COMPLEXITY FOR MODULES OVER FINITE CHEVALLEY GROUPS AND CLASSICAL LIE ALGEBRAS

ZONGZHU LIN AND DANIEL K. NAKANO

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 the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = \overline{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 between 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 characteristics) (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,

(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 irreducible in both categories). Humphreys’ idea was to use this fact to realize the restricted indecomposable projective 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)

Date: March 1999.

1991 Mathematics Subject Classification. Primary 17B55, 20G; Secondary 17B50.

Research of the first author was supported in part by NSF grant DMS-9401389.

Research of the second author was supported in part by NSF grant DMS-9800960.

This paper will appear in Inventiones Mathematicae.

[Jan1]. 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 false 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 Figure 1 commutes. The

[figure: mod(B); mod(𝔫^{+}); mod(U(𝔽_{p})); gr; res; res]

Figure 1

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(𝔽_{q}) with q = p^{r}. It turns 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 C_{p} 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 C_{p}, the converse to Parshall conjecture for modules in C_{p} 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 = \overline{𝔽_{p}} and split over 𝔽_{p}. Without a 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 Δ 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. Let H be an affine algebraic group scheme over k and let H_{1} = ker(F r). Here F r : H → H^{(1)} is the Frobenius map. There is a categorical equivalence between restricted Lie(H)-modules and H_{1}-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 {e_{α}, h_{α_{i}} : α ∈ Φ, α_{i} ∈ Δ}. For each α, β ∈ Φ, let [e_{α}, e_{β}] = c_{αβ}e_{α+β}. The c_{αβ} are the ℤ-valued structure constants. Recall that each positive root β can be written as β = ∑_{α∈Δ} n_{α}α with n_{α} ∈ ℕ. The height of β is defined to be ht(β) = ∑_{α∈Δ} 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 ≠ 2 if Φ has a subroot system of type B_{n}, C_{n}, F_{4}, and p ≠ 2, 3 if Φ has a component of type G_{2}. When p is excellent, then c_{α,β} ≢ 0 mod p if α + β is a root by Chevalley’s theorem (cf. [Hum1, Theorem 25.2(c)]).

1.5. Acknowledgments. The authors would like to thank Christopher P. Bendel and Jens C. 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 (t_{1}, t_{2}, …, t_{N}) → ∏_{i=1}^{N} x_{α_{i}}(t_{i}) is an isomorphism of algebraic varieties defined over 𝔽_{p} where N is the number of positive roots.

(ii) The Frobenius morphisms F r : 𝔸^{N} → (𝔸^{N})^{(1)} is compatible under x with the restric-

tion to U of the Frobenius morphism F r : G → G^{(1)}. (iii) For any α, β ∈ Φ^{+} there exist integers c_{ij} ∈ ℤ such that

[x_{α}(s), x_{β}(t)] = x_{α+β}(c_{αβ}st) ∏_{i+j>2,i,j>0} x_{iα+jβ}(c_{ij}s^{i}t^{j}).

By convention, x_{iα+jβ} is the trivial homomorphism if iα + jβ is not a root. Fix r ≥ 1 and set q = p^{r}. Let G(𝔽_{q}) = G^{F r^{r}} be the 𝔽_{q}-rational points. The group

G(𝔽_{q}) is a finite group of Lie type. Furthermore, U(𝔽_{q}) = U^{F r^{r}} is a p-Sylow subgroup of G(𝔽_{q}). Let {Γ_{n}} be the lower central series of U(𝔽_{q}) defined setting Γ_{1} = U(𝔽_{q}) and Γ_{i+1} = [U(𝔽_{q}), Γ_{i}]. Set gr U(𝔽_{q}) = ⊕_{n≥1}Γ_{n}/Γ_{n+1}. Following Lazard [La], gr U(𝔽_{q}) is a graded Lie algebra over the field 𝔽_{p} (also see Jennings [Jen]). The bracket is defined by [x̄, ȳ] = \overline{[x, y]} ∈ Γ_{n+m}/Γ_{n+m+1} for all x̄ ∈ Γ_{n}/Γ_{n+1} and ȳ ∈ Γ_{m}/Γ_{m+1} where x ∈ Γ_{n} and y ∈ Γ_{m} are pre-images of x̄ and ȳ, respectively. By [ABS, Lemma 4], we have Γ_{n} = ∏_{ht(α)≥n} x_{α}(𝔽_{q}). Furthermore, Γ_{n}/Γ_{n+1} ≅ ∏_{ht(α)=n} x_{α}(𝔽_{q}) as Abelian groups and as vector spaces over 𝔽_{p}. In fact, both are vector spaces over 𝔽_{q} and the isomorphism is 𝔽_{q}-linear.

