PACIFIC JOURNAL OF MATHEMATICS Vol. 192, No. 2, 2000

RESTRICTIONS OF Ω_{m}(q)-MODULES TO ALTERNATING GROUPS

William J. Husen

We consider the restriction of an irreducible FΩ_{m}(q)-mo-dule M to a subgroup H where F^{∗}(H) ≅ A_{n} and where F is algebraically closed with (char(F), q) ≠ 1. Given certain restrictions on the highest weight of M, we show that if m > n^{6}, then M↓_{H} is reducible.

1. Introduction.

In the study of the maximal subgroups of classical groups, the following question arises: Given an absolutely irreducible module M for K and a subgroup H, when does M↓_{H} remain absolutely irreducible? In this article K ≅ Ω_{m}(q) is the commutator subgroup of an m-dimensional orthogonal group over F_{q}, and F^{∗}(H) ≅ A_{n} is the alternating group of degree n. We treat the case that the field of definition of M has characteristic dividing q.

Let F be an algebraically closed field containing F_{q}, the field with q elements, such that char(F) > 3. Then K < \overline{K} where \overline{K} ≅ Ω_{m}(F) and we may assume that M is a FK-module. By [6, Theorem 43], every absolutely irreducible FK-module is the restriction of an irreducible F\overline{K}-module of the same weight. So we may assume that M = M(λ) is an irreducible F\overline{K}- module with highest weight λ. Let ℓ = ⌊m/2⌋ be the Lie rank of \overline{K} and let \{λ_{i}\} be the fundamental dominant weights of \overline{K}. The labeling of these weights corresponds to the labeling of the Dynkin diagrams for \overline{K} as given in [3].

Hypothesis 1.1. Assume the following are true:

(1) If m is even, then λ = (∑_{i=1}^{ℓ−2} a_{i}λ_{i}) + a_{ℓ−1}(λ_{ℓ−1} + λ_{ℓ}); a_{i} ∈ Z, a_{i} ≥ 0.

(2) If m is odd, then λ = (∑_{i=1}^{ℓ−1} a_{i}λ_{i}) + 2a_{ℓ}λ_{ℓ}; a_{i} ∈ Z, a_{i} ≥ 0.

(3) If μ_{i} = ∑_{j=i}^{ℓ−1} a_{j}, m even or if μ_{i} = ∑_{j=i}^{ℓ} a_{j}, m odd then

(a) μ_{1} < p = char(F_{q});

(b) 1 < ∑ μ_{i} = k < ℓ.

Conditions (1) and (2) imply that M is not a faithful module for any proper covering group of \overline{K}. We now state our main result:

Theorem 1.2. Assume that H, K and M = M(λ) are as above with n, m ≥ 10 and (q, 6) = 1. Suppose further that λ satisfies Hypothesis 1.1. If m > n^{6}, then M↓_{H} is reducible.

Our strategy is to produce a small subspace in M with a large stabilizer in H and then, using Frobenius reciprocity, produce an upper bound for dim(M). We produce a lower bound for dim(M) as an F\overline{K}-module using the length of the Weyl group orbit of a subdominant weight in M. The result then follows by comparing these two bounds.

2. A construction of \overline{W} (λ).

In this section we construct the Weyl module \overline{W} (λ) of \overline{K} with highest weight λ. Then M is a homomorphic image of this module. Our construction proceeds by first constructing the Weyl module W (λ) for a complex Lie group G of the same type and rank as \overline{K}, then we use Kostant’s Z-form to produce \overline{W} (λ). For notational convenience we assume that \{λ_{i}\} are the fundamental dominant weights for G as well as for \overline{K}, and accordingly, assume that λ is a dominant weight of G.

Let V be a complex, m-dimensional vector space possessing a non-degene-rate orthogonal form f( , ) and let B be a basis for V so that

B = \{ \begin{matrix}\{e_{i}, f_{i} | 1 ≤ i ≤ ℓ\} & if m is even \\ \{e_{i}, f_{i}, x | 1 ≤ i ≤ ℓ\} & if m is odd \end{matrix}

with f(e_{i}, e_{j}) = f(f_{i}, f_{j}) = f(x, e_{i}) = f(x, f_{i}) = 0, f(e_{i}, f_{j}) = δ_{i,j} and f(x, x) = 2. We then define G = Ω(V) and let T be the maximal torus of G with respect to B. Set V_{e} = ⟨e_{i} | 1 ≤ i ≤ ℓ⟩ and V_{f} = ⟨f_{i} | 1 ≤ i ≤ ℓ⟩.

