Minimal inductive systems of modular representations for naturally embedded algebraic and finite groups of type A

A. A. Baranov and I. D. Suprunenko

Institute of Mathematics National Academy of Sciences of Belarus Surganova 11, Minsk, 220072, Belarus

baranov@im.bas-net.by suprunenko@im.bas-net.by

Abstract

The article is devoted to the classification of the minimal and minimal nontrivial inductive systems of modular representations for naturally embedded algebraic and finite groups of type A and related locally finite groups. It occurs that the minimal systems consist of the trivial representations for the relevant groups and the minimal nontrivial ones are connected with Frobenius twists of the standard representations and their duals. These results are applied to the description of the maximal ideals in group algebras of the locally finite groups SL_{∞} and SU_{∞} in describing characteristic.

It is also proved that for an arbitrary classical algebraic group, the restriction of an irreducible module with highest weight large enough to a naturally embedded finite Chevalley group of the same type, but a smaller rank contains the regular module.

1 Introduction

Let 𝔽 be any field of characteristic p > 0, 𝕂 be an algebraically closed field of characteristic p, and ℕ be the set of natural numbers. Let

Γ_{1} → Γ_{2} → ⋯ → Γ_{n} → …  (1)

be a sequence of embeddings where either all Γ_{i} are finite or all of them are algebraic groups over 𝕂 and for algebraic Γ_{i} all embeddings in (1) are rational. Throughout the paper Irr 𝔽Γ_{n} (or Irr Γ_{n} if 𝔽 is fixed) is the set of equivalence classes of irreducible (rational for Γ_{n} algebraic) Γ_{n}-modules over 𝔽 (𝔽 = 𝕂 for Γ_{n} algebraic); Irr M is the set of composition factors of a module M; M^{∗} is the dual module of M; τ is the trivial one-dimensional module for any group. For groups H ⊂ G and a G-module M denote by M↓H the restriction of M to H. Let Φ_{n} be a nonempty finite subset of Irr Γ_{n}. We say that the collection Φ = {Φ_{n}}_{n∈ℕ} is an inductive system for the sequence (1) if

⋃_{M∈Φ_{n+1}} Irr(M↓Γ_{n}) = Φ_{n}

for all n ∈ ℕ (see also the definition in Section 5). The system Φ is called trivial if Φ_{n} = {τ} for all n; otherwise Φ is nontrivial.

Set G_{n} = SL_{n+1}(𝕂) (the algebraic group of type A_{n}). Let Fr be the Frobenius morphism of G_{n} associated with raising elements of 𝕂 to the pth power. For a morphism ρ of G_{n} and M ∈ Irr G_{n} denote by ρ(M) the G_{n}-module affording the representation G_{n} → ρ(G_{n}) → GL(M). Let V_{n} be the standard G_{n}-module and V_{n}^{∗} be its dual. Set

L_{n}^{j} = {Fr^{j}(V_{n}), τ} and R_{n}^{j} = {Fr^{j}(V_{n}^{∗}), τ} (j = 0, 1, 2, …).

Let 𝔄 be the sequence of the natural embeddings

G_{1} → G_{2} → ⋯ → G_{n} → ….

It is clear that L^{j} = {L_{n}^{j}}_{n∈ℕ} and R^{j} = {R_{n}^{j}}_{n∈ℕ} are inductive systems for 𝔄. The following theorem shows that they exhaust the minimal nontrivial inductive systems.

Theorem 1.1 Let Φ be a nontrivial inductive system for 𝔄. Then Φ contains one of the systems L^{j} or R^{j} (j ≥ 0). In particular, any inductive system for 𝔄 contains the trivial one.

The first result of the article on finite groups concerns not only groups of type A. We prove that the restriction of an irreducible representation φ of a classical algebraic group over 𝕂 to a Chevalley group of the same type and a smaller rank over a finite subfield of 𝕂 contains the regular representation provided the highest weight of φ is large enough. Let Γ_{1} ⊂ … ⊂ Γ_{n} ⊂ … be the sequence of the naturally embedded simply connected classical groups Γ_{i} = Γ_{i}(𝕂) of rank i with the root systems of the same type Γ = A, B, C, or D. Transfer the notation Fr to the groups Γ_{i}. Denote by ω_{j}^{i}, 1 ≤ j ≤ i, the fundamental weights of the group Γ_{i} labeled in the standard way. Let M ∈ Irr Γ_{n} and let λ be the highest weight of M. Represent λ in the form λ = ∑_{j=0}^{u} p^{j}λ_{j} where the weights λ_{j} are p-restricted. Let λ_{j} = ∑_{i=1}^{n} a_{ij}ω_{i}^{n}. Set S(M) = max{∑_{i=1}^{n} a_{ij} | 0 ≤ j ≤ u}.

Theorem 1.2 Let H_{m} ⊂ Γ_{m} be a finite ordinary or twisted Chevalley group of rank m which is the fixed subgroup of some Frobenius morphism of Γ_{m} and let l = |H_{m}|−1. Then for each n > m + 1 and each M ∈ Irr Γ_{n} with S(M) ≥ p + p^{2} + … + p^{l} + (p − 1)(m + 1) the restriction M↓H_{m} contains the regular 𝕂H_{m}-module. In particular, Irr(M↓H_{m}) = Irr 𝕂H_{m}.

Remark 1.3 It is clear that Theorem 1.2 can be transferred to irreducible 𝕂representations of finite Chevalley groups associated with Γ_{n} which contain H_{m}.

Now let q = p^{s} and let

H_{1} → H_{2} → ⋯ → H_{n} → …  (2)

be the sequence of finite groups H_{n} = SL_{n+1}(q) or SU_{n+1}(q^{2}) (all the groups are of the same type) naturally embedded each into another. Set H = \underrightarrow{lim} H_{n}. Then H is isomorphic to one of the groups SL_{∞}(q) or SU_{∞}(q^{2}). One can consider these groups as the groups of infinite matrices of the form

A = diag(A_{n}, 1, 1, …)

where A_{n} ∈ SL_{n}(q) or SU_{n}(q^{2}). So we can define an automorphism Fr of H as above. We say that Ψ is an inductive system for H if Ψ is an inductive system for the sequence (2). Let M be an 𝔽H-module. One can define the corresponding inductive system Φ = Φ(M) as follows. Set Φ_{n} = Irr(M↓H_{n}). One easily checks that Φ = {Φ_{n}}_{n∈ℕ} is an inductive system for H. Let 𝔽 contain the field of order q for H = SL_{∞}(q) and the field of order q^{2} for H = SU_{∞}(q^{2}). We say that 𝔽 is a splitting field for H. Denote by V the natural module for H over 𝔽 and by V^{∗} the dual to V. Set L^{j} = Φ(Fr^{j}(V)), and R^{j} = Φ(Fr^{j}(V^{∗})), 0 ≤ j < s. We keep the notation L^{j} and R^{j} used for algebraic groups since the relevant inductive systems can be obtained by restriction from algebraic groups.

