PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Xxxx XXXX, Pages 000–000 S 0002-9939(XX)0000-0

ON AN ASYMPTOTIC BEHAVIOR OF ELEMENTS OF ORDER p IN IRREDUCIBLE REPRESENTATIONS OF THE CLASSICAL

ALGEBRAIC GROUPS WITH LARGE ENOUGH HIGHEST WEIGHTS

I.D. SUPRUNENKO

Abstract. The behavior of the images of a fixed element of order p in irreducible representations of a classical algebraic group in characteristic p with highest weights large enough with respect to p and this element is investigated. More precisely, let G be a classical algebraic group of rank r over an algebraically closed field K of characteristic p > 2. Assume that an element x ∈ G of order p is conjugate to that of an algebraic group of the same type and rank m < r naturally embedded into G. Next, an integer function σ_{x} on the set of dominant weights of G and a constant c_{x} that depend only upon x, and a polynomial d of degree one are defined. It is proved that the image of x in the irreducible representation of G with highest weight ω contains more than d(r − m) Jordan blocks of size p if m and r − m are not too small and σ_{x}(ω) ≥ p − 1 + c_{x}.

Asymptotic lower estimates for the number of Jordan blocks of size p in the images of a fixed element of order p in irreducible representations of a classical algebraic group in characteristic p with highest weights large enough with respect to p and this element are obtained. More precisely, let G be a classical algebraic group of rank r over an algebraically closed field K of characteristic p > 2. Assume that an element x ∈ G of order p is conjugate to that of an algebraic group G_{m} of the same type and rank m < r naturally embedded into G. Set

d(r − m) = \begin{cases}r − m & for G = A_{r}(K), \\ 2r − 2m + 1 & for G = B_{r}(K), \\ 2r − 2m & for G = C_{r}(K) or D_{r}(K). \end{cases}

Let Δ_{x} be the labelled Dynkin diagram of the conjugacy class containing x in the sense of Bala and Carter [1] and let c_{x} be the sum of the labels at Δ_{x} for G ≠ A_{r}(K) and the half of this sum for G = A_{r}(K). For brevity, throughout the article we refer to Δ_{x} as to the labelled Dynkin diagram of x. Next, an integer function σ_{x} on the set of dominant weights of G that depends only upon Δ_{x} is defined. For p-restricted weights σ_{x} coincides with the canonical homomorphism determined by Δ_{x}. It is proved that the image of x in the irreducible representation of G with

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors January , 2000 .

1991 Mathematics Subject Classification. Primary 20G05; Secondary 20G40.

Key words and phrases. Classical groups, representations, Jordan blocks.

This research has been supported by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the State program “Mathematical structures” and by the Belarus Basic Research Foundation, Project F 98-180.

highest weight ω contains more than d(r − m) Jordan blocks of size p if m and r − m are not too small and σ_{x}(ω) ≥ p − 1 + c_{x}.

We need some more notation to formulate the main results. Let ω_{i} and α_{i} be the fundamental weights and the simple roots of G (with respect to a fixed maximal torus T) labelled as in [2]. Denote by δ_{i}, 1 ≤ i ≤ r, the label on Δ_{x} corresponding to its ith node. We have 0 ≤ δ_{i} ≤ 2. In what follows ℤ is the set of integers, X = X(G) is the set of weights of G, X^{+} ⊂ X is the set of dominant weights, Irr = Irr(G) is the set of irreducible rational representations of G (considered up to the equivalence), ω(φ) is the highest weight of a representation φ. There exists a uniquely determined homomorphism τ_{x} : X → ℤ such that τ_{x}(α_{i}) = δ_{i}. The weight μ ∈ X^{+} is called p-restricted if μ = ∑_{i=1}^{r} a_{i}ω_{i} with all a_{i} < p. Each weight ω ∈ X^{+} can be represented in the form ∑_{j=0}^{s} p^{j}ω^{j} where ω^{j} are p-restricted. Set σ_{x}(ω) = τ_{x}(∑_{j=0}^{s} ω^{j}). Now we can state our main result.

Theorem 1. Let φ ∈ Irr and σ_{x}(ω(φ)) ≥ p − 1 + c_{x}. Assume that m > 1 for G = B_{r}(K) or D_{r}(K), r − m > 1 for G = A_{r}(K), > 2 for G = B_{r}(K), and > 3 for G = D_{r}(K). Then the element φ(x) has more than d(r − m) Jordan blocks of size p.

Proposition 2 below shows that one cannot weaken the inequality for σ_{x}(ω(φ)) in Theorem 1 and that the estimates obtained are asymptotically exact.

Proposition 2. Let φ ∈ Irr and ω = ω(φ) = aω_{1} with a < p. Assume that m and r are such as in Theorem 1 and x is a regular unipotent element in G_{m}. Set c = m for G = A_{r}(K), 2m for G = B_{r}(K), 2m − 1 for G = C_{r}(K), and 2m − 2 for G = D_{r}(K). Suppose that p > c. Then |x| = p, σ_{x}(ω) = ac and c_{x} = c. There exist constants N_{G}(a, m, p) and Q_{G}(a, m, p) that depend upon the type of G, a, m, and p and do not depend upon r such that φ(x) contains at most N_{G}(a, m, p) Jordan blocks of size p if p < ac < p + c − 1 and at most d(r − m) + Q_{G}(a, m, p) such blocks if ac = p + c − 1.

Put l = [(m + 2)/2] for G = A_{r}(K) and l = m otherwise. By Lemma 3 below, c_{x} = ∑_{i=1}^{l} δ_{i} and c_{x} ≤ p − 1. Hence c_{x} ≤ 2l.

For φ ∈ Irr define the weight ω̄(φ) as follows: write down the p-adic expansion for the weight ω = ω(φ) considered before the statement of Theorem 1 and set ω̄(φ) = ∑_{j=0}^{s} ω^{j}. So σ_{x}(ω(φ)) = τ_{x}(ω̄(φ)).

ON AN ASYMPTOTIC BEHAVIOR OF ELEMENTS OF ORDER p 3

The study of an asymptotic behavior of elements of order p in representations of the classical groups in characteristic p was begun by the author in [12] where a notion of a p-large representation was introduced. In our present notation a representation φ ∈ Irr is p-large if and only if σ_{x}(ω(φ)) ≥ p for a long root element x ∈ G (an equivalent definition from [12]: the value of ω̄(φ) on the maximal root is ≥ p). The common goal of [12] and the present article is to investigate the behavior of elements of order p in irreducible representations in characteristic p for a fixed p and r → ∞ and to discover asymptotic regularities which are specific for prime characteristics but do not (or almost do not) depend upon p. Such properties can find applications in recognizing representations and linear groups. According to [12, Theorem 1.1], the image of any element of order p in a p-large representation has at least f(r) Jordan blocks of size p where

f(r) = \begin{cases}2r − 2 & for G = A_{r}(K), \\ 6r − 7 & for G = B_{r}(K), p = 3, \\ 8r − 10 & for G = B_{r}(K), p > 3, \\ 4r − 4 & for G = C_{r}(K), \\ 4r − 8 & for G = D_{r}(K), p = 2, \\ 6r − 10 & for G = D_{r}(K), p = 3, \\ 8r − 16 & for G = D_{r}(K), p > 3. \end{cases}

In [12, Theorem 1.3] for p > 2 and all types of the classical groups examples of representations φ ∈ Irr such