ON THE EXISTENCE OF MOCK INJECTIVE MODULES FOR ALGEBRAIC GROUPS

WILLIAM D. HARDESTY, DANIEL K. NAKANO, AND PAUL SOBAJE

Abstract. Let G be an affine algebraic group scheme over an algebraically closed field k of characteristic p > 0, and let G_{r} denote the r-th Frobenius kernel of G. Motivated by recent work of Friedlander, the authors investigate the class of mock injective G-modules, which are defined to be those rational G-modules that are injective on restriction to G_{r} for all r ≥ 1. In this paper the authors provide necessary and sufficient conditions for the existence of non-injective mock injective G-modules, thereby answering a question raised by Friedlander. Furthermore, the authors investigate the existence of non-injective mock injectives with simple socles. Interesting cases are discovered that show that this can occur for reductive groups, but will not occur for their Borel subgroups.

1. Introduction

1.1. Let k be an algebraically closed field and G be an affine algebraic group scheme over k, defined over the prime subfield 𝔽_{p}. The 𝔽_{p}-structure on G admits a Frobenius morphism F : G → G. Let G_{r} denote the kernel of the rth iteration of the Frobenius map, F^{r}, and let G(𝔽_{q}) denote the F^{r}-fixed point subgroup of G(k), where q = p^{r}. A fundamental problem in the representation theory of G is to determine the extent to which the finite subgroup schemes G_{r} and G(𝔽_{q}) detect the cohomology of G. For semi-simple groups, the relationship between G, G_{r}, and G(𝔽_{q}) cohomology has been studied extensively (see [And1, And2], [BNP1, BNP2, BNP3, BNP4, BNP5], [CPS], [CPSvdK], and [N] for a survey).

Recently this problem has arisen in a new context. In [F1], Friedlander defined a support theory for rational G-modules when G has a particular “exponential” structure, a condition met by many important classes of groups. The aforementioned work is a generalization of the support variety theory that has been developed over the last several decades for finite group schemes and related algebraic objects, of which G_{r} and G(𝔽_{q}) are particular examples. Among the many important features of support varieties in the finite group scheme setting is their ability to detect the injectivity of a module (which is equivalent to detecting projectivity in this case). A natural question in [F1, F2] that is investigated is whether or not the newly introduced rational supports detect injectivity for rational G-modules.

In [F2], it is shown that a G-module M has trivial rational support precisely when M is injective over every Frobenius kernel of G. Such modules are dubbed to be “mock injective.” Friedlander then proved that mock injectivity is in general a weaker condition than injectivity. In particular, he established the existence of non-injective mock injective modules for G in cases when G is not a reductive group using the following argument. Embed G into GL_{n} for some n. The coordinate algebra k[GL_{n}] is injective over every finite subgroup scheme of GL_{n}, hence over G_{r} for every r. On the other hand, k[GL_{n}] is injective over the reduced subgroup scheme G if and only if G is reductive. Consequently, there exists non-injective mock injective G-modules. Such modules are referred to as “proper mock injective modules.” The example above leads to a clear question: for arbitrary G, do there exist proper mock injective modules? In this paper we provide a definitive answer to this question via the construction of new families of examples of proper mock injective modules.

Date: April 26, 2016.

2010 Mathematics Subject Classification. Primary 20G05, 20J06; Secondary 18G05.

Research of the first author was partially supported by NSF RTG grant DMS-1344994.

Research of the second author was partially supported by NSF grant DMS-1402271.

Research of the third author was partially supported by NSF RTG grant DMS-1344994.

1.2. A key tool in our study of injective G-modules is the induction functor [J, §I.3]. If H ≤ G, then any injective H-module induces up to an injective G-module. A particularly useful converse to this, which we will exploit in this paper, is if H is a finite subgroup scheme of G, then inducing the trivial H-module k up to G produces an injective G-module if and only if k is injective over H. Induced modules of this form, when G is semi-simple and H is a finite Chevalley subgroup, have featured prominently in the work of Bendel, Nakano, and Pillen (see, for example, [BNP3, §2.1]).

The paper is organized as follows. In the following section (Section 2), for arbitrary affine group schemes G we generalize the BNP-induced modules by inducing from finite subgroup schemes to construct mock injective modules. With these constructions, it is shown when proper mock injective modules arise. This leads us to give a necessary and sufficient condition on G (cf. Theorem 2.2.1) for the existence of proper mock injective modules. Our investigation leads to several natural questions which are presented in this section. Finally, we demonstrate that our constructions (for mock injectives) in this section can be used to show that a formation of the known Parshall Conjecture (cf. [LN, D]) for infinitely generated modules is false.