Let 𝔫_{ℤ}^{+} be the Lie ring generated by the Chevalley basis {e_{α} | α ∈ Φ^{+}} in 𝔤. Set 𝔫_{𝔽_{q}}^{+} = 𝔫_{ℤ}^{+} ⊗_{ℤ} 𝔽_{q}.

Proposition . As Lie algebras over 𝔽_{q}, gr U(𝔽_{q}) ≅ 𝔫_{𝔽_{q}}^{+}.

Proof. Observe that \{\overline{x_{α}(1)} | α ∈ Φ^{+}, ht(α) = n\} is a basis for the 𝔽_{q}-vector space Γ_{n}/Γ_{n+1}. By (iii) above, we have [\overline{x_{α}(1)}, \overline{x_{β}(1)}] = c_{α,β}\overline{x_{α+β}(1)}. Hence, the homomorphism of Abelian groups ϕ : gr U(𝔽_{q}) → 𝔫_{𝔽_{q}}^{+} extending ϕ(\overline{x_{α}(1)}) = e_{α} defines an isomorphism of Lie algebras over 𝔽_{q}.

From this point on the two 𝔽_{q}-Lie algebras gr U(𝔽_{q}) and 𝔫_{𝔽_{q}}^{+} will be identified.

2.2. We now consider the adjoint representation of the algebraic group U on 𝔫_{k}^{+} = Lie(U). Set F_{n} = ⊕_{ht(α)≥n}ke_{α}. Then 𝔫_{k}^{+} = F_{1} ⊇ F_{2} ⊇ ⋯ defines a filtration of U-submodules of 𝔫_{k}^{+}. It is straight forward to check that (1 − x)F_{n} ⊆ F_{n+i} for all x ∈ Γ_{i}.

Lemma . Suppose that p is excellent. Then (i) Γ_{i} = {x ∈ U(𝔽_{q}) | (x − 1)F_{n} ⊆ F_{n+i}, ∀n}; (ii) x^{p} ∈ Γ_{ip} for all x ∈ Γ_{i}.

Proof. We can assume that the root system is irreducible. Suppose that x ∈ Γ_{n_{0}} \ Γ_{n_{0}+1} such that (x − 1)F_{n} ⊆ F_{n+i} for all n. If n_{0} < i, we write x = x′y′ such that x′ = ∏_{ht(β)=n_{0}} x_{β}(t_{β}) ≠ 1 and y′ ∈ Γ_{n_{0}+1}. Thus (y′ − 1)F_{n} ⊆ F_{n+n_{0}+1} for all n. In End_{k}(𝔫_{k}^{+}) we can write x − 1 = (x′ − 1) + x′(y′ − 1). Consequently, for any n,

(x′ − 1)F_{n} ⊆ (x − 1)F_{n} + x′(y′ − 1)F_{n} ⊆ F_{n+i} + x′F_{n+n_{0}+1} ⊆ F_{n+n_{0}+1}.

Since x′ ≠ 1, we can write x′ = x_{β_{1}}(t_{1})x_{β_{2}}(t_{2}) ⋯ x_{β_{r}}(t_{r}) with β_{j} ∈ Φ^{+}, ht(β_{j}) = n_{0}, and t_{j} ≠ 0. If β_{1} is the highest root in Φ, then x = x′ = x_{β_{1}}(t_{1}) is in the center of U(𝔽_{p}) and x^{p} = 1. If β_{1} is not the highest root, there exists a simple root α such that α + β_{1} ∈ Φ^{+} by [Hum1, 10.4] (Φ is irreducible). We take n = ht(α) and write

x′ − 1 = (x_{β_{1}}(t_{1}) − 1)x_{1} + (x_{β_{2}}(t_{2}) − 1)x_{2} + ⋯ (x_{β_{r}}(t_{r}) − 1)

for some x_{i} ∈ U(𝔽_{q}). Since x_{i}e_{α} − e_{α} ∈ F_{n+1} and x_{β_{j}}(t_{j})e_{α} − e_{α} − t_{j}[e_{β_{j}}, e_{α}] ∈ F_{n+n_{0}+1}, we have

