Theorem 4.3. THEOREM B. The support ¨ arieties and the complexities of the restricted Verma modules for uŽW, x . are gi¨ en by
¡W
F Ž < W < V Ž l , 0. . s
~
1
¢W ¡p y 2 2
cW Ž V Ž l , 0 . . s
~
¢p y 3
for l / for l s
py1 2 py1
for l / for l s
2 py1 2 py1 2
Proof. Let x g W2 . Then k Ž 1 m 1l . [ k Ž ey1 m 1l . : V Ž l , 0 .
kx
.
.
REPRESENTATIONS OF THE WITT ALGEBRA
463
Since dim k V Ž l, 0. s p, it follows that V Ž l, 0.< k x cannot be projective. Hence, W2 : F Ž< W < V Ž l, 0. .. Also, note that Proposition 4.1Ža. implies F Ž < W < V Ž l , 0. . : x g W0 < x w px s 0 4 s W1 . Now let x s a e1 q y where 0 / a g k and y g W2 . Since dim k V Ž l, 0. s p, V Ž l, 0. is projective as a uŽ kx, x .-module if and only if x py 1 V Ž l, 0. / 0. By using wFe1, III.3.1x, one can see that 0
V Ž l , 0. s
[
isy Ž py1 .
k Ž eyi y1 m 1l .
is a graded uŽW, 0.-module. Therefore, x py 1 V Ž l, 0. / 0 if and only if e1py 1 V Ž l, 0. / 0. The latter condition holds if and only if l s p y 1r2 Ži.e., V Ž l, 0. is the Steinberg module for s ( s l Ž2... Hence, F Ž< W < V Ž l , 0. . s W1 for l / Ž p y 1.r2, and F Ž< W < V Ž l, 0. . s W2 for l s Ž p y 1.r2. 4.3. w r Ž x . s 0x. We now compute the support varieties of the simple uŽW, x .-modules when r Ž x . s 0. Observe that the result is somewhat similar to the restricted case as the proof is dependent on looking at s ( s l Ž2.. THEOREM. If r Ž x . s 0, then the support ¨ arieties and the complexities of the simple uŽW, x .-modules are gi¨ en by
¡W
F Ž < W < SŽ l . . s
~
1
¢W ¡p y 2 2
cW Ž S Ž l . . s
~
¢p y 3
for l / for l s for l / for l s
py1 2 py1 2 py1 2 py1 2
.
The reader should be made aware that wFa, Theorem 5.1Ž3.x claims that F Ž< W < SŽ l . . s W1 for all l g LŽ x . in the case when r Ž x . s 0. This p 1. < s is isomorphic to the Steinberg module statement is false because S Ž y 2 p 1. < k e1 is projecfor s ( s l Ž2. Žsee wFP1, Proposition 2.3x.. Therefore, S Ž y 2 tive, and thus e1 f F Ž< W < S Ž 2 . .. By the same reasoning as above, the first inclusion in wFa, Theorem 5.1Ž1.x does not hold for r Ž x . s 0. py 1
464
FELDVOSS AND NAKANO
Proof. Observe that SŽ l.< W 0 ( V Ž l, 0.< W 0 for every l g LŽ x .. Therefore, by Proposition 4.1Žb. we have F Ž < W < SŽ l. . s F Ž < W0 < SŽ l . . s F Ž < W0 < V Ž l , 0. . s F Ž < W < V Ž l , 0. . . The result now follows by Theorem 4.2B. 4.4. w r Ž x . s 1x. We will now prove that the support varieties of the simple uŽW, x .-modules for r Ž x . G 1 begin to show some generic behavior. THEOREM. If r Ž x . s 1, then the support ¨ arieties and the complexities of the simple uŽW, x .-modules are gi¨ en by F Ž< W < SŽ l . . s W2 and cW Ž SŽ l.. s p y 3 for all l g LŽ x .. Proof. By the same reasoning as in Theorem 4.2B, W2 : F Ž< W < SŽ l. . : W1. Suppose that x s a e1 q y where 0 / a g k and y g W2 . The modules 0
S Ž l. s
[
y Ž py1 .
k Ž eyi y1 m 1l .
are graded uŽW0 , x .-modules. Since SŽ l. is p-dimensional and graded, it follows that S Ž l.< k x is projective if and only if SŽ l.< k e1 is projective. But x Ž e0 . / 0, so that x < s is regular semisimple and SŽ l.< s is projective. This implies that x f F Ž< W < SŽ l. . and F Ž< W < SŽ l. . s W2 . 4.5. w1 - r Ž x . - p y 1, r Ž x . evenx. The following result is already contained in wG, Theorem 4.16x and also in wFa, Theorem 5.1Ž3.x. We include a proof following the ideas in wGx since it will be used in an essential way in the proof of Theorem 4.6. THEOREM. If 1 - r Ž x . - p y 1 and r Ž x . is e¨ en, then the support ¨ ariety and the complexity of the simple uŽW, x .-module S are gi¨ en by F Ž< W < S . s Wr Ž x .q1 and cW Ž S . s p y r Ž x . y 2.
Proof. Let r [ r Ž x .. Since r is even, s s rr2. The inclusion Wrq1 : F Ž< W < S . was shown in wFa, Theorem 5.1Ž1.x. In order to prove the reverse inclusion, we consider x f Wrq1. By Proposition 4.1Ža., F Ž < W < S . : Ws , and py 2 therefore we may assume that x g Ws . Hence x s Ý ism j i e i with j m / 0 and m G s G 1. Since Wr is a p-ideal of W0 and x < W r s 0, it follows from Engel’s theorem that Wr acts trivially on every simple uŽW0 , x .-module which is therefore a simple uŽW0rWr , x .-module where x Ž e . [ x Ž e . for every e g W0 .
REPRESENTATIONS OF THE WITT ALGEBRA
465
Suppose that m - r y 1. Then 1 F r y 1 y m F p y 2, and furthermore, r y 1 y 2 m k 0 Žmod p . because r is even. This implies that x Žw x, e ry1ym x. s j m ? Ž r y 1 y 2 m. ? x Ž e ry1 . / 0. Consider the p-subalgebra H [ kX [ kY [ kZ of W0rWr where X [ x, Y [ e ry1ym , and Z [ w x, e ry1ym x . Since 1 F m F r y 2 and r G 2, we have w X, Z x s 0, w Y, Z x s 0, w X, Y x s Z, and X w px s Y w px s Z w px s 0. Hence, H is a threedimensional Heisenberg algebra with trivial restriction map. But x Ž Z . / 0, so Žup to isomorphism. there is a unique simple uŽ H , x .-module SH represented by uŽ H , x . muŽ k Y[k Z, x . kx . It is easy to see that SH < k X is projective. Therefore, M0 < k x is projective for every finite-dimensional uŽW0 , x .-module M0 . By wFP1, Proposition 7.1Ža.x, x f F Ž< W < S .. Suppose now that m s r y 1. Consider the p-subalgebra M [ kT [ kX of W0rWr where T [ e0 and X [ x. Since 2 F r F p y 2, we have w T, X x s Ž r y 1. ? X / 0, T w px s T, and X w px s 0. So M is the two-dimensional non-abelian Lie algebra. But x Ž X . s j ry1 ? x Ž e ry1 . / 0, so there exists a simple module of dimension p. Therefore uŽ M , x . is semisimple. Hence as above, M0 < k x is projective for every finite-dimensional uŽW0 , x .-module M0 , thus x f F Ž< W < S .. Finally, suppose that m s r. By wC, Satz 3x there is only one isomorphism class of simple uŽW0 , x .-modules represented by S0 [ uŽW0 , x . muŽW s , x . kx and therefore S ( uŽW, x . muŽW 0 , x . S0 . Since x acts trivially on S0 and r G 2, one can show by induction that n x py 1 ? Ž ey1 m s0 .