In Section 3, we explore the situation when a given mock injective module has finitely many composition factors in its socle. In the case when G is reductive, we construct a proper mock injective with the socle isomorphic to the trivial module. On the other hand, using general results about mock injectives for parabolics, we prove that for a Borel subgroup of a reductive group, any mock injective with finite-dimensional socle must be injective. Finally we prove that for G reductive, a mock injective G-module with a simple G-socle is injective over G if and only if it has a good filtration.

1.3. Acknowledgements. The authors would like to thank Eric Friedlander for initiating the interesting study of mock injectivity which inspired many of our results, and for providing helpful feedback on this paper.

2. Conditions on G for the Existence of Proper Mock Injective Modules

2.1. Let G be an arbitrary affine group scheme and k be an algebraically closed field. The coordinate algebra k[G], which can be realized as ind_{1}^{G}k, is an important example of an induced module for G. More generally, we find that induction from finite subgroup schemes of G can produce G-modules with interesting properties. Such subgroup schemes are always exact in G [J, I.5.13(b)].

Proposition 2.1.1. Let H be a finite subgroup scheme of G.

(a) ind_{H}^{G} k is an injective G-module if and only if k is an injective H-module.

(b) If F restricts to an automorphism of H, then ind_{H}^{G} k is injective over G_{r} for every r > 0.

Proof. Since H is exact in G, we may apply generalized Frobenius reciprocity [J, Corollary I.4.6] to obtain isomorphisms

Ext_{G}^{i}(k, ind_{H}^{G} k) ≅ Ext_{H}^{i}(k, k)

for every i ≥ 0. If ind_{H}^{G} k is injective over G, then the groups on the left side of the isomorphism are 0 for every i > 0. This implies that the cohomological variety of the finite group scheme H is trivial, hence that k is an injective H-module. Conversely, if k is injective over H, then ind_{H}^{G} k is injective over G, since induction always takes injective modules to injective modules [J, Proposition I.3.9]. This proves the first statement.

Let r > 0, and let I be an injective G-module. Then I is injective over H. Furthermore, if F restricts to an automorphism of H, then I^{(r)} is also injective over H. By the tensor identity, we have

I^{(r)} ⊗ ind_{H}^{G} k ≅ ind_{H}^{G} I^{(r)}.

The module ind_{H}^{G} I^{(r)} is, by earlier remarks, injective over G, hence is injective over G_{r}. On the other hand, over G_{r}, ind_{H}^{G} k is a summand of ind_{H}^{G} I^{(r)}, as is evident from the left hand side of the isomorphism above. This shows that ind_{H}^{G} k is injective over G_{r}. □

If H has the property that every simple H-module comes from a G-module, then we can also completely determine which G-modules can will tensor with ind_{H}^{G} k to produce injective G-modules. This condition is satisfied whenever H is unipotent, and also when H is a Frobenius kernel or finite Chevalley subgroup of a semi-simple and simply connected group.

Proposition 2.1.2. Let H be a finite subgroup of G for which every simple H-module comes from a G-module. Then for any G-module M, the module M ⊗ ind_{H}^{G} k is injective over G if and only if M is injective over H.

Proof. By the tensor identity,

M ⊗ ind_{H}^{G} k ≅ ind_{H}^{G} M.

If M is injective over H, then ind_{H}^{G} M is injective over G. Conversely, if ind_{H}^{G} M is injective over G, then for every G-module N, we find that

0 = Ext_{G}^{i}(N, ind_{H}^{G} M) ≅ Ext_{H}^{i}(N, M).

Since every simple H-module comes from some G-module, this immediately implies that M is injective over the finite group scheme H. □

2.2. By using these induced modules from the previous subsection, we are now in position to give a complete picture of which affine algebraic groups over k have proper mock injective modules.

Theorem 2.2.1. Let G be an algebraic group over k which is defined and split over a finite subfield 𝔽_{q} of k. Then G has proper mock injective modules if and only if either G^{0} is not a torus or G/G^{0} has order divisible by p.

Proof. If p does not divide the order of G(𝔽_{q}), then it must be the case that G^{0} is a torus, and that G/G^{0} is a finite group of order not divisible by p. To see this, we note that the second statement is clear, while the first follows from the fact that every element in G(𝔽_{q}) would have to be semi-simple, since unipotent elements have order which is a power of p. In this case, G is linearly reductive [Na], hence there are no non-injective G-modules of any kind, and therefore there cannot be any proper mock injective G-modules.