Suppose that λ satisfies hypothesis 1.1 and d = max{i | μ_{i} ≠ 0} so that μ = (μ_{1}, …, μ_{d}) is a proper partition of k. Let T be the tableau of shape μ with entries t_{i,j} = j + ∑_{s<i} μ_{s}. Define the following subgroups of the symmetric group S_{k}:

R_{μ} = {σ ∈ S_{k} | σ(t_{i,j}) lies in the same row as t_{i,j} for all i, j}
C_{μ} = {σ ∈ S_{k} | σ(t_{i,j}) lies in the same column as t_{i,j} for all i, j}

and elements of CS_{k}:

r_{μ} = ∑_{σ∈R_{μ}} σ  and  c_{μ} = ∑_{σ∈C_{μ}} sgn(σ)σ

RESTRICTIONS OF Ω_{m}(q)-MODULES 299

Define κ_{i,j} : V^{⊗k} → V^{⊗(k−2)} by κ_{i,j}(v_{l_{1}} ⊗ ⋯ ⊗ v_{l_{k}}) = f(v_{l_{i}}, v_{l_{j}})(v_{l_{1}} ⊗ ⋯ ⊗ v̂_{l_{i}} ⊗ ⋯ ⊗ v̂_{l_{j}} ⊗ ⋯ ⊗ v_{l_{k}}) for 1 ≤ i < j ≤ k and set

K = ⋂_{i,j} ker(κ_{i,j}).

S_{k} acts on V^{⊗k} by place permutation, specifically:

σ(v_{i_{1}} ⊗ ⋯ ⊗ v_{i_{k}}) = v_{i_{σ^{−1}(1)}} ⊗ ⋯ ⊗ v_{i_{σ^{−1}(k)}}.

This action commutes with the diagonal action of G on V^{⊗k}.

Given v ∈ V^{⊗k}, we define one additional element r_{μ}^{v} of the group algebra CS_{k} as follows: Let R_{μ}^{v} = {σ ∈ R_{μ} | σ(v) = v} and let {s_{i}} be a left transversal for R_{μ}^{v} in R_{μ}. Define r_{μ}^{v} = ∑_{i} s_{i} Notice that r_{μ}(v) = |R_{μ}^{v}|r_{μ}^{v}(v).

By [2, Theorem 19.22], W (λ) = c_{μ}r_{μ} (V^{⊗k}) ∩ K is the Weyl module for G with highest weight λ. Since V is a complex vector space, c_{μ}r_{μ} (V^{⊗k}) = ⟨c_{μ}r_{μ}^{v}(v) | v ∈ V^{⊗k}⟩.

Define V_{Z} = Z[B] and let \overline{V} = V_{Z} ⊗_{Z} F. Then \overline{f}( , ) = f( , ) ⊗ 1_{F} is a non-degenerate orthogonal form on \overline{V}. Without loss of generality, we may assume that \overline{K} = Ω(\overline{V}). Moreover if \overline{e}_{i} = e_{i} ⊗ 1_{F}, \overline{f}_{i} = f_{i} ⊗ 1_{F} and \overline{x} = x ⊗ 1_{F}, then

\overline{B} = \{ \begin{matrix}\{\overline{e}_{i}, \overline{f}_{i} | 1 ≤ i ≤ ℓ\} & if m is even \\ \{\overline{e}_{i}, \overline{f}_{i}, \overline{x} | 1 ≤ i ≤ ℓ\} & if m is odd \end{matrix}

is a standard basis for \overline{V} with respect to \overline{f}( , ). We identify r_{μ} and c_{μ} with the elements r_{μ} ⊗ 1_{F} and c_{μ} ⊗ 1_{F} of FS_{k}.

Suppose that L ⊂ End(V) is the adjoint module for G so that L is a complex Lie algebra of type D_{ℓ} or B_{ℓ}. Let Δ = {r_{1}, …, r_{ℓ}} be the set of simple roots corresponding to the torus T and let Φ be the root system generated by Δ. Set Δ_{0} = {r_{1}, …, r_{ℓ−1}} and let Φ_{0} ⊂ Φ be the subset generated by Δ_{0}. Using the setup of [1, §11.2], {ϵ_{r}, h_{r_{i}} | r ∈ Φ, 1 ≤ i ≤ ℓ} is a Chevalley basis for L and {ϵ_{r}, h_{r_{i}} | r ∈ Φ_{0}, 1 ≤ i ≤ ℓ − 1} is a Chevalley basis for L_{0} ⊂ L where L_{0} is a Lie algebra of type A_{ℓ−1}. Let G_{0} < N_{G}(V_{e} ⊕ V_{f}) such that G_{0} ≅ SL_{ℓ}(C). Then, by [1, Theorem 11.3.2], G = ⟨exp(ζϵ_{r}) | r ∈ Φ, ζ ∈ C⟩ and G_{0} = ⟨exp(ζϵ_{r}) | r ∈ Φ_{0}, ζ ∈ C⟩. Note that neither G nor G_{0} is the adjoint group for L or L_{0}, respectively. We may consider V_{e} to be the natural module for G_{0}. Under this identification, V_{f} is the dual of V_{e}.

