Minimal inductive systems of modular representations for naturally

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 [email protected] [email protected]

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 Γ1 → Γ2 → · · · → Γn → . . .

(1)

be a sequence of embeddings where either all Γi are finite or all of them are algebraic groups over K and for algebraic Γi all embeddings in (1) are rational. Throughout the paper Irr FΓn (or Irr Γn if F is fixed) is the set of equivalence classes of irreducible (rational for Γn algebraic) Γn -modules over F (F = K for Γn 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 Φn be a nonempty finite subset

1

of Irr Γn . We say that the collection Φ = {Φn }n∈N is an inductive system for the sequence (1) if [ Irr(M ↓Γn ) = Φn M ∈Φn+1

for all n ∈ N (see also the definition in Section 5). The system Φ is called trivial if Φn = {τ } for all n; otherwise Φ is nontrivial. Set Gn = SLn+1 (K) (the algebraic group of type An ). Let Fr be the Frobenius morphism of Gn associated with raising elements of K to the pth power. For a morphism ρ of Gn and M ∈ Irr Gn denote by ρ(M ) the Gn -module affording the representation Gn → ρ(Gn ) → GL(M ). Let Vn be the standard Gn -module and Vn∗ be its dual. Set Ljn = {Frj (Vn ), τ } and

Rjn = {Frj (Vn∗ ), τ }

(j = 0, 1, 2, . . . ).

Let A be the sequence of the natural embeddings G1 → G2 → · · · → Gn → . . . . It is clear that Lj = {Ljn }n∈N and Rj = {Rjn }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 Lj or Rj (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 Γ1 ⊂ . . . ⊂ Γn ⊂ . . . be the sequence of the naturally embedded simply connected classical groups Γi = Γi (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 Γi . Denote by ωji , 1 ≤ j ≤ i, the fundamental weights of the group Γi labeled in the standard way. Let Irr Γn and let λ be the highest weight of M . Pu M ∈ j λ where the weights λ are p-restricted. Let p Represent λ in the form λ = j j=0 P j P λj = ni=1 aij ωin . Set S(M ) = max{ ni=1 aij | 0 ≤ j ≤ u}. Theorem 1.2 Let Hm ⊂ Γm be a finite ordinary or twisted Chevalley group of rank m which is the fixed subgroup of some Frobenius morphism of Γm and let l = |Hm |−1. Then for each n > m + 1 and each M ∈ Irr Γn with S(M ) ≥ p + p2 + . . . + pl + (p − 1)(m + 1) the restriction M ↓Hm contains the regular KHm -module. In particular, Irr(M ↓Hm ) = Irr KHm . Remark 1.3 It is clear that Theorem 1.2 can be transferred to irreducible Krepresentations of finite Chevalley groups associated with Γn which contain Hm .

2

Now let q = ps and let H1 → H2 → · · · → Hn → . . .

(2)

be the sequence of finite groups Hn = SLn+1 (q) or SUn+1 (q 2 ) (all the groups are of the same type) naturally embedded each into another. Set H = lim Hn . 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(An , 1, 1, . . . ) where An ∈ SLn (q) or SUn (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 Φn = Irr(M ↓Hn ). One easily checks that Φ = {Φn }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 Lj = Φ(Frj (V )), and Rj = Φ(Frj (V ∗ )), 0 ≤ j < s. We keep the notation Lj and Rj 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 Lj or Rj (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 AnnFH Frj (V ) and AnnFH Frj (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. 3

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 Cn and p > 2 Zalesskii and Suprunenko [10] n have described an inductive system Ψ with Ψn = {ϕ(ωn−1 + 12 (p − 3)ωnn ), ϕ( 21 (p − n 1)ωn )}, n = 1, 2, . . . . The authors [1] have obtained the branching rules for modular fundamental representations of Cn and classified the inductive systems consisting of such and trivial representations. In particular, there exists a minimal inductive p−1 system Rp−1 = {Rp−1 = {ϕ(ωjn ) | n + 1 − j ≤ p − 1}. n }, n = 1, 2, . . . , with Rn