If p divides the order of G(𝔽_{q}), then k is a non-injective G(𝔽_{q})-module. By the previous proposition, the G-module ind_{G(𝔽_{q})}^{G} k is both non-injective over G, while also being injective over G_{r} for all r. Therefore, it is a proper mock injective G-module. □

Remark 2.2.2. If G is connected reductive and not a torus then the mock injective modules described above do not have a good filtration. One can see this because

Ext_{G}^{i}(k, ind_{G(𝔽_{q})}^{G} k) ≅ Ext_{G(𝔽_{q})}^{i}(k, k) ≠ 0

for some i > 0. Here, the trivial module should be viewed as the Weyl module of highest weight 0.

2.3. We will now show that it is even easier to exhibit non-injective G-modules which are mock injective with respect to the fixed-point subgroups of powers of the Frobenius morphism.

Proposition 2.3.1. Let I be an injective G-module, and r > 0. Then I^{(r)} is injective over every subgroup G(𝔽_{q}), but is not injective over G.

Proof. Since G(𝔽_{q}) is exact in G, I is injective on restriction to G(𝔽_{q}). The Frobenius map defines an automorphism of G(𝔽_{q}), hence I^{(r)} is also injective over it. Finally, this module is trivial for G_{r}, thus cannot be injective over G. □

2.4. A natural question at this point is whether or not a non-injective G-module can be injective over all Frobenius kernels and finite Chevalley subgroups. We at least observe that taking a tensor product of modules from our two classes of examples above cannot produce such an example. That is, if H is a finite subgroup scheme such that the Frobenius morphism restricts to an automorphism of H, then I^{(r)} is injective over H, and so we have

I^{(r)} ⊗ ind_{H}^{G}k ≅ ind_{H}^{G} I^{(r)},

which is then injective over G. In fact, one can go a step further and ask the following question.

Question 2.4.1. Let G be a reductive group. If M is a G-module which is injective over every finite subgroup scheme of G, is M then injective over G?

For non-reductive groups, Friedlander’s argument can be immediately used to show that the answer to this question is “no.” That is, embed such a group G into some GL_{n}. The coordinate algebra of GL_{n} is then non-injective for G, but is injective over every finite subgroup scheme since they are all exact in GL_{n}.

2.5. Parshall Conjecture and infinite-dimensional modules. In this subsection we assume that G is a reductive algebraic group scheme over a field k of any positive characteristic. In 1986, Parshall [P] conjectured that a finite-dimensional rational G-module which is projective over G_{1} is then projective over G(𝔽_{p}). This conjecture was proven by Lin and Nakano (cf. [LN]). Drupieski [D] later generalized this result by proving that, for any r ≥ 1, a finite-dimensional rational G-module that is projective over G_{r} is projective over G(𝔽_{q}).

Now one could also ask what happens to the statement of Parshall’s Conjecture when the finite-dimensionality assumption is removed. As in the case with mock injectives, infinitedimensional rational G-modules can exhibit new and interesting structures. We will show the natural analogues to the above conjectures are false by using the infinite-dimensional rational G-module M which is projective over G_{r} (as defined in Section 2) and by demonstrating that this module is not projective over G(𝔽_{q}).

Proposition 2.5.1. Let r ≥ 1. Then the G-module ind_{G(𝔽_{p})}^{G} k is projective over G_{r}, but is not projective over G(𝔽_{q}).

Proof. Let M = ind_{G(𝔽_{p})}^{G} k. Proposition 2.1.1 estabilishes the mock injectivity of M, so it remains only to show that M is not projective over G(𝔽_{q}). Since kG(𝔽_{q}) is a free kG(𝔽_{p})- module, every injective G(𝔽_{q})-module is injective over G(𝔽_{p}), thus it suffices to show that M is not projective over G(𝔽_{p}).

The subgroup G(𝔽_{p}) acts on G via the right regular action. Then M ≅ k[G]^{G(𝔽_{p})}, where

k[G]^{G(𝔽_{p})} = {f ∈ k[G] | f(gh) = f(g) for all g ∈ G(k), h ∈ G(𝔽_{p})}.

This allows us to define a G(𝔽_{p})-module homomorphism ev : M → k given by ev(f) = f(1) for any f ∈ M. Moreover, we also have a G-module homomorphism ψ : k → M given by ψ(a) = a (the constant function) for any a ∈ k. The composition ev ∘ ψ = 1_{k}, and hence there exists a G(𝔽_{p}) decomposition M = k ⊕ N for some G(𝔽_{p})-module N. It follows immediately that M is not projective over G(𝔽_{p}).