Theorem 1.4 Let 𝔽 be a splitting field for H = SL_{∞}(q) or SU_{∞}(q^{2}). Then each nontrivial inductive system for H over 𝔽 contains either L^{j} or R^{j} (0 ≤ j < s). In particular, any inductive system for H contains the trivial one.

The notion of an inductive system has been introduced by Zalesskii in [11] and has been developed in [12]. Inductive systems yield an asymptotic version of the branching rules for relevant embeddings. Classical branching rules for algebraic groups in characteristic 0 have found numerous applications. It is quite easy to deduce an analog of Theorem 1.1 from these rules. Since one cannot expect to find their explicit modular analogs in the general case, it is worth to seek for an asymptotic version. Moreover, inductive systems can be applied to the study of ideals in group algebras of locally finite groups. It is proved in [12] that there exists a bijective correspondence between the inductive systems for a locally finite group and the semiprimitive ideals of the corresponding group algebra. This enabled the authors to obtain some results on the ideals of 𝔽H.

Theorem 1.5 Let H = SL_{∞}(q) or SU_{∞}(q^{2}) and 𝔽 be a splitting field for H. Then any proper ideal of 𝔽H is contained in the augmentation ideal Aug(𝔽H). Moreover, the annihilators Ann_{𝔽H} Fr^{j}(V) and Ann_{𝔽H} Fr^{j}(V^{∗}) (0 ≤ j < s), are exactly all distinct maximal ideals of Aug(𝔽H).

Theorems 1.4 and 1.5 are particular cases of Theorems 5.3 and 5.5, respectively, which for an arbitrary locally finite field 𝕃 and a field 𝔽 with char 𝔽 = char 𝕃 describe the minimal nontrivial inductive systems and the maximal ideals in Aug(𝔽H) for H = SL_{∞}(𝕃) or SU_{∞}(𝕃). To note this, see Remark 3.3 as well.

For other classical groups the question on the minimal inductive systems seems substantially more difficult. Analogs of Theorems 1.1 and 1.4 are not valid there. Indeed, for the groups of types B and D there are inductive systems consisting of the spinor and semispinor representations, respectively. Denote by φ(ω) the irreducible representation of a fixed algebraic group with highest weight ω. For the natural embeddings of the groups of type C_{n} and p > 2 Zalesskii and Suprunenko [10] have described an inductive system Ψ with Ψ_{n} = \{φ(ω_{n−1}^{n} + \frac{1}{2}(p − 3)ω_{n}^{n}), φ(\frac{1}{2}(p − 1)ω_{n}^{n})}, n = 1, 2, . . .. The authors [1] have obtained the branching rules for modular fundamental representations of C_{n} and classified the inductive systems consisting of such and trivial representations. In particular, there exists a minimal inductive system R^{p−1} = {R_{n}^{p−1}}, n = 1, 2, …, with R_{n}^{p−1} = {φ(ω_{j}^{n}) | n + 1 − j ≤ p − 1}.

2 Inductive systems for algebraic groups

In this section the minimal nontrivial inductive systems for the sequence 𝔄 of the natural embeddings G_{1} → G_{2} → ⋯ → G_{n} → … are determined and Theorem 1.2 is proved. Let α_{1}, …, α_{n} be the simple roots of G_{n}. We identify these roots with the relevant simple roots of G_{n+1}. Denote by X_{n}^{+} the set of dominant weights of G_{n} and let ω_{1}^{n}, …, ω_{n}^{n} (or ω_{1}, …, ω_{n} if n is fixed) be the fundamental weights of G_{n}. Recall that X_{n}^{+} = {a_{1}ω_{1} +⋯+a_{n}ω_{n} | a_{i} ∈ ℕ_{0}} where ℕ_{0} is the set of nonnegative integers. Set X_{n}^{q} = {a_{1}ω_{1}+⋯+a_{n}ω_{n} | 0 ≤ a_{i} < q}. The weights in X_{n}^{q} are called q-restricted. Denote by ⟨·, ·⟩ the canonical pairing on the set of weights of G_{n}. In what follows M(λ) is the irreducible G_{n}-module with highest weight λ. Recall that V_{n} = M(ω_{1}^{n}) and V_{n}^{∗} = M(ω_{n}^{n}). For λ = a_{1}ω_{1} + ⋯ + a_{n}ω_{n} ∈ X_{n}^{+} set δ(λ) = a_{1} + ⋯ + a_{n}. The following lemmas are used in the proof of Theorem 1.1.

Lemma 2.1 For each λ ∈ X_{n}^{+} and each root α of G_{n} we have

δ(λ) = max⟨μ, α⟩

where μ runs over the weights of M(λ).

Proof. Let β be the maximal root of G_{n}. It is well known that β = α_{1} + ⋯ + α_{n}. Since the Weyl group acts transitively on the set of roots, max⟨μ, α⟩ = max⟨μ, β⟩ for every root α. As β is a dominant weight, ⟨α_{i}, β⟩ ≥ 0. This forces δ(λ) = ⟨λ, β⟩ = max⟨μ, β⟩, as required.

Lemma 2.2 Let k, m, n ∈ ℕ be such that n ≥ m(k + 1). Assume that λ ∈ X_{n}^{+}\{0} and δ(λ) ≤ k. Then τ ∈ Irr(M(λ)↓G_{m}). Moreover, if λ is p-restricted, then Irr(M(λ)↓G_{m}) contains V_{m} or V_{m}^{∗}.

Proof. Denote by H(i_{1}, …, i_{k}) the subgroup generated by the root subgroups associated with the roots ±α_{i_{1}}, …, ±α_{i_{k}}. Let λ = a_{1}ω_{1}^{1} + ⋯ + a_{n}ω_{n}^{n}. Obviously, at most k of the coefficients a_{i} are nonzero. Now one can observe that there exists i such that a_{i} ≠ 0 and either a_{i+1} = ⋯ = a_{i+m} = 0, or a_{i−m} = ⋯ = a_{i−1} = 0. Put a = a_{i}, H = H(i, i + 1, …, i + m) in the first case and H = H(i − m, …, i − 1, i) in the second one. It is clear that H is conjugate to G_{m+1} in G_{n}. Hence Irr(M(λ)↓G_{m+1}) = Irr(M(λ)↓H). Now Smith’s theorem [7] yields that Irr(M(λ)↓G_{m+1}) contains M(aω_{1}^{m+1}) or M(aω_{m+1}^{m+1}). Similar arguments show that τ ∈ Irr(M(λ)↓G_{m}).

Now assume that λ is p-restricted. Then a < p and M(aω_{1}^{m+1}) is isomorphic to the ath symmetric power S^{a}(V_{m+1}) of V_{m+1} (see, for instance, [6, (1.14)]). Let e be a highest weight vector of V_{m+1} and f ∈ V_{m+1} be a nonzero vector fixed by G_{m}. It is not difficult to see that the vector ef^{a−1} ∈ S^{a}(V_{m+1}) generates the standard G_{m}-module. Therefore Irr(M(aω_{1}^{m+1})↓G_{m}) contains V_{m}. Considering the dual modules, we get that Irr(M(aω_{m+1}^{m+1})↓G_{m}) contains V_{m}^{∗}. This yields our claim for Irr(M(λ)↓G_{m}).

