Complexity for modules over finite Chevalley groups ...

3 downloads 34 Views 138KB Size Report
Aug 5, 1999 - Acknowledgements. The authors would like to thank Jonathan Al- perin, Christopher Bendel, James Humphreys, and Jens Jantzen for making.
Invent. math. 138, 85–101 (1999) Digital Object Identifier (DOI) 10.1007/s002229900002  Springer-Verlag 1999

Complexity for modules over finite Chevalley groups and classical Lie algebras Zongzhu Lin1,? , Daniel K. Nakano2,?? 1 2

Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA (e-mail: [email protected]) Department of Mathematics and Statistics, Utah State University, Logan, UT 84322, USA (e-mail: [email protected])

Oblatum 14-IX-1998 & 16-III-1999 / Published online: 5 August 1999

1. Introduction 1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field F p of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(F p ) of F p -rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(F p )-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = F p ) of Lie(G) and those of G(F p ). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(F p ). More precisely, ? ??

Research was supported in part by NSF grant DMS-9401389 Research was supported in part by NSF grant DMS-9800960

Mathematics Subject Classification (1991): 17B55, 20G, 17B50

86

Z. Lin, D.K. Nakano

(i) Are the projective indecomposable modules for Lie(G) (or G(F p )) obtained by restriction of certain G-modules? (ii) If M is a G-module, when does the projectivity of the module M when restricted to one of Lie(G) or G(F p ) imply the projectivity of M when restricted to the other? Partial answers to these two questions were obtained by Humphreys [Hum3]. One example of a module, which is projective upon restriction to both Lie(G) and G(F p ), is the Steinberg module, St (this module is also simple in both categories). Humphreys’ idea was to use this fact to realize the restricted projective indecomposable Lie(G)-module Q 1 (λ) as a G-direct summand of St ⊗V for a G-module V . This approach requires that p is sufficiently large and in this case the projectivity of Q 1 (λ) as a G(F p )-module immediately follows. Ballard [Bal] showed that the Lie(G)-projective cover Q 1 (λ) of the simple module L(λ) has G-module structure when p ≥ 3(h − 1). Jantzen improved the condition on p to p ≥ 2(h − 1) [Jan1, II 11.11]. Although there have been numerous attempts to prove the existence of G-structures on the projective Lie(G)modules when p < 2(h − 1), no proof or counterexample has been found. Donkin [Don] has conjectured that the restriction of certain tilting modules (which are indeed G-modules) should be the projective covers of the simple Lie(G)-modules. In 1987, Parshall [P] conjectured that for any G-module M, projectivity of M when restricted to Lie(G) implies the projectivity of M when restricted to G(F p ). Parshall’s conjecture was the first step in a hope to find a relationship between the support varieties of M over Lie(G) and over G(F p ). In this paper we provide a comparison theorem relating the complexities of M over Lie(G) and over G(F p ) (Theorem 3.4) and Parshall’s conjecture is proved as a direct consequence. The converse to Parshall’s conjecture is also discussed. The answer is affirmative if the weights of M are not too large and negative in general. 1.2. Our approach is quite different than ones previously used. Let M be a B-module where B is a Borel subgroup of G. Moreover, let U be the unipotent radical of B and n+ = Lie(U). Work of Quillen and Jennings is employed to construct a functor gr from (filtered) mod(U(F p )) to mod(n+ ). Later we show that this functor behaves well with respect to the restriction functors, that is, the diagram given in Fig. 1 commutes. The ambient algebraic group B is used in an essential way, but this approach has the advantage that homological information is retained between mod(n+ ) and mod(U(F p )) via the May spectral sequence. In order to pass cohomological data from the category of G(F p )-modules to Lie(G)-modules one can use cohomological facts relating Sylow subgroups in the ambient group, in addition to the work of Friedlander and Parshall [FP2]. 1.3. The paper is organized as follows. In Section 2, we construct the p-central series of the Sylow p-subgroups of G(Fq ) with q = pr . It turns

Complexity of modules

87 mod(B)

, @ res , @@ , , @R  mod(n ) mod(U(F )) gr res +

p

Fig. 1.

out that the p-central series is exactly the lower central series. Then we use Quillen’s isomorphism [Q] to give an algebra isomorphism between the restricted enveloping algebra u(n+ ) and the graded algebra gr kU(F p ) associated to the radical filtration of the group algebra kU(F p ). In Section 3, we first construct the May spectral sequence of a filtered algebra. This spectral sequence is used to compare the complexities of G-module M over Lie(G) and G(F p ). Parshall’s conjecture is derived as a consequence. In Section 4, we discuss the converse of Parshall’s conjecture. This is accomplished by first classifying the indecomposable modules in the category Cp of SL 2 modules with weights less than p( p − 1), by using the representations of quivers. Donkin and Reiten [DR] have shown that this category is of finite representation type. By computing the dimensions of indecomposable modules in Cp , the converse to Parshall conjecture for modules in Cp is proved for SL 2. The proof of the converse to Parshall conjecture for general groups uses the SL 2 result and results of Friedlander and Parshall [FP2]. As a consequence, a test for projectivity over Lie(G) of a G-module when restricted to G(F p ) is proven. Several projectivity tests are then given for modules over G(F p ). 1.4. Notation and Conventions. All results in this paper hold for reductive algebraic groups G that are defined over k = F p and split over F p . Without loss of generality we will simply assume that the group G is semisimple and simply connected. The following notation and conventions will be fixed throughout the paper. Let Φ be a root system with Φ+ (resp. Φ− ) being the fixed set of positive (resp. negative) roots. Moreover, let ∆ be a base consisting of simple roots. Let T be a fixed split maximal torus such that (G, T ) defines the root system Φ. Let B be a Borel subgroup containing T corresponding to the positive roots and U be the unipotent radical of B. For an affine algebraic group scheme H over k, let H1 = ker(Fr). Here Fr : H → H (1) is the Frobenius map. There is a categorical equivalence between restricted Lie(H)-modules and H1 -modules. We will use this fact extensively in Section 3 for H = G, B, and U. Let g be the complex semisimple Lie algebra with root system Φ. The Lie algebra has a Chevalley basis {eα , h αi : α ∈ Φ, αi ∈ ∆}. For each α, β ∈ Φ, let [eα , eβ ] = cαβ eα+β . The cαβ are the Z-valued structure constants. Recall