□

3. Mock Injective Modules With Finite Socles

3.1. Throughout this section, let G be a connected reductive algebraic scheme group over k. We investigate the relationship between mock injective modules and injective modules when we impose additional finiteness conditions on our modules. Specifically, we are concerned in this section with modules that have finite socles. This inquiry was motivated by Friedlander’s observation that for a unipotent group U, there are proper mock injective U-modules that embed in k[U], hence have a trivial socle. Thus we ask whether there exist proper mock injective modules with a simple socle, or at least a finite socle, over other types of algebraic groups.

We find that there do exist proper mock injective modules for reductive groups which have a simple socle by producing an example of one for SL_{2}. However, for the Borel subgroups of a reductive group, there do not even exist proper mock injective modules with a finite socle. This is established by proving a more general result for mock injective modules of a parabolic subgroup P_{J} of G which are injective over the Levi factors of P_{J}.

3.2. Unipotent Groups. Let U denote a connected unipotent group over k which is defined over 𝔽_{p}. We give an alternate proof here to Friedlander’s result showing that unipotent groups can have lots of proper mock injective modules with finite socle.

Proposition 3.2.1. Let r ≥ 1, and q = p^{r}. Then the proper mock injective module M_{r} = ind_{U(𝔽_{q})}^{U} k satisfies soc_{U}(M_{r}) = k.

Proof. The only simple U-module is the trivial module k. Now by Frobenius reciprocity, Hom_{U}(k, M_{r}) ≅ Hom_{U(𝔽_{q})}(k, k). It follows that soc_{U}M_{r} ≅ k. □

3.3. Actions of the center. We begin by recalling the following fact about actions involving the center of a connected reductive algebraic group. Note that the center in this case consists of semi-simple elements in G (cf. [S, Corollary 7.6.4(iii)]).

Lemma 3.3.1. Let Z = Z(G) be the center of a reductive algebraic group G and M be any rational G-module. If χ ∈ X(Z) is a central character then

M = M_{χ} ⊕ N

where M_{χ} = {m ∈ M | z · m = χ(z)m for all z ∈ Z}.

Moreover, if χ, χ′ ∈ X(Z) satisfies χ ≠ χ′ and M, N are G-modules satisfies M = M_{χ} and N = N_{χ′}, then Hom_{G}(M, N) = 0.

Let Δ denote the basis of simple roots for the root system corresponding to G and let J ⊆ Δ. Let P_{J} be a parabolic subgroup of G, with unipotent radical U_{J} and Levi factor L_{J}. If Z_{J} = Z(L_{J}) be the center of L_{J}, then it can be verified that

X(Z_{J}) = X(T)/ℤJ.

Thus, every central character is of the form χ = λ + ℤJ. If we let π : X(T) → X(Z_{J}) denote the quotient map, then π(ℤΦ) = ℤΦ/ℤJ ≅ ℤI where I = Δ\J. It follows that any L_{J}-module M whose weights all lie in ℤΦ, has a central character decomposition of the form

M = ⨁_{χ∈ℤI} M_{χ}.

This statement particularly applies to k[U_{J}].

Lemma 3.3.2. The module k[U_{J}] has a central character decomposition of the form

k[U_{J}] = ⨁_{χ∈ℕI} k[U_{J}]_{χ}

where dim(k[U_{J}]_{χ}) < ∞ for all χ.

Proof. First, recall that k[U_{J}] is a polynomial ring given by

k[U_{J}] = k[x_{γ}|γ ∈ Φ^{+}\Φ_{J}^{+}]

where each x_{γ} has weight γ. For any weight λ ∈ X(T), the weight space k[U_{J}]_{λ} is spanned by monomials of the form ∏_{γ∈Φ^{+}\Φ_{J}^{+}} x_{γ}^{n_{γ}} where λ = ∑_{γ} n_{γ}γ with n_{γ} ≥ 0 is any expression for λ. Since there are only finitely many ways to express λ as a non-negative integral combination of elements in Φ^{+}\Φ_{J}^{+}, we can conclude that the weight spaces are finite-dimensional.

Fix a central character χ ∈ X(Z_{J}) which acts non-trivially on k[U_{J}]. By the remark preceding this lemma and the above description of the weight spaces of k[U_{J}], it follows that there exists a unique weight λ ∈ ℕI satisfying

χ = λ + ℤJ.

Hence,

k[U_{J}]_{χ} = ⨁_{μ∈ℤJ} k[U_{J}]_{λ+μ}.

So it will be sufficient to show that for any λ ∈ ℤI, there are only finitely many μ ∈ ℤJ such that k[U_{J}]_{λ+μ} ≠ 0. In fact, it can be immediately deduced that any such μ must satisfy μ ∈ ℕJ (i.e. it must be a non-negative linear combination of elements in Φ_{J}^{+}).