Lemma 2.3 Let k, m, n, and λ = a_{1}ω_{1}^{1} + ⋯ + a_{n}ω_{n}^{n} be such as in Lemma 2.2. Let i be the maximal power of p such that p^{i} divides all the coefficients a_{j}. Then Irr(M(λ)↓G_{m}) contains M(p^{i}ω_{1}^{m}) or M(p^{i}ω_{m}^{m}).

Proof. One can write λ = p^{i}λ_{1} + p^{i+1}λ_{2} where λ_{1} ∈ X_{n}^{p}\{0} and λ_{2} ∈ X_{n}^{+}. By Steinberg’s tensor product theorem [8],

M(λ) = Fr^{i}(M(λ_{1})) ⊗ Fr^{i+1}(M(λ_{2})).

Obviously, δ(λ_{1}), δ(λ_{2}) ≤ k. By Lemma 2.2, Irr(M(λ_{1})↓G_{m}) contains either V_{m} or V_{m}^{∗}, and Irr(M(λ_{2})↓G_{m}) contains τ. It remains to observe that Fr^{i}(V_{m}) = M(p^{i}ω_{1}^{m}) and Fr^{i}(V_{m}^{∗}) = M(p^{i}ω_{m}^{m}).

Let Ψ_{n} be a finite subset of Irr G_{n} and Ψ = {Ψ_{n}}_{n∈ℕ}. Set δ(Ψ_{n}) = max{δ(λ) | M(λ) ∈ Ψ_{n}} and

δ(Ψ) = min{δ(Ψ_{n}) | n ∈ ℕ}.

Lemma 2.4 Let Φ = {Φ_{n}}_{n∈ℕ} be an inductive system for 𝔄. Then δ(Φ_{n}) = δ(Φ) for all n. In particular, δ(λ) ≤ δ(Φ) if M(λ) ∈ Φ_{n}.

Proof. It suffices to show that δ(Φ_{n}) = δ(Φ_{n+1}) for all n. Let λ and ρ be such that M(λ) ∈ Φ_{n+1}, M(ρ) ∈ Φ_{n}, and δ(λ) = δ(Φ_{n+1}) and δ(ρ) = δ(Φ_{n}). By Lemma 2.1, there exists a weight μ of M(λ) such that

δ(λ) = ⟨μ, α_{1} + ⋯ + α_{n}⟩ = ⟨ν, α_{1} + ⋯ + α_{n}⟩

where ν is the restriction of the weight μ to G_{n}. Hence

δ(Φ_{n+1}) = δ(λ) ≤ δ(Irr(M(λ)↓G_{n})) ≤ δ(Φ_{n}).

Since Φ_{n} = Irr(Φ_{n+1}↓G_{n}), there exist weights λ_{1} and λ_{2} of G_{n+1} such that ρ is the restriction of λ_{1} to G_{n} and λ_{1} is a weight of M(λ_{2}) ∈ Φ_{n+1}. Hence

δ(Φ_{n}) = δ(ρ) = ⟨ρ, α_{1} + ⋯ + α_{n}⟩ = ⟨λ_{1}, α_{1} + ⋯ + α_{n}⟩ ≤ δ(λ_{2}) ≤ δ(Φ_{n+1}),

so δ(Φ_{n}) = δ(Φ_{n+1}).

Proof of Theorem 1.1. Clearly, Fr^{j}(V_{n}) = M(p^{j}ω_{1}^{n}) and Fr^{j}(V_{n}^{∗}) = M(p^{j}ω_{n}^{n}) for all n. Fix any m ∈ ℕ. Set k = δ(Φ). Take any n such that n ≥ m(k+1) and any nonzero λ such that M(λ) ∈ Φ_{n}. By Lemma 2.4, δ(λ) ≤ k. Therefore by Lemma 2.3, Φ_{m} contains M(p^{i}ω_{1}^{m}) or M(p^{i}ω_{m}^{m}) for some i = i(m) ≥ 0. Observe that p^{i} ≤ k. Now it is clear that there exists j ≥ 0 and an infinite sequence m_{1} < m_{2} < ⋯ < m_{r} < … such that either for each r the set Φ_{m_{r}} contains M(p^{j}ω_{1}^{m_{r}}), or each of them contains M(p^{j}ω_{m_{r}}^{m_{r}}). Applying Smith’s theorem [7], we obtain the required assertion.

Now we prove a lemma that holds for arbitrary classical groups. Transfer to other classical groups the notation for roots and weight systems introduced at the beginning of the section for the groups G_{n}. Until the end of this section Γ_{1} ⊂ … Γ_{n} ⊂ … is the sequence of the simply connected classical groups Γ_{i} = Γ_{i}(𝕂) with the root systems of the same type where the group Γ_{i} of rank i is identified with the subgroup H(2, 3, …, i + 1) ⊂ Γ_{i+1}. So the roots α_{2}, …, α_{i+1} of Γ_{i+1} yield a basis of the root system for Γ_{i}. Similarly the roots α_{n−m+1}, …, α_{n} yield a basis of the root system for the group Γ_{m} embedded into Γ_{n}. Below we consider the weights of restrictions of Γ_{n}-modules to Γ_{m} with respect to this basis.

Lemma 2.5 Let λ = a_{1}ω_{1}^{n} + … + a_{n}ω_{n}^{n} ∈ X_{n}^{p} and m < n. Then for each integer b with 0 ≤ b ≤ a_{1} + a_{2} + … + a_{n−m} the restriction M(λ)↓Γ_{m} has a composition factor with highest weight λ_{b} = (a_{n−m+1} + b)ω_{1}^{m} + a_{n−m+2}ω_{2}^{m} + … + a_{n}ω_{m}^{m}.

Proof. For a root α of Γ_{n} and t ∈ 𝕂 let x_{α}(t) ∈ Γ_{n} and X_{α} ⊂ Γ_{n} be the root element and the root subgroup of Γ_{n} associated with α and t. Set M = M(λ). Let M_{ν} ⊂ M be the weight subspace of weight μ. By [2, Proposition 5.13], there exist X_{α,d} ∈ End M (d = 0, 1, 2, …) with the following properties: X_{α,d} = 0 for d large enough, X_{α,0} = 1, x_{α}(t) = ∑_{d=0}^{∞} t^{d}X_{α,d} (in End M), X_{α,d}M_{ν} ⊂ M_{ν+dα} for all weights ν of M. Let b be such as in the assertion of the lemma. Set s = n − m + 1. If b > a_{s−1}, represent b in the form b = c + a_{s−k+1} + … + a_{s−1} where 0 ≤ c ≤ a_{s−k}; if b ≤ a_{s−1}, put c = b and k = 1. Let v ∈ M be a nonzero highest weight vector. Now we shall construct a vector u_{b} ∈ M that generates an indecomposable Γ_{m}- module with highest weight λ_{b}. First we define the integers d_{1}, …, d_{k} as follows: d_{k} = c, d_{j} = a_{s−j} + d_{j+1} for 1 ≤ j < k. Set u_{b} = X_{−α_{s−1},d_{1}} … X_{−α_{s−k},d_{k}}v. Observe that d_{1} = b. By [9, Lemma 2.9], the vector u_{b} is nonzero and is fixed by the root subgroups X_{α_{i}} for i ≠ n − m.