88

Z. Lin, D.K. Nakano

P that each positive root β can be written P as β = α∈∆ n α α with n α ∈ N. The height of β is defined to be ht(β) = α∈∆ n α . For a given root system Φ, we say that p is excellent if p is at least the length of all root strings (cf. [Hum1, 9.4]). For irreducible root systems, p 6 = 2 if Φ has a subroot system of type Bn , Cn , F4 , and p 6 = 2, 3 if Φ has a component of type G 2 . When p is excellent, then cα,β 6 ≡ 0 mod p if α + β is a root by Chevalley’s theorem (cf. [Hum1, Theorem 25.2(c)]). 1.5. Acknowledgements. The authors would like to thank Jonathan Alperin, Christopher Bendel, James Humphreys, and Jens Jantzen for making useful comments on an earlier draft of this paper. 2. Quillen’s isomorphism theorem for finite Chevalley groups 2.1. The following facts are standard (see [St,Hum2]). Let N be the number of positive roots. (i)

With a fixed order of positive roots that is compatible with height (i.e., if ht(α) < ht(β) then α < β), the map x : A N → U defined by QN (t1 , t2 , . . . , t N ) → i=1 xαi (ti ) is an isomorphism of algebraic varieties defined over F p where xα : A → G is a root subgroup. (ii) The Frobenius morphisms Fr : A N → (A N )(1) is compatible under x with the restriction to U of the Frobenius morphism Fr : G → G (1) . (iii) For any α, β ∈ Φ+ there exist integers cij ∈ Z such that Y  xiα+ jβ cij si t j . [xα (s), xβ (t)] = xα+β (cαβ st) i+ j>2,i, j>0

By convention, xiα+ jβ is the trivial homomorphism if iα + jβ is not a root. r

Fix r ≥ 1 and set q = pr . Let G(Fq ) = G Fr be the Fq -rational points. r The group G(Fq ) is a finite group of Lie type. Furthermore, U(Fq ) = U Fr is a p-Sylow subgroup of G(Fq ). Let {Γn } be the lower central series of U(Fq ) defined by setting Γ1 = U(Fq ) and Γi+1 = [U(Fq ), Γi ]. Set gr U(Fq ) = ⊕n≥1 Γn /Γn+1 . Following Lazard [La], gr U(Fq ) is a graded Lie algebra over the field F p (also see Jennings [Jen]). The bracket is defined by [x, ¯ y] ¯ = [x, y] ∈ Γn+m /Γn+m+1 for all x¯ ∈ Γn /Γn+1 and y¯ ∈ Γm /Γm+1 where x ∈ Γn and y ∈ Γm areQ pre-images of x¯ and y¯, respectively. By [ABS, Q Lemma 4], we have Γn = ht(α)≥n xα (Fq ). Furthermore, Γn /Γn+1 ∼ = ht(α)=n xα (Fq ) as Abelian groups and as vector spaces over F p . In fact, both are vector spaces over Fq and the isomorphism is Fq -linear. + Let n+ Z be the Lie ring generated by the Chevalley basis {eα | α ∈ Φ } + in g. Set n+ Fq = nZ ⊗Z Fq . Proposition. As Lie algebras over Fq , gr U(Fq ) ∼ = n+ Fq .

Complexity of modules

89

Proof. Observe that {xα (1) | α ∈ Φ+ , ht(α) = n} is a basis for the Fq vector space Γn /Γn+1 . By (iii) above, we have [xα (1), xβ (1)] = cα,β xα+β (1). Hence, the homomorphism of Abelian groups φ : gr U(Fq ) → n+ Fq extend-

ing φ(xα (1)) = eα defines an isomorphism of Lie algebras over Fq .

t u

From this point on the two Fq -Lie algebras gr U(Fq ) and n+ Fq will be identified. 2.2. We now consider the adjoint representation of the algebraic group + U on n+ k = Lie(U). Set Fn = ⊕ht(α)≥n keα . Then nk = F1 ⊇ F2 ⊇ · · · defines a filtration of U-submodules of n+ k . It is straightforward to check that (1 − x)Fn ⊆ Fn+i for all x ∈ Γi . Lemma. Suppose that p is excellent. Then (i) Γi = {x ∈ U(Fq ) | (x − 1)Fn ⊆ Fn+i , ∀n}; (ii) x p ∈ Γi p for all x ∈ Γi . Proof. We can assume that the root system is irreducible. Suppose that 1)Fn ⊆ Fn+i for all n. If n 0 < i, we x ∈ Γn0 \ Γn0 +1 such that (x − Q write x = x 0 y0 such that x 0 = ht(β)=n0 xβ (tβ ) 6 = 1 and y0 ∈ Γn0 +1 . Thus (y0 − 1)Fn ⊆ Fn+n0 +1 for all n. In Endk (n+ k ) we can write x − 1 = (x 0 − 1) + x 0 (y0 − 1). Consequently, for any n, (x 0 − 1)Fn ⊆ (x − 1)Fn + x 0 (y0 − 1)Fn ⊆ Fn+i + x 0 Fn+n0 +1 ⊆ Fn+n0 +1 . Since x 0 6 = 1, we can write x 0 = xβ1 (t1 )xβ2 (t2 ) · · · xβr (tr ) with β j ∈ Φ+ , ht(β j ) = n 0 , and t j 6 = 0. If β1 is the highest root in Φ, then x = x 0 = xβ1 (t1 ) is in the center of U(F p ) and x p = 1. If β1 is not the highest root, there exists a simple root α such that α + β1 ∈ Φ+ by [Hum1, 10.4] (Φ is irreducible). We take n = ht(α) and write x 0 − 1 = (xβ1 (t1 ) − 1)x1 + (xβ2 (t2 ) − 1)x2 + · · · (xβr (tr ) − 1) for some xi ∈ U(Fq ). Since xi eα − eα ∈ Fn+1 and xβ j (t j )eα − eα − t j [eβ j , eα ] ∈ Fn+n0 +1 , we have (2.2.1)