s
½
Ž p y 1 . !? 1 m w x, ey1 x 0
py 1
? s0
if n s p y 1 Ž 4.5.1. if 0 F n - p y 1
and every s0 g S0 . From 2 F r F p y 2, we conclude that w x, ey1 x p ? s0 s x Žw x, ey1 x. p ? s0 for ever s0 g S0 and x Žw x, ey1 x. s yj r ? Ž r q 1. ? x Ž e ry1 . / 0. Hence Ž4.5.1. implies that x py 1S ( S0 . Moreover, by a straightforward computation, one can see that S k x ( S0 . Then it follows from wFe2, Propositions 2.6 and 5.1x that S < k x is projective, and thus x f F Ž< W < S .. 4.6. w1 - r Ž x . - p y 1, r Ž x . oddx. The following theorem provides inclusions for the support variety F Ž< W < S .. The result is slightly stronger than the results given in wFa, Theorem 5.1Ž1.x with a more direct proof. THEOREM. If 1 - r Ž x . - p y 1, r Ž x . is odd and S is the simple uŽ g, x .-module, then Wr Ž x .q1 : F Ž < W < S . : a Ž x q y . : a g k, y g ke r 4 [ Wr Ž x .q1 ,
466
FELDVOSS AND NAKANO
where x s e s q Ý ry1 issq1 j i ? e i and the s elements j sq1 , . . . , j ry1 g k are uniquely determined by the s linear equations obtained from x Žw x, e i x. s 0 for 0 F i F s y 1. Moreo¨ er, p y r Ž x . y 2 F cW Ž S . F p y r Ž x . y 1. Proof. Let r [ r Ž x .. The first inclusion is just wFa, Theorem 5.1Ž1.x. In order to prove the second inclusion, we consider x f Wrq1. As in the proof py 2 of Theorem 4.5, we can assume that x g Ws , i.e., x s Ý ism j i ? e i with j m / 0 and m G s G 1. Assume m - r y 1. Then the proof of Theorem 4.5 shows, except when m s Ž r y 1.r2 s s, that x f Ž< W < S .. In the latter case, we may assume that x s e s q Ý rissq1 j i ? e i . It remains to show that if x Žw x, e i x. / 0 for some 0 F i F s y 1, then x f F Ž< W < S .. Suppose that x Žw x, e0 x. / 0. Consider the p-subalgebra M of W0rWr generated by T [ e0 and X [ x. Then T w px s T, and the p-ideal I of M generated by X is abelian and has trivial restriction map because I : WsrWr and s G 1. Since x Žw T, X x. s x Žw e0 , x x. / 0, wSF, Corollary 5.7.6x shows that Žup to isomorphism. there is a unique simple uŽ M , x .-module S M represented by uŽ M , x . muŽ I , x . kx . Since S M m ky x is a restricted kX-module, wFe2, Propositions 2.6 and 5.1x in conjunction with Lemma 2.2x imply that S M < k X is projective which shows as in the proof of Theorem 4.5 that x f F Ž< W < S .. Suppose now that x Žw x, e j x. / 0 for some 1 F j F s y 1 Žif s G 2.. Consider the p-subalgebra N of W0rWr generated by X [ x and Y [ e j. Then N is nilpotent with trivial restriction map. Let I denote the p-ideal of N generated by Y. Since x Žw X, Y x. s x Žw x, e j x. / 0, wSF, Corollary 5.7.6x shows that Žup to isomorphism. there is a unique simple uŽ N , x .module SN represented by uŽ N , x . muŽ I , x . kx . It is easy to see that Ž SN m ky x . k X ( k Ž X py1 m 1x . m 1y x ( X py1 Ž SN m ky x .. Hence we obtain as above that x f F Ž< W < S .. If m s r y 1 or m s r, then one can use the same argument as in the proof of Theorem 4.5 to conclude that x f F Ž< W < S . which finishes the proof of the theorem. The conjugation result in Subsection 3.1 shows that for the odd case it is enough to consider the characters x a for some a g k to calculate the support varieties. We prove for the character of smallest possible odd height that the support variety of the simple module is W x. EXAMPLE. Consider x s x 0 for r [ r Ž x . s 3 Ži.e., s s 1.. Then x Ž e2 . s 1 and x Ž e i . s 0 for every i / 2. According to Proposition 3.2Žc., we
REPRESENTATIONS OF THE WITT ALGEBRA
467
have W x s ke1 [ W4 . Let x [ a e1 q y with a g k and y g W4 . Therefore, n
x ? Ž e0n m 1x . s
ny n Ý Ž y1.
ns0
ž nn / e Ž ad e . n 0
0
ny n
Ž x . m 1x s 0
because Žad e0 . m Ž x . g W1 and x ŽŽad e0 . m Ž x .. s 0 for every non-negative integer m. Moreover, ey1Ž e0py 1 y 1. m 1x g S k x. Hence, dim k S k x G p q 1 which implies that S < k x is not projective. We can conclude that W x : F Ž< W < S .. It remains to show that F Ž< W < S . : W x. According to Theorem 4.6, this will follow by showing that S < k x is projective for x [ e1 q j 3 e3 q y where 0 / j 3 g k and y g W4 . Since S0k e 0 s k Ž e0py 1 y 1., a direct computation shows that S k x ( S0 , thus S < k x is projective. 4.7. w r Ž x . s p y 1x. The final case to consider is when r Ž x . s p y 1. THEOREM. If r Ž x . s p y 1, then the support ¨ arieties of the simple uŽW, x .-modules are gi¨ en as follows. Ža. If W x is a torus, then the support ¨ ariety of any simple uŽW, x .module is 04 . Žb. If W x is p-unipotent, then the support ¨ ariety is 04 for all simple uŽW, x .-modules except for L Ž see Theorem 2.6Žb... Moreo¨ er, F Ž< W < L . s W x. Proof. Part Ža. immediately follows from Theorem 2.6Ža. because in this case uŽW, x . is semisimple. For part Žb., without a loss of generality we may assume that x Ž e i . s 0 for y1 F i F p y 3 and x Ž e py2 . / 0 by using the argument after Theorem 1.3C. By a direct calculation one can show that W x s key1 . Consider the modules M [ uŽW, x . muŽ by , 0. k and N [ uŽW, 0. muŽ by , 0. k. By comparison the by-actions one can see that M < by( N < by. Now suppose that ey1 f F Ž< W < M .. Then N is projective over uŽ key1 , 0., and thus projective over uŽW, 0. by the universal property of induction. Since there is an epimorphism from N onto k, there is an epimorphism from N onto the projective cover P Ž0. of k. In view of Theorem 2.2A, this is a contradiction because dim k P Ž0. s 2 p py 2 ) dim k N. Therefore, ey1 g F Ž< W < M .. Since all simple uŽW, x .-modules except for L are projective, it follows that F Ž
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Furthermore, dim F Ž< W < L . s 1 by Theorem 2.6Žb.. Since L is simple and hence indecomposable, F Ž< W < L . must be a line. Consequently, F Ž< W < L . s key1 . 4.8. Gottman wGx conjectured that for every finite-dimensional restricted Lie algebra g and every x g gU the following holds:
D
FŽ < g < M . s FŽ < g < k . l g x.
M simple
For the case when x s 0, the above statement is always satisfied. Premet wP4x has shown that for a Lie algebra g of a connected reductive algebraic group G with G Ž1. simply connected and p good for the root system of G the above conjecture holds. One main ingredient in his proof is that under the above assumptions g satisfies Richardson’s property which according to wP4, Lemma 3.7x is not the case for the Witt algebra W. Furthermore, he provides a counterexample to the conjecture wP3, 3.6x. Namely, for an appropriate character x of a toral extension of the three-dimensional Heisenberg algebra g, one has
D
FŽ < g < M . FŽ < g < k . l g x.
M simple
The results in this section in conjunction with Proposition 3.2 verify this conjecture in all cases for W except when the character has odd height larger than one. An interesting problem would be to provide conditions on the Lie algebra and the character which would ensure the validity of this conjecture.
ACKNOWLEDGMENTS The first author thanks the Department of Mathematics and Statistics of Utah State University for the hospitality during his visit in Spring 1997 when this paper was written. Moreover, he gratefully acknowledges the financial support by the DFG. The authors thank the referee for several useful comments.
REFERENCES wBenx wBWx wCx
D. J. Benson, ‘‘Representations and Cohomology. II. Cohomology of Groups and Modules,’’ Cambridge Univ. Press, Cambridge, UK, 1991. R. E. Block and R. L. Wilson, Classification of the restricted simple Lie algebras, J. Algebra 114 Ž1988., 115]259. ¨ H.-J. Chang, Uber Wittsche Lie-Ringe, Abh. Math. Sem. Uni¨ . Hamburg 14 Ž1941., 151]184.
REPRESENTATIONS OF THE WITT ALGEBRA
wFax
469
R. Farnsteiner, Representations of blocks associated to induced modules of restricted Lie algebras, Math. Nachr. 179 Ž1996., 57]88. wFe1x J. Feldvoss, ‘‘Homologische Aspekte der Darstellungstheorie modularer Lie-Algebren,’’ Doctoral Dissertation, University of Hamburg, 1989. wFe2x J. Feldvoss, On the cohomology of restricted Lie algebras, Comm. Algebra 19, No. 10 Ž1991., 2865]2906. wFP1x E. M. Friedlander and B. J. Parshall, Modular representations theory of Lie algebras, Amer. J. Math. 110 Ž1988., 1055]1094. wFP2x E. M. Friedlander and B. J. Parshall, Deformations of Lie algebra representations, Amer. J. Math. 112 Ž1990., 375]395. wGx J. E. Gottman, ‘‘Characters on Restricted Lie Algebras,’’ Ph.D. thesis, Northwestern University, 1993. wHN1x R. R. Holmes and D. K. Nakano, Brauer-type reciprocity for a class of graded associative algebras, J. Algebra 144 Ž1991., 117]126. wHN2x R. R. Holmes and D. K. Nakano, Block degeneracy and Cartan invariants for graded Lie algebras of Cartan type, J. Algebra 161 Ž1993., 155]170. wKWx V. Kac and B. Weisfeiler, Coadjoint action of a semi-simple algebraic group and the center of the enveloping algebra in characteristic p, Indag. Math. 38 Ž1976., 136]151. wKo1x N. A. Koreshkov, On irreducible representations of Hamiltonian algebras of dimension p 2 y 2, Iz¨ . Vyssh. Uchebn. Za¨ ed. Mat. 22, No. 10 Ž1978., 37]46. wKo2x N. A. Koreshkov, On irreducible representations of the Lie p-algebra W2 , Iz¨ . Vyssh. Uchebn. Za¨ ed. Mat. 24, No. 4 Ž1980., 39]46. wLNx Z. Lin and D. K. Nakano, Algebraic group actions in the cohomology theory of Lie algebras of Cartan type, J. Algebra 179 Ž1996., 852]888. wN1x D. K. Nakano, Projective modules over Lie algebras of Cartan type, Mem. Amer. Math. Soc. 98, No. 470 Ž1992.. wN2x D. K. Nakano, Representation theory of Lie algebras of Cartan type, in ‘‘Proc. of Conference on MonsterrLie Algebras, Ohio State University,’’ in press. wP1x A. A. Premet, The theorem on restriction of invariants and nilpotent elements in Wn , Math. USSR-Sb. 73 Ž1992., 135]159. wP2x A. A. Premet, Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture, In¨ ent. Math. 121 Ž1995., 79]117. wP3x A. A. Premet, Support varieties of non-restricted modules over Lie algebras of reductive groups, J. London Math. Soc., in press. wP4x A. A. Premet, Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms, preprint, 1996. wSx H. Strade, Representations of the Witt algebra, J. Algebra 49 Ž1977., 595]605. wSFx H. Strade and R. Farnsteiner, ‘‘Modular Lie Algebras and Their representations,’’ Dekker, New York, 1988. wWilx R. L. Wilson, Automorphisms of graded Lie algebras of Cartan type, Comm. Algebra 3, No. 7 Ž1995., 591]613. wWKx B. Yu. Weisfeiler and V. G. Kac, Irreducible representations of Lie p-algebras, Functional Anal. Appl. 5 Ž1971., 111]17.