The properties of the operators X_{α,d} imply that u_{b} has the weight λ_{b} with respect to Γ_{m}. Now it is clear that u_{b} generates an indecomposable Γ_{m}-module with highest weight λ_{b} and so M↓Γ_{m} has the required composition factor.

Proof of Theorem 1.2. Set p + p^{2} + … + p^{l} = y. Let λ = ∑_{j=0}^{u} p^{j}λ_{j} with λ_{j} = ∑_{i=1}^{n} a_{ij}ω_{i}^{n} ∈ X_{p}^{n} be the highest weight of M. Our assumptions yield that ∑_{i=1}^{n−m−1} a_{it} ≥ y for some t, 0 ≤ t ≤ u. Hence, by Lemma 2.5 and Steinberg’s tensor product theorem [8], M(λ_{t})↓Γ_{m+1} has a composition factor of the form M(yω_{1}^{m+1})⊗ M(μ) with p-restricted μ. The same theorem implies that

M(yω_{1}^{m+1})↓Γ_{m} = (τ ⊕ M(pω_{1}^{m})) ⊗ (τ ⊕ M(p^{2}ω_{1}^{m})) ⊗ ⋯ ⊗ (τ ⊕ M(p^{l}ω_{1}^{m})).

Now it is clear that M(yω_{1}^{m+1})↓H_{m} = M′ where M′ = (τ ⊕ M_{1}) ⊗ (τ ⊕ M_{2}) ⊗ … ⊗ (τ ⊕ M_{l}) for some irreducible 𝕂H_{m}-modules M_{1}, …, M_{l}. Hence (Fr^{t}(M′) ⊗ N) is a quotient of a submodule of M↓H_{m} for some H_{m}-module N. By [4, ch.3, Corollary 2.17], the regular H_{m}-module 𝕂H_{m} is a direct summand of M′ as each nonidentity element of H_{m} acts nontrivially on the modules M_{j}, 1 ≤ j ≤ l. So 𝕂H_{m} is a direct summand of Fr^{t}(M′) as well since Fr^{t}(𝕂H_{m}) ≅ 𝕂H_{m}. Let Q be an irreducible submodule of N. By [3, Corollary 10.20], 𝕂H_{m} ⊗ Q is isomorphic to the direct sum of dim Q copies of 𝕂H_{m}. Now the projectivity of 𝕂H_{m} implies that 𝕂H_{m} is a submodule of M↓H_{m}.

3 Inductive systems over splitting fields

In this section q = p^{s}, s ∈ ℕ, and H_{1} → H_{2} → ⋯ → H_{n} → … is the sequence of finite groups H_{n} = SL_{n+1}(q) or SU_{n+1}(q^{2}) (≅ ^{2}A_{n}(q)) naturally embedded each into another; H = \underrightarrow{lim} H_{n}; 𝔽 is a splitting field for H. Here the minimal nontrivial inductive systems for H over 𝔽 are determined.

Assume that 𝕂 is the algebraic closure of 𝔽, so H_{n} ⊂ G_{n} for all n. By Steinberg’s theorem [8], the restrictions to H_{n} of the q-restricted irreducible G_{n}-modules yield the complete set of inequivalent irreducible 𝕂H_{n}-modules. Since 𝔽 is a splitting field for H_{n}, the extension of the ground field yields a natural bijection from Irr 𝔽H_{n} to Irr 𝕂H_{n}. Therefore the irreducible 𝔽H_{n}-modules can be parametrized by q-restricted dominant weights of G_{n}. For such a weight λ we denote by N(λ) the corresponding irreducible 𝔽H_{n}-module, so

M(λ)↓𝕂H_{n} ≅ N(λ) ⊗_{𝔽} 𝕂.  (3)

Therefore it will cause no confusion if we use the same symbol Irr H_{n} both for Irr 𝔽H_{n} and Irr 𝕂H_{n}.

Remark 3.1 Sometimes we shall consider modules N(λ) with λ = q^{l}λ_{0} and qrestricted λ_{0}. In this case M(λ)↓𝕂H_{n} ≅ N(μ) ⊗_{𝔽} 𝕂 with q-restricted μ and we set N(λ) = N(μ). Here μ = λ_{0} except the case where H = SU_{n+1}(q^{2}) and l is odd. In the exceptional case μ = −w_{0}(λ_{0}) where w_{0} is the longest element of the Weyl group.

Lemma 3.2 Let m < n, λ ∈ X_{n}^{q}, μ ∈ X_{m}^{q}, and M(μ) ∈ Irr(M(λ)↓G_{m}). Then N(μ) ∈ Irr(N(λ)↓H_{m}).

Proof. In view of (3),

Irr(N(λ) ⊗_{𝔽} 𝕂↓𝕂H_{m}) = Irr((M(λ)↓𝕂G_{m})↓𝕂H_{m}) ∋ M(μ)↓𝕂H_{m} = N(μ) ⊗_{𝔽} 𝕂.

Since 𝔽 is a splitting field for H_{m}, the set Irr(N(λ)↓H_{m}) contains N(μ).

Proof of Theorem 1.4. Let L^{j} = {L_{n}^{j}}_{n∈ℕ} and R^{j} = {R_{n}^{j}}_{n∈ℕ} (0 ≤ j < s) be the inductive systems for H defined earlier. One easily observes that L_{n}^{j} = {N(p^{j}ω_{1}^{n}), τ} and R_{n}^{j} = {N(p^{j}ω_{n}^{n}), τ}.