Observe now that we can write

Φ^{+}\Φ_{J}^{+} = Φ_{I}^{+} ∪ ((Φ_{I}^{+} + Φ_{J}^{+}) ∩ Φ^{+}).

So k[U_{J}]_{λ+μ} ≠ 0 if and only if we can write

λ + μ = ∑_{α∈Φ_{I}^{+}} r_{α}α + ∑_{α+β∈(Φ_{I}^{+}+Φ_{J}^{+})∩Φ^{+}} t_{α+β}(α + β)

= ∑_{α∈Φ_{I}^{+}} (r_{α} + ∑_{{β∈Φ_{J}^{+}|α+β∈Φ^{+}}} t_{α+β}) α + ∑_{β∈Φ_{J}^{+}} (  ∑_{{α∈Φ_{I}^{+}|α+β∈Φ^{+}}} t_{α+β}) β.

where for all α ∈ Φ_{I}^{+} and β ∈ Φ_{J}^{+},

n_{α} = r_{α} + ∑_{{β∈Φ_{J}^{+}|α+β∈Φ^{+}}} t_{α+β} ≥ 0

m_{β} = ∑_{{α∈Φ_{I}^{+}|α+β∈Φ^{+}}} t_{α+β} ≥ 0

and λ = ∑ n_{α}α and β = ∑ m_{β}β. Since the r_{α} and t_{α+β} are also required to be non-negative integers, it follows that there are only finitely many μ ∈ ℕJ^{+} for which this is possible. Therefore, k[U_{J}]_{χ} is finite-dimensional. □

3.4. We begin this section with a standard fact about composition factor multiplicities.

Lemma 3.4.1. Let M be a finite-dimensional G-module and λ, μ ∈ X(T)_{+}. Then

dim Hom_{G}(L(μ), I(λ) ⊗ M) = [L(μ) ⊗ M^{∗} : L(λ)] < ∞.

Consequently, every indecomposable summand of I(λ) ⊗ M occurs with finite multiplicity.

Proof. By adjointness,

Hom_{G}(L(μ), I(λ) ⊗ M) = Hom_{G}(L(μ) ⊗ M^{∗}, I(λ)).

The injectivity of I(λ) implies that the functor Hom_{G}(−, I(λ)) is exact on short exact sequences. Thus, dim Hom_{G}(L(μ) ⊗ M^{∗}, I(λ)) counts the number of times the composition factor L(λ) appears L(μ) ⊗ M^{∗} (i.e., [L(μ) ⊗ M^{∗} : L(λ)]). □

Remark 3.4.2. Although not necessary for our main result, one might wonder if the socle of I(λ) ⊗ M is always finite-dimensional for a finite-dimensional module M. This would be equivalent to showing that [L(μ) ⊗ M^{∗} : L(λ)] = 0 for all but finitely many μ ∈ X(T)_{+}. It turns out this is false in general, as one can see by taking G = SL_{2}(\overline{𝔽_{2}}), and setting λ = 0 and M = L(1). Then it can verified that there are infinitely many μ ∈ X(T)_{+} = ℕ such that [L(μ) ⊗ L(1) : L(0)] ≠ 0, and thus soc_{G}(I(0) ⊗ L(1)) is infinite-dimensional.

For any weight λ ∈ X(T) which is dominant for L_{J}, the injective hull for L_{P_{J}} (λ) (i.e., the inflation of L_{J}(λ) to P_{J}) is given by

I_{P_{J}} (λ) = ind_{L_{J}}^{P_{J}} I_{L_{J}} (λ) ≅ I_{L_{J}} (λ) ⊗ k[U_{J}]

where L_{J} acts by conjugation on k[U_{J}]. This can be verified by using the fact that induction take injectives to injectives and Frobenius reciprocity to show that the socle of I_{P_{J}} (λ) is L_{P_{J}} (λ).

We would like to show that all of the indecomposable summands of I_{P_{J}} (λ)|_{L_{J}} occur with finite multiplicity. Unfortunately, the module k[U_{J}] is not a finite-dimensional L_{J}-module, so we cannot immediately apply Lemma 3.4.1. We can remedy this situation with the use of the central characters for L_{J}.

Proposition 3.4.3. If I is any injective P_{J}-module with a finite-dimensional socle, then

I|_{L_{J}} = ⨁_{ν∈X(T)} I_{L_{J}} (ν)^{n_{ν}}

where n_{ν} < ∞ for all ν ∈ X(T).

Proof. Without loss of generality, one can assume that I = I_{P_{J}} (λ) for some λ ∈ X(T) and hence I = I_{L_{J}} (λ) ⊗ k[U_{J}]. It will be sufficient to show that for any weight μ ∈ X(T) dominant for L_{J},

dim Hom_{L_{J}} (L_{J}(μ), I_{L_{J}} (λ) ⊗ k[U_{J}]) < ∞.