JOURNAL OF ALGEBRA ARTICLE NO.
Representation Theory of the Witt Algebra Jorg ¨ FeldvossU Mathematisches Seminar der Uni¨ ersitat, ¨ Bundesstrasse 55, D-20146, Hamburg, Germany
and Daniel K. Nakano† Department of Mathematics and Statistics, Utah State Uni¨ ersity, Logan, Utah 84322 Communicated by Georgia Benkart Received April 24, 1997
1. INTRODUCTION 1.1. Let g be a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic p ) 0. Every simple g-module is finite-dimensional and therefore admits a character x g gU Žsee wSF, Theorem 5.2.5x.. Conversely, for any such linear form x g gU , there exists a finite-dimensional algebra uŽ g, x . which is a quotient of the universal enveloping algebra UŽ g . and whose simple modules are exactly the simple modules for g with character x . Because of this fact, these algebras play an important role in studying the representations of g. In particular, the restricted g-representations coincide with the uŽ g, 0.-modules. Every finite-dimensional restricted simple Lie algebra over an algebraically closed field of characteristic p ) 7 is either classical or of Cartan type wBWx. For a classical Lie algebra g, the representation theory of uŽ g, x . was first studied by Kac and Weisfeiler wWD, KWx, and further * Research of the first author was supported by a Habilitandenstipendium of the DFG. E-mail: [email protected]. † Research of the second author was supported in part by NSF Grant DMS-9500715 and Utah State University Grant 84986. E-mail: [email protected]. 447 0021-8693r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.
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developed by Friedlander and Parshall wFP1, FP2x. Recently, Premet wP2x proved a conjecture of Kac and Weisfeilder on the dimension of the simple uŽ g, x .-modules by using support varieties. On the other hand, for restricted Lie algebras of Cartan type a classification of the simple g-modules is given for g s W Ž1, 1., H Ž2, 1., and W Ž2, 1. wC, Ko1, Ko2x. Unlike the situation for classical Lie algebras, it is not even known in general how to compute the number of simple uŽ g, x .-modules. In this paper we study the representation and cohomology theory of uŽ g, x . for the toral rank one Cartan type Lie algebra g s W Ž1, 1.. The dimensions of the projective indecomposable uŽ g, x .-modules are determined as well as the multiplicities of their composition factors. We also compute the orbits of characters under the action of the automorphism group. This provides a complete picture of the isomorphism classes of the reduced enveloping algebras. Moreover, support varieties for simple uŽ g, x .-modules are computed. We hope our results lend some insight into the general picture of the representation theory of uŽ g, x . for other restricted Lie algebras of Cartan type. 1.2. Notation. Let k be an algebraically closed field of characteristic p ) 3, and let W [ W Ž1, 1. be the p-dimensional Z-graded restricted simple Witt algebra with basis e i < y 1 F i F p y 24 . The Lie bracket and p-mapping are given by
w e i , e j x s Ž j y i . ? e iqj
; y 1 F i , j F p y 2,
where e iqj [ 0 if i q j f y1, . . . , p y 24 and e wi px [ d i0 ? e i for any y1 F i F p y 2. It will be useful to consider the p-subalgebras Wi [ ke i [ ??? [ ke py2
; y 1 F i F p y 2.
In this way W is a Z-filtered Lie algebra. The subalgebra W0 is strongly solvable with maximal torus ke0 and p-unipotent radical W1. Moreover, Wi is p-unipotent for every 1 F i F p y 2 because W1w px s 0. We will also need the p-subalgebra s [ key1 [ ke0 [ ke1 which is isomorphic to s l Ž2. with its Borel subalgebras b "[ ke0 [ ke " 1. Let x g W U . The centralizer of x in W is defined as W x s x g W < x Ž w x, W x . s 0 4 . It is easy to verify that W x is a p-subalgebra of W. In order to classify the simple uŽW, x .-modules, Strade wSx defines the height r Ž x . of x by r Ž x . [ min i < y 1 F i F p y 2 and x < W i s 0 4
REPRESENTATIONS OF THE WITT ALGEBRA
449
if x Ž e py 2 . s 0 and r Ž x . [ p y 1 if x Ž e py2 . / 0. The height of x is a very useful invariant for studying the representation and cohomology theory of uŽW, x .. 1.3. Simple W-Modules. We quickly review Chang’s description of the simple uŽW, x .-modules Žsee also wSx.. First consider x g W U with r Ž x . s y1, 0, 1. Observe that r Ž x . s y1 if and only if x s 0. In all these cases x vanishes on the p-unipotent radical W1 of W0 . In particular, W1 acts trivially on every simple uŽW0 , x .-module. Hence the isomorphism classes of simple uŽW0 , x .-modules are in one-to-one correspondence with the weights L Ž x . [ l g k < l p y l s x Ž e0 .
p
4.
If r Ž x . s y1, 0, then LŽ x . coincides with the prime field of k. The uŽW0 , x .-module corresponding to the weight l is given by kl [ k with action e0 ? 1 [ l ? 1,
e i ? 1 [ 0 ;1 F i F p y 2.
As in the case of Lie algebras of classical type Žsee wFP1, Proposition 1.5x., every simple uŽW, x .-module is a homomorphic image of a x-reduced Verma module , x. V Ž l , x . [ Ind uŽW uŽW 0 , x . kl s u Ž W , x . muŽW 0 , x . kl
for some l g LŽ x .. Except for the restricted case all reduced Verma modules are simple, and the Žunique. maximal submodules of the restricted Verma modules can be determined by a straightforward computation Žcf. wFe1, III.3.2x.. With the exception of V Ž0, x . ( V Ž p y 1, x . for r Ž x . s 0 Žsee wC, Hilfssatz 7x., the reduced Verma modules are isomorphic if and only if the corresponding weights coincide. Hence the isomorphism classes of simple uŽW, x .-modules are in one-to-one correspondence with LŽ x . unless r Ž x . s 0. THEOREM A wC, Hauptsatz 2X x.
Let x g W U with r Ž x . s y1, 0, 1.
Ži. If r Ž x . s y1, then there are p isomorphism classes of simple Ž u W, x .-modules. These modules are represented by the one-dimensional tri¨ ial W-module, the Ž p y 1.-dimensional module V Ž p y 1, x .r Soc uŽW , x .V Ž p y 1, x ., and the p-dimensional modules V Ž l, x . for l s 1, 2, . . . , p y 2. Žii. If r Ž x . s 0, then there are p y 1 isomorphism classes of simple uŽW, x .-modules each of dimension p and represented by V Ž l, x . for l s 0, 1, . . . , p y 2.
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Žiii. If r Ž x . s 1, then there are p isomorphism classes of simple uŽW, x .-modules each of dimension p and represented by V Ž l, x . for l g LŽ x .. The reader should be made aware that the statement that there are p isomorphism classes of uŽW, x .-modules when r Ž x . s 0 given in wS, p. 601x is not correct. The simple uŽW, x .-modules in the case of r Ž x . s y1, 0, 1 will be denoted by S Ž l. if l is the eigenvalue of e0 on SŽ l.W1 . For the other cases when 1 - r Ž x . - p y 1, the simple uŽW, x .-modules are given by the following result. THEOREM B wC, Hauptsatz 1; Sx. Let x g W U with 1 - r Ž x . - p y 1. Then up to isomorphism there is a unique simple uŽW, x .-module S represented by u Ž W , x . muŽW s , x . kx , where s [ w r Ž x .r2x and kx is the one-dimensional simple uŽWx , x .-module. In particular, dim k S s p sq1. Finally, in the case r Ž x . s p y 1, every simple uŽW, x .-module is also a simple uŽW0 , x .-module and thus has dimension p py 1r2 wC, Satz 3, 5; S, pp. 601]602x. Moreover, Chang determined the number of isomorphism classes of simple uŽW, x .-modules in this case wC, Hauptsatz 3X x Žsee also wS, pp. 602]604x.. THEOREM C. sional.