Let Φ = {Φ_{n}}_{n∈ℕ} where Φ_{n} = {N(λ_{1}^{n}), …, N(λ_{k_{n}}^{n})} and λ_{i}^{n}, 1 ≤ i ≤ k_{n}, are q-restricted. Set Ψ_{n} = {M(λ_{1}^{n}), …, M(λ_{k_{n}}^{n})} and δ_{n} = max{δ(λ_{i}^{n}) | i = 1, …, k_{n}}. The proof will be divided into two subcases. Case 1: sup{δ_{n} | n ∈ ℕ} = k < ∞. Fix m, n ∈ ℕ such that n ≥ m(k + 1). By Lemma 2.3, Irr(Ψ_{n}↓G_{m}) contains M(p^{i}ω_{1}^{m}) or M(p^{i}ω_{m}^{m}) for some i = i(m) < s. Now Lemma 3.2 implies that Ψ_{m} contains N(p^{i}ω_{1}^{m}) or N(p^{i}ω_{m}^{m}). So there exists j < s and an infinite sequence m_{1} < m_{2} < ⋯ < m_{r} < … such that either for each r the set Φ_{m_{r}} contains N(p^{j}ω_{1}^{m_{r}}), or each of them contains N(p^{j}ω_{m_{r}}^{m_{r}}). This completes Case 1 since L^{j} and R^{j} are inductive systems. Case 2: sup{δ_{n} | n ∈ ℕ} = ∞. We shall prove that Φ_{n} = Irr H_{n}, so Φ contains all the systems L^{j} and R^{j}. Fix m ∈ ℕ. Set l = |H_{m}| − 1 and c = p + p^{2} + … + p^{l} + (p − 1)(m + 1). Since sup{δ_{n} | n ∈ ℕ} = ∞, there exist n > m + 1 and M = M(λ) ∈ Ψ_{n} such that δ(λ) ≥ (1+p+…+p^{s−1})c. Recall that for each M(λ) ∈ Ψ_{n} the weight λ is q-restricted and so can be represented in the form λ = λ_{0}+pλ_{1}+⋯+p^{s−1}λ_{s−1} where λ_{j} ∈ X_{n}^{p}. Observe that δ(λ) = δ(λ_{0})+pδ(λ_{1})+⋯+p^{s−1}δ(λ_{s−1}). Therefore s(M) ≥ c in the notation of Theorem 1.2. Now Theorem 1.2 yields that Irr(M↓H_{m}) = Irr H_{m} and completes the proof.

Remark 3.3 By Remark 3.1, one can consider the inductive systems L^{j} and R^{j} for arbitrary j = 0, 1, 2, … Note that R^{j} ≅ L^{j+s} for H = SU_{∞}(q^{2}). Hence any nontrivial inductive system for H = SU_{∞}(q^{2}) contains one of the systems L^{j} (0 ≤ j < 2s).

4 Nonsplitting fields

We shall heavily use the following fact.

Lemma 4.1 Let 𝔽 be a field of characteristic p and 𝔼 be any extension field of 𝔽. Let G be a finite group and M be an irreducible 𝔽G-module. Then M ⊗_{𝔽} 𝔼 is a direct sum of irreducible 𝔽G-modules no two of which are isomorphic. If 𝔼 is a finite Galois extension of 𝔽 and a splitting field for G, then there is a bijection between Irr 𝔽G and Gal(𝔼/𝔽)-orbits in Irr 𝔼G; M ⊗_{𝔽} 𝔼 is a direct sum of all modules in the corresponding orbit O(M); and O(M) is the set of all N ∈ Irr 𝔼G such that the 𝔽G-module N is a sum of copies of M.

Proof. This follows from [3, Corollaries 7.11 and 7.19] and the fact that the 𝔽Gmodule M ⊗_{𝔽} 𝔼 is a sum of copies of M.

Set t = s for H = SL_{∞}(p^{s}) and t = 2s for H = SU_{∞}(p^{2s}); 𝔽_{H} = 𝔽_{p^{t}} (the field of order p^{t}). As above, the irreducible 𝔽_{H}H_{n}-modules can be parametrized by q-restricted weights λ and will be denoted by N(λ). In this section we assume that 𝔽 is not a splitting field for H, i.e. 𝔽_{H} ⊄ 𝔽. Then 𝔽_{H} ∩ 𝔽 is a finite field 𝔽_{0} = 𝔽_{p^{u}}. Note that u < t and u|t. We identify 𝔽_{H} with the isomorphic subfield of the algebraic closure 𝔽̄ of 𝔽. Let 𝔽_{1} = 𝔽_{H}𝔽 be the composite of 𝔽_{H} and 𝔽. It is well known that 𝔽_{1} is a Galois extension of 𝔽 and the Galois group Gal(𝔽_{1}/𝔽) is isomorphic to Σ = Gal(𝔽_{H}/𝔽_{0}) ≅ ℤ_{t/u} (the cyclic group of order t/u); moreover, we can obtain this isomorphism restricting the automorphisms of 𝔽_{1} to 𝔽_{H}.

The following proposition allows us to reduce the study of inductive systems over 𝔽 to the similar problem for 𝔽_{0}.

Proposition 4.2 The map f : M ↦ M̄ = M ⊗_{𝔽_{0}} 𝔽 yields a bijection between Irr 𝔽_{0}H_{n} and Irr 𝔽H_{n}. Moreover, for n > 1

Irr(M̄↓H_{n−1}) = {N̄ | N ∈ Irr(M↓H_{n−1})}.

Proof. First we claim that M̄ is irreducible. Indeed, by Lemma 4.1, M̄ is a direct sum of k irreducible inequivalent H_{n}-modules. Since 𝔽_{1} is a Galois extension of 𝔽 and is a splitting field for H_{n}, the same lemma implies that M̄ ⊗_{𝔽}𝔽_{1} is the direct sum of all modules involved in some different k Gal(𝔽_{1}/𝔽)-orbits in Irr 𝔽_{1}H_{n}. On the other hand, M̄⊗_{𝔽}𝔽_{1} ≅ (M⊗_{𝔽_{0}}𝔽_{H})⊗_{𝔽_{H}}𝔽_{1}. By Lemma 4.1, M⊗_{𝔽_{0}}𝔽_{H} = M_{1}⊕⋯⊕M_{l} where M_{1}, …, M_{l} constitute a Σ-orbit in Irr 𝔽_{H}H_{n}. Since 𝔽_{H} is a splitting field for H_{n}, every M_{i} remains irreducible after a field extension. As each element of Gal(𝔽_{1}/𝔽) is completely determined by its action on 𝔽_{H} and irreducible 𝔽_{1}H_{n}-representations can be realized over 𝔽_{H}, one can identify the action of Gal(𝔽_{1}/𝔽) on Irr 𝔽_{1}H_{n} with that of Σ on Irr 𝔽_{H}H_{n}. Hence M_{1} ⊗_{𝔽_{H}} 𝔽_{1}, …, M_{l} ⊗_{𝔽_{H}} 𝔽_{1} constitute a Gal(𝔽_{1}/𝔽)-orbit in Irr 𝔽_{1}H_{n}. This implies k = 1, so M̄ is irreducible.

The same arguments with field extensions show that the map f is injective. By Lemma 4.1, | Irr 𝔽_{0}H_{n}| = | Irr 𝔽H_{n}| since the relevant Galois groups have equal numbers of orbits on Irr 𝔽_{H}H_{n} and Irr 𝔽_{1}H_{n}, respectively. Hence f is a bijection. Since taking field extensions commutes with restricting to subgroups, the irreducibility of M̄ yields the assertion of the lemma for restrictions.

Let Φ = {Φ_{n}}_{n∈ℕ} be an inductive system for H over 𝔽_{0}. Set Φ̄_{n} = {M̄ | M ∈ Φ_{n}}; Φ̄ = {Φ̄_{n}}_{n∈ℕ}.

Corollary 4.3 The map Φ → Φ̄ yields a bijection between the sets of inductive systems for H over 𝔽_{0} = 𝔽 ∩ 𝔽_{H} and 𝔽, respectively.