(x′ − 1)e_{α} = ∑_{j=1}^{r} t_{j}c_{β_{j},α}e_{β_{j}+α} + e  (2.2.1)

for some e ∈ F_{n+n_{0}+1}. Since t_{1}c_{β_{1},α} ≠ 0, we have (x′ − 1)e_{α} ∉ F_{n+n_{0}+1} by a basis consideration. This contradicts the fact that (x′ − 1)F_{n} ⊆ F_{n+n_{0}+1}. Hence, n_{0} ≥ i and (i) follows. (ii) follows from (i) and the fact that (x^{p} − 1)F_{n} = (x − 1)^{p}F_{n} ⊆ F_{n+ip} for all n.

The Lie algebra gr U(𝔽_{q}) has a restricted structure by setting \overline{x_{α}(1)}^{[p]} = \overline{x_{α}(1)^{p}} (see [La]).

Proposition . The isomorphism between gr U(𝔽_{q}) and 𝔫_{𝔽_{q}}^{+} (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 {e_{α} : α ∈ Φ^{+}}, the isomorphism in Proposition 2.1 preserves the p-maps.

Although we will not use this, we remark that all results in the last two subsections hold with 𝔽_{q} 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(𝔽_{q}) is a cocommutative Hopf algebra with counit ϵ : kU(𝔽_{q}) → k defined by ϵ(g) = 1 for all g ∈ U(𝔽_{q}). Let I = Ker(ϵ) be the augmentation ideal. The filtration kU(𝔽_{q}) = I^{0} ⊇ ⋯ ⊇ I ⊃ I^{2} ⋯ is the radical filtration of kU(𝔽_{q}). Define gr kU(𝔽_{q}) = ⊕_{n=0}^{∞}I^{n}/I^{n+1}. Then gr kU(𝔽_{q}) 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(𝔽_{q}) ≅ u(𝔫_{𝔽_{q}}^{+} ⊗_{𝔽_{p}} k).

Proof. By Quillen’s theorem [Q, Thm. 1], we have an isomorphism of algebras gr kU(𝔽_{q}) ≅ u(gr U(𝔽_{q}) ⊗_{𝔽_{p}} k). The theorem follows from Proposition 2.2.

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 Γ′_{n} = {x ∈ U(𝔽_{q}) | (x − 1) ∈ I^{n}}. Then Γ′_{n} is a normal subgroup of U(𝔽_{q}) and satisfying [Γ′_{n}, Γ′_{m}] ⊆ Γ′_{n+m} and (Γ′_{n})^{p} ⊆ Γ′_{np}. By the definition, we have Γ_{n} ⊆ Γ′_{n}. By [La, Thm 6.1] and [Jen], we have Γ′_{n} = Γ_{n} for all n.

2.4. Filtrations of modules. The algebra A = kU(𝔽_{q}).is a filtered algebra with filtration filtration defined 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 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.

For the Borel subgroup B containing T given in Section 1.4, observe that all simple B-modules are 1-dimensional and denoted by k_{λ} with λ ∈ X(T). The injective hull of k_{λ} is Y (λ) = k[U] ⊗ k_{λ}, which has weights in λ − ℕΦ^{+}. For each indecomposable finitedimensional 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}M_{λ+γ}. For the sake of convenience, we can shift the degree of the filtration such that F^{0}M/F^{1}M ≠ 0. Then the resulting filtration is defined to be F^{i}M = ∑_{ht(μ−λ)−t≥i} M_{μ} 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_{k}^{+}) ≅ gr kU(𝔽_{p}). On the other hand, M is a restricted 𝔫_{k}^{+}-module by restriction. The following proposition provide an isomorphism of gr M and M as restricted 𝔫_{k}^{+}-modules.