2

Inductive systems for algebraic groups

In this section the minimal nontrivial inductive systems for the sequence A of the natural embeddings G1 → G2 → · · · → Gn → . . . are determined and Theorem 1.2 is proved. Let α1 , . . . , αn be the simple roots of Gn . We identify these roots with the relevant simple roots of Gn+1 . Denote by Xn+ the set of dominant weights of Gn and let ω1n , . . . , ωnn (or ω1 , . . . , ωn if n is fixed) be the fundamental weights of Gn . Recall that Xn+ = {a1 ω1 + · · · + an ωn | ai ∈ N0 } where N0 is the set of nonnegative integers. Set Xnq = {a1 ω1 +· · ·+an ωn | 0 ≤ ai < q}. The weights in Xnq are called q-restricted. Denote by h·, ·i the canonical pairing on the set of weights of Gn . In what follows M (λ) is the irreducible Gn -module with highest weight λ. Recall that Vn = M (ω1n ) and Vn∗ = M (ωnn ). For λ = a1 ω1 + · · · + an ωn ∈ Xn+ set δ(λ) = a1 + · · · + an . The following lemmas are used in the proof of Theorem 1.1. Lemma 2.1 For each λ ∈ Xn+ and each root α of Gn we have δ(λ) = maxhµ, αi where µ runs over the weights of M (λ). Proof. Let β be the maximal root of Gn . It is well known that β = α1 + · · · + αn . Since the Weyl group acts transitively on the set of roots, maxhµ, αi = maxhµ, βi for every root α. As β is a dominant weight, hαi , βi ≥ 0. This forces δ(λ) = hλ, βi = maxhµ, βi, as required. Lemma 2.2 Let k, m, n ∈ N be such that n ≥ m(k + 1). Assume that λ ∈ Xn+ \{0} and δ(λ) ≤ k. Then τ ∈ Irr(M (λ)↓Gm ). Moreover, if λ is p-restricted, then Irr(M (λ)↓Gm ) contains Vm or Vm∗ . Proof. Denote by H(i1 , . . . , ik ) the subgroup generated by the root subgroups associated with the roots ±αi1 , . . . , ±αik . Let λ = a1 ω11 + · · · + an ωnn . Obviously, at most k of the coefficients ai are nonzero. Now one can observe that there exists i such that ai 6= 0 and either ai+1 = · · · = ai+m = 0, or ai−m = 4

· · · = ai−1 = 0. Put a = ai , 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 Gm+1 in Gn . Hence Irr(M (λ)↓Gm+1 ) = Irr(M (λ)↓H). Now Smith’s theorem [7] yields m+1 that Irr(M (λ)↓Gm+1 ) contains M (aω1m+1 ) or M (aωm+1 ). Similar arguments show that τ ∈ Irr(M (λ)↓Gm ). Now assume that λ is p-restricted. Then a < p and M (aω1m+1 ) is isomorphic to the ath symmetric power S a (Vm+1 ) of Vm+1 (see, for instance, [6, (1.14)]). Let e be a highest weight vector of Vm+1 and f ∈ Vm+1 be a nonzero vector fixed by Gm . It is not difficult to see that the vector ef a−1 ∈ S a (Vm+1 ) generates the standard Gm -module. Therefore Irr(M (aω1m+1 )↓Gm ) contains Vm . Considering the m+1 dual modules, we get that Irr(M (aωm+1 )↓Gm ) contains Vm∗ . This yields our claim for Irr(M (λ)↓Gm ). Lemma 2.3 Let k, m, n, and λ = a1 ω11 + · · · + an ωnn be such as in Lemma 2.2. Let i be the maximal power of p such that pi divides all the coefficients aj . Then m ). Irr(M (λ)↓Gm ) contains M (pi ω1m ) or M (pi ωm Proof. One can write λ = pi λ1 + pi+1 λ2 where λ1 ∈ Xnp \{0} and λ2 ∈ Xn+ . By Steinberg’s tensor product theorem [8], M (λ) = Fri (M (λ1 )) ⊗ Fri+1 (M (λ2 )). Obviously, δ(λ1 ), δ(λ2 ) ≤ k. By Lemma 2.2, Irr(M (λ1 )↓Gm ) contains either Vm or Vm∗ , and Irr(M (λ2 )↓Gm ) contains τ . It remains to observe that Fri (Vm ) = M (pi ω1m ) m ). and Fri (Vm∗ ) = M (pi ωm Let Ψn be a finite subset of Irr Gn and Ψ = {Ψn }n∈N . Set δ(Ψn ) = max{δ(λ) | M (λ) ∈ Ψn } and δ(Ψ) = min{δ(Ψn ) | n ∈ N}. Lemma 2.4 Let Φ = {Φn }n∈N be an inductive system for A. Then δ(Φn ) = δ(Φ) for all n. In particular, δ(λ) ≤ δ(Φ) if M (λ) ∈ Φn . Proof. It suffices to show that δ(Φn ) = δ(Φn+1 ) for all n. Let λ and ρ be such that M (λ) ∈ Φn+1 , M (ρ) ∈ Φn , and δ(λ) = δ(Φn+1 ) and δ(ρ) = δ(Φn ). By Lemma 2.1, there exists a weight µ of M (λ) such that δ(λ) = hµ, α1 + · · · + αn i = hν, α1 + · · · + αn i where ν is the restriction of the weight µ to Gn . Hence δ(Φn+1 ) = δ(λ) ≤ δ(Irr(M (λ)↓Gn )) ≤ δ(Φn ). Since Φn = Irr(Φn+1 ↓Gn ), there exist weights λ1 and λ2 of Gn+1 such that ρ is the restriction of λ1 to Gn and λ1 is a weight of M (λ2 ) ∈ Φn+1 . Hence δ(Φn ) = δ(ρ) = hρ, α1 + · · · + αn i = hλ1 , α1 + · · · + αn i ≤ δ(λ2 ) ≤ δ(Φn+1 ), 5