Assume that U(L) is the universal enveloping algebra of L. From [3, §26], Kostant’s Z-form U_{Z}(L) is the Z-span of {ϵ_{r}^{m}/m! | r ∈ Φ, m ∈ Z^{+}}. Given any vector v of weight λ in W (λ), U_{Z}(L)v ⨂_{Z} F = \overline{W} (λ) where \overline{W} (λ) is the Weyl module for \overline{K} with highest weight λ. By the previous remarks, U_{Z}(L_{0}) ⊂ U_{Z}(L), which implies that U_{Z}(L_{0})v ⨂_{Z} F ⊂ \overline{W} (λ).

Define v_{μ_{i}} = ⨂_{j=1}^{μ_{i}} e_{i} and v_{μ} = ⨂_{i=1}^{d} v_{μ_{i}}.

Lemma 2.1. We have:

(1) c_{μ}(v_{μ}) is a vector of weight λ in W (λ);

(2) U_{Z}(L_{0})c_{μ}(v_{μ}) = c_{μ}r_{μ} (V_{e}^{⊗k}) ∩ Z[e_{1}, …, e_{ℓ}]^{⊗k}.

Proof. First note that R_{μ}^{v_{μ}} = R_{μ} so that r_{μ}^{v_{μ}}(v_{μ}) = v_{μ} and that c_{μ}(v_{μ}) ≠ 0. This implies that c_{μ}(v_{μ}) ∈ c_{μ}r_{μ} (V^{⊗k}). It is clear that c_{μ}(v_{μ}) ∈ K so we have c_{μ}(v_{μ}) ∈ W (λ). Let t ∈ T and write t = diag(t_{1}, …, t_{ℓ}, t_{1}^{−1}, …, t_{ℓ}^{−1}) or t = diag(t_{1}, …, t_{ℓ}, t_{1}^{−1}, …, t_{ℓ}^{−1}, t′) depending on the parity of m. Then

tv = c_{μ}(tc_{μ}(v_{μ})) = c_{μ} (⨂_{i=1}^{d} t_{i}^{μ_{i}}v_{μ_{i}}) = (∏_{i=1}^{d} t_{i}^{μ_{i}}) c_{μ}(v_{μ}).

From the definition of μ it follows that c_{μ}(v_{μ}) is a vector of weight λ and so (1) follows. With the identification of V_{e} with the natural module of G_{0}, we see by [2, Theorem 15.15] that c_{μ}r_{μ} (V_{e}^{⊗k}) is the Weyl module for G_{0} corresponding to the partition μ of k via the Schur functor. The argument above restricted to t ∈ T ∩ G_{0} shows that c_{μ}(v_{μ}) is a highest weight vector in c_{μ}r_{μ} (V_{e}^{⊗k}). In particular U(L_{0})c_{μ}(v_{μ}) = c_{μ}r_{μ} (V_{e}^{⊗k}). Using the proof of [4, Theorem 8.3.1], we have

U_{Z}(L_{0})c_{μ}(v_{μ}) = c_{μ}r_{μ} (V_{e}^{⊗k}) ∩ Z[e_{1}, …, e_{ℓ}]^{⊗k}

which completes our proof. □

Lemma 2.2. Suppose \overline{v} = \overline{v}_{i_{1}} ⊗ ⋯ ⊗ \overline{v}_{i_{k}} where \{\overline{v}_{i}\} is a collection of mutually orthogonal, linearly independent singular vectors. Then:

(1) If sgn(σ_{c})σ_{c}σ_{r}(\overline{v}) ≠ −\overline{v} for all σ_{c} ≠ 1 ∈ C_{μ}, σ_{r} ∈ R_{μ}, then c_{μ}r\overline{v} ; μ(\overline{v}) ≠ 0;

(2) c_{μ}r\overline{v} ; μ(\overline{v}) ∈ \overline{W} (λ).

Proof. Since \overline{v} is a summand of c_{μ}r\overline{v} ; μ(\overline{v}) and all other summands of c_{μ}r\overline{v} ; μ(\overline{v}) have the form sgn(σ_{c})σ_{c}σ_{r}(\overline{v}), part (1) must hold. There is g ∈ \overline{K} such that g(\overline{v}_{i_{j}}) = α_{i_{j}} \overline{e}_{i_{j}} such that α_{i_{j}} ≠ 0 for all 1 ≤ i ≤ k. If w = e_{i_{1}} ⊗ ⋯ ⊗ e_{i_{k}},

then r\overline{v} ; μ = r_{μ}^{w}. As

c_{μ}r_{μ}^{w}(w) ∈ c_{μ}r_{μ} (V_{e}^{⊗k}) ∩ Z[e_{1}, …, e_{ℓ}]^{⊗k},

Lemma 2.1 implies that c_{μ}r_{μ}^{w}(w) ∈ U_{Z}(L)v. Writing \overline{w} = α_{i_{1}} \overline{e}_{i_{1}}⊗⋯⊗α_{i_{k}} \overline{e}_{i_{k}}, we then have

c_{μ}r\overline{w} ; μ(\overline{w}) ∈ U_{Z}(L)v ⨂_{Z} F = \overline{W} (λ).