Proposition . Let A = kU(𝔽_{q}) and M be a B-module.

(a) Each F^{i}M is a U-submodule and upon restriction to U(𝔽_{q}), 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.

Proof. For each positive root α, we have (x_{α}(t) − 1)M_{μ} ⊆ ∑_{i>0} M_{μ+iα}. Consequently, F^{i}M is invariant under each root subgroup x_{α}(k). Since ht(α) > 0 and the augmentation ideal I is generated by {x_{α}(t) − 1 | t ∈ 𝔽_{q}, 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 sends x ∈ Γ_{n} to \overline{x − 1} in I^{n}/I^{n+1}. The element x = x_{α}(t) acts on M as 1+te_{α} +∑_{i>1} t^{i}e_{α}^{(i)}. The linear transformation \overline{x − 1} : F^{i}M/F^{i+1}M → F^{i+ht(α)}M/F^{i+ht(α)+1}M is the same as te_{α} because

∑_{i>1} t^{i}e_{α}^{(i)}F^{n}M ⊆ F^{n+2 ht(α)}M ⊆ F^{n+ht(α)+1}M.

Note that F^{i}M/F^{i+1}M = ⊕_{ht(μ−λ)−t=i}M_{μ} (for some λ and t which defines the filtration) and the 𝔫_{k}^{+}-module structure on M is determined by the action of e_{α} : M_{μ} → M_{μ+α}. Therefore, under the isomorphism M → gr M, the action of \overline{(x_{α}(t) − 1)} is the same as the action of te_{α} on M.

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 a spectral sequence E^{1} = Tor_{i+j}^{gr A}(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 B_{n}(N, M) = N ⊗ A^{⊗n} ⊗ M with differential d : B_{n}(N, M) → B_{n−1}(N, M) defined by

d(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ⊗ a_{n+1}) = ∑_{i=0}^{n}(−1)^{i}a_{0} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1} (3.1.1)

Then Tor_{n}^{A}(N, M) = H_{n}(B_{•}(N, M)). Observe that N^{∗} = Hom_{k}(N, k) is a left A-module. Set C^{n}(N, M) = B_{n}(N, M)^{∗}. Then Ext_{A}^{n}(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

F_{i}B_{n}(N, M) = ∑_{Σ_{s=0}^{n+1}i_{s}≥−i}(N^{i_{0}} ⊗ A^{i_{1}} ⊗ ⋯ ⊗ A^{i_{n}} ⊗ M^{i_{n+1}}),  (3.1.2)

F^{i}C^{n}(N, M) = (B_{n}(N, M)/F_{i−1}B_{n}(N, M))^{∗} .  (3.1.3)

Then F_{i}B_{•}(N, M) ⊆ F_{i+1}(B_{•}(N, M)) and F^{i+1}C^{•}(N, M) ⊆ F^{i}C^{•}(N, M). The filtered complexes yield spectral sequences E_{i,j}^{1} = H_{i+j}(F_{i}B(N, M)/F_{j−1}B(N, M)) and E_{1}^{i,j} = Ext_{gr A}^{i+j}(gr M, (gr N)^{∗})_{i}, respectively. If the filtrations of A, N, M are bounded, then the spectral sequences E_{i,j}^{r} and E_{r}^{i,j} converge to Tor_{i+j}^{A}(N, M) and Ext_{A}^{i+j}(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

F_{i}B_{n}(N, M)/F_{i−1}B_{n}(N, M) ≅ B_{n}(gr N, gr M)_{−i},  (3.1.4)
F^{i}C^{•}(N, M)/F^{i+1}C^{•}(N, M) ≅ (B_{•}(gr N, gr M)_{−i})^{∗} ≅ C^{•}(gr N, gr M)_{i}. (3.1.5)

Note that the differentials of the complexes B_{•}(gr N, gr M) and C^{•}(gr N, gr M) are homogeneous of degree 0. Thus Tor_{n}^{gr A}(gr N, gr M) and Ext_{gr A}^{n}(gr M, (gr N)^{∗}) have natural graded structures. Computing the E^{1} and E_{1} pages for these two spectral sequences, we get converging spectral sequences