so δ(Φn ) = δ(Φn+1 ). Proof of Theorem 1.1. Clearly, Frj (Vn ) = M (pj ω1n ) and Frj (Vn∗ ) = M (pj ωnn ) 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) ∈ Φn . By Lemma 2.4, δ(λ) ≤ k. Therefore by Lemma 2.3, Φm m ) for some i = i(m) ≥ 0. Observe that pi ≤ k. Now it contains M (pi ω1m ) or M (pi ωm is clear that there exists j ≥ 0 and an infinite sequence m1 < m2 < · · · < mr < . . . such that either for each r the set Φmr contains M (pj ω1mr ), or each of them contains mr ). Applying Smith’s theorem [7], we obtain the required assertion. M (pj ωm r 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 Gn . Until the end of this section Γ1 ⊂ . . . Γn ⊂ . . . is the sequence of the simply connected classical groups Γi = Γi (K) with the root systems of the same type where the group Γi of rank i is identified with the subgroup H(2, 3, . . . , i + 1) ⊂ Γi+1 . So the roots α2 , . . . , αi+1 of Γi+1 yield a basis of the root system for Γi . Similarly the roots αn−m+1 , . . . , αn yield a basis of the root system for the group Γm embedded into Γn . Below we consider the weights of restrictions of Γn -modules to Γm with respect to this basis. Lemma 2.5 Let λ = a1 ω1n + . . . + an ωnn ∈ Xnp and m < n. Then for each integer b with 0 ≤ b ≤ a1 + a2 + . . . + an−m the restriction M (λ)↓Γm has a composition factor m. with highest weight λb = (an−m+1 + b)ω1m + an−m+2 ω2m + . . . + an ωm Proof. For a root α of Γn and t ∈ K let xα (t) ∈ Γn and Xα ⊂ Γn be the root element and the root subgroup of Γn associated with α and t. Set M = M (λ). Let Mν ⊂ M be the weight subspace of weight µ. By [2, Proposition 5.13], there exist Xα,d ∈ End M (d = 0, 1, 2, P . . . ) with the following properties: Xα,d = 0 for d d large enough, Xα,0 = 1, xα (t) = ∞ d=0 t Xα,d (in End M ), Xα,d Mν ⊂ Mν+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 > as−1 , represent b in the form b = c + as−k+1 + . . . + as−1 where 0 ≤ c ≤ as−k ; if b ≤ as−1 , put c = b and k = 1. Let v ∈ M be a nonzero highest weight vector. Now we shall construct a vector ub ∈ M that generates an indecomposable Γm module with highest weight λb . First we define the integers d1 , . . . , dk as follows: dk = c, dj = as−j + dj+1 for 1 ≤ j < k. Set ub = X−αs−1 ,d1 . . . X−αs−k ,dk v. Observe that d1 = b. By [9, Lemma 2.9], the vector ub is nonzero and is fixed by the root subgroups Xαi for i 6= n − m. The properties of the operators Xα,d imply that ub has the weight λb with respect to Γm . Now it is clear that ub generates an indecomposable Γm -module with highest weight λb and so M ↓Γm has the required composition factor. Pu j Proof P of Theorem 1.2. Set p + p2 + . . . + pl = y. Let λ = j=0 p λj with λj = n aij ωin ∈ Xpn be the highest weight of M . Our assumptions yield that Pn−m−1i=1 ait ≥ y for some t, 0 ≤ t ≤ u. Hence, by Lemma 2.5 and Steinberg’s tensor i=1 product theorem [8], M (λt )↓Γm+1 has a composition factor of the form M (yω1m+1 )⊗ M (µ) with p-restricted µ. The same theorem implies that M (yω1m+1 )↓Γm = (τ ⊕ M (pω1m )) ⊗ (τ ⊕ M (p2 ω1m )) ⊗ · · · ⊗ (τ ⊕ M (pl ω1m )). 6