Finally, as \overline{W} (λ) is a F\overline{K}-module, g^{−1}c_{μ}r\overline{w} ; μ(\overline{w}) = c_{μ}r\overline{v} ; μ(\overline{v}) ∈ \overline{W} (λ). □

Though W (λ) is a irreducible module for G, \overline{W} (λ) may not be an irreducible module for \overline{K}; however, it does possess a unique maximal submodule by [6, Lemma 80] which we denote by Rad(\overline{W} (λ)). Moreover, M ≅ \overline{W} (λ)/Rad(\overline{W} (λ)).

We now discuss the orthogonal forms on V^{⊗k} and W (λ). Suppose v, w ∈ V^{⊗k} where v = v_{1} ⊗ ⋯ ⊗ v_{k} and w = w_{1} ⊗ ⋯ ⊗ w_{k}. We define f^{k}( , ) by

f^{k}(v, w) = ∏_{i=1}^{k} f(v_{i}, w_{i}).

f^{k}( , ) is a non-degenerate, G-invariant orthogonal form on V^{⊗k}. This form is also invariant under the action of S_{k}. Note that

f^{k}[c_{μ}(v), c_{μ}(w)] = ∑_{σ∈C_{μ}} sgn(σ)f^{k}[σ(v), c_{μ}(w)]
= ∑_{σ∈C_{μ}} sgn(σ)f^{k}[v, σ^{−1}c_{μ}(w)]
= ∑_{σ∈C_{μ}} f^{k}[v, c_{μ}(w)]
= |C_{μ}|f^{k}[v, c_{μ}(w)].

We define f_{μ}^{k}( , ) on c_{μ} (V^{⊗k}) by

f_{μ}^{k}[c_{μ}(v), c_{μ}(w)] = f^{k}[v, c_{μ}(w)].

By a similar argument as above, we see that f^{k}[v, c_{μ}(w)] = f^{k}[w, c_{μ}(v)], so this form is symmetric. Since f^{k}( , ) is bilinear and G-invariant, f_{μ}^{k}( , ) is also bilinear and G-invariant. Therefore f_{μ}^{k}( , ) is a G-invariant orthogonal

form on W (λ) ⊂ c_{μ} (V^{⊗k}). As before, \overline{f}^{k}( , ) = f^{k}( , ) ⊗ 1_{F} is a \overline{K}- invariant orthogonal form on \overline{V}^{⊗k} and \overline{f}_{μ}^{k}( , ) = f_{μ}^{k}( , )⊗1_{F} is a \overline{K}-invariant orthogonal form on \overline{W} (λ). This form is possibly degenerate. We denote the radical of this form as \overline{W} (λ)^{⊥}. The following lemma is generally known, although we present a proof:

Lemma 2.3. Rad(\overline{W} (λ)) = \overline{W} (λ)^{⊥}.

Proof. Define \overline{v}_{−μ_{i}} = ⨂_{j=1}^{μ_{i}} \overline{f}_{i} and \overline{v}_{−μ} = ⨂_{i=1}^{d} \overline{v}_{−μ_{i}}. Noting that r_{μ}^{\overline{v}_{−μ}} = 1, c_{μ}(v_{−μ}) ≠ 0 ∈ \overline{W} (λ) by Lemma 2.2. A similar argument as in the proof of Lemma 2.1 shows that c_{μ}(v_{−μ}) is a vector of weight −λ. Hypothesis 1.1 implies that d < ℓ. In particular, there is an element ω_{0} of the Weyl group of \overline{K} such that ω_{0}[c_{μ}(v_{−μ})] = c_{μ}(v_{μ}). This means that M = M(λ) must be selfdual. Clearly we have that \overline{W} (λ)^{⊥} ⊂ Rad(\overline{W} (λ)) and that \overline{W} (λ)/\overline{W} (λ)^{⊥} is non-degenerate, so this latter module is also self-dual. Since M is selfdual and is a homomorphic image of \overline{W} (λ)/\overline{W} (λ)^{⊥}, \overline{W} (λ)/\overline{W} (λ)^{⊥} must possess a submodule isomorphic to M. Since M ≅ \overline{W} (λ)/Rad(\overline{W} (λ)) and Rad(\overline{W} (λ)) does not possess a constituent which is isomorphic to M, we must have Rad(\overline{W} (λ)) = \overline{W} (λ)^{⊥} and our result follows. □

Lemma 2.4. Let \{\overline{v}_{i}, \overline{w}_{i} | 1 ≤ i ≤ k\} be a hyperbolic basis for some 2kdimensional subspace of \overline{V}. Set \overline{v} = \overline{v}_{1} ⊗ ⋯ ⊗ \overline{v}_{k} and \overline{w} = \overline{w}_{1} ⊗ ⋯ ⊗ \overline{w}_{k}. Then:

(1) c_{μ}r_{μ}(\overline{v}) ≠ 0, c_{μ}r_{μ}(\overline{w}) ≠ 0;

(2) c_{μ}r_{μ}(\overline{v}), c_{μ}r_{μ}(\overline{w}) ∈ \overline{W} (λ);

(3) \overline{f}_{μ}^{k}[c_{μ}r_{μ}(\overline{v}), c_{μ}r_{μ}(\overline{w})] ≠ 0.