E_{i,j}^{1} = Tor_{i+j}^{gr A}(gr N, gr M)_{−i} ⇒_{i} Tor_{i+j}^{A}(N, M),  (3.1.6)
E_{1}^{i,j} = Ext_{gr A}^{i+j}(gr M, (gr N)^{∗})_{i} ⇒_{i} Ext_{A}^{i+j}(M, N^{∗}).  (3.1.7)

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 ≅ u(𝔫^{+}) ≅ Dist(U_{1}) by Quillen’s isomorphism. Moreover, gr M ≅ M as Dist(U_{1})- modules. As a special case of the construction given in Section 3.1, one obtains the following spectral sequence.

Theorem . Let M be a finite dimensional rational B-module. Then there exists a spectral sequence

E_{1}^{i,j} = H^{i+j}(U_{1}, 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/J_{M} where J_{M} 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 = {V_{n} : 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 dim_{k} V_{n} ≤ 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) An A-module M is projective if and only if c_{A}(M) = 0. (ii) An A-module M is periodic if and only if c_{A}(M) = 1. (iii) c_{A}(M) = dim |A|_{M} for all M ∈ mod(A). (iv) 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 Gmodules (resp. B-modules) upon restriction to G_{1} (resp. B_{1}) and G(𝔽_{p}) (resp. B(𝔽_{p})).

Lemma . Let G be a connected reductive algebraic group. Then for any rational G-module M,

c_{U_{1}}(M) = c_{B_{1}}(M) = \frac{1}{2}c_{G_{1}}(M).

Proof. Observe that the algebraic group G acts on the variety |G_{1}|_{k} and |G_{1}|_{M} is a Ginvariant closed subvariety of |G_{1}|_{M}. By [FP2, (1.2) Thm.], |B_{1}|_{M} is a closed subvariety of |G_{1}|_{M} such that |G_{1}|_{M} = G · |B_{1}|_{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 G-orbits 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}) ∩ 𝔫_{k}^{+}) (see [Hum4, 6.7]). Hence,

dim Ψ(|G_{1}|_{M}) = 2 dim(Ψ(|G_{1}|_{M}) ∩ 𝔫_{k}^{+}) = 2 dim Ψ(|U_{1}|_{M}).  (3.4.1)

Now the lemma follows because Ψ(|B_{1}|_{M}) = Ψ(|G_{1}|_{M}) ∩ Lie(B) = Ψ(|U_{1}|_{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_{B_{1}}(M).

(b) If M is a rational G-module then c_{G(𝔽_{p})}(M) ≤ \frac{1}{2}c_{G_{1}}(M).

Proof. (a). Let M be a rational B-module and consider the spectral sequence given in Theorem 3.2:

E_{1}^{i,j} = H^{i+j}(U_{1}, M)_{i} ⇒ H^{i+j}(kU(𝔽_{p}), M).

For each n ≥ 0 set V^{n} = ⊕_{i,j:i+j=n}E_{1}^{i,j}. Then

V^{n} = ⊕_{i≤0} ⊕_{j:j=n−i} H^{i+j}(U_{1}, M)_{i} = H^{n}(U_{1}, M).

Since E_{∞} is a subquotient of E_{1}, it follows that

c_{U_{1}}(M) = r(V^{•}) ≥ r(H^{•}(U(𝔽_{p}), M)) = c_{U(𝔽_{p})}(M).  (3.4.2)

Furthermore, U(𝔽_{p}) is a p-Sylow subgroup of B(𝔽_{p}), thus c_{B(𝔽_{p})}(M) = c_{U(𝔽_{p})}(M). But, by Lemma 3.4, c_{B_{1}}(M) = c_{U_{1}}(M). Hence, c_{B(𝔽_{p})}(M) ≤ c_{B_{1}}(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 c_{G(𝔽_{p})}(M) = c_{B(𝔽_{p})}(M) = c_{U(𝔽_{p})}(M). The result now follows from part (a) and the lemma above.

Corollary . Let M be a finite-dimensional rational B-module.

(a) If M is a projective B_{1}-module then M is a projective B(𝔽_{p})-module.