Let x g W U with r Ž x . s p y 1. Then W x is one-dimen-
Ži. If W x is a torus, then there are p isomorphism classes of simple uŽW, x .-modules each of dimension p py 1r2 . Žii. If W x is p-unipotent, then there are p y 1 isomorphism classes of simple uŽW, x .-modules each of dimension p py 1r2 . It follows from the proof of the corollary on p. 604 in wSx that dim kW x s 1. Since W x is a p-subalgebra of W, it is either a torus or p-unipotent. A direct computation shows that W x s ke with e f W0 Žsee wG, Lemma 4.14Ž3.x.. One can now apply wP1, Lemma 4x with another straightforward computation Žsee the proof of wG, Lemma 4.15x. to prove that W x is p-unipotent if and only if there exists an automorphism h of W such that x Žh Ž e i .. s 0 for y1 F i F p y 3 and x Žh Ž e py2 .. / 0. In light of wC, Hauptsatz 3X x, the latter statement holds if and only if there are p y 1 isomorphism classes of simple uŽW, x .-modules.
REPRESENTATIONS OF THE WITT ALGEBRA
451
2. PROJECTIVE MODULES AND CARTAN INVARIANTS This section will be devoted to determining the projective indecomposable modules and to calculating their composition factors for the algebras uŽW, x .. 2.1. The projective indecomposable modules of a finite-dimensional associative k-algebra are in one-to-one correspondence with the simple modules via the map P Ž S . ª P Ž S .rRad P Ž S . ( S. The map P Ž S . ª P Ž S .rRad P Ž S . ( S is essential, so P Ž S . is the projective cover of S. If S is a simple module, then w M : S x will denote the number of times S appears as a composition factor of M. Moreover, if L is a set of representatives for the simple modules we will let w M x s ÝS g L w M : S xw S x denote the formal sum of composition factors of M in the Grothendieck group. In particular, w P Ž S . : SX x for S, SX g L are called Cartan in¨ ariants. In the case r Ž x . s y1, 0, 1 we will denote by P Ž l. the projective cover of the simple uŽW, x .-module SŽ l. for any l g LŽ x .. 2.2. w r Ž x . s y1x. For g a finite-dimensional restricted simple Lie algebra of Cartan type the projective indecomposable modules and Cartan invariants for uŽ g, 0. can be determined up to knowing information given by the Lusztig conjecture Žsee wN1, HN1, HN2x.. In particular, it was shown that uŽ g, 0. has one block. Since r Ž x . s y1 is precisely the case when x s 0 we have by wN1, Proposition 3.3.1x. THEOREM A. The dimensions of the projecti¨ e indecomposable uŽW, 0.modules and their composition factors are gi¨ en as follows: Ži. dim k P Ž l. s 2 p py 2 for l s 0, p y 1, and dim k P Ž l. s p py 2 for l / 0, p y 1. py4 Ž .x. Žii. w P Ž l.x s 4 p py4 w S Ž0.x q ŽÝmpy2 S m q 4 p py4 w S Ž p y s1 2 p 1.x for l s 0, p y 1. 2 py4 w Ž .x. Žiii. w P Ž l.x s 2 p py 4 w SŽ0.x q ŽÝmpy S m q 2 p py 4 w SŽ p y 1.x s1 p for l / 0, p y 1. In particular, uŽW, 0. has one block. The simple uŽW, 0.-modules SŽ l. have simple socles upon restriction to uŽ s, 0.. Let Ps Ž l. be the projective cover of the socle of S Ž l.< s . The next result provides an explicit realization of the projective indecomposable uŽW, 0.-modules. THEOREM B. The projecti¨ e indecomposable uŽW, 0.-modules can be realized as follows: Ži. Žii.
, 0. uŽW , 0. P Ž l. ( Coind uŽW uŽ b y , 0. kl ( Ind uŽ b y , 0. kl q1 for l / 0, p y 1. uŽW , 0. P Ž l. ( Ind uŽ s , 0. Ps Ž l. for l s 0, p y 1.
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Proof. The first isomorphism in Ži. follows by wN1, Theorem 1.3.6x, because V Ž l, 0. is a simple uŽW, 0.-module for l / 0, p y 1. Moreover, the second isomorphism is an immediate consequence of wFe2, Corollary 1.2x. Now let l s 0, p y 1. According to Theorem 2.2A, dim k P Ž l. s , 0. Ž . Ž . 2 p py 2 . Furthermore, Ind uŽW uŽ s , 0. Ps l is a projective u W, 0 -module of py 2 dimension 2 p because Ps Ž l. is a projective uŽ s, 0.-module of dimen, 0. Ž . sion 2 p. The result will follow if Ind uŽW uŽ s , 0. Ps l has simple head isomorphic to SŽ l.. By Frobenius reciprocity and the knowledge of the composition factors of SŽ l. as a uŽ s, 0.-module, we have for l s 0, p y 1, , 0. Hom uŽW , 0. Ž Ind uŽW uŽ s , 0. Ps Ž l . , S Ž m . . ( Hom uŽ s , 0. Ž Ps Ž l . , S Ž m . .
(
½
k, 0,
msl m / l.
, 0. Ž . Ž . Ž . Hence, Ind uŽW uŽ s , 0. Ps l has simple head isomorphic to S l , and part ii follows.
2.3. w r Ž x . s 0x. We will now show that the representation theory of uŽW, x . for r Ž x . s 0 is somewhat similar to that of the case when r Ž x . s y1. THEOREM. If r Ž x . s 0, then the dimensions of the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. dim k P Ž0. s 2 p py 2 and dim k P Ž l. s p py 2 for all l / 0. 2 py4 w Ž .x Žb. w P Ž0.x s 4 p py 4 w SŽ0.x q Ýmpy S m . s1 2 p py 4 py 2 py4 Žc. w P Ž l.x s 2 p w SŽ0.x q Ýms1 p w SŽ m .x for l / 0. In particular, uŽW, x . has one block. Proof. Let l g 0, 1, 2, . . . , p y 24 and let Ps Ž l. be the projective cover of SŽ l.< s Žwhich still remains simple over s .. Note that x Ž ey1 . / 0 and x Ž e0 . s 0 s x Ž e1 ., so that x < s is a regular nilpotent character for s. This means that for any m g 0, 1, . . . , p y 14 , SŽ m .< s is simple and S Ž m .< s ( SŽ mX .< s where mX s p y 1 y m wFP1, Proposition 2.3x. We remark that the isomorphism between s and s l 2 is given by F ¬ ey1 , H ¬ 2 e0 , and E ¬ ey1 . , x. Ž . Ž . Consider the module QŽ l. [ Ind uŽW uŽ s , x . Ps l . Since Ps l is a projective uŽ s, x .-module, it follows that QŽ l. is a projective uŽW, x .-module. Furthermore, , x. Hom uŽW , x . Ž Ind uŽW uŽ s , x . Ps Ž l . , S Ž m . . ( Hom uŽ s , x . Ž Ps Ž l . , S Ž m . .
(
½
k, 0,
m s l , lX otherwise.
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REPRESENTATIONS OF THE WITT ALGEBRA
The preceding calculation shows that
¡P Ž 0. ,
Q Ž l. (
~P
ž
py1 2
/
ls0 py1 ls 2 py1 l / 0, . 2
,
¢P Ž l. [ P Ž l . , X
Ž 2.3.1.
For l / Ž p y 1.r2, dim k Ps Ž l. s 2 p and dim k Ps Ž p y 1.r2 s p wFP1, Proposition 2.3x. Therefore, we have verified part Ža. for l s 0, Ž p y 1.r2. Now observe that x < W1 s 0, so uŽW1 , x . is a local algebra. This implies that p dim kW1 s p py 2 divides the dimension of every projective uŽW, x .module. Combining this fact with Ž2.3.1. shows that dim k P Ž l. s p py 2 for l / 0, Ž p y 1.r2. If l s 0, Ž p y 1.r2, then P Ž l. ( QŽ l.. By using transitivity and exactness of induction along with the fact that w uŽW0 , x . muŽ bq , x . k py1r2 x s Ým g LŽ x . p py 4 w km x we have py1 py1 , x. P s Ind uŽW uŽ s , x . Ps 2 2
ž
/
ž
/
, x. s Ind uŽW k uŽ b q , x . py 1r2
s u Ž W , x . muŽW 0 , x . Ž u Ž W0 , x . muŽ bq , x . k py1r2 . s
Ý
m gL Ž x .
p py 4 u Ž W , x . muŽW 0 , x . km py2
s 2 p py 4 S Ž 0 . q
Ý
p py4 S Ž m . .
ms1 uŽW , x . , x. Ž . w x wFP1, Proposition Moreover, w P Ž0.x s wInd uŽW uŽ s , x . Ps 0 s 2 Ind uŽ b q , x . k 0 x 2.3 . The argument above can now be used to show that
py2
P Ž 0 . s 4 p py 4 S Ž 0 . q
Ý
2 p py4 S Ž m . .
ms1
This completes the computation of w P Ž0.x and w P Ž p y 1.r2x. Now suppose that l / 0, p y 1r2. Since uŽW, x . is a symmetric algebra, we have for m s 0, p y 1r2, P Ž l. : S Ž m . s P Ž m . : S Ž l.
¡2 p ~ s ¢p
py 4
py 4
,
,
ms0 py1 ms . 2
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FELDVOSS AND NAKANO
Therefore, in order to finish the proof, it suffices to show that w P Ž l. : S Ž m .x s p py 4 for l, m / 0, p y 1r2. For l g LŽ0. with l / 0, p y 1, Let S Ž l. [ V Ž l, 0. be the simple uŽW, 0.-module corresponding to the weight l. Furthermore, let P Ž l. be the projective cover of S Ž l.. Suppose that P Ž l. m P Ž m .U ( P Ž l. m P Ž m .U as uŽW, 0.-modules for l, m / 0, p y 1. Then by Theorem 2.2A, p py 4 s P Ž l . : S Ž m . s dim k Hom uŽW , 0. Ž P Ž m . , P Ž l . . s dim k Hom uŽW , 0. Ž k, P Ž l . m P Ž m . s dim k Hom uŽW , 0. Ž k, P Ž l . m P Ž m .