Now it is clear that M (yω1m+1 )↓Hm = M ′ where M ′ = (τ ⊕ M1 ) ⊗ (τ ⊕ M2 ) ⊗ . . . ⊗ (τ ⊕ Ml ) for some irreducible KHm -modules M1 , . . . , Ml . Hence (Frt (M ′ ) ⊗ N ) is a quotient of a submodule of M ↓Hm for some Hm -module N . By [4, ch.3, Corollary 2.17], the regular Hm -module KHm is a direct summand of M ′ as each nonidentity element of Hm acts nontrivially on the modules Mj , 1 ≤ j ≤ l. So KHm is a direct summand of Frt (M ′ ) as well since Frt (KHm ) ∼ = KHm . Let Q be an irreducible submodule of N . By [3, Corollary 10.20], KHm ⊗ Q is isomorphic to the direct sum of dim Q copies of KHm . Now the projectivity of KHm implies that KHm is a submodule of M ↓Hm .

3

Inductive systems over splitting fields

In this section q = ps , s ∈ N, and H1 → H2 → · · · → Hn → . . . is the sequence of finite groups Hn = SLn+1 (q) or SUn+1 (q 2 ) (∼ = 2 An (q)) naturally embedded each into another; H = lim Hn ; 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 Hn ⊂ Gn for all n. By Steinberg’s theorem [8], the restrictions to Hn of the q-restricted irreducible Gn -modules yield the complete set of inequivalent irreducible KHn -modules. Since F is a splitting field for Hn , the extension of the ground field yields a natural bijection from Irr FHn to Irr KHn . Therefore the irreducible FHn -modules can be parametrized by q-restricted dominant weights of Gn . For such a weight λ we denote by N (λ) the corresponding irreducible FHn -module, so M (λ)↓KHn ∼ = N (λ) ⊗F K.

(3)

Therefore it will cause no confusion if we use the same symbol Irr Hn both for Irr FHn and Irr KHn . Remark 3.1 Sometimes we shall consider modules N (λ) with λ = q l λ0 and qrestricted λ0 . In this case M (λ)↓KHn ∼ = N (µ) ⊗F K with q-restricted µ and we set N (λ) = N (µ). Here µ = λ0 except the case where H = SUn+1 (q 2 ) and l is odd. In the exceptional case µ = −w0 (λ0 ) where w0 is the longest element of the Weyl group. q Lemma 3.2 Let m < n, λ ∈ Xnq , µ ∈ Xm , and M (µ) ∈ Irr(M (λ)↓Gm ). Then N (µ) ∈ Irr(N (λ)↓Hm ).

Proof. In view of (3), Irr(N (λ) ⊗F K↓KHm ) = Irr((M (λ)↓KGm )↓KHm ) ∋ M (µ)↓KHm = N (µ) ⊗F K. Since F is a splitting field for Hm , the set Irr(N (λ)↓Hm ) contains N (µ). Proof of Theorem 1.4. Let Lj = {Ljn }n∈N and Rj = {Rjn }n∈N (0 ≤ j < s) be the inductive systems for H defined earlier. One easily observes that Ljn = {N (pj ω1n ), τ } and Rjn = {N (pj ωnn ), τ }. 7