In view of Corollary 4.3, we restrict our attention to the modules and the inductive systems over 𝔽_{0} = 𝔽_{p^{u}}. One easily observes that the orbit of N(p^{j}ω_{1}^{n}), 0 ≤ j < u, under Σ is the set {N(p^{j+ku}ω_{1}^{n}) | k = 0, 1, …, t/u − 1}. The orbit of N(p^{j}ω_{n}^{n}) has a similar form.

Denote by N_{n}^{j} and N_{n}^{∗j} the H_{n}-modules N(p^{j}ω_{1}^{n}) and N(p^{j}ω_{n}^{n}), respectively, considered over the field 𝔽_{0}. Since we are interested in inductive systems for H = \underrightarrow{lim} H_{n}, we may and shall assume hereinafter that n > 1.

Lemma 4.4 (i) The modules N_{n}^{j} and N_{n}^{∗j} (0 ≤ j < s) are irreducible.

(ii) For H_{n} = SL_{n+1}(q) the modules N_{n}^{i} ≅ N_{n}^{j} (N_{n}^{∗i} ≅ N_{n}^{∗j}) if and only if i ≡ j (mod u); N_{n}^{i} ≅ N̸_{n}^{∗j} for all i and j.

(iii) For H_{n} = SU_{n+1}(q^{2}) the modules N_{n}^{i} ≅ N_{n}^{j} (N_{n}^{∗i} ≅ N_{n}^{∗j}) if and only if i ≡ j (mod u); N_{n}^{i} ≅ N_{n}^{∗j} if and only if i ≡ s + j (mod u).

Proof. (i) By [5, Lemma 4.3.2], N_{n}^{0} is irreducible. This implies the irreducibility of N_{n}^{j} and N_{n}^{∗j} (0 ≤ j < s) since the relevant linear groups coincide.

(ii) It follows from Lemma 4.1 and (i) that N_{n}^{i} ⊗_{𝔽_{0}} 𝔽_{H} (0 ≤ i < s) is a direct sum of modules in Irr 𝔽_{H}H_{n} that constitute a Σ-orbit and this orbit consists of all N(p^{k}ω_{1}^{n}) with 0 ≤ k < s and k ≡ i (mod u). Since Irr 𝔽_{0}H_{n} is parametrized by Σ-orbits on Irr 𝔽_{H}H_{n}, we obtain the required assertion.

(iii) The arguments are quite similar to those of item (ii). We only have to take into account that the 𝔽_{H}H_{n}-modules N(p^{k+s}ω_{1}^{n}) and N(p^{k}ω_{n}^{n}) are isomorphic.

Set L̃_{n}^{j} = {N_{n}^{j}, τ}, R̃_{n}^{j} = {N_{n}^{∗j}, τ}, L̃^{j} = {L̃_{n}^{j}}_{n∈ℕ}, and R̃^{j} = {R̃_{n}^{j}}_{n∈ℕ}. Note that L̃^{j} and R̃^{j} are in fact the systems L^{j} and R^{j} for H considered as the systems of modules over 𝔽_{0}. It is not difficult to see that L̃^{j} and R̃^{j} are inductive systems.

Lemma 4.5 Let π(j) be the residue of j modulo u.

(i) For H = SL_{∞}(q) the inductive systems L̃^{0}, …, L̃^{u−1}, R̃^{0}, …, R̃^{u−1} are pairwise nonequivalent; L̃^{j} ≅ L̃^{π(j)} and R̃^{j} ≅ R̃^{π(j)}.

(ii) For H = SU_{∞}(q^{2}) the inductive systems L̃^{0}, …, L̃^{u−1} are pairwise nonequivalent; L̃^{j} ≅ L̃^{π(j)} and R̃^{j} ≅ L̃^{π(j+s)}.

Proof. This follows from Lemma 4.4.

Theorem 4.6 Let Ψ be a nontrivial inductive system for H = SL_{∞}(q) or SU_{∞}(q^{2}) over the field 𝔽_{0} = 𝔽_{p^{u}} ⊂ 𝔽_{H} (𝔽_{0} ≠ 𝔽_{H}). Then Ψ contains one of the following systems:

L̃^{0}, …, L̃^{u−1}, R̃^{0}, …, R̃^{u−1} for H = SL_{∞}(q);

L̃^{0}, …, L̃^{u−1} for H = SU_{∞}(q^{2}).

Proof. Set

Ψ̄_{n} = ⋃_{M∈Ψ_{n}} Irr(M ⊗_{𝔽_{0}} 𝔽_{H}).

Obviously, Ψ̄ = {Ψ̄_{n}}_{n∈ℕ} is a nontrivial inductive system for H over 𝔽_{H}. Therefore by Theorem 1.4, Ψ̄ contains either L^{j} or R^{j} for some j < s. Now Lemma 4.1 implies that for each n ∈ ℕ the set Ψ̄_{n} contains the Σ-orbit of the module N(p^{j}ω_{1}^{n}) or N(p^{j}ω_{n}^{n}). Therefore it follows from the proof of Lemma 4.4 that Ψ_{n} contains either N_{n}^{j}, or N_{n}^{∗j}, so Ψ contains either L̃^{j}, or R̃^{j}. Now Lemma 4.5 yields the theorem.

Let σ be an automorphism of the field 𝔽_{0}. For an irreducible 𝔽_{0}H_{n}-module N denote by N^{σ} the module obtained from N by applying σ to all matrix entries of the relevant matrix representation. Define an action of σ on Irr 𝔽H_{n} = {N ⊗_{𝔽_{0}} 𝔽 | N ∈ Irr 𝔽_{0}H_{n}} (see Proposition 4.2) via (N ⊗_{𝔽_{0}} 𝔽)^{σ} = N^{σ} ⊗_{𝔽_{0}} 𝔽. Similarly, one can define Ψ^{σ} for an inductive system Ψ for 𝔽H. Set L̃ = L̃^{0} and R̃ = R̃^{0}. Let θ ∈ Aut 𝔽_{0} raise elements to the power p. Observe that L̃^{i} = L̃^{θ^{i}} and R̃^{i} = R̃^{θ^{i}} for i = 0, 1, …, u − 1.

Corollary 4.7 Let Ψ be a nontrivial inductive system for H = SL_{∞}(q) or SU_{∞}(q^{2}) over a field 𝔼 ⊂ 𝔽_{H}. Then Ψ contains one of the following systems:

L̃^{σ} or R̃^{σ} (σ ∈ Aut 𝔼) for H = SL_{∞}(q);

L̃^{σ} (σ ∈ Aut 𝔼) for H = SU_{∞}(q^{2}).

Proof. The case 𝔼 ≠ 𝔽_{H} follows from Theorem 4.6; for 𝔼 = 𝔽_{H} the corollary follows from Theorem 1.4 and Remark 3.3.