0

(x − 1)eα =

r X

t j cβ j ,α eβ j +α + e

j=1

for some e ∈ Fn+n0 +1 . Since t1 cβ1 ,α 6 = 0, we have (x 0 − 1)eα 6 ∈ Fn+n0 +1 by a basis consideration. This contradicts the fact that (x 0 − 1)Fn ⊆ Fn+n0 +1 . Hence, n 0 ≥ i and (i) follows. (ii) follows from (i) and the fact that t u (x p − 1)Fn = (x − 1) p Fn ⊆ Fn+i p for all n. The Lie algebra gr U(Fq ) has a restricted structure by setting xα (1) xα (1) p (see [La]).

[ p]

=

90

Z. Lin, D.K. Nakano

Proposition. The isomorphism between gr U(Fq ) and n+ Fq (given in 2.1) is an isomorphism of restricted Lie algebras. Proof. This is clear because the p-maps are uniquely determined by the images of a basis. For the standard Chevalley basis {eα : α ∈ Φ+ }, the isomorphism in Proposition 2.1 preserves the p-maps. t u Although we will not use this, we remark that all results in the last two subsections hold with Fq being replaced by any field k provided that the characteristic is excellent. This can be seen from the argument provided above. 2.3. The group algebra kU(Fq ) is a cocommutative Hopf algebra with counit  : kU(Fq ) → k defined by (g) = 1 for all g ∈ U(Fq ). Let I = Ker() be the augmentation ideal. The filtration kU(Fq ) = I 0 ⊇ I ⊇ · · · ⊇ n n+1 I 2 · · · is the radical filtration of kU(Fq ). Define gr kU(Fq ) = ⊕∞ . n=0 I /I Then gr kU(Fq ) is a cocommutative Hopf algebra (see [Br]). For a restricted Lie algebra n, denote by u(n) the restricted enveloping algebra of n. Theorem (Quillen). As finite-dimensional algebras gr kU(Fq ) and u(n+ Fq ⊗F p k) are isomorphic. Proof. By Quillen’s theorem [Q, Thm. 1], we have an isomorphism of algebras gr kU(Fq ) ∼ = u(gr U(Fq ) ⊗F p k). The theorem follows from Proposition 2.2. t u Remark. For a given p-group Γ, Jennings [Jen] studied the associated graded algebra gr kΓ. He constructed a basis for this algebra using the lower central p-series of Γ. This basis is precisely the PBW basis of the restricted enveloping algebra u(gr Γ ⊗F p k) as constructed by Jacobson [Jac]. Incidentally, both articles appeared in the same issue of the Transactions of AMS in 1941. The algebra isomorphism was not established until 1968 by Quillen. The associated Lie ring gr Γ for any group Γ and prime number p was constructed as early as 1940 by H. Zassenhaus [Z]. The authors acknowledge Leonard Scott for drawing their attention to Jennings’ work. Set Γ0n = {x ∈ U(Fq ) | (x − 1) ∈ I n }. Then Γ0n is a normal subgroup of U(Fq ) and satisfying [Γ0n , Γ0m ] ⊆ Γ0n+m and (Γ0n ) p ⊆ Γ0n p . By definition, we have Γn ⊆ Γ0n . By [La, Thm 6.1] and [Jen], we have Γ0n = Γn for all n. 2.4. Filtrations of modules. The algebra A = kU(Fq ) is a filtered algebra with filtration defined by F n A = I n . A filtered A-module is an A-module M with a filtration · · · ⊇ F i M ⊇ F i+1 M ⊇ · · · of A-submodules of M such that I n F i M ⊆ F i+n M for all n and i. The associated graded vector space gr M = ⊕i F i M/F i+1 M is a graded gr A-module such that elements of F n A/F n+1 A act as linear transformations from F i M/F i+1 M to F i+n M/F i+n+1 M.