Let Φ = {Φn }n∈N where Φn = {N (λn1 ), . . . , N (λnkn )} and λni , 1 ≤ i ≤ kn , are q-restricted. Set Ψn = {M (λn1 ), . . . , M (λnkn )} and δn = max{δ(λni ) | i = 1, . . . , kn }. The proof will be divided into two subcases. Case 1: sup{δn | n ∈ N} = k < ∞. Fix m, n ∈ N such that n ≥ m(k + 1). By m ) for some i = i(m) < s. Lemma 2.3, Irr(Ψn ↓Gm ) contains M (pi ω1m ) or M (pi ωm i m m ). So there exists Now Lemma 3.2 implies that Ψm contains N (p ω1 ) or N (pi ωm j < s and an infinite sequence m1 < m2 < · · · < mr < . . . such that either for mr ). This each r the set Φmr contains N (pj ω1mr ), or each of them contains N (pj ωm r completes Case 1 since Lj and Rj are inductive systems. Case 2: sup{δn | n ∈ N} = ∞. We shall prove that Φn = Irr Hn , so Φ contains all the systems Lj and Rj . Fix m ∈ N. Set l = |Hm | − 1 and c = p + p2 + . . . + pl + (p − 1)(m + 1). Since sup{δn | n ∈ N} = ∞, there exist n > m + 1 and M = M (λ) ∈ Ψn such that δ(λ) ≥ (1+p+. . .+ps−1 )c. Recall that for each M (λ) ∈ Ψn the weight λ is q-restricted and so can be represented in the form λ = λ0 +pλ1 +· · ·+ps−1 λs−1 where λj ∈ Xnp . Observe that δ(λ) = δ(λ0 )+pδ(λ1 )+· · ·+ps−1 δ(λs−1 ). Therefore s(M ) ≥ c in the notation of Theorem 1.2. Now Theorem 1.2 yields that Irr(M ↓Hm ) = Irr Hm and completes the proof. Remark 3.3 By Remark 3.1, one can consider the inductive systems Lj and Rj for arbitrary j = 0, 1, 2, . . . Note that Rj ∼ = Lj+s for H = SU∞ (q 2 ). Hence any nontrivial inductive system for H = SU∞ (q 2 ) contains one of the systems Lj (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∞ (ps ) and t = 2s for H = SU∞ (p2s ); FH = Fpt (the field of order pt ). As above, the irreducible FH Hn -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. FH 6⊂ F. Then FH ∩ F is a finite field F0 = Fpu . Note that u < t and u|t. We identify FH with the isomorphic subfield ¯ of F. Let F1 = FH F be the composite of FH and F. It of the algebraic closure F is well known that F1 is a Galois extension of F and the Galois group Gal(F1 /F) is 8

isomorphic to Σ = Gal(FH /F0 ) ∼ = Zt/u (the cyclic group of order t/u); moreover, we can obtain this isomorphism restricting the automorphisms of F1 to FH . The following proposition allows us to reduce the study of inductive systems over F to the similar problem for F0 . ¯ = M ⊗F F yields a bijection between Proposition 4.2 The map f : M 7→ M 0 Irr F0 Hn and Irr FHn . Moreover, for n > 1 ¯ ↓Hn−1 ) = {N ¯ | N ∈ Irr(M ↓Hn−1 )}. Irr(M ¯ is irreducible. Indeed, by Lemma 4.1, M ¯ is a direct Proof. First we claim that M sum of k irreducible inequivalent Hn -modules. Since F1 is a Galois extension of F and ¯ ⊗F F1 is the direct sum of is a splitting field for Hn , the same lemma implies that M all modules involved in some different k Gal(F1 /F)-orbits in Irr F1 Hn . On the other ¯ ⊗ F F1 ∼ hand, M = (M ⊗F0 FH )⊗FH F1 . By Lemma 4.1, M ⊗F0 FH = M1 ⊕· · ·⊕Ml where M1 , . . . , Ml constitute a Σ-orbit in Irr FH Hn . Since FH is a splitting field for Hn , every Mi remains irreducible after a field extension. As each element of Gal(F1 /F) is completely determined by its action on FH and irreducible F1 Hn -representations can be realized over FH , one can identify the action of Gal(F1 /F) on Irr F1 Hn with that of Σ on Irr FH Hn . Hence M1 ⊗FH F1 , . . . , Ml ⊗FH F1 constitute a Gal(F1 /F)-orbit in ¯ is irreducible. Irr F1 Hn . This implies k = 1, so M The same arguments with field extensions show that the map f is injective. By Lemma 4.1, | Irr F0 Hn | = | Irr FHn | since the relevant Galois groups have equal numbers of orbits on Irr FH Hn and Irr F1 Hn , respectively. Hence f is a bijection. Since taking field extensions commutes with restricting to subgroups, the irreducibility of ¯ yields the assertion of the lemma for restrictions. M ¯ n = {M ¯ |M ∈ Let Φ = {Φn }n∈N be an inductive system for H over F0 . Set Φ ¯ = {Φ ¯ n }n∈N . Φn }; Φ ¯ yields a bijection between the sets of inductive Corollary 4.3 The map Φ → Φ systems for H over F0 = F ∩ FH and F, respectively. In view of Corollary 4.3, we restrict our attention to the modules and the inductive systems over F0 = Fpu . One easily observes that the orbit of N (pj ω1n ), 0 ≤ j < u, under Σ is the set {N (pj+ku ω1n ) | k = 0, 1, . . . , t/u − 1}. The orbit of N (pj ωnn ) has a similar form. Denote by Nnj and Nn∗j the Hn -modules N (pj ω1n ) and N (pj ωnn ), respectively, considered over the field F0 . Since we are interested in inductive systems for H = lim Hn , we may and shall assume hereinafter that n > 1. −→ Lemma 4.4 (i) The modules Nnj and Nn∗j (0 ≤ j < s) are irreducible. ∗j j (ii) For Hn = SLn+1 (q) the modules Nni ∼ = Nn ) if and only if i ≡ j = Nn (Nn∗i ∼ ∗j 6 Nn for all i and j. (mod u); Nni ∼ = ∗j j (iii) For Hn = SUn+1 (q 2 ) the modules Nni ∼ = Nn ) if and only if i ≡ j = Nn (Nn∗i ∼ ∗j (mod u); Nni ∼ = Nn if and only if i ≡ s + j (mod u). 9