U
U
.
.
s P Ž l. : S Ž m . . This shows that our problem reduces to proving that P Ž l. m P Ž m .U ( P Ž l. m P Ž m .U as uŽW, 0.-modules for all l, m / 0, p y 1. , 0. Ž Ž .U . by Theorem Observe that P Ž l. m P Ž m .U ( Ind uŽW uŽ b y , 0. kl q1 m P m y 2.2BŽi.. Let i be the injective uŽ b , 0. homomorphism from klq1 m P Ž m .U , 0. Ž Ž .U .. We will construct a uŽ by, 0. homomorphism ¨ Ind uŽW uŽ b y , 0. kl q1 m P m U s from klq1 m P Ž m . to P Ž l. m P Ž m .U in the following way. Since Soc uŽ by , 0. S Ž l. ( klq1 , there exists an injective uŽ by, 0. homomorphism r : klq1 ¨ S Ž l.. Let s 1 be the uŽ by, 0. homomorphism defined by r m id from klq1 m P Ž m .U to S Ž l. m P Ž m .U . Next we have S Ž l.< W 0 ( SŽ l.< W 0 and P Ž m .< W 0 ( P Ž m .< W 0 . This implies that for any simple uŽW, 0.-module L, U
U
Hom uŽW , 0. Ž S Ž l . m P Ž m . , L . ( Hom uŽW 0 , 0. Ž l m P Ž m . , L . U
( Hom uŽW 0 , 0. Ž l m P Ž m . , L . U
( Hom uŽW , 0. Ž S Ž l . m P Ž m . , L . . Therefore, S Ž l. m P Ž m .U ( SŽ l. m P Ž m .U because both modules are projective with the same projective summands appearing Žusing the computation above.. Let s 2 denote the uŽW, 0. isomorphism between these modules. Let a be the uŽW, x . homomorphism from P Ž l. ¸ P Ž l.rRadŽ P Ž l.. ( S Ž l.. Since S Ž l. is a projective uŽ by, x .-module, there exists a uŽ by, x . monomorphism b from S Ž l. into P Ž l. such that a ( b s id. Set s 3 [ b m id. Finally, let s [ s 3 ( s 2 ( s 1. The universal property of induction shows , 0. Ž that there exists a uŽW, 0. homomorphism C from Ind uŽW uŽ b y , 0. kl q1 m U U P Ž m . . to P Ž l. m P Ž m . . The dimensions of the two modules are the same. Therefore, to show that they are isomorphic, it suffices to prove that C is surjective.
455
REPRESENTATIONS OF THE WITT ALGEBRA
First we claim that Im s 3 generates P Ž l. m P Ž m .U as a uŽW, 0.-module. This will follow by showing that Im b generates P Ž l. as a uŽW, x .module. Consider b ( r : klq1 ¨ P Ž l. as a uŽ t, 0.-module homomorphisms. Let iX be the injective uŽ t, 0. homomorphism from klq1 to 0 , 0. Ž . Ind uŽW uŽ t , 0. kl q1 . The universal property of induction yields a u W0 , 0 -map d 0 , 0. k Ž . Ž . Ž l. as a from Ind uŽW to P l . Since P l is the projective cover of S uŽ t , 0. l q1 uŽW0 , 0.-module with S Ž l.rRad SŽ l. ( klq1 , it follows that a : P Ž l. ª SŽ l. is an essential epimorphism. Now observe that b ( r s d ( iX , thus a ( b ( r s a ( d ( iX . But, a ( b s id, so r s a ( d ( iX . Suppose that Im b does not generate P Ž l. as a uŽW0 , 0.-module. Therefore, d cannot be surjective which implies that Im a ( d ( iX : Rad SŽ l.. This is a contradiction because Im r is not contained in Rad S Ž l.. Hence, Im b generates P Ž l. as a uŽW0 , 0.-module, and also as a uŽW, x .-module. To complete the proof observe that S Ž l. is generated by Im r as a Ž u W, 0.-module. Therefore, S Ž l. m P Ž m .U is generated by Im s 1 as a uŽW, 0.-module. Since s 2 is an isomorphism and Im s 3 generates P Ž l. m P Ž m .U as a uŽW, 0.-module, it follows that Im s generates P Ž l. m P Ž m .U as a uŽW, 0.-module. Hence, C is surjective. 2.4. w r Ž x . s 1x. We will show that for r Ž x . s 1 all the projective indecomposable uŽW, x .-modules have the same dimension and the same composition factors. THEOREM. If r Ž x . s 1, then the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. sp
py 2
.
, x. Ž . Ž . P Ž l. ( Ind uŽW uŽ b q , x . kl for e¨ ery l g L x . In particular, dim k P l
Žb. w P Ž l.x s Ým g LŽ x . p py 4 w SŽ m .x for e¨ ery l g LŽ x .. In particular, uŽW, x . has one block. Proof. Observe that x Ž e0 . / 0 and x Ž e1 . s 0 so that uŽ s, x . is semisimple. This means that there are p pairwise non-isomorphic simple uŽ s, x .-modules given by SŽ l.< s , for l g LŽ x . wFP1, Corollary 2.2x. By transitivity of induction and Frobenius reciprocity we have , x. Hom uŽW , x . Ind uŽW uŽ b q , x . kl , S Ž m .
ž
/
, x. uŽ s , x . ( Hom uŽW , x . Ind uŽW uŽ s , x . Ind uŽ b q , x . kl , S Ž m .
ž
ž
( Hom uŽ s , x . Ž S Ž l . , S Ž m . . (
½
k, 0,
msl m / l.
/
/
456
FELDVOSS AND NAKANO
, x. uŽW , x . Ž . Therefore, Ind uŽW uŽ b q , x . kl has a simple head S l . Moreover, Ind uŽ b q , x . kl is uŽ s , x . projective as a uŽW, x .-module because Ind uŽ bq , x . kl is projective as a uŽ s, x .-module and induction takes projective modules to projective mod, x. ules. Hence, P Ž l. ( Ind uŽW uŽ b q , x . kl. In order to prove the second assertion, we can use transitivity and exactness of induction to see that
P Ž l . s u Ž W , x . muŽ bq , x . kl s u Ž W , x . muŽW 0 , x . Ž u Ž W0 , x . muŽ bq , x . kl . s
Ý
p py 4 u Ž W , x . muŽW 0 , x . km
Ý
p py 4 S Ž m . .
m gL Ž x .
s
m gL Ž x .
Hence, the second assertion holds and the above formula shows that uŽW, x . has one block. 2.5. w1 - r Ž x . - p y 1x. Let S be the unique simple uŽW, x .-module given in Theorem 1.3B, and let P be its projective cover. THEOREM. If 1 - r Ž x . - p y 1, then the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. dim k P s p py sy1 and w P x s p py 2 sy2 w S x where s [ w r Ž x .r2x. Žb. If r s 2, 3, then P ( uŽW, x . muŽ s , x . St where St is a p-dimensional projecti¨ e simple uŽ s, x .-module. Proof. Since p p s dim k uŽW, x . s Ždim k S . ? Ždim k P ., the first assertion follows immediately from Theorem 1.3B. Furthermore, uŽW, x . muŽ s , x . St is a projective uŽW, x .-module because St is a projective uŽ s, x .-module. Also, if r s 2, 3, then this module has the same dimension as P because St is p-dimensional. Note that for all x < s there always exists such a p-dimensional projective simple module wFP1, Sect. 2x. It would be nice to have an explicit realization of the projective indecomposable uŽW, x .-modules for 3 - r Ž x . - p y 1. 2.6. w r Ž x . s p y 1x. The following theorem is a re-interpretation of an argument given in wFa, p. 83x by using Theorem 1.3C. THEOREM. If r Ž x . s p y 1, then the projecti¨ e indecomposable modules and the Cartan in¨ ariants of uŽW, x . are gi¨ en as follows: Ža. If W x is a torus, then e¨ ery simple uŽW, x .-module is projecti¨ e. In particular, uŽW, x . is semisimple.
REPRESENTATIONS OF THE WITT ALGEBRA
457
Žb. If W x is p-unipotent, then e¨ ery simple uŽW, x .-module with one exception is projecti¨ e. The remaining simple module L has a uniserial projecti¨ e co¨ er of dimension 2 p py 1r2 with two composition factors both isomorphic to L. In particular, uŽW, x . has p y 1 blocks.
3. ORBITS OF CHARACTERS 3.1. Conjugation Results In this section we will compute the orbits of characters under the action of the automorphism group AutŽW .. For classical Lie algebras one can identify the Lie algebra with its linear dual via the Killing form. This identification allows one to obtain a Jordan]Chevalley decomposition for characters. For the nilpotent characters, there are finitely many orbits under the action of the ambient algebraic group provided the characteristic of the field is good. We will see that this is not the case. The number of coadjoint orbits under AutŽW . for the Witt algebra will depend primarily on the parity of the height of the character. We should also remark that if x and x X are characters in the same AutŽW .-orbit, then the algebras uŽW, x . and uŽW, x X . are isomorphic. Therefore, knowledge of conjugacy classes provides information about the module theory of the algebras. Consider the following family of characters x a of height r where a g k. If r is odd and r / 1, let s s Ž r y 1.r2 and define 0, i / r y 1, s x a Ž e i . s 1, Ž 3.1.1. isry1 a, i s s.