Complexity of modules

91

For the Borel subgroup B containing T given in Section 1.4, observe that all simple B-modules are 1-dimensional and denoted by kλ with λ ∈ X(T). The injective hull of kλ is Y(λ) = k[U] ⊗ kλ , which has weights in λ − NΦ+ . For each indecomposable finite-dimensional B-module M, there is a weight λ ∈ X(T) such that all other weights of M are in λ − NΦ+ . The height function ht can be extended to the set NΦ, such that ht(γ) is the sum of the coefficients of simple roots in the expression of γ as a linear combination of simple roots. We define F i M = ⊕γ ∈NΦ,ht(γ)≥i Mλ+γ . For the sake of convenience, we can shift the degree of the filtration such 0 1 i that P F M/F M 6 = 0. Then the resulting filtration is defined to be F M = ht(µ−λ)−t≥i Mµ for some t ∈ Z which depends on λ. It follows from the definition of the filtration on M that there is a natural isomorphism gr M ∼ = M as vector spaces. For an arbitrary finite dimensional B-module M, we write M as a direct sum of indecomposable B-modules. Then define the filtration of M as the direct sum of the filtrations of each indecomposable component. Thus we have a restricted n+ k -module structure on gr M via ∼ the algebra isomorphism u(n+ ) gr kU(F = p ). On the other hand, M is k a restricted n+ -module by restriction. The following proposition provides k + an isomorphism of gr M and M as restricted nk -modules. Proposition. Let A = kU(Fq ) and M be a B-module. (a) Each F i M is a U-submodule and upon restriction to U(Fq ) the aforementioned filtration defines a filtered A-module. (b) If q = p, the natural linear isomorphism M → gr M is an isomorphism of restricted n+ k -modules under the algebra isomorphism given in Theorem 2.3. P Proof. For each positive root α, we have (xα (t) − 1)Mµ ⊆ i>0 Mµ+iα . Consequently, F i M is invariant under each root subgroup xα (k). Since ht(α) > 0 and the augmentation ideal I is generated by {xα (t) − 1 | t ∈ Fq , α ∈ Φ+ } as a two sided ideal in A, we have IF i M ⊆ F i+1 M. Thus (a) holds. To show (b), we recall that the Quillen isomorphism sendsPx ∈ Γn to x − 1 in I n /I n+1 . The element x = xα (t) acts on M as 1 + teα + i>1 t i e(i) α . The linear transformation x − 1 : F i M/F i+1 M → F i+ht(α) M/F i+ht(α)+1 M is the same as teα because X

n n+2 ht(α) t i e(i) M ⊆ F n+ht(α)+1 M. α F M ⊆ F

i>1

Note that F i M/F i+1 M = ⊕ht(µ−λ)−t=i Mµ (for some λ and t which defines the filtration) and the n+ k -module structure on M is determined by the action of eα : Mµ → Mµ+α . Therefore, under the isomorphism M → gr M, the action of (xα (t) − 1) is the same as the action of teα on M. t u

92

Z. Lin, D.K. Nakano

3. May spectral sequence and some consequences 3.1. For a filtered algebra A with filtration A = F 0 A ⊇ F 1 A ⊇ F 2 A ⊇ · · · and filtered left (and right) A-modules M (and N) with decreasing filtration · · · ⊇ F i M ⊇ F i+1 M ⊇ F i+2 M ⊇ · · · , May [May] defined gr A a spectral sequence E 1 = Tori+ j (gr N, gr M) converging to Tor•A (N, M). We will use a slightly different setup by constructing a zero graded version. All tensor products will be taken over the field k unless otherwise specified. The standard chain complex is defined by Bn (N, M) = N ⊗ A⊗n ⊗ M with differential d : Bn (N, M) → Bn−1 (N, M) defined by (3.1.1) d(a0 ⊗ a1 ⊗ · · · ⊗ an ⊗ an+1 ) =

n X (−1)i a0 ⊗ · · · ⊗ ai ai+1 ⊗ · · · ⊗ an+1 . i=0

Then TornA (N, M) = Hn (B•(N, M)). Observe that N ∗ = Homk (N, k) is a left A-module. Set C n (N, M) = Bn (N, M)∗ . Then ExtnA (M, N ∗ ) = H n (C • (N, M)). Set Y i = F i Y for Y = A, N, M. The complexes B• (N, M) and C • (N, M) have natural filtered structures defined by X (3.1.2) Fi Bn (N, M) = (N i0 ⊗ Ai1 ⊗ · · · ⊗ Ain ⊗ M in+1 ), n+1 Σs=0 is ≥−i

(3.1.3)

F i C n (N, M) = (Bn (N, M)/Fi−1 Bn (N, M))∗ .

Then Fi B• (N, M) ⊆ Fi+1 (B•(N, M)) and F i+1 C • (N, M) ⊆ F i C • (N, M). The filtered complexes yield respectively spectral sequences with first pages E i,1 j = Hi+ j (Fi B(N, M)/F j−1 B(N, M)), E 1 = Extgr A (gr M, (gr N)∗ )i . i, j

i+ j

If the filtrations of A, N, M are bounded, then the spectral sequences E i,r j i, j i+ j A ∗ and Er converge to Tori+ j (N, M) and Ext A (M, N ), respectively. For the graded algebra gr A, the complexes B• (gr N, gr M) and C • (gr N, gr M) are naturally graded with homogeneous parts (3.1.4) (3.1.5)

Fi Bn (N, M)/Fi−1 Bn (N, M) ∼ = Bn (gr N, gr M)−i , i • i+1 • F C (N, M)/F C (N, M) ∼ = (B• (gr N, gr M)−i )∗ ∼ = C • (gr N, gr M)i .

Note that the differentials of the complexes B• (gr N, gr M) and A C • (gr N, gr M) are homogeneous of degree 0. Thus Torgr n (gr N, gr M) n and Extgr A (gr M, (gr N)∗ ) have natural graded structures. Computing the

Complexity of modules

93

E 1 and E 1 pages for these two spectral sequences, we get converging spectral sequences (3.1.6)

gr A

A E i,1 j = Tori+ j (gr N, gr M)−i ⇒i Tori+ j (N, M),

(3.1.7) E 1 = Extgr A (gr M, (gr N)∗ )i ⇒i Ext A (M, N ∗ ). i, j

i+ j

i+ j

3.2. Let A = kU(F p ) and M be a finite-dimensional rational B-module. Let M = F 0 M ⊇ F 1 M ⊇ · · · be the filtration of M defined in Section 2.4. Under this filtration we have gr A ∼ = Dist(U1 ) by Quillen’s iso= u(n+ ) ∼ ∼ morphism. Moreover, gr M = M as Dist(U1 )-modules. As a special case of the construction given in Section 3.1, one obtains the following result. Theorem. Let M be a finite dimensional rational B-module. Then there exists a spectral sequence i, j