Proof. (i) By [5, Lemma 4.3.2], Nn0 is irreducible. This implies the irreducibility of Nnj and Nn∗j (0 ≤ j < s) since the relevant linear groups coincide. (ii) It follows from Lemma 4.1 and (i) that Nni ⊗F0 FH (0 ≤ i < s) is a direct sum of modules in Irr FH Hn that constitute a Σ-orbit and this orbit consists of all N (pk ω1n ) with 0 ≤ k < s and k ≡ i (mod u). Since Irr F0 Hn is parametrized by Σ-orbits on Irr FH Hn , 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 FH Hn -modules N (pk+s ω1n ) and N (pk ωnn ) are isomorphic. ˜ j = {R ˜ jn }n∈N . Note ˜ jn = {Nn∗j , τ }, L˜j = {L˜jn }n∈N , and R Set L˜jn = {Nnj , τ }, R j j j j ˜ are in fact the systems L and R for H considered as the systems that L˜ and R ˜ j are inductive systems. of modules over F0 . It is not difficult to see that L˜j and R Lemma 4.5 Let π(j) be the residue of j modulo u. ˜ 0, . . . , R ˜ u−1 are pair(i) For H = SL∞ (q) the inductive systems L˜0 , . . . , L˜u−1 , R j π(j) j π(j) ∼ ∼ ˜ ˜ ˜ ˜ wise nonequivalent; L = L and R = R . (ii) For H = SU∞ (q 2 ) the inductive systems L˜0 , . . . , L˜u−1 are pairwise nonequiv˜j ∼ alent; L˜j ∼ = L˜π(j+s) . = L˜π(j) and R 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 F0 = Fpu ⊂ FH (F0 6= FH ). Then Ψ contains one of the following systems: ˜ 0, . . . , R ˜ u−1 for H = SL∞ (q); L˜0 , . . . , L˜u−1 , R 0 u−1 L˜ , . . . , L˜ for H = SU∞ (q 2 ). Proof. Set ¯n = Ψ

[

Irr(M ⊗F0 FH ).

M ∈Ψn

¯ = {Ψ ¯ n }n∈N is a nontrivial inductive system for H over FH . Therefore Obviously, Ψ ¯ contains either Lj or Rj for some j < s. Now Lemma 4.1 implies by Theorem 1.4, Ψ ¯ n contains the Σ-orbit of the module N (pj ω n ) or that for each n ∈ N the set Ψ 1 j n N (p ωn ). Therefore it follows from the proof of Lemma 4.4 that Ψn contains either ˜ j . Now Lemma 4.5 yields the theorem. Nnj , or Nn∗j , so Ψ contains either L˜j , or R Let σ be an automorphism of the field F0 . For an irreducible F0 Hn -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 FHn = {N ⊗F0 F | N ∈ Irr F0 Hn } (see Proposition 4.2) via (N ⊗F0 F)σ = N σ ⊗F0 F. Similarly, one ˜ =R ˜ 0 . Let can define Ψσ for an inductive system Ψ for FH. Set L˜ = L˜0 and R i i θ i ˜ ˜ ˜ ˜ θi for θ ∈ Aut F0 raise elements to the power p. Observe that L = L and R = R 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 ⊂ FH . Then Ψ contains one of the following systems: ˜ σ (σ ∈ Aut E) for H = SL∞ (q); L˜σ or R σ ˜ L (σ ∈ Aut E) for H = SU∞ (q 2 ). 10

Proof. The case E 6= FH follows from Theorem 4.6; for E = FH 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 L1 ⊂ L2 ⊂ · · · ⊂ Li ⊂ . . .