Proof. Parts (1) and (2) follow from Lemma 2.2 since r\overline{v} ; μ = r\overline{w} ; μ = r_{μ} and the \overline{v}_{i} are distinct, similarly for \overline{w}_{i}. If σ_{1}, σ_{2} ∈ S_{k}, then

\overline{f}^{k}[σ_{1}(\overline{v}), σ_{2}(\overline{w})] = ∏_{i=1}^{k} \overline{f}[\overline{v}_{σ_{1}^{−1}(i)}, \overline{w}_{σ_{2}^{−1}(i)}] = \{ \begin{matrix}1 & if σ_{1} = σ_{2} \\ 0 & otherwise. \end{matrix}

Recall that R_{μ} ∩ C_{μ} = 1. Then we have

\overline{f}_{μ}^{k}[c_{μ}r_{μ}(\overline{v}), c_{μ}r_{μ}(\overline{w})] = \overline{f}^{k}[r_{μ}(\overline{v}), c_{μ}r_{μ}(\overline{w})]
= ∑_{σ∈R_{μ}} \overline{f}^{k}[σ(\overline{v}), c_{μ}r_{μ}(\overline{w})]

= ∑_{σ∈R_{μ}} \overline{f}^{k}[σ(\overline{v}), σ(\overline{w})]
= |R_{μ}|.

Part (3) then follows as |R_{μ}| = ∏_{i=1}^{d} μ_{i}! and μ_{i} < char(F_{q}) for all i. □

Lemma 2.5. M possesses a vector of weight λ_{k}.

Proof. Let \{\overline{e}_{i}, \overline{f}_{i} | 1 ≤ i ≤ k\} be a subset of our standard basis \overline{B} for \overline{V}. By part (2) of Lemma 2.4, c_{μ}r_{μ}(\overline{e}_{1} ⊗ ⋯ ⊗ \overline{e}_{k}) ∈ \overline{W} (λ). An argument similar to that used in Lemma 2.1 shows that c_{μ}r_{μ}(\overline{e}_{1} ⊗ ⋯ ⊗ \overline{e}_{k}) is a vector of weight λ_{k}. Hence λ_{k} is a subdominant weight of λ. Condition (3) of Hypothesis 1.1 insures that λ is p-restricted. Therefore using the results of [5], M possesses a vector of weight λ_{k}. □

3. Elementary abelian 3-subgroup E_{k}.

Assume that k ≤ n/3 − 2 and recall that F^{∗}(H) possesses a subgroup H_{0} isomorphic to S_{n−2}. Let

E_{k} ≅ ⟨(123), (456), …, (3k − 2, 3k − 1, 3k)⟩ < A_{n}

be a subgroup of H_{0} generated by commuting 3-cycles in F^{∗}(H) so that E_{k} is an elementary abelian 3-group of rank k. Then

N_{k} = N_{H_{0}}(E_{k}) ≅ S_{3} ≀ S_{k} × S_{n−3k−2}
C_{k} = C_{H_{0}}(E_{k}) ≅ E_{k} × S_{n−3k−2}

and let H_{k} < C_{k} so that H_{k} ≅ S_{n−3k−2}. Note that C_{N_{k}}(H_{k}) ≅ S_{3} ≀ S_{k} and this subgroup controls fusion in E_{k}. Let σ ≠ 1 ∈ E_{k} and assume that σ is the product of k_{1} disjoint 3-cycles. Then C_{N_{k}}(σ) ≅ Z_{3}≀S_{k_{1}} ×S_{3}≀S_{k−k_{1}} ×S_{n−3k−2} which implies |σ^{N_{k}}| = 2^{k_{1}} \binom{k}{k_{1}}.

Let φ ∈ E_{k}^{∗} = Hom(E_{k}, F^{∗}). The group N_{k} acts on this group by φ^{g} : σ ↦ φ(g^{−1}σg) for g ∈ N_{k}, σ ∈ E_{k}. We abuse notation slightly and define φ^{−1} by φ^{−1} : σ ↦ φ(σ^{−1}) for all σ ∈ E_{k}. Recall that In_{N_{k}}(φ) = {g ∈ N_{k}|φ^{g} = φ} is the inertia group of φ in N_{k} and note that H_{k} ∈ In_{N_{k}}(φ).