E 1 = H i+ j (U1 , M)i ⇒ H i+ j (kU(F p ), M). 3.3. Complexity and support varieties. This subsection provides a quick review of the notions of complexity and support varieties. The reader is referred to [Ben] for further details and proofs of these results. Let A be a finite-dimensional cocommutative Hopf algebra over k and mod(A) be the category of finite-dimensional A-modules. The cohomology ring R := H 2• (A, k) is finitely generated and for M ∈ mod(A), H • (A, M) is a finitely generated module over R. Let M ∈ mod(A) and set |A| M to be the maximal ideal spectrum of R/JM where JM is the annihilator ideal of R on H • (A, M ⊗ M ∗ ). The variety |A| M is called the cohomological support variety of M. Let V = {Vn : n = 0, 1, · · · } be a sequence of finite-dimensional vector spaces over k. The rate of growth r(V) of V is defined to be the smallest positive integer c such that dimk Vn ≤ D · n c−1 for some constant D and n = 0, 1, · · · . For M ∈ mod(A) the complexity of M denoted by c A (M) is the rate of growth of the minimal projective resolution P• of M. An A-module M is periodic if and only if it admits a periodic projective resolution. The basic facts given below will be used in the upcoming sections. (i) (ii) (iii) (iv)

An A-module M is projective if and only if c A (M) = 0. A non-projective A-module M is periodic if and only if c A (M) = 1. c A (M) = dim |A| M for all M ∈ mod(A). If A has the trivial module k as its only simple module then c A (M) = r({H • (A, M)}).

3.4. The following result provides a comparison between the complexity of rational G-modules (resp. B-modules) upon restriction to G 1 (resp. B1 ) and G(F p ) (resp. B(F p)).

94

Z. Lin, D.K. Nakano

Lemma. Let G be a connected reductive algebraic group. Then for any rational G-module M, 1 cU1 (M) = c B1 (M) = cG 1 (M). 2 Proof. Observe that the algebraic group G acts on the variety |G 1 |k and |G 1 | M is a G-invariant closed subvariety of |G 1 |k . By [FP2, (1.2) Thm.], |B1 | M is a closed subvariety of |G 1 | M such that |G 1 | M = G · |B1 | M . Also, Friedlander and Parshall [FP2, §1] constructed a finite G-equivariant map Ψ : |G 1 |k → Lie(G) with the adjoint G-action on Lie(G) and Ψ(|G 1 | M ) ⊆ N , the variety of nilpotent elements. By the work of Holt and Spaltenstein [HS, Thm. 1, Thm. 2], N has only finitely many Gorbits and for each orbit O, one has dim O = 2 dim(O ∩ n+ k ). Therefore, there is an orbit O ⊆ Ψ(|G 1 | M ) such that dim Ψ(|G 1 | M ) = dim O and + dim(O ∩ n+ k ) = dim(Ψ(|G 1 | M ) ∩ nk ) (see [Hum4, 6.7]). Hence, (3.4.1)

dim Ψ(|G 1 | M ) = 2 dim(Ψ(|G 1 | M ) ∩ n+ k ) = 2 dim Ψ(|U1 | M ).

Now the lemma follows because Ψ(|B1| M ) = Ψ(|G 1 | M ) ∩ Lie(B) = u t Ψ(|U1 | M ). Theorem. Let G be a connected reductive algebraic group and B be a Borel subgroup of G. (a) If M is a rational B-module then c B(F p ) (M) ≤ c B1 (M). (b) If M is a rational G-module then cG(F p ) (M) ≤ 12 cG 1 (M). Proof. (a). Let M be a rational B-module and consider the spectral sequence given in Theorem 3.2: i, j

E 1 = H i+ j (U1 , M)i ⇒ H i+ j (kU(F p ), M). i, j

For each n ≥ 0 set V n = ⊕i, j:i+ j=n E 1 . Then V n = ⊕i≤0 ⊕ j: j=n−i H i+ j (U1 , M)i = H n (U1 , M). Since E ∞ is a subquotient of E 1 , it follows that (3.4.2)

cU1 (M) = r(V • ) ≥ r(H • (U(F p ), M)) = cU(F p ) (M).

Furthermore, U(F p ) is a p-Sylow subgroup of B(F p ), thus c B(F p ) (M) = cU(F p ) (M). But, by Lemma 3.4, c B1 (M) = cU1 (M). Hence, c B(F p ) (M) ≤ c B1 (M). (b). Now let M be a rational G-module. Since, U(F p ) is a p-Sylow subgroup of G(F p ), we have cG(F p ) (M) = c B(F p ) (M) = cU(F p ) (M). The result now follows from part (a) and the lemma above. t u Corollary. Let M be a finite-dimensional rational B-module. (a) If M is a projective B1 -module then M is a projective B(F p)-module.