(4)

Denote by SL∞ (L) the union of the groups SL∞ (L1 ) ⊂ SL∞ (L2 ) ⊂ · · · ⊂ SL∞ (Li ) ⊂ . . . Let L have an automorphism α of order 2. Then α fixes all Li . Removing a finite number of subfields in sequence (4) if necessary, one can assume that α acts nontrivially on L1 . Denote by SU∞ (L) the union of the groups SU∞ (L1 ) ⊂ SU∞ (L2 ) ⊂ · · · ⊂ SU∞ (Li ) ⊂ . . . 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 l and only if L has a subfield of order p2 and has no subfields of order p2 for l large enough. The groups SL∞ (L) and SU∞ (L) are locally finite. Recall that a set S {Gα }α∈A of finite subgroups of a locally finite group G is called a local system if G = α∈A Gα and for each pair α, β ∈ A there is γ ∈ A such that Gα , Gβ ⊂ Gγ . Set α ≤ β if Gα ⊂ Gβ . Then A is a directed set and G = lim Gα . Let Φα be a finite subset of −→ Irr FGα . The collection Φ = {Φα }α∈A is called an inductive system for G if [ Irr(M ↓Gα ) = Φα M ∈Φβ

for all α < β in A. In this section we shall denote by H one of the groups SL∞ (L) or SU∞ (L), by Hi the subgroup SL∞ (Li ) or SU∞ (Li ), and by Hn,i the subgroup SLn (Li ) or SUn (Li ), respectively. Observe that the set {Hn,i }n,i∈N is a local system for H, i.e. a directed j set of subgroups of H with lim Hn,i = H. Set Ei = F ∩ Li . Denote by Nn,i and −→ ∗j j n j n Nn,i the Hn,i -modules N (p ω1 ) and N (p ωn ), respectively, considered over the field Ei (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 Ek+1 Hn,k -module can be represented in the form S ⊗Ek Ek+1 where S ∈ Irr Ek Hn,k as Lk is a splitting field for Hn,k and Lk ∩ Ek+1 = Ek . 11

Lemma 5.2 Let M be an irreducible Ek+1 Hn,k+1 -module and Irr(M ↓Hn,k ) = {S1 ⊗Ek Ek+1 , . . . , Sl ⊗Ek Ek+1 } where St ∈ Irr Ek Hn,k , 1 ≤ t ≤ l. Then Irr(M ⊗Ek+1 F↓Hnk ) = {S1 ⊗Ek F, . . . , Sl ⊗Ek F}. In particular, j j Irr(Nn,k+1 ⊗Ek+1 F↓Hnk ) = {Nnk ⊗Ek F}; ∗j ∗j ⊗Ek F}. ⊗Ek+1 F↓Hnk ) = {Nnk Irr(Nn,k+1

Proof. Let F1 be the composite of the fields F and Lk+1 . Set Σ1 = Gal(F1 /F). ¯ = Recall that Gal(F1 /F) ∼ = Gal(Lk+1 /Ek+1 ). Put H k = Hn,k , H k+1 = Hn,k+1 , M (M ⊗Ek+1 F) ⊗F F1 , S¯t = (St ⊗Ek F) ⊗F F1 . By Lemma 4.1, the set Irr(M ⊗Ek+1 ¯ ↓H k ) = F↓H k ) is completely determined by the Σ1 -orbits of the elements of Irr(M l ¯ ∪t=1 Irr(St ). Proposition 4.2 implies that S1 ⊗Ek F are nonequivalent irreducible FH k -modules. Hence by Lemma 4.1, Irr(S¯t ) is an Σ1 -orbit on Irr F1 H k . This yields j ∗j the first assertion of our lemma. It remains to observe that Nn,k+1 and Nn,k+1 are k+1 irreducible Ek+1 H -modules by Lemma 4.4(i) and consider their restrictions to Hk. ∗0 0 ⊗ Set Ln,i = {Nn,i Ei F, τ }, Rn,i = {Nn,i ⊗Ei F, τ }, L = {Ln,i }n,i∈N , R = {Rn,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 σi = σ|Ei , then σi ∗σi Lσ = {{Nn,i ⊗Ei F, τ } | n, i ∈ N} and Rσ = {{Nn,i ⊗Ei F, τ } | n, i ∈ N} are inducj ∗j σi ∗σi tive systems. Here Nn,i = Nn,i and Nn,i = Nn,i if σi raises elements to the pj 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 = {Ψn,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: Lσ or Rσ (σ ∈ Aut E) for H = SL∞ (L); Lσ (σ ∈ Aut E) for H = SU∞ (L). In particular, Ψ contains the trivial inductive system. Proof. Denote by Ψi the set {Ψn,i }n∈N . Clearly, Ψi is a nontrivial inductive system S for Hi = n∈N Hn,i . Using Corollary 4.3, we shall identify inductive systems for Hi over Ei with those over F. By Corollary 4.7, Ψi contains one of the following systems: ˜ σi (σi ∈ Aut Ei ) for Hi = SL∞ (Li ); L˜σi (σi ∈ Aut Ei ) for Hi = SU∞ (Li ). L˜σi or R Assume, for definiteness, that L˜σi ⊂ Ψi for Hi = SL∞ (Li ) and infinitely many i. Then by Lemma 5.2, L˜ρ ⊂ Ψi−1 where ρ = σi |Ei−1 . Therefore we shall assume that ˜ σi ⊂ Ψi for for each i there exists σi ∈ Aut Ei such that L˜σi ⊂ Ψi . The case where R