½
If r is even or r s 1, then set
x 0 Ž ei . s THEOREM.
x0.
Ža.
½
0, 1,
i/ry1 i s r y 1.
Ž 3.1.2.
Let x g W U , r [ r Ž x ., and G [ AutŽW ..
If r is e¨ en with 0 F r F p y 3 or r s 1, then x is G-conjugate to
Žb. If r is odd with 3 F r F p y 2, then x is G-conjugate to x a for some a g k. Moreo¨ er, x a < a g k 4 forms a complete set of inequi¨ alent representati¨ es for the G-orbits of characters of height r. Proof. The automorphism group G s T h U where T is a one-dimensional torus and U is a unipotent algebraic group wWilx. Recall that G acts on W U by Ž g ? x .Ž w . s x Ž gy1 ? w . where g g G, x g W U , and w g W. Let x g W U , r s r Ž x ., and 0 F r - p y 1. By using the action of the
458
FELDVOSS AND NAKANO
torus T we may assume that x Ž e ry1 . s 1, thus reducing the problem to finding u g U such that u ? x s x a for some a g k. This is equivalent to showing that for some a g k there exists u g U such that x aŽ u ? w . s x Ž w . for all w g W. An automorphism u g U arises from an automorphism su g AutŽ k w x xr Ž x p .. where suŽ1. s 1 and suŽ x . s x q b 2 x 2 q b 3 x 3 q ??? qbpy1 x py1 , bj g k for j s 2, 3, . . . , p y 1 Žset b1 s 1.. For any w g W, u ? w s suy1 wsu . One can express py2
u ? ej s ej q
a kj e k
Ý ksjq1
j for j s y1, 0, 1, . . . , r y 2. Moreover, observe that x aŽ u ? e j . s a ry1 qa? j a s . On the other hand,
Ž suy1 e j su . Ž x . s Ž u ? e j . Ž x . py2
ž
s ej q
Ý
/
a kj e k Ž x .
ksjq1 py2
sx
jq1
q
Ý
a kj x kq1 .
ksjq1
Therefore, x aŽ u ? e j . can be found by taking the coefficient of the x r-term in suy1 e j suŽ x . and adding it to a times the coefficient of the x sq 1-term. One can prove by induction that
Ž suy1 e j su . Ž x . s Ž suy1 Ž x . .
jq1
= 1 q 2 b 2 suy1 Ž x . q ??? q Ž p y 1 . bpy1 Ž suy1 Ž x . .
ž
py 2
/
Ž 3.1.3. for j s y1, 0, 1, . . . , r y 2. Let a j [ x Ž e j . for j s y1, 0, 1, . . . , r y 2 and suy1 Ž x . s x q c 2 x 2 q ??? qc py1 x py1 Žset c1 s 1.. Now assume that r s 1. Equation Ž3.13. shows that ay1 s 2 b 2 . Therefore, x is G-conjugate to x 0 . To handle both the even and odd cases simultaneously we abide by the following conventions Žfor r / 1.. If r Ž x . is even, then set s [ r y 2 and a [ 0. On the other hand, if r Ž x . is odd, then set s [ Ž r y 1.r2. Then by using Ž3.13. there exists f j Ž a, b 2 , . . . , bryŽ jq1. , c 2 , . . . , c ryŽ jq1. . g k for j s y1, 0, . . . , s and f j Ž b 2 , . . . , bryŽ jq1. ,
REPRESENTATIONS OF THE WITT ALGEBRA
459
c 2 , . . . , c ryŽ jq1. . g k for j s s q 1, . . . , r y 2 without constant term such that a j s Ž r y j . bryj q Ž j q 1 . c ryj q f j Ž a, b 2 , . . . , bryŽ jq1. , c 2 , . . . , c ryŽ jq1. . ;
Ž 3.1.4. for j s y1, 0, . . . , s. a j s Ž r y j . bryj q Ž j q 1 . c ryj q f j Ž b 2 , . . . , bryŽ jq1. , c 2 , . . . , c ryŽ jq1. .
Ž 3.1.5. that j s s q 1, . . . , r y 2. Furthermore, by using the fact that suŽ suy1 Ž x .. s x, it follows that there exist g j Ž b 2 , . . . , bjy1 , c 2 , . . . , c jy1 . g k without constant term such that bj s yc j q g n Ž b 2 , . . . , bjy1 , c 2 , . . . , C jy1 . .
Ž 3.1.6.
for j s 2, 3, . . . , p y 1. Suppose that r is even. We can recursively find b 2 , c2 , b 3 , c 3 , . . . , bpy1 , c py 1 satisfying the equations above as long as r y j / j q 1, or j / r y 1r2. This condition is always satisfied because r is even, thus x is G-conjugate to x 0 . On the other hand, suppose that r is odd and r / 1. Then s s Ž r y2 1 . . In particular, by similar reasoning as above we can find b 2 , . . . , bsy1 , c 2 , . . . , c sy 1 satisfying Eq. Ž3.1.5. for a sq1 , a sq2 , . . . , a ry2 . The equation involving a s given in Ž3.1.4. is a s s Ž r y s . brys q Ž s q 1 . c rys q a q h Ž b 2 , . . . , brysy1 , c 2 , . . . , c rysy1 . .
Ž 3.1.7. With an appropriate choice of a g k, this equation can be satisfied by some brys and c rys . One can then recursively find b 2 , . . . , bpy1 , c 2 , . . . , c py1 which satisfy Ž3.1.4.. Hence, x is G-conjugate to x a . Finally suppose that x a is G-conjugate to x aX . One can set x s x aX and use Eqs. Ž3.1.5. and Ž3.1.6.. Since a ry2 s a ry3 s ??? s a sq1 s 0, it follows that bj s c j s 0 for j s 2, 3, . . . , r y s y 1. This will imply that brys s yc rys , and by Ž3.1.7., x aX Ž e s . s a s s a s x aŽ e s .. Hence, x a s x aX and the proof of the theorem is complete. 3.2. Centralizers. Let x g W U . Observe that the definition of r Ž x . implies that Wr Ž x .q1 : W x. The next proposition calculates the centralizers for the canonical orbit representatives of characters as listed in Theorem 3.1.
460
FELDVOSS AND NAKANO
PROPOSITION. Ža. Žb. Žc.
Let 0 / x g W U .
If r Ž x . is e¨ en and 0 F r Ž x . F p y 3, then W x s Wr Ž x .q1. Let x [ x 0 with r s 1. Then W x s ke0 [ W2 . Let a g k, 3 F 4 [ r Ž x a . F p y 2 odd, and s [ r y 1r2. Then W
xa
s
½
k Ž e1 q 12 a ? e2 q 38 a2 ? e 3 . [ W4 ,
ss1
k Ž e2 y a ? e ry1 . [ Wrq1 ,
s ) 1.
1 2
Proof. Ža. If r Ž x . is even, the argument in wS, p. 604x shows that W x s Wr Ž x .q1. For Žb. and Žc., according to the proof of wG, Lemma 4.14.Ž2.x, W x arWrq1 is one-dimensional with basis x s Ý riss j i e i Ž j i s 0 for i - s or i ) r . which satisfies the equations 0 s x a Ž w x, e j x . s
r
Ý j i Ž j y i . xa Ž eiqj . iss
for all y1 F j F s y 1. From the definition of x 0 , it immediately follows that j 1 s 0 if s s 0. Furthermore, by the definition of x a ,
Ž 2 j y s . a ? j sy j q Ž 2 j y r q 1 . ? j ry1yj s 0 for all y1 F j F s y 1. In particular, the first summand is non-zero only if j F 0. Hence,
Ž s q 2 . a ? j sq 1 q Ž r q 1 . ? j r s 0, sa ? j s q Ž r y 1 . ? j ry1 s 0,
Ž 2 j y r q 1 . ? j ry1yj s 0 for all 1 F j F s y 1. If s s 1, the assertion follows from the first two equations. If s G 2, then one can conclude from 3 F r F p y 2 that 2 j y r q 1 k 0 Žmod p . for every 1 F j F s y 1. Then we obtain from the third equation that j j s 0 for all s q 1 F j F r y 2. Since r F p y 2, the first equation yields j r s 0. Finally, the result follows by using the second equation and s s Ž r y 1.r2.
4. SUPPORT VARIETIES 4.1. Let g be a finite-dimensional restricted Lie algebra and x g gU . n Ž . For any two uŽ g, x .-modules M and N, let Ext uŽ g , x . M, N be the nth extension group of M by N. The commutative ring R [ Ext 2uŽ?g , 0.Ž k, k . is
REPRESENTATIONS OF THE WITT ALGEBRA
461
a finitely generated k-algebra. Furthermore, for any finite-dimension ? Ž . is a finitely generated R-module uŽ g, x .-module M, Ext uŽ g , x . M, M wFP1, Sect. 6x. Let < g < M [ SpecŽ RrJM . where JM is the annihilator of R ? Ž . on Ext uŽ g , x . M, M . There exists a natural finite morphism F : < g < k ª g of varieties. The image of < g < M under F is given by
x g g < x w px s 0 and
M < k x is not projective4 j 0 4
wFP1, Theorem 6.4x. The goal of this section will be to compute F Ž< W < M . for simple W-modules M. In particular, this enables us by using cW Ž M . s dim F Ž< W < M . wFP1, Remark 6.3x to determine the complexity cW Ž M . for a simple W-module M. Since many of the simple W-modules are induced, the following proposition will be used extensively. PROPOSITION. Let h be a p-subalgebra of a finite-dimensional restricted Lie algebra g and x g gU . Moreo¨ er, let M be a finite-dimensional uŽ h, x .ˆ [ Ind uŽuŽgh ,, xx .. M. Then module and M Ža. Žb.