If φ ∈ E_{k}^{∗} is non-trivial, then the previous remarks concerning the action of N_{k} on E_{k} imply that [N_{k} : In_{N_{k}}(φ)] = 2^{k_{1}} \binom{k}{k_{1}} for some k_{1}, 1 ≤ k_{1} ≤ k and that φ^{−1} ∈ φ^{N_{k}}. Since \binom{k}{k_{1}} ≥ k unless k = k_{1}, in which case 2^{k_{1}} ≥ 2k, we have [N_{k} : In_{N_{k}}(φ)] ≥ 2k.

4. Decomposition of \overline{V} ↓_{E_{k}} and C_{k}-invariant subspace of \overline{W} (λ).

We continue to assume that k ≤ n/3−2 and we now consider the restriction of \overline{V} to E_{k}. Since char(F) ≠ 3, we have \overline{V} ↓_{E_{k}} ≅ ⨁_{φ∈E_{k}^{∗}} \overline{V}_{φ} where \overline{V}_{φ} is the homogeneous component of φ. Let \overline{v}_{1} ∈ \overline{V}_{φ_{1}} and \overline{v}_{2} ∈ \overline{V}_{φ_{2}}. Then (g\overline{v}_{1}, g\overline{v}_{2}) = φ_{1}(g)φ_{2}(g)(\overline{v}_{1}, \overline{v}_{2}) for all g ∈ E_{k}. If φ_{1}^{−1} ≠ φ_{2} then (\overline{v}_{1}, \overline{v}_{2}) = 0 which implies \overline{V}_{φ_{1}} ⊥ \overline{V}_{φ_{2}} when φ_{1}^{−1} ≠ φ_{2}. Since \overline{V} is non-degenerate, dim(\overline{V} _{φ_{1}}^{⊥}) = dim(\overline{V}) − dim(\overline{V}_{φ_{1}}) and it follows that \overline{V}_{φ} ⊕ \overline{V}_{φ^{−1}} must be non-degenerate and therefore possesses a hyperbolic basis.

Pick φ ≠ 1 so that \overline{V}_{φ} ≠ 0. Since g\overline{V}_{φ} = \overline{V}_{φ^{g}} for g ∈ N_{k}, we may consider \overline{V}_{φ} to be an FIn_{N_{k}}(φ)-module. Let E_{k−1}^{∗} be a maximal subgroup of E_{k}^{∗} which does not contain φ. Define O_{+} = φE_{k−1}^{∗} ∩ φ^{N_{k}} and O_{−} = φ^{−1}E_{k−1}^{∗} ∩ φ^{N_{k}} so that O_{+} ∪ O_{−} = φ^{N_{k}} and |O_{+}| = |O_{−}| ≥ k. Moreover φ_{i} ∈ O_{+} if and only if φ_{i}^{−1} ∈ O_{−}. We assume that O_{+} = {φ_{i}} and that O_{−} = {φ_{i}^{−1}}. Then (⨁_{φ_{i}∈O_{+}} \overline{V}_{φ_{i}}) ⨁ (⨁_{φ_{i}^{−1}∈O_{−}} \overline{V}_{φ_{i}^{−1}}) is an FN_{k}-submodule of \overline{V} ↓_{N_{k}}. If

φ′ ∈ φ^{N_{k}} then, as C_{N_{k}}(H_{k}) also controls fusion in E_{k}^{∗}, there is a g ∈ C_{N_{k}}(H_{k}) such that g\overline{V}_{φ} = \overline{V}_{φ′}. In particular \overline{V}_{φ} ≅ \overline{V}_{φ′} as FH_{k}-modules. Define D = dim(\overline{V}_{φ}) so that D = dim(\overline{V}_{φ_{i}}) for all i.

Given the above decomposition, we form the following:

\overline{V}_{+} = ⨂_{i=1}^{k} \overline{V}_{φ_{i}}  and  \overline{V}_{−} = ⨂_{i=1}^{k} \overline{V}_{φ_{i}^{−1}}.

Recall that D = dim(\overline{V}_{φ_{i}}) and assume that \{\overline{v}_{i,j}, \overline{w}_{i,j} | 1 ≤ j_{i} ≤ D\} is a hyperbolic basis for \overline{V}_{φ_{i}} ⊕ \overline{V}_{φ_{i}^{−1}}. Define \overline{v}^{j_{1},... ,j_{k}} = ⨂_{i=1}^{k} \overline{v}_{i,j_{i}} and \overline{w}^{j_{1},... ,j_{k}} =

⨂_{i=1}^{k} \overline{w}_{i,j_{i}}. Then \{\overline{v}^{j_{1},... ,j_{k}}, \overline{w}^{j_{1},... ,j_{k}} | 1 ≤ j_{i} ≤ D\} forms a hyperbolic basis for \overline{V}_{+} ⊕ \overline{V}_{−}. If σ ∈ S_{k}, then σ(\overline{v}^{j_{1},... ,j_{k}}) = \overline{v}^{j_{1},... ,j_{k}} if and only if σ = 1 since the V_{φ_{i}} are distinct. Moreover, r_{μ}^{\overline{v}^{j_{1},... ,j_{k}}} = r_{μ} for all \overline{v}^{j_{1},... ,j_{k}} ∈ \overline{V}_{+}. Similarly for \overline{w}^{j_{1},... ,j_{k}} ∈ V_{−}.