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all i can be considered quite similarly. Let ∆i be the set of all θ ∈ Aut Ei such that L˜θ ⊂ Ψi . For i < j set ∆ij = {θ|Ei | θ ∈ ∆j } ⊂ Aut Ei . It follows from Lemma 5.2 that ∆ij ⊂ ∆i . Note that ∆i ⊃ ∆i,i+1 ⊃ ∆i,i+2 ⊃ . . . ¯ i = T∞ ¯ Therefore one can define ∆ j=i+1 ∆ij . Observe that ∆i is nonempty and there ¯ i = ∆ik for all k ≥ c. Take any k > c(i), c(i + 1). exists c = c(i) ∈ N such that ∆ ¯ ¯ i . Therefore there exists a sequence of We have ∆i+1 |Ei = ∆i+1,k |Ei = ∆ik = ∆ automorphisms θi ∈ ∆i such that θi+1 |Ei = θi for all i. This sequence determines a unique automorphism σ of E such that σ|Ei = θi . Now it is clear that Lσ is contained in Ψ. Let G be a locally finite group and {Gα }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↓Gα ). 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 7→ 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 MΦ be the semiprimitive ideal corresponding to an inductive system Φ. Then for each proper ideal I of FG we have I ⊂ MΦ(I) and the quotient MΦ(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 AnnFH V σ and AnnFH (V ∗ )σ (σ ∈ Aut E) for H = SL∞ (L); AnnFH 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 ↓Hn,i ) = {τ } and Hn,i is perfect at least for n > 2, M ∪ FHn,i = Aug FHn,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 13

among those properly lying in Aug(FH)) are exactly the ideals MΦ where Φ runs over the minimal nontrivial inductive systems for H. Let W be the natural H-module over the field L and W0 be the module W considered over the field E. Recall that V = W0 ⊗E F and V σ = W0σ ⊗E F for every σ ∈ Aut E. Let I = AnnFH V σ and J = AnnFH (V ∗ )σ . Obviously, Φ(I) = Lσ and Φ(J) = Rσ (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.

References [1] Baranov, A.A.; Suprunenko, I.D. Branching rules for modular fundamental representations of symplectic groups. Bull. London Math. Soc. 2000, 32 (4), 409–420. [2] Borel, A. Properties and linear representations of Chevalley groups. In Seminar on Algebraic Groups and Related Finite Groups; Lecture Notes in Math. vol. 131; SpringerVerlag, 1970; 1-55. [3] Curtis, Ch.; Reiner, I. Methods of representation theory—with applications to finite groups and orders, vol. I; Wiley: New York, 1990. [4] Feit, W. The representation theory of finite groups; North-Holland Publ. Comp.: Amsterdam, New York, Oxford; 1982. [5] Kleidman, P.; Liebeck, M. The subgroup structure of the finite classical groups; London Math. Soc. Lecture Notes, no 129; Cambridge Univ. Press, 1990. [6] Seitz, G.M. The maximal subgroups of classical algebraic groups; Mem. Amer. Math. Soc. no 365; 1987. [7] Smith, S. Irreducible modules and parabolic subgroups. J. Algebra 1982, 75, 286–289. [8] Steinberg, R. Representations of algebraic groups. Nagoya Math. J. 1963, 22, 33-56. [9] Suprunenko, I.D. On Jordan blocks of elements of order p in irreducible representations of classical groups with p-large highest weights. J. Algebra 1997, 191, 589-627. [10] Suprunenko, I.D.; Zalesskii, A.E. Representations of dimensions (pn ∓ 1)/2 of a symplectic group of degree 2n over a finite field (in Russian). Vestsi AN BSSR, ser. fiz.-mat. n. 1987, (6), 9–15. [11] Zalesskii, A. E. Group rings of locally finite groups and representation theory. In Proceedings of the International Conference on Algebra, Novosibirsk, 1989; Contemporary Math. vol. 131, part 1; Amer. Math. Soc., 1992; 453–472. [12] Zalesskii, A.E. Group rings of simple locally finite groups. In Finite and locally finite groups; Hartley, B., Seitz, G.M., Borovik, A.V., Bryant, R.M., Eds.; NATO ASI Series, C, vol. 471; Kluwer, 1995; 219–246.

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