5 Finitary groups over locally finite fields

Throughout this section 𝔽 and 𝕃 are fields of characteristic p and 𝕃 is locally finite. One can represent 𝕃 as the union of finite subfields

𝕃_{1} ⊂ 𝕃_{2} ⊂ ⋯ ⊂ 𝕃_{i} ⊂ …  (4)

Denote by SL_{∞}(𝕃) the union of the groups

SL_{∞}(𝕃_{1}) ⊂ SL_{∞}(𝕃_{2}) ⊂ ⋯ ⊂ SL_{∞}(𝕃_{i}) ⊂ …

Let 𝕃 have an automorphism α of order 2. Then α fixes all 𝕃_{i}. Removing a finite number of subfields in sequence (4) if necessary, one can assume that α acts nontrivially on 𝕃_{1}. Denote by SU_{∞}(𝕃) the union of the groups

SU_{∞}(𝕃_{1}) ⊂ SU_{∞}(𝕃_{2}) ⊂ ⋯ ⊂ SU_{∞}(𝕃_{i}) ⊂ …

One can observe that this construction does not depend upon the choice of sequence (4).

Remark 5.1 It is not difficult to see that 𝕃 has an automorphism of order 2 if and only if 𝕃 has a subfield of order p^{2} and has no subfields of order p^{2^{l}} for l large enough.

The groups SL_{∞}(𝕃) and SU_{∞}(𝕃) are locally finite. Recall that a set {G_{α}}_{α∈A} of finite subgroups of a locally finite group G is called a local system if G = ⋃_{α∈A} G_{α} and for each pair α, β ∈ A there is γ ∈ A such that G_{α}, G_{β} ⊂ G_{γ}. Set α ≤ β if G_{α} ⊂ G_{β}. Then A is a directed set and G = \underrightarrow{lim} G_{α}. Let Φ_{α} be a finite subset of Irr 𝔽G_{α}. The collection Φ = {Φ_{α}}_{α∈A} is called an inductive system for G if

⋃_{M∈Φ_{β}} Irr(M↓G_{α}) = Φ_{α}

for all α < β in A.

In this section we shall denote by H one of the groups SL_{∞}(𝕃) or SU_{∞}(𝕃), by H_{i} the subgroup SL_{∞}(𝕃_{i}) or SU_{∞}(𝕃_{i}), and by H_{n,i} the subgroup SL_{n}(𝕃_{i}) or SU_{n}(𝕃_{i}), respectively. Observe that the set {H_{n,i}}_{n,i∈ℕ} is a local system for H, i.e. a directed set of subgroups of H with \underrightarrow{lim} H_{n,i} = H. Set 𝔼_{i} = 𝔽 ∩ 𝕃_{i}. Denote by N_{n,i}^{j} and N_{n,i}^{∗j} the H_{n,i}-modules N(p^{j}ω_{1}^{n}) and N(p^{j}ω_{n}^{n}), respectively, considered over the field 𝔼_{i} (cf. the notation before Lemma 4.4). In the lemma below j = 0, 1 …, but one takes into account that the relevant modules can coincide for different j. Observe that by Proposition 4.2, each irreducible 𝔼_{k+1}H_{n,k}-module can be represented in the form S ⊗_{𝔼_{k}} 𝔼_{k+1} where S ∈ Irr 𝔼_{k}H_{n,k} as 𝕃_{k} is a splitting field for H_{n,k} and 𝕃_{k} ∩ 𝔼_{k+1} = 𝔼_{k}.

Lemma 5.2 Let M be an irreducible 𝔼_{k+1}H_{n,k+1}-module and

Irr(M↓H_{n,k}) = {S_{1} ⊗_{𝔼_{k}} 𝔼_{k+1}, …, S_{l} ⊗_{𝔼_{k}} 𝔼_{k+1}}

where S_{t} ∈ Irr 𝔼_{k}H_{n,k}, 1 ≤ t ≤ l. Then

Irr(M ⊗_{𝔼_{k+1}} 𝔽↓H_{nk}) = {S_{1} ⊗_{𝔼_{k}} 𝔽, …, S_{l} ⊗_{𝔼_{k}} 𝔽}.

In particular,

Irr(N_{n,k+1}^{j} ⊗_{𝔼_{k+1}} 𝔽↓H_{nk}) = {N_{nk}^{j} ⊗_{𝔼_{k}} 𝔽};
Irr(N_{n,k+1}^{∗j} ⊗_{𝔼_{k+1}} 𝔽↓H_{nk}) = {N_{nk}^{∗j} ⊗_{𝔼_{k}} 𝔽}.

Proof. Let 𝔽_{1} be the composite of the fields 𝔽 and 𝕃_{k+1}. Set Σ_{1} = Gal(𝔽_{1}/𝔽). Recall that Gal(𝔽_{1}/𝔽) ≅ Gal(𝕃_{k+1}/𝔼_{k+1}). Put H^{k} = H_{n,k}, H^{k+1} = H_{n,k+1}, M̄ = (M ⊗_{𝔼_{k+1}} 𝔽) ⊗_{𝔽} 𝔽_{1}, S̄_{t} = (S_{t} ⊗_{𝔼_{k}} 𝔽) ⊗_{𝔽} 𝔽_{1}. By Lemma 4.1, the set Irr(M ⊗_{𝔼_{k+1}} 𝔽↓H^{k}) is completely determined by the Σ_{1}-orbits of the elements of Irr(M̄↓H^{k}) = ∪_{t=1}^{l} Irr(S̄_{t}). Proposition 4.2 implies that S_{1} ⊗_{𝔼_{k}} 𝔽 are nonequivalent irreducible 𝔽H^{k}-modules. Hence by Lemma 4.1, Irr(S̄_{t}) is an Σ_{1}-orbit on Irr 𝔽_{1}H^{k}. This yields the first assertion of our lemma. It remains to observe that N_{n,k+1}^{j} and N_{n,k+1}^{∗j} are irreducible 𝔼_{k+1}H^{k+1}-modules by Lemma 4.4(i) and consider their restrictions to H^{k}.

Set L_{n,i} = {N_{n,i}^{0} ⊗_{𝔼_{i}} 𝔽, τ}, R_{n,i} = {N_{n,i}^{∗0} ⊗_{𝔼_{i}} 𝔽, τ}, L = {L_{n,i}}_{n,i∈ℕ}, R = {R_{n,i}}_{n,i∈ℕ}, and 𝔼 = 𝔽 ∩ 𝕃. Using Lemma 5.2, one easily observes that L and R are inductive systems for H. Furthermore, if σ ∈ Aut 𝔼 and σ_{i} = σ|_{𝔼_{i}}, then L^{σ} = {{N_{n,i}^{σ_{i}} ⊗_{𝔼_{i}} 𝔽, τ} | n, i ∈ ℕ} and R^{σ} = {{N_{n,i}^{∗σ_{i}} ⊗_{𝔼_{i}} 𝔽, τ} | n, i ∈ ℕ} are inductive systems. Here N_{n,i}^{σ_{i}} = N_{n,i}^{j} and N_{n,i}^{∗σ_{i}} = N_{n,i}^{∗j} if σ_{i} raises elements to the p^{j}th power.