F Ž< g < Mˆ . : F Ž< h < M .. F Ž< g < Mˆ . s F Ž< h < Mˆ ..
Proof. Consider the commutative diagram r
? Ž ˆ ˆ. Ext uŽ g , x . M, M
6
i
? ˆ. Ž Ext uŽ h , x . M, M
6
6
S ?Ž h U .
6
S ?Ž g U .
where the right-hand vertical arrow is the isomorphism given by using Frobenius reciprocity. Here, i is the map induced by the inclusion h ¨ g ? Ž ˆ ˆ. and r is composition S ?Ž gU . ª H 2 ?Ž uŽ g ., k . ª Ext uŽ ˆ g , x . M, M . Let IM be the kernel of r and V Ž IMˆ . be the associated variety. Then F Ž< g < Mˆ . s ?Ž U . ? ˆ. Ž V Ž IMˆ .. Now let K M , Mˆ be the annihilator of Ext uŽ h , x . M, M in S h . The commutativity of the diagram shows that iy1 Ž K M , Mˆ . : IMˆ . Therefore, F Ž< g < Mˆ . : V Ž iy1 Ž K M , Mˆ ... Furthermore, by an argument given in wBen, p. 181x, it follows that V Ž iy1 Ž K M , Mˆ . . : F Ž < h < M . l F Ž < h < Mˆ . . Hence, the proof of part Ža. is complete. The inclusions above also show that F Ž< g < Mˆ . : F Ž< h < Mˆ .. By wFP1, Proposition 7.1Ža.x, F Ž< h < Mˆ . : F Ž< g < Mˆ ., thus F Ž< g < Mˆ . s F Ž< h < Mˆ .. A more general statement of part Žb. of the above proposition can be found in wFa, Proposition 3.4x which in view of wFP1, Proposition 7.1Ža., Žb.x
462
FELDVOSS AND NAKANO
is an immediate consequence of our result. Part Ža. is a stronger version of wFa, Corollary 3.5x because there are no additional hypotheses on the Lie algebras g and h Žsee also wFP1, Remark 7.5x.. 4.2. w r Ž x . s y1x. The computation of the support varieties for all simple restricted modules of the Witt algebra was first done in wLN, Theorem 5.4x. This result is given below. THEOREM A. The support ¨ arieties and the complexities of the simple uŽW, 0.-modules are gi¨ en by
¡AutŽ W . ? e
y1
F Ž < W < SŽ l. . s
~W
1
¢W ¡p y 1 2
cW Ž S Ž l . . s
for l s 0, p y 1 py1 for l / 0, , py1 2 py1 for l s 2 for l s 0, p y 1 py1 for l / 0, , py1 2 py1 for l s . 2
~p y 2
¢p y 3
The computation of the support varieties for all restricted Verma modules V Ž l, 0. is provided in the following theorem. This result will be used in the proof of Theorem 4.3. THEOREM B. The support ¨ arieties and the complexities of the restricted Verma modules for uŽW, x . are gi¨ en by
¡W
F Ž < W < V Ž l , 0. . s
~
1
¢W ¡p y 2 2
cW Ž V Ž l , 0 . . s
~
¢p y 3
for l / for l s
py1 2 py1
for l / for l s
2 py1 2 py1 2
Proof. Let x g W2 . Then k Ž 1 m 1l . [ k Ž ey1 m 1l . : V Ž l , 0 .
kx
.
.
REPRESENTATIONS OF THE WITT ALGEBRA
463
Since dim k V Ž l, 0. s p, it follows that V Ž l, 0.< k x cannot be projective. Hence, W2 : F Ž< W < V Ž l, 0. .. Also, note that Proposition 4.1Ža. implies F Ž < W < V Ž l , 0. . : x g W0 < x w px s 0 4 s W1 . Now let x s a e1 q y where 0 / a g k and y g W2 . Since dim k V Ž l, 0. s p, V Ž l, 0. is projective as a uŽ kx, x .-module if and only if x py 1 V Ž l, 0. / 0. By using wFe1, III.3.1x, one can see that 0
V Ž l , 0. s
[
isy Ž py1 .
k Ž eyi y1 m 1l .
is a graded uŽW, 0.-module. Therefore, x py 1 V Ž l, 0. / 0 if and only if e1py 1 V Ž l, 0. / 0. The latter condition holds if and only if l s p y 1r2 Ži.e., V Ž l, 0. is the Steinberg module for s ( s l Ž2... Hence, F Ž< W < V Ž l , 0. . s W1 for l / Ž p y 1.r2, and F Ž< W < V Ž l, 0. . s W2 for l s Ž p y 1.r2. 4.3. w r Ž x . s 0x. We now compute the support varieties of the simple uŽW, x .-modules when r Ž x . s 0. Observe that the result is somewhat similar to the restricted case as the proof is dependent on looking at s ( s l Ž2.. THEOREM. If r Ž x . s 0, then the support ¨ arieties and the complexities of the simple uŽW, x .-modules are gi¨ en by
¡W
F Ž < W < SŽ l . . s
~
1
¢W ¡p y 2 2
cW Ž S Ž l . . s
~
¢p y 3
for l / for l s for l / for l s
py1 2 py1 2 py1 2 py1 2
.
The reader should be made aware that wFa, Theorem 5.1Ž3.x claims that F Ž< W < SŽ l . . s W1 for all l g LŽ x . in the case when r Ž x . s 0. This p 1. < s is isomorphic to the Steinberg module statement is false because S Ž y 2 p 1. < k e1 is projecfor s ( s l Ž2. Žsee wFP1, Proposition 2.3x.. Therefore, S Ž y 2 tive, and thus e1 f F Ž< W < S Ž 2 . .. By the same reasoning as above, the first inclusion in wFa, Theorem 5.1Ž1.x does not hold for r Ž x . s 0. py 1
464
FELDVOSS AND NAKANO
Proof. Observe that SŽ l.< W 0 ( V Ž l, 0.< W 0 for every l g LŽ x .. Therefore, by Proposition 4.1Žb. we have F Ž < W < SŽ l. . s F Ž < W0 < SŽ l . . s F Ž < W0 < V Ž l , 0. . s F Ž < W < V Ž l , 0. . . The result now follows by Theorem 4.2B. 4.4. w r Ž x . s 1x. We will now prove that the support varieties of the simple uŽW, x .-modules for r Ž x . G 1 begin to show some generic behavior. THEOREM. If r Ž x . s 1, then the support ¨ arieties and the complexities of the simple uŽW, x .-modules are gi¨ en by F Ž< W < SŽ l . . s W2 and cW Ž SŽ l.. s p y 3 for all l g LŽ x .. Proof. By the same reasoning as in Theorem 4.2B, W2 : F Ž< W < SŽ l. . : W1. Suppose that x s a e1 q y where 0 / a g k and y g W2 . The modules 0
S Ž l. s
[
y Ž py1 .
k Ž eyi y1 m 1l .
are graded uŽW0 , x .-modules. Since SŽ l. is p-dimensional and graded, it follows that S Ž l.< k x is projective if and only if SŽ l.< k e1 is projective. But x Ž e0 . / 0, so that x < s is regular semisimple and SŽ l.< s is projective. This implies that x f F Ž< W < SŽ l. . and F Ž< W < SŽ l. . s W2 . 4.5. w1 - r Ž x . - p y 1, r Ž x . evenx. The following result is already contained in wG, Theorem 4.16x and also in wFa, Theorem 5.1Ž3.x. We include a proof following the ideas in wGx since it will be used in an essential way in the proof of Theorem 4.6. THEOREM. If 1 - r Ž x . - p y 1 and r Ž x . is e¨ en, then the support ¨ ariety and the complexity of the simple uŽW, x .-module S are gi¨ en by F Ž< W < S . s Wr Ž x .q1 and cW Ž S . s p y r Ž x . y 2.
Proof. Let r [ r Ž x .. Since r is even, s s rr2. The inclusion Wrq1 : F Ž< W < S . was shown in wFa, Theorem 5.1Ž1.x. In order to prove the reverse inclusion, we consider x f Wrq1. By Proposition 4.1Ža., F Ž < W < S . : Ws , and py 2 therefore we may assume that x g Ws . Hence x s Ý ism j i e i with j m / 0 and m G s G 1. Since Wr is a p-ideal of W0 and x < W r s 0, it follows from Engel’s theorem that Wr acts trivially on every simple uŽW0 , x .-module which is therefore a simple uŽW0rWr , x .-module where x Ž e . [ x Ž e . for every e g W0 .