(b) If M is a periodic B_{1}-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 = B_{1} or B(𝔽_{p}). The result immediately follows by (a) of the preceding theorem.

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 = c_{G_{1}}(M) ≥ 2c_{G(𝔽_{p})}(M) by Theorem 3.4, 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 indecomposable projective 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 Gstructure. 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 C_{p} be the full subcategory of G-modules with all weights λ satisfying ⟨λ, β^{∨}⟩ < p(p − 1) for the highest root β of each component. We will use the notation C_{p}(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 C_{p}.

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 C_{p} have highest weights λ ∈ X(T) satisfying ⟨λ, α^{∨}⟩ < p(p − 1). Observe that this condition prevents the Frobenius twist of a projective G_{1}-module from being in C_{p}. The category C_{p} is a highest weight category in the sense of [CPS]. Therefore, C_{p} has enough projective and injective objects. Let P (λ) be the projective cover of L(λ) in C_{p}, and H^{0}(λ) and V (λ) be the induced and Weyl modules of highest weight λ respectively.

For 0 ≤ λ^{0} < p − 1, define for each i ∈ ℕ,

λ_{i} = \begin{cases}λ^{0} + ip & if i is even, \\ p − 2 − λ^{0} + ip & if i is odd. \end{cases}  (4.1.1)

Then the simple module L(λ_{i}) is in C_{p} 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 = λ^{0} + pλ^{1} < p^{2} − p.

(a) For λ^{0} = p − 1, then P (λ) = St ⊗L(λ^{1})^{(1)} ≅ H^{0}(λ) ≅ V (λ) is irreducible if λ^{1} ≤ p − 2;

(b) For λ^{0} < p − 1, then P (λ_{i}) for i < p − 2 is also the injective hull of L(λ_{i}) in C_{p} and

Rad_{G} P (λ_{i})/ Soc_{G} P (λ_{i}) ≅ \begin{cases}L(λ_{i−1}) ⊕ L(λ_{i+1}) & if 0 < i < p − 2, \\ L(λ_{1}) & if i = 0. \end{cases}  (4.1.2)

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 C_{p}.

The blocks in C_{p} are classified as follows. For λ^{0} = p − 1, then each L(λ^{0} + ip) itself is a block for i = 0, 1, ⋯, p − 2. For λ^{0} < p − 1, let B(λ^{0}) be the block consisting of simple modules L(λ_{i}) for i = 0, 1, ⋯, p − 2.

4.2. In this subsection, we classify all indecomposable modules in C_{p} for G = SL_{2}. As a consequence, one sees that the corresponding quasi-hereditary 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 Rad^{l} M = 0 and Rad^{l−1} M ≠ 0. For example, M has Loewy length 1 if and only if M is semisimple.

By Lemma 4.1, all modules in C_{p} 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 C_{p}. 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(λ^{0}) for some 0 ≤ λ^{0} < p − 1. Let us fix such λ^{0} and set P = ⊕_{i=0}^{p−2}P (λ_{i}). Let A = End_{G}(P) be the basic algebra for the block B(λ_{0}). Then the category of finite dimensional A-modules is Morita equivalent to the block B(λ^{0}) in C_{p}. The quiver Γ of the basic algebra A is given in Figure 2. The simple module L(λ_{i}) corresponds to X_{i} and s = p − 2. Note that an

[figure: •; X_{0}; •; X_{1}; •; X_{2}; •; X_{s−1}; •; X_{s}; α_{1}; β_{1}; α_{2}; β_{2}; ⋯; α_{s}; β_{s}] Figure 2

indecomposable A-module has Loewy length at most two if and only if it is a module for the algebra B = A/(Rad A)^{2}. The algebra B is constructed in [Pi, 8.1] to be 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 Γ^{s} 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 C_{p}, 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:

\begin{aligned}L(λ_{s}) & & & & L(λ_{s+2}) & & ⋯ & & & L(λ_{t}) \\ & ⧹ & & ⧸ & & ⧹ & & & ⧸ \\ & & L(λ_{s+1}) & & & & ⋯ & L(λ_{t−1}) \end{aligned} .  (4.2.1)