By parts (1) and (2) of Lemma 2.4, and as \overline{V}_{±} are both totally singular, c_{μ}r_{μ} (\overline{V}_{±}) ⊂ \overline{W} (λ). By part (3) of Lemma 2.4, \overline{f}_{μ}^{k}[c_{μ}r_{μ}(\overline{v}^{j_{1},... ,j_{k}}), c_{μ}r_{μ}(\overline{w}^{j_{1},... ,j_{k}})] ≠ 0. Whenever (j_{1}, …, j_{k}) ≠ (j′_{1}, …, j′_{k}), we have that \overline{f}_{μ}^{k}[c_{μ}r_{μ}(\overline{v}^{j_{1},... ,j_{k}}), c_{μ}r_{μ}(\overline{w}^{j′_{1},... ,j′_{k}})] = 0. Therefore \{c_{μ}r_{μ}(\overline{v}^{j_{1},... ,j_{k}}), c_{μ}r_{μ}(\overline{w}^{j_{1},... ,j_{k}}) | 1 ≤ j_{i} ≤ D\} is a hyperbolic basis for

c_{μ}r_{μ} (\overline{V}_{+}) ⨁ c_{μ}r_{μ} (\overline{V}_{−}) .

Lemma 4.1. We have:

(1) \overline{V}_{±} ≅ c_{μ}r_{μ}(\overline{V}_{±}) as FC_{k}-modules;

(2) If k is even, then C_{k} stabilizes a 1-dimensional subspace of M;

(3) If k is odd, then C_{k} stabilizes a D-dimensional subspace of M.

Proof. Given the hyperbolic basis \{\overline{v}^{j_{1},... ,j_{k}}, \overline{w}^{j_{1},... ,j_{k}} | 1 ≤ j_{i} ≤ D\} for \overline{V}_{+} ⊕ \overline{V}_{−}, it is clear that the map \overline{v}^{j_{1},... ,j_{k}} ↦ c_{μ}r_{μ}(\overline{v}^{j_{1},... ,j_{k}}) is a C_{k}-invariant bijection. Therefore \overline{V}_{+} ≅ c_{μ}r_{μ}(\overline{V}_{+}) as FC_{k}-modules. The case for \overline{V}_{−} follows by a similar argument, proving part (1). Suppose that k is even and recall that \overline{V}_{φ_{i}} ≅ \overline{V}_{φ_{j}} and \overline{V}_{φ_{i}^{−1}} ≅ \overline{V}_{φ_{j}^{−1}} as FH_{k}-modules. As H_{k} ≅ S_{n−3k−2} and all irreducible FS_{n−2k−2} are self-dual, H_{k} stabilizes a 1-dimensional subspace of \overline{V}_{φ_{i}} ⊗ \overline{V}_{φ_{j}}. It follows by induction that H_{k} stabilizes a 1dimensional subspace of \overline{V}_{+}. If k is odd, then the same argument leads to a D-dimensional subspace being stabilized by H_{k}. As E_{k} acts as scalars on \overline{V}_{±}, these spaces are, in fact, stabilized by C_{k}. Using part (1), C_{k} stabilizes a subspace \overline{W}_{0} of one of these dimensions in \overline{W} (λ). Since c_{μ}r_{μ} (\overline{V}_{+}) ⨁ c_{μ}r_{μ} (\overline{V}_{−}) possesses a hyperbolic basis, \overline{W}_{0} ∩ \overline{W} (λ)^{⊥} = 0. If we let

M_{0} = (\overline{W}_{0} + \overline{W} (λ)^{⊥}) /\overline{W} (λ)^{⊥}

then Lemma 2.3 implies that M_{0} ⊂ \overline{W} (λ)/\overline{W} (λ)^{⊥} ≅ M, hence (2) and (3). □

5. Proof of Theorem 1.2.

We are now in a position to prove Theorem 1.2:

Since M possesses a vector \overline{v}_{λ_{k}} of weight λ_{k} by Lemma 2.5, we can produce a lower bound for dim(M) as follows: Let Weyl(\overline{K}) be the Weyl group of \overline{K} and recall that ℓ is the Lie rank of \overline{K}. We compute C_{Weyl(\overline{K})}(λ_{k}) using [3,

§13.1], and compute |λ_{k}^{Weyl(\overline{K})}|, whence

dim(M) ≥ |λ_{k}^{Weyl(\overline{K})}| = 2^{k}\binom{ℓ}{k}. (1)