Since I_{L_{J}} (λ) and L_{J}(μ) are indecomposable L_{J}-modules, then Z_{J} acts on former by the character [λ] = λ + ℤJ and the latter by [μ] = μ + ℤJ. Therefore,

Hom_{L_{J}} (L_{J}(μ), I_{L_{J}} (λ) ⊗ k[U_{J}]) = Hom_{L_{J}} (L_{J}(μ), I_{L_{J}} (λ) ⊗ k[U_{J}]_{μ−λ})

because any L_{J}-homomorphism must preserve the X(Z_{J}) weight spaces. The proposition immediately follows from Lemmas 3.3.2 and 3.4.1. □

3.5. Parabolic subgroups. Let F : P_{J} → P_{J} be the Frobenius morphism and (P_{J})_{r}L_{J} = (F^{r})^{−1}(L_{J}). The following result investigate P_{J}-modules which are injective over (P_{J})_{r}L_{J} = (F^{r})^{−1}(L_{J}) and have finite socles.

Theorem 3.5.1. If M is a P_{J}-module with a finite-dimensional socle which is injective over (P_{J})_{r}L_{J} for all r ≥ 1, then M is injective over P_{J}.

Proof. It can be deduced from the finite-dimensionality of soc_{P_{J}} (M) that the P_{J}-injective hull of M is of the form I ⊗k[U_{J}] where I is an injective L_{J}-module with a finite-dimensional socle, and we get an embedding

ι : M ↪ I ⊗ k[U_{J}].

By Proposition 3.4.3,

(I ⊗ k[U_{J}])|_{L_{J}} = ⨁_{ν∈X(T)} I_{L_{J}} (ν)^{n_{ν}}

with n_{ν} < ∞ for all ν.

By assumption, M|_{(P_{J})_{r}L_{J}} is injective for all r, and thus there exist embeddings over L_{J} (or (P_{J})_{r}L_{J}) of the form ϕ_{r} : I ⊗ k[(U_{J})_{r}] ↪ M. For each r, let M_{r} = im(ϕ_{r}) ⊆ M ⊆ I ⊗ k[U_{J}]. It will be sufficient to show that I ⊗ k[U_{J}] = ⋃_{r} M_{r}.

The explicit descriptions of the coordinate algebras for U and U_{r} can be used to show that k[U_{J}] = ⋃_{r} k[(U_{J})_{r}] where each k[(U_{J})_{r}] can be canonically embedded inside k[U_{J}] as (P_{J})_{r}L_{J}-module. We can deduce that I ⊗ k[U_{J}] = ⋃_{r} I ⊗ k[(U_{J})_{r}]. Thus, for any ν satisfying n_{ν} > 0, there must exist some r ≥ 1 such that I_{L_{J}} (ν)^{n_{ν}} | M_{r} ⊆ M. This forces M = I ⊗ k[U_{J}] and therefore M is injective as a P_{J}-module. □

3.6. Borel subgroups. The aforementioned theorem in the previous section has a particularly nice formulation in the case when P_{J} = B where B is a Borel subgroup of G.

Corollary 3.6.1. Let M be any mock injective B-module which has a finite-dimensional socle, then M is an injective B-module.

Proof. In this situation, B = P_{∅} and T = L_{∅}. The proof then follows by applying Theorem 3.5.1 and observing that a module M is injective over B_{r}T if and only if it is injective over B_{r}. □

3.7. Reductive groups. In this section we show that the analogue of Proposition 3.6.1 does not hold for reductive groups. Specifically, we will give an example of a module for G = SL_{2}(k), with k = \overline{𝔽_{2}}, which is proper mock injective and has a simple socle. Let

W = \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})\} ⊆ SL_{2}(𝔽_{2})

denote the Weyl group. Since 2 | |W |, then there exist non-trivial extensions of the trivial representation for W. This implies that the module ind_{W}^{G} k = k[G]^{W} is proper mock injective by the same argument as in Theorem 2.2.1. Furthermore, the normalizer of the torus, N_{G}(T) = W T, is given by

W T = \{(\begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix}) , (\begin{matrix}0 & a \\ a^{−1} & 0 \end{matrix}) | a ∈ k^{∗}\} .

The following theorem demonstrates that M = ind_{W T}^{G} k is proper mock injective and has a simple socle.

Proposition 3.7.1. Let M = ind_{W T}^{G} k. Then (a) soc_{G}M = k,

(b) M is proper mock injective.