Complexity of modules

95

(b) If M is a periodic B1-module then M is a periodic or a projective B(F p )-module. Proof. Note that c D (M) = 1 holds if and only if M is periodic for D = B1 or B(F p ). The result immediately follows by (a) of the preceding theorem. t u 3.5. Parshall’s conjecture. We can now answer affirmatively the question presented by Parshall in 1987 [P, §5.3]. Corollary. Let G be a connected reductive algebraic group and M be a finite-dimensional rational G-module. If M is a projective G 1 -module then M is a projective G(F p )-module. Proof. Let M be a finite dimensional rational G-module which is projective upon restriction to G 1 . Since 0 = cG 1 (M) ≥ 2cG(F p ) (M) by Theorem 3.4, t u M must be projective over G(F p ). 3.6. A natural question is whether the converse to Parshall’s conjecture holds, i.e., a finite-dimensional G-module that is projective upon restriction to G(F p ) is also projective as G 1 -module. One might immediately reject this converse based on two reasons. First, G(F p ) has projective modules that are proper G(F p )-direct summands of a projective indecomposable G 1 -modules Q 1 (λ) (at least for p ≥ 2(h − 1)). The reader is referred to [Jan2] for details. But, the projective G(F p )-module obtained in this way may not admit a G-structure. Second, one can consider the projective G 1 -module Q 1 (λ), that extends to a G-module structure. The Frobenius twist Q 1 (λ)(1) remains projective upon restriction to G(F p ). However, the module Q 1 (λ)(1) is a direct sum of trivial modules upon restriction to G 1 and is clearly not projective as G 1 -module. 4. Projective modules with G-structures 4.1. Let Cp be the full subcategory of G-modules with all weights λ satisfying hλ, β ∨ i < p( p − 1) for the highest root β of each component. We will use the notation Cp (G) if there is more than one group under consideration. In this section we will show the converse to Parshall’s conjecture is true for all G-modules in Cp . We first consider G = SL 2. The set of T -characters X(T) is identified with Z. A weight λ is dominant if λ ≥ 0. All irreducible G-modules L(λ) in Cp have highest weights λ ∈ X(T) satisfying hλ, α∨ i < p( p − 1). Observe that this condition prevents the Frobenius twist of a projective G 1 -module from being in Cp . The category Cp is a highest weight category in the sense of [CPS]. Therefore, Cp has enough projective and injective objects. Let P(λ) be the projective cover of L(λ) in Cp , and H 0 (λ) and V(λ) be the induced and Weyl modules of highest weight λ respectively.

96

Z. Lin, D.K. Nakano

For 0 ≤ λ0 < p − 1, define for each i ∈ N,  0 λ + ip if i is even, λi = (4.1.1) p − 2 − λ0 + i p if i is odd. Then the simple module L(λi ) is in Cp if and only if i < p − 1. The following lemma can be easily derived from [Jan1] by using known facts about extensions of simple modules and translation functors for SL 2. Lemma. Let λ = λ0 + pλ1 < p2 − p. (a) For λ0 = p − 1, then P(λ) = St ⊗L(λ1)(1) ∼ = V(λ) is = H 0 (λ) ∼ 1 irreducible if λ ≤ p − 2; (b) For λ0 < p − 1, then P(λi ) for i < p − 2 is also the injective hull of L(λi ) in Cp and (4.1.2)

 L(λi−1 ) ⊕ L(λi+1 ) ∼ RadG P(λi )/ SocG P(λi ) = L(λ1 )

if 0 < i < p − 2, if i = 0.

Furthermore, for i = p − 2, P(λ p−2 ) = V(λ p−2 ) while H 0 (λ p−2 ) is the injective hull of L(λ p−2 ) in Cp . The blocks in Cp are classified as follows. For λ0 = p − 1, then each L(λ0 +i p) itself is a block for i = 0, 1, · · · , p−2. For λ0 < p−1, let B(λ0 ) be the block consisting of simple modules L(λi ) for i = 0, 1, · · · , p − 2. 4.2. In this subsection, we classify all indecomposable modules in Cp for G = SL 2. As a consequence, one sees that the corresponding quasihereditary algebra as in [CPS] is of finite representation type. Donkin and Reiten in [DR] observed this fact, but a classification of all indecomposable modules was not provided. The Loewy length of a module is the smallest length of a filtration of the module such that the subquotients are semisimple. A module M has Loewy length l if and only if Radl M = 0 and Radl−1 M 6 = 0. For example, M has Loewy length 1 if and only if M is semisimple. By Lemma 4.1, each module in Cp has Loewy length at most 3. Projective modules with Loewy length 3 are also injective. Therefore, any indecomposable module M of Loewy length 3 has to be projective and injective in Cp . Any indecomposable module of Loewy length 1 is irreducible. If M has Loewy length 2 and is indecomposable, then M has to be in a block B(λ0 ) for some 0 ≤ λ0 < p − 1. Let us fix such λ0 and set p−2 P = ⊕i=0 P(λi ). Let A = EndG (P) be the basic algebra for the block B (λ0 ). Then the category of finite dimensional A-modules is Morita equivalent to the block B(λ0 ) in Cp . The quiver Γ of the basic algebra A is given in Fig. 2. The simple module L(λi ) corresponds to X i and s = p − 2. Note that an indecomposable A-module has Loewy length at most two if and

Complexity of modules



• X0

α1 β1

97

- • X1

α2 β2

-•



· · ··

• X s−1

X2

αs βs

-• Xs

Fig. 2.

only if it is a module for the algebra B = A/(Rad A)2 . The algebra B is constructed in [Pi, 8.1] as the quotient algebra of the path algebra of the quiver Γ modulo paths of length 2. All indecomposable B-modules can be classified by using the corresponding separated quiver Γs of the quiver Γ as in [Pi, Lemma 8.3] and their dimension vectors are positive roots of the root system of type A p−1 . The following proposition summarizes the above argument. Proposition. If M is an indecomposable module in Cp , then one of the following holds: (i) M is irreducible; (ii) M is both projective and injective (if and only if M has Loewy length 3); (iii) M (or its dual) has the module structure: L(λs ) (4.2.1)



L(λs+1 )



L(λs+2 )



··· · · · L(λt−1)



L(λt )

.