RESTRICTIONS OF Ω_{m}(q)-MODULES 305

Case 1. First suppose that k ≥ n/3 − 1. We assume that dim(\overline{V}) ≥ 2n^{4}, so ℓ ≥ n^{4}. Since dim(M) ≤ \sqrt{|H|} ≤ \sqrt{n!}, Eq. (1) implies that 2^{k}\binom{ℓ}{k} ≤ \sqrt{n!}. Trivially, 2^{n^{4}/2} > \sqrt{n!} for all n ≥ 1, so that k < n^{4}/2 ≤ ℓ/2. Using the fact that \binom{ℓ}{k_{1}} < \binom{ℓ}{k_{2}} if k_{1} < k_{2} < ℓ/2, \binom{ℓ}{k} will be minimal when k = n/3 − 1 and ℓ = n^{4}. Note also that \binom{ℓ}{k} = ∏_{i=1}^{k} \frac{(ℓ−i+1)}{(k−i+1)} ≥ \frac{(ℓ−k+1)^{k}}{k^{k}}. We have:

2^{n/3−1}\binom{n^{4}}{n/3 − 1} < \sqrt{n!},

2^{n/3−1} \frac{(n^{4} − n/3 + 2)^{n/3−1}}{(n/3)^{n/3−1}} < (n^{1/2})^{n−1},

2^{n/3−1}(n^{3} − 1)^{n/3−1} < n^{(n−1)/2},

n^{n−3} < n^{(n−1)/2},
n − 3 < (n − 1)/2,
n < 5.

This contradicts our assumption that n ≥ 10, so that dim(\overline{V}) ≤ 2n^{4} or k < n/3 − 1. Case 2. We assume that k < n/3 − 1 and that k is odd. Lemma 4.1 and Frobenius reciprocity imply dim(M) ≤ D[H : C_{k}]. Since D ≥ \frac{ℓ}{2k} and [H : C_{k}] = \frac{n!}{2(3^{k})(n−3k−2)!}, we have dim(M) ≤ \frac{ℓ}{2k} \frac{n!}{3^{k}(n−3k−2)!}. Combining with (1) we get:

2^{k}\binom{ℓ}{k} ≤ \frac{ℓ}{2k} \frac{n!}{2(3^{k})(n − 3k − 2)!},

2^{k}\binom{ℓ − 1}{k − 1} < \frac{n^{3k+2}}{3^{k−1}} ,

2^{k} \frac{(ℓ − k + 1)^{k−1}}{(k − 1)^{k−1}} < \frac{n^{3(k−1)}n^{5}}{3^{k−1}} ,

2\frac{ℓ − k}{k − 1} < \frac{n^{3}}{3} n^{5/(k−1)}.

Observing that (k − 1)n^{5/(k−1)} < n^{3} when k ≥ 3 and n ≥ 10, we have

2ℓ < \frac{n^{6} + 2n}{3} < n^{6}.

Case 3. Finally we assume that k < n/3 − 1 and that k is even. Again Lemma 4.1 and Frobenius reciprocity imply that dim(M) ≤ [H : C_{k}] ≤ \frac{n!}{2(3^{k})(n−3k−2)!}. Combining with (1) we get:

2^{k}\binom{ℓ}{k} ≤ \frac{n!}{3^{k}(n − 3k − 2)!},

2^{k} \frac{(ℓ − k + 1)^{k}}{k^{k}} < \frac{n^{3k+2}}{3^{k}} = \frac{n^{3k}}{3^{k}} n^{2},

2\frac{ℓ − k}{k} < \frac{n^{3}}{3} n^{2/k},

2ℓ < \frac{n^{5} + 3n}{9}.

In all cases, 2ℓ < n^{6}, which implies that dim(\overline{V}) ≤ n^{6}. This completes the proof of Theorem 1.2. □

Acknowledgements. The author would like to thank Prof. Kay Magaard and Prof. Robert L. Griess for their help during the preparation of this article, and the referee for suggested improvements, particularly for Lemma 2.5.

References

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[2] W. Fulton and J. Harris, Representation theory, Springer-Verlag, 1991.

[3] J.E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, 1972.

[4] G.D. James and A. Kerber, The Representation Theory of the Symmetric Groups, Encyclopedia of Math. and its Appl., Vol. 16, Addison-Wesley, 1981.

[5] A. Premet, Weights of infinitesimally irreducible representations of Chevalley groups over a field of prime characteristic, Math. USSR Sbornik, 61 (1988), 167-183 (English translation).

[6] R. Steinberg, Lectures on Chevalley Groups, Yale University Mathematics Department, 1968.

Received June 16, 1998 and revised November 11, 1998. This research was done at Wayne State University, Detroit MI, 48202.

Ohio State University Columbus, OH 43210 E-mail address: husen@math.ohio-state.edu

This paper is available via http://nyjm.albany.edu:8000/PacJ/2000/192-2-7.html.