Proof. (a) In this situation, one can identify X(T) = ℤ and ℤ[X(T)] = ℤ[q, q^{−1}]. We will first show by induction that the simple module L(λ) does not contain a vector of weight 0 for any λ > 0. For the base case, observe that

ch L(1) = q + q^{−1}

and thus doesn’t contain a 0-weight vector. Now suppose that λ ≥ 2 and that the result holds for all μ < λ. If λ is odd, then all the weights appearing in L(λ) must also be odd, thus 0 cannot appear as a weight. Conversely, if λ is even, then by Steinberg’s tensor product theorem L(λ) = L(λ/2)^{(1)} and we can apply the inductive hypothesis to λ/2 < λ.

Therefore, (a) follows because for any λ > 0,

Hom_{G}(L(λ), M) ≅ Hom_{W T} (L(λ), k) ⊆ Hom_{T} (L(λ), k) = 0.

and

Hom_{G}(k, M) ≅ Hom_{W T} (k, k) ≅ k.

(b) Next we show that M is proper mock injective. Since W T/T ≅ W, it follows from a Lyndon-Hochschild-Serre spectral sequence argument that for some i > 0,

Ext_{G}^{i}(k, M) ≅ Ext_{W T}^{i} (k, k) ≅ H^{i}(W, k^{T}) = Ext_{W}^{i} (k, k) ≠ 0.

Therefore, M is not injective, and in fact it doesn’t even have a good filtration.

It remains to prove that M is mock injective, this will follow by showing that M is a summand of the mock injective module ind_{W}^{G} k. To see why this is true, first observe that

ind_{W}^{G} k = ind_{W T}^{G} ind_{W}^{W T} k.

Let N = ind_{W}^{W T} k, then it can be verified that N = k[T] where T acts as the left regular representation and W ≅ W T/T acts by conjugation. Now k[T] is an injective T -module and the zero weight space under this action appears exactly once. Furthermore, the zero weight space is a summand of k[T] under W. It follows that ind_{W}^{G} k = M ⊕ ind_{W T}^{G} S, and therefore M is mock injective. □

3.8. Mock injectives with good filtrations. As exhibited above, for G reductive, a proper mock injective G-module may have a simple socle. We will now show that this cannot occur if we also assume that M has a good filtration.

Let M be a mock injective G-module with simple G-socle L(λ), and choose r_{0} such that λ ∈ X_{r_{0}}(T). For any r ≥ r_{0} it follows that L_{r}(λ) is the only iso-class appearing in soc_{G_{r}}(M), and the only iso-class appearing in soc_{G_{r}} I(λ). As a consequence, the only G_{r}-summands appearing in each of these modules is Q_{r}(λ). Since M is injective over G_{r}, there is a splitting I(λ) → M. This shows that I(λ)/M is isomorphic over G_{r} to a direct sum of the modules Q_{r}(λ), hence that L_{r}(λ) is the only iso-class appearing in soc_{G_{r}}(I(λ)/M). Since this holds for all r ≥ r_{0}, it follows that L(λ) is the only iso-class appearing in soc_{G}(I(λ)/M). Indeed, if L(μ) appears in soc_{G}(I(λ)/M) for some μ ≠ λ, we could pick r large so that μ ∈ X_{r}(T), and then L_{r}(μ) would appear in the G_{r}-socle of I(λ)/M, a contradiction. Thus L(λ) is the only iso-class of simple G-module appearing in soc_{G}(I(λ)/M).

We recall that for any G-module N, the injectivity of I(λ) and the short exact sequence

0 → M → I(λ) → I(λ)/M → 0

gives rise to the exact sequence

0 → Hom_{G}(N, M) → Hom_{G}(N, I(λ)) → Hom_{G}(N, I(λ)/M) → Ext_{G}^{1}(N, M) → 0.

From this we can deduce the following.

Theorem 3.8.1. If M is a mock injective G-module with a simple socle, then M is injective if and only if it has a good filtration.

Proof. The “only if” direction is clear since injective modules always have a good filtration. On the other hand, assume that M has a good filtration, and embed M into its injective hull I(λ). By [J, II.4.18], any good filtration of I(λ) has H^{0}(λ) appearing with multiplicty one.