Remark. Let l be an odd positive integer and ξ be a primitive l-th root of 1 in C. The simple modules for the quantum group Uξ (sl2 ) are denoted by L l (λ) with nonnegative integers λ. Write λ = λ0 + lλ1 with 0 ≤ λ0 ≤ l − 1. The blocks of Uξ (sl2 ) are determined by the strong linkage class as follows. If λ0 = l − 1 then L l (l − 1 + lλ1 ) is injective and projective. If λ0 < l − 1, the block containing L ξ (λ0 ) has simple modules L ξ (λi ) (i = 0, 1, · · · ) with λi defined as in Section 4.1. For a fixed positive integer i 0 , one can consider the highest weight category of modules with composition factors of the form L(λi ) with i < i 0 . Now the argument in Section 4.2 also classifies all finite dimensional indecomposable Uξ (sl2 )-modules. All modules of Loewy length 3 are projective and injective while all indecomposable modules of Loewy length 2 are as described in (4.2.1). For the quantum group case, there is no restriction on t. 4.3. For 0 < λ0 < p − 1, by the Steinberg tensor product theorem, we have dim L(λ0 + i p) = (λ0 + 1)(i + 1) and dim L( p − λ0 − 2 + i p) = ( p − λ0 − 1)(i + 1). In particular, we have dim L(λs ) + dim L(λs+1) ≡ (−1)s+1 (λ0 + 1) (mod p). If M is an indecomposable module in Cp of Loewy length 2, using Proposition 4.2, dim M (mod p) can be computed easily according to t − s + 1 (the number of composition factors of M) and s are even or odd. For example, when s and t − s + 1 both are odd (t is thus odd), then t+s+2 0 t−s 0 dim M ≡ (λ +1)+( p−1−λ0 )(t+1) ≡ − (λ +1) (mod p). 2 2

98

Z. Lin, D.K. Nakano

In all cases, we have dim M 6 ≡ 0 (mod p). Theorem. Let G = SL 2 and M be a finite-dimensional module in C p (SL 2). If M is a projective G(F p )-module then M is a projective G 1 -module. Proof. We can assume that M is indecomposable in Cp . If M|G(F p ) is projective, then M|U(F p ) is projective. Since U(F p ) is cyclic of order p and kU(F p ) is a local ring, then all projective modules are free and thus have dimension divisible by p. If M is simple, then M ∼ = L( p − 1 + i p) ∼ = St ⊗L(i)(1), which is projective when restricted to G 1 . If M has Loewy length 3 then M|G 1 is projective by Lemma 4.1. If M has Loewy length 2, the above computation shows that dim M is not divisible by p, so M cannot be projective t u over G(F p ) or G 1 .

4.4. For higher rank groups G, we can assume the root system is irreducible. For any simple long root α, let G α be the semisimple subgroup of G generated by the root subgroups xα and x−α . Then G α is isomorphic to SL 2. It is immediate by the definition of C p that for any M in C p (G), M|G α is in Cp (G α ). The group G α is also defined over F p . In fact, for any simple G-module L(λ) in Cp (G), if L α (µ) (for some µ ∈ X(T) with hµ, α∨ i ≥ 0) is a composition factor of L(λ)|G α (i.e., an irreducible factor for a minimal Levi subgroup), then µ is a weight of L(λ). The highest root β has to be a long root if the root system has two different root lengths. Consequently, there is a Weyl group element w ∈ W such that w(α) = β. Since w(µ) is also a weight of L(λ), we can write w(µ) = λ − γ with γ ∈ NΦ+ . Therefore, by [Hum1, 10.4] (4.4.1)

hµ, α∨ i = hλ, β ∨ i − hγ, β ∨ i ≤ hλ, β ∨ i < p( p − 1).

Theorem. If M is a G-module in Cp , such that M|G(F p ) is projective, then M|G 1 is also projective. Proof. Note that the projectivity of M|G(F p ) implies that M|G α (F p ) is also projective for a long simple root α. Therefore, by Theorem 4.3, M|(G α )1 is also projective. In particular, M| is projective for the 1-dimensional restricted Lie subalgebra kxα of Lie(G). By [FP2, (2.4b) Prop.], M|G 1 is projective. t u Remark. Alperin and Mason [AM2] have shown that in the case of a type A, D, E root system, a simple G(Fq )-module V is free when restricted to a root subgroup xα (Fq ) if and only if V is isomorphic to the Steinberg module Str where q = pr . Note that all simple G(Fq )-modules are G-modules. Consequently, our result shows that the Steinberg module is the only projective simple module over G(F p ) (which is well-known).

Complexity of modules

99

4.5. Recall that [FP2, (2.4b) Prop.] states that a G-module is projective as G 1 -module if and only if M| is projective for all roots α and if and only if M| is projective for a long root α. The proof of this result uses the fact that the support variety of M|G 1 is G-invariant. In general we do not have a G-action on the support variety of M|G(F p ) . However, we do have the following result for the finite group G(F p ). Theorem. If M is a G-module in C p , then the following are equivalent: (a) M|G(F p ) is projective for the group G(F p ); (b) M|xα (F p ) is projective for all roots α; (c) M|xα (F p ) is projective for a long root α. Proof. Clearly (a) ⇒ (b) ⇒ (c) holds for any G(F p )-module M. We will now prove that (c) ⇒ (a). By using (4.4.1), the condition for M in C p is equivalent to the condition that |hλ, α∨ i| < p( p − 1) for all weights λ of M and all long roots α. This condition is independent of the choice of the set of positive roots. Therefore, we can choose a basis of the root system such that α is a simple root. Since M|G α (F p ) is projective, the proof of Theorem 4.4 shows that M|G 1 is projective. Thus M|G(F p ) is projective by Corollary 3.5. t u Remark. Chouinard [Ch] proved that a module for a finite group is projective if and only if its restriction to each elementary p-subgroup is projective. This was generalized by Alperin and Evens in a theorem involving complexities of modules and further developed using a stratification theorem on support varieties by Avrunin and Scott. In Theorem 4.5, xα (F p ) is elementary of rank 1. 4.6. Theorem 4.4 can be used to prove the following result involving tensor products of G(F p )-modules. We do not know any direct way to prove it without using the result for Lie algebras. Corollary. Let M and V be rational G-modules such that M ⊗ V is in Cp . If (M ⊗ V)|G(F p ) is projective, then either M|G(F p ) is projective or V |G(F p ) is projective. Proof. By Theorem 4.4, M|G 1 ⊗ V |G 1 = (M ⊗ V)|G 1 is projective. Consequently, by [FP2, (2.4c) Prop.], either M|G 1 or V |G 1 is projective. The corollary now follows from Corollary 3.5. t u 4.7. By Theorem 3.4(b), for any G-module M, (4.7.1)