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Suppose that m - r y 1. Then 1 F r y 1 y m F p y 2, and furthermore, r y 1 y 2 m k 0 Žmod p . because r is even. This implies that x Žw x, e ry1ym x. s j m ? Ž r y 1 y 2 m. ? x Ž e ry1 . / 0. Consider the p-subalgebra H [ kX [ kY [ kZ of W0rWr where X [ x, Y [ e ry1ym , and Z [ w x, e ry1ym x . Since 1 F m F r y 2 and r G 2, we have w X, Z x s 0, w Y, Z x s 0, w X, Y x s Z, and X w px s Y w px s Z w px s 0. Hence, H is a threedimensional Heisenberg algebra with trivial restriction map. But x Ž Z . / 0, so Žup to isomorphism. there is a unique simple uŽ H , x .-module SH represented by uŽ H , x . muŽ k Y[k Z, x . kx . It is easy to see that SH < k X is projective. Therefore, M0 < k x is projective for every finite-dimensional uŽW0 , x .-module M0 . By wFP1, Proposition 7.1Ža.x, x f F Ž< W < S .. Suppose now that m s r y 1. Consider the p-subalgebra M [ kT [ kX of W0rWr where T [ e0 and X [ x. Since 2 F r F p y 2, we have w T, X x s Ž r y 1. ? X / 0, T w px s T, and X w px s 0. So M is the two-dimensional non-abelian Lie algebra. But x Ž X . s j ry1 ? x Ž e ry1 . / 0, so there exists a simple module of dimension p. Therefore uŽ M , x . is semisimple. Hence as above, M0 < k x is projective for every finite-dimensional uŽW0 , x .-module M0 , thus x f F Ž< W < S .. Finally, suppose that m s r. By wC, Satz 3x there is only one isomorphism class of simple uŽW0 , x .-modules represented by S0 [ uŽW0 , x . muŽW s , x . kx and therefore S ( uŽW, x . muŽW 0 , x . S0 . Since x acts trivially on S0 and r G 2, one can show by induction that n x py 1 ? Ž ey1 m s0 .
s
½
Ž p y 1 . !? 1 m w x, ey1 x 0
py 1
? s0
if n s p y 1 Ž 4.5.1. if 0 F n - p y 1
and every s0 g S0 . From 2 F r F p y 2, we conclude that w x, ey1 x p ? s0 s x Žw x, ey1 x. p ? s0 for ever s0 g S0 and x Žw x, ey1 x. s yj r ? Ž r q 1. ? x Ž e ry1 . / 0. Hence Ž4.5.1. implies that x py 1S ( S0 . Moreover, by a straightforward computation, one can see that S k x ( S0 . Then it follows from wFe2, Propositions 2.6 and 5.1x that S < k x is projective, and thus x f F Ž< W < S .. 4.6. w1 - r Ž x . - p y 1, r Ž x . oddx. The following theorem provides inclusions for the support variety F Ž< W < S .. The result is slightly stronger than the results given in wFa, Theorem 5.1Ž1.x with a more direct proof. THEOREM. If 1 - r Ž x . - p y 1, r Ž x . is odd and S is the simple uŽ g, x .-module, then Wr Ž x .q1 : F Ž < W < S . : a Ž x q y . : a g k, y g ke r 4 [ Wr Ž x .q1 ,
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where x s e s q Ý ry1 issq1 j i ? e i and the s elements j sq1 , . . . , j ry1 g k are uniquely determined by the s linear equations obtained from x Žw x, e i x. s 0 for 0 F i F s y 1. Moreo¨ er, p y r Ž x . y 2 F cW Ž S . F p y r Ž x . y 1. Proof. Let r [ r Ž x .. The first inclusion is just wFa, Theorem 5.1Ž1.x. In order to prove the second inclusion, we consider x f Wrq1. As in the proof py 2 of Theorem 4.5, we can assume that x g Ws , i.e., x s Ý ism j i ? e i with j m / 0 and m G s G 1. Assume m - r y 1. Then the proof of Theorem 4.5 shows, except when m s Ž r y 1.r2 s s, that x f Ž< W < S .. In the latter case, we may assume that x s e s q Ý rissq1 j i ? e i . It remains to show that if x Žw x, e i x. / 0 for some 0 F i F s y 1, then x f F Ž< W < S .. Suppose that x Žw x, e0 x. / 0. Consider the p-subalgebra M of W0rWr generated by T [ e0 and X [ x. Then T w px s T, and the p-ideal I of M generated by X is abelian and has trivial restriction map because I : WsrWr and s G 1. Since x Žw T, X x. s x Žw e0 , x x. / 0, wSF, Corollary 5.7.6x shows that Žup to isomorphism. there is a unique simple uŽ M , x .-module S M represented by uŽ M , x . muŽ I , x . kx . Since S M m ky x is a restricted kX-module, wFe2, Propositions 2.6 and 5.1x in conjunction with Lemma 2.2x imply that S M < k X is projective which shows as in the proof of Theorem 4.5 that x f F Ž< W < S .. Suppose now that x Žw x, e j x. / 0 for some 1 F j F s y 1 Žif s G 2.. Consider the p-subalgebra N of W0rWr generated by X [ x and Y [ e j. Then N is nilpotent with trivial restriction map. Let I denote the p-ideal of N generated by Y. Since x Žw X, Y x. s x Žw x, e j x. / 0, wSF, Corollary 5.7.6x shows that Žup to isomorphism. there is a unique simple uŽ N , x .module SN represented by uŽ N , x . muŽ I , x . kx . It is easy to see that Ž SN m ky x . k X ( k Ž X py1 m 1x . m 1y x ( X py1 Ž SN m ky x .. Hence we obtain as above that x f F Ž< W < S .. If m s r y 1 or m s r, then one can use the same argument as in the proof of Theorem 4.5 to conclude that x f F Ž< W < S . which finishes the proof of the theorem. The conjugation result in Subsection 3.1 shows that for the odd case it is enough to consider the characters x a for some a g k to calculate the support varieties. We prove for the character of smallest possible odd height that the support variety of the simple module is W x. EXAMPLE. Consider x s x 0 for r [ r Ž x . s 3 Ži.e., s s 1.. Then x Ž e2 . s 1 and x Ž e i . s 0 for every i / 2. According to Proposition 3.2Žc., we
REPRESENTATIONS OF THE WITT ALGEBRA
467
have W x s ke1 [ W4 . Let x [ a e1 q y with a g k and y g W4 . Therefore, n
x ? Ž e0n m 1x . s
ny n Ý Ž y1.
ns0
ž nn / e Ž ad e . n 0
0
ny n
Ž x . m 1x s 0
because Žad e0 . m Ž x . g W1 and x ŽŽad e0 . m Ž x .. s 0 for every non-negative integer m. Moreover, ey1Ž e0py 1 y 1. m 1x g S k x. Hence, dim k S k x G p q 1 which implies that S < k x is not projective. We can conclude that W x : F Ž< W < S .. It remains to show that F Ž< W < S . : W x. According to Theorem 4.6, this will follow by showing that S < k x is projective for x [ e1 q j 3 e3 q y where 0 / j 3 g k and y g W4 . Since S0k e 0 s k Ž e0py 1 y 1., a direct computation shows that S k x ( S0 , thus S < k x is projective. 4.7. w r Ž x . s p y 1x. The final case to consider is when r Ž x . s p y 1. THEOREM. If r Ž x . s p y 1, then the support ¨ arieties of the simple uŽW, x .-modules are gi¨ en as follows. Ža. If W x is a torus, then the support ¨ ariety of any simple uŽW, x .module is 04 . Žb. If W x is p-unipotent, then the support ¨ ariety is 04 for all simple uŽW, x .-modules except for L Ž see Theorem 2.6Žb... Moreo¨ er, F Ž< W < L . s W x. Proof. Part Ža. immediately follows from Theorem 2.6Ža. because in this case uŽW, x . is semisimple. For part Žb., without a loss of generality we may assume that x Ž e i . s 0 for y1 F i F p y 3 and x Ž e py2 . / 0 by using the argument after Theorem 1.3C. By a direct calculation one can show that W x s key1 . Consider the modules M [ uŽW, x . muŽ by , 0. k and N [ uŽW, 0. muŽ by , 0. k. By comparison the by-actions one can see that M < by( N < by. Now suppose that ey1 f F Ž< W < M .. Then N is projective over uŽ key1 , 0., and thus projective over uŽW, 0. by the universal property of induction. Since there is an epimorphism from N onto k, there is an epimorphism from N onto the projective cover P Ž0. of k. In view of Theorem 2.2A, this is a contradiction because dim k P Ž0. s 2 p py 2 ) dim k N. Therefore, ey1 g F Ž< W < M .. Since all simple uŽW, x .-modules except for L are projective, it follows that F Ž
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Furthermore, dim F Ž< W < L . s 1 by Theorem 2.6Žb.. Since L is simple and hence indecomposable, F Ž< W < L . must be a line. Consequently, F Ž< W < L . s key1 . 4.8. Gottman wGx conjectured that for every finite-dimensional restricted Lie algebra g and every x g gU the following holds:
D
FŽ < g < M . s FŽ < g < k . l g x.
M simple
For the case when x s 0, the above statement is always satisfied. Premet wP4x has shown that for a Lie algebra g of a connected reductive algebraic group G with G Ž1. simply connected and p good for the root system of G the above conjecture holds. One main ingredient in his proof is that under the above assumptions g satisfies Richardson’s property which according to wP4, Lemma 3.7x is not the case for the Witt algebra W. Furthermore, he provides a counterexample to the conjecture wP3, 3.6x. Namely, for an appropriate character x of a toral extension of the three-dimensional Heisenberg algebra g, one has
D
FŽ < g < M . FŽ < g < k . l g x.
M simple
The results in this section in conjunction with Proposition 3.2 verify this conjecture in all cases for W except when the character has odd height larger than one. An interesting problem would be to provide conditions on the Lie algebra and the character which would ensure the validity of this conjecture.
ACKNOWLEDGMENTS The first author thanks the Department of Mathematics and Statistics of Utah State University for the hospitality during his visit in Spring 1997 when this paper was written. Moreover, he gratefully acknowledges the financial support by the DFG. The authors thank the referee for several useful comments.
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