For each V (μ) we have that Ext_{G}^{1}(V (μ), M) = 0, giving a short exact sequence

0 → Hom_{G}(V (μ), M) → Hom_{G}(V (μ), I(λ)) → Hom_{G}(V (μ), I(λ)/M) → 0

showing that Hom_{G}(V (μ), I(λ)/M) < ∞. By [J, II.4.17], I(λ)/M has a good filtration. If I(λ)/M ≠ 0, then L(λ) ⊆ I(λ)/M. Consequently, H^{0}(λ) ⊆ I(λ)/M and H^{0}(λ) ⊆ M, so it occurs with multiplicity 2 in a good filtration of I(λ), a contradiction. Thus, I(λ)/M = 0. □

Finally, although M in general cannot be shown to have a good filtration (for this would contradict the example for SL_{2} given earlier), we can at least say that M starts out with the beginnings of a good filtration.

Theorem 3.8.2. If M is a mock injective module with a simple socle, then H^{0}(λ) is a submodule of M.

Proof. By [J, Lemma II.4.15], it suffices to show that

Ext_{G}^{1}(V (μ), M) = 0

for all μ < λ. The sequence

0 → Hom_{G}(V (μ), M) → Hom_{G}(V (μ), I(λ)) → Hom_{G}(V (μ), I(λ)/M) → Ext_{G}^{1}(V (μ), M) → 0

shows that Hom_{G}(V (μ), I(λ)/M) surjects onto Ext_{G}^{1}(V (μ), M). However, L(λ) is not a composition factor of V (μ) since λ > μ. Therefore, Hom_{G}(V (μ), I(λ)/M) = 0, and so Ext_{G}^{1}(V (μ), M) must also be 0. □

References

[And1] H.H. Andersen, Extensions of modules for algebraic groups, Amer. J. Math., 106, (1984), 498-504.

[And2] H.H. Andersen, Extensions of simple modules for finite Chevalley groups, J. Algebra, 111, (1987), 388-403.

[BNP1] C.P. Bendel, D.K. Nakano, C. Pillen, On comparing the cohomology of algebraic groups, finite Chevalley groups and Frobenius kernels, J. Pure and Applied Algebra, 163, (2001), 119-146.

[BNP2] C.P. Bendel, D.K. Nakano, C. Pillen, Extensions for finite Chevalley groups II, Transactions of the AMS, 354, (2002), 4421-4454.

[BNP3] C.P. Bendel, D.K. Nakano, C. Pillen, Extensions for finite Chevalley groups I, Advances in Math., 183, (2004), 380–408.

[BNP4] C. Bendel, D. Nakano, C. Pillen, Extensions for finite groups of Lie type II: Filtering the truncated induction functor, Contemp. Math., 413, (2006), 1-23.

[BNP5] C.P. Bendel, D.K. Nakano, C. Pillen, Extensions for finite Chevalley groups III: Rational and Generic Cohomology, Advances in Math., 262, (2014), 484-519.

[CPS] E. Cline, B.J. Parshall, L. Scott, On injective modules for infinitesimal algebraic groups I, J. London Math. Soc., 31, (1985), 277-291.

[CPSvdK] E. Cline, B. Parshall, L. Scott, W. van der Kallen, Rational and generic cohomology, Invent. Math., 39, (1977), 143-163.

[D] C. Drupieski, On projective modules for Frobenius kernels and finite Chevalley groups, Bull. London Math. Soc., 45, (2013), no. 4, 715-720.

[F1] E. Friedlander, Support varieties for rational representations, Compos. Math., 151, (2015), no. 4, 765-792.

[F2] E. Friedlander, Filtrations, 1-parameter subgroups, and rational injectivity, preprint.

[J] J. Jantzen, Representations of Algebraic Groups, 2nd ed. Mathematical Surveys and Monographs, 107, American Mathematical Society, 2003.

[LN] Z. Lin, D. Nakano, Complexity for modules over finite Chevalley groups and classical Lie algebras, Invent. Math., 138, (1999), 85-101.

[Na] M. Nagata, Complete reducibility of rational representations of a matric group, J. Math. Kyoto Univ., 1,(1961/1962), 87-99.

[N] D.K. Nakano, Cohomology of algebraic groups, finite groups, and Lie algebras: Interactions and Connections, Lie Theory and Representation Theory, Survey of Modern Mathematics Volume II (edited by N. Hu, B. Shu, J.P. Wang), International Press, 2012, pp. 151-176.

[P] B. Parshall, Cohomology of algebraic groups, The Arcata conference on representations of finite groups (Arcata, CA, 1986), Proceedings of Symposia in Pure Mathematics, 47, American Mathematical Society, (1987), 233-248.

[S] T. A. Springer, Linear Algebraic Groups, Progress in Math. 9, Birkhäuser, 1998.

Department of Mathematics, University of Georgia, Athens, GA 30602

   E-mail address: hardes1@uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602

   E-mail address: nakano@math.uga.edu

Department of Mathematics, University of Georgia, Athens, GA 30602

   E-mail address: sobaje@uga.edu