Theorem 5.3 Let 𝔽 and 𝕃 be fields of characteristic p, 𝕃 be locally finite, H = SL_{∞}(𝕃) or SU_{∞}(𝕃), and 𝔼 = 𝔽 ∩ 𝕃. Let Ψ = {Ψ_{n,i}}_{n,i∈ℕ} be a nontrivial inductive system for H = SL_{∞}(𝕃) or SU_{∞}(𝕃) over 𝔽. Then Ψ contains one of the following systems:

L^{σ} or R^{σ} (σ ∈ Aut 𝔼) for H = SL_{∞}(𝕃);

L^{σ} (σ ∈ Aut 𝔼) for H = SU_{∞}(𝕃). In particular, Ψ contains the trivial inductive system.

Proof. Denote by Ψ_{i} the set {Ψ_{n,i}}_{n∈ℕ}. Clearly, Ψ_{i} is a nontrivial inductive system for H_{i} = ⋃_{n∈ℕ} H_{n,i}. Using Corollary 4.3, we shall identify inductive systems for H_{i} over 𝔼_{i} with those over 𝔽. By Corollary 4.7, Ψ_{i} contains one of the following systems: L̃^{σ_{i}} or R̃^{σ_{i}} (σ_{i} ∈ Aut 𝔼_{i}) for H_{i} = SL_{∞}(𝕃_{i}); L̃^{σ_{i}} (σ_{i} ∈ Aut 𝔼_{i}) for H_{i} = SU_{∞}(𝕃_{i}). Assume, for definiteness, that L̃^{σ_{i}} ⊂ Ψ_{i} for H_{i} = SL_{∞}(𝕃_{i}) and infinitely many i. Then by Lemma 5.2, L̃^{ρ} ⊂ Ψ_{i−1} where ρ = σ_{i}|_{𝔼_{i−1}}. Therefore we shall assume that for each i there exists σ_{i} ∈ Aut 𝔼_{i} such that L̃^{σ_{i}} ⊂ Ψ_{i}. The case where R̃^{σ_{i}} ⊂ Ψ_{i} for all i can be considered quite similarly. Let Δ_{i} be the set of all θ ∈ Aut 𝔼_{i} such that L̃^{θ} ⊂ Ψ_{i}. For i < j set

Δ_{ij} = {θ|_{𝔼_{i}} | θ ∈ Δ_{j}} ⊂ Aut 𝔼_{i}.

It follows from Lemma 5.2 that Δ_{ij} ⊂ Δ_{i}. Note that

Δ_{i} ⊃ Δ_{i,i+1} ⊃ Δ_{i,i+2} ⊃ …

Therefore one can define Δ̄_{i} = ⋂_{j=i+1}^{∞} Δ_{ij}. Observe that Δ̄_{i} is nonempty and there exists c = c(i) ∈ ℕ such that Δ̄_{i} = Δ_{ik} for all k ≥ c. Take any k > c(i), c(i + 1). We have Δ̄_{i+1}|_{𝔼_{i}} = Δ_{i+1,k}|_{𝔼_{i}} = Δ_{ik} = Δ̄_{i}. Therefore there exists a sequence of automorphisms θ_{i} ∈ Δ_{i} such that θ_{i+1}|_{𝔼_{i}} = θ_{i} for all i. This sequence determines a unique automorphism σ of 𝔼 such that σ|_{𝔼_{i}} = θ_{i}. Now it is clear that L^{σ} is contained in Ψ.

Let G be a locally finite group and {G_{α}}_{α∈A} be a local system of G. Let I be a (two-sided) proper ideal of the group algebra 𝔽G. Then the quotient 𝔽G/I can be considered as an 𝔽G-module. Set

Φ(I)_{α} = Irr(𝔽G/I↓G_{α}).

One can easily check that the collection Φ(I) = {Φ(I)_{α}}_{α∈A} is an inductive system for G. Recall that an ideal I of an algebra R is called semiprimitive if the Jacobson radical Rad(R/I) = 0. The following Zalesskii’s result reduces the problem of describing lattices of ideals in 𝔽G to inductive systems.

Theorem 5.4 ([12, 1.25]) The map I ↦ Φ(I) is a bijection between the poset of proper semiprimitive ideals of 𝔽G and the poset of inductive systems for G over 𝔽. Let M_{Φ} be the semiprimitive ideal corresponding to an inductive system Φ. Then for each proper ideal I of 𝔽G we have I ⊂ M_{Φ(I)} and the quotient M_{Φ(I)}/I is locally nilpotent.

Theorem 1.5 (see Introduction) is a particular case of the following theorem (see also Remark 3.3).

Theorem 5.5 Let 𝔽 and 𝕃 be fields of characteristic p, 𝕃 be locally finite, H = SL_{∞}(𝕃) or SU_{∞}(𝕃), and 𝔼 = 𝔽 ∩ 𝕃. Let V be the natural H-module and V^{σ} be the module V twisted by σ ∈ Aut 𝔼. Then any proper ideal of 𝔽H is contained in the augmentation ideal Aug(𝔽H). Moreover, the annihilators

Ann_{𝔽H}V^{σ} and Ann_{𝔽H}(V^{∗})^{σ} (σ ∈ Aut 𝔼) for H = SL_{∞}(𝕃);

Ann_{𝔽H}V^{σ} (σ ∈ Aut 𝔼) for H = SU_{∞}(𝕃) are exactly all distinct maximal ideals of the algebra Aug(𝔽H).

Proof. Let M be an ideal of 𝔽H such that Φ(M) is the trivial inductive system for H. Since Irr(𝔽H/M↓H_{n,i}) = {τ} and H_{n,i} is perfect at least for n > 2, M ∪ 𝔽H_{n,i} = Aug 𝔽H_{n,i} for all n and i. This implies that M = Aug 𝔽H. Therefore in view of Theorem 5.4, the maximal ideals of Aug(𝔽H) (= the ideals of 𝔽H that are maximal among those properly lying in Aug(𝔽H)) are exactly the ideals M_{Φ} where Φ runs over the minimal nontrivial inductive systems for H.

Let W be the natural H-module over the field 𝕃 and W_{0} be the module W considered over the field 𝔼. Recall that V = W_{0} ⊗_{𝔼} 𝔽 and V^{σ} = W_{0}^{σ} ⊗_{𝔼} 𝔽 for every σ ∈ Aut 𝔼. Let I = Ann_{𝔽H}V^{σ} and J = Ann_{𝔽H}(V^{∗})^{σ}. Obviously, Φ(I) = L^{σ} and Φ(J) = R^{σ} (see Theorem 5.3) since the annihilators of V^{σ} and 𝔽H/I (respectively, (V^{∗})^{σ} and 𝔽H/J) coincide. It remains to observe that the ideals I and J are semiprimitive (as V^{σ} and (V^{∗})^{σ} are completely reducible) and to apply Theorems 5.3 and 5.4.

Acknowledgements

Both authors have been supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the program “Mathematical structures” and by the Belarus Basic Research Foundation, Project F 98-180. The first author has been also supported by Alexander von Humboldt Foundation.

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