1 cG(F p ) (M) ≤ cG 1 (M). 2

Examples mentioned in Section 3.6 show that the equality does not hold in general. But it seems to be reasonable to ask this question for M in the category Cp . Indeed, by Theorem 4.4 and Corollary 3.5, cG(F p ) (M) = 0 if

100

Z. Lin, D.K. Nakano

and only if cG 1 (M) = 0 provided that M is in Cp . However, if M = k is the trivial G-module, cG(F p ) (M) is the maximal rank of elementary Abelian p-subgroups. But, cG 1 (M) is the number of roots (for sufficiently large prime p) and the equality does not hold. An interesting question would be to find an equation relating cG(F p ) (M) and cG 1 (M) for G-modules M in Cp . References [A] [AM1]

J.L. Alperin, Diagrams for modules. J. Pure Appl. Algebra 16 (1980), 111–119 J.L. Alperin, G. Mason, On simple modules for SL(2, q). Bull. London Math. Soc. 25 (1993), no. 1, 17–22 [AM2] J.L. Alperin, G. Mason, Partial Steinberg modules for finite groups of Lie type. Bull. London Math. Soc. 25 (1993), no. 6, 553–557 [ABS] H. Azad, M. Barry, G. Seitz, On the structure of parabolic groups. Comm. Algebra, 18 (2) (1990), 551–562 [Baj] A.M. Bajer, The May spectral sequence for a finite p-group stops. J. Algebra 167 (1994), 448–459 [Bal] J.W. Ballard, Injective modules for restricted enveloping algebras. Math. Zeit. 163 (1978), 57–63 [Ben] D.J. Benson, Representations and cohomology. II. Cohomology of groups and modules. Cambridge Studies in Advanced Mathematics, 31. Cambridge University Press, Cambridge, 1991 [Br] W. Browder, On differential Hopf algebras. Trans. Amer. Math. Soc. 107 (1963), 153–176 [Ch] L.G. Chouinard, Projectivity and relative projectivity over group rings. J. Pure Appl. Algebra 7 (1976), no. 3, 287–302 [CPS] E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391 (1988), 85–99 [CPSvan] E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic cohomology. Inv. Math. 39 (1977), 143–163 [Don] S. Donkin, On tilting modules for algebraic groups. Math. Zeit. 212 (1993), 39–60 [DR] S. Donkin, I. Reiten, On Schur algebras and related algebras. V. Some quasihereditary algebras of finite type. J. Pure Appl. Algebra 97 (1994), no. 2, 117–134 [FP1] E.M. Friedlander, B.J. Parshall, On the cohomology of algebraic and related finite groups. Invent. Math. 74 (1983), 85–117 [FP2] E.M. Friedlander, B.J. Parshall, Support varieties for restricted Lie algebras. Invent. Math. 86 (1986), 553–562 [FP3] E.M. Friedlander, B.J. Parshall, Geometry of p-unipotent Lie algebras. J. Algebra 109 (1987), 25–45 [HS] D.F. Holt, N. Spaltenstein, Nilpotent orbits of exceptional Lie algebras over algebraically closed fields of bad characteristics. J. Austral. Math. Soc. Ser. A 38 (1985), 330–350 [Hum1] J.E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer 1972 [Hum2] J.E. Humphreys, Linear Algebraic Groups, Springer 1975 [Hum3] J.E. Humphreys, Ordinary and Modular Representations of Chevalley Groups, Lecture Notes in Math. 528, Springer 1976 [Hum4] J.E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups Mathematical Surveys and Monographs 43. Amer. Math. Soc., Providence, 1995 [Jac] N. Jacobson, Restricted Lie algebras of characteristic p. Trans. AMS 50, (1941), 15–25

Complexity of modules [Jan1] [Jan2]

[Jen] [La] [May] [MM] [P] [Pi] [Q] [St] [Z]

101

J.C. Jantzen, Representations of Algebraic Groups. Academic Press, 1987 J.C. Jantzen, Representations of Chevalley groups in their own characteristic. The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), 127–146, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI. 1987 S.A. Jennings, The structure of the group ring of a p-group over a modular field. Trans. AMS 50, (1941), 175–185 M. Lazard, Sur les groupes nilpotents et les anneaux de Lie. Ann. Ecole Norm. Sup´er. 71 (1954), 101–190 J.P. May, The cohomology of Lie algebras and of Hopf algebras. J. Algebra, 3 (1966), 123–146 J. Milnor, J.C. Moore, On the structure of Hopf algebras. Ann. Math. (2) 81 (1965), 211–264 B.J. Parshall, Cohomology of algebraic groups, The Arcata Conference on Representations of Finite Groups. (Arcata, Calif., 1986), 233–248, Proc. Sympos. Pure Math., 47, Part 1, Amer. Math. Soc., Providence, RI, 1987 R.S. Pierce, Associative Algebras, Graduate Texts in Mathematics 88. Springer, New York/Heidelberg/Berlin. 1982 D. Quillen, On the associated graded ring of a group ring. J. Algebra 10 (1968), 411–418 R. Steinberg, Lectures on Chevalley Groups. Yale University, 1967 H. Zassenhaus, Ein Verfahren, jeder endlichen p-Gruppe einen Lie-Ring mit der Charakteristik p zuzuordnen. Abh. Math. Sem. Univ. Hamburg, 13 (1940), 200–207