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 U_{ξ}(𝔰𝔩_{2}) are denoted by L_{l}(λ) with nonnegative integers λ. Write λ = λ^{0} + lλ^{1} with 0 ≤ λ^{0} ≤ l − 1. The blocks of U_{ξ}(𝔰𝔩_{2}) are determined by the strong linkage class as follows. If λ^{0} = l−1 then L_{l}(l−1+lλ^{1}) is injective and projective. If λ^{0} < l − 1, the block containing L_{ξ}(λ^{0}) has simple modules L_{ξ}(λ_{i}) (i = 0, 1, ⋯) with λ_{i} 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(λ_{i}) with i < i_{0}. Now the argument in Section 4.2 also classifies all finite dimensional indecomposable U_{ξ}(𝔰𝔩_{2})-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 < λ^{0} < p−1, by the Steinberg tensor product theorem, we have dim L(λ^{0}+ip) = (λ^{0} + 1)(i + 1) and dim L(p − λ^{0} − 2 + ip) = (p − λ^{0} − 1)(i + 1). In particular, we have dim L(λ_{s}) + dim L(λ_{s+1}) ≡ (−1)^{s+1}(λ^{0} + 1) (mod p). If M is an indecomposable module in C_{p} 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

dim M ≡ \frac{t − s}{2}(λ^{0} + 1) + (p − 1 − λ^{0})(t + 1) ≡ −\frac{t + s + 2}{2}(λ^{0} + 1) (mod p).

In all cases, we have dim M ≢ 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 C_{p}. 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 + ip) ≅ 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 over G(𝔽_{p}) or G_{1}.

4.4. For 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 x_{α} 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 C_{p}(G_{α}). The group G_{α} is also defined on 𝔽_{p}. In fact, for any irreducible G-module L(λ) in C_{p}(G), if L_{α}(μ) (for some μ ∈ X with ⟨μ, α^{∨}⟩ ≥ 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]

⟨μ, α^{∨}⟩ = ⟨λ, β^{∨}⟩ − ⟨γ, β^{∨}⟩ ≤ ⟨λ, β^{∨}⟩ < p(p − 1).  (4.4.1)

Theorem . If M is a G-module in C_{p}, 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}} also projective. In particular, M|_{<x_{α}>} is projective for the 1-dimensional restricted Lie subalgebra kx_{α} of Lie(G). By [FP2, (2.4b) Prop.], M|_{G_{1}} is projective.

Remark . Alperin and Mason [AM2] have shown that in the case of a type A, D, E root system, a simple G(𝔽_{q})-module V is free when restricted to a root subgroup x_{α}(𝔽_{q}) if and only if V is isomorphic to the Steinberg module St_{r} where q = p^{r}. Note that all simple G(𝔽_{q})-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).

4.5. Recall that [FP2, (2.4b) Prop.] states that a G-module is projective as G_{1}-module if and only if M|_{<x_{α}>} is projective for all roots α and if and only if M|_{<x_{α}>} 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|_{x_{α}(𝔽_{p})} is projective for all roots α;

(c) M|_{x_{α}(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 |⟨λ, α^{∨}⟩| < p(p−1) for all weight 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.

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, x_{α}(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 C_{p}. 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.

4.7. By Theorem 3.4(b), for any G-module M,

c_{G(𝔽_{p})}(M) ≤ \frac{1}{2}c_{G_{1}}(M).  (4.7.1)

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 C_{p}. Indeed, by Theorem 4.4 and Corollary 3.5, c_{G(𝔽_{p})}(M) = 0 if and only if c_{G_{1}}(M) = 0 provided that M is in C_{p}. However, if M = k is the trivial G-module, c_{G(𝔽_{p})}(M) is the maximal rank of elementary Abelian subgroups. But, c_{G_{1}}(M) is the number of roots (for sufficiently large prime p) and equality does not hold. An interesting question would be to find an equation relating c_{G(𝔽_{p})}(M) and c_{G_{1}}(M) for G-modules M in C_{p}.

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Department of Mathematics, Kansas State University, Manhattan, KS 66506

   E-mail address: zlin@math.ksu.edu

Department of Mathematics and Statistics, Utah St. University, Logan, UT 84322

   E-mail address: nakano@sunfs.math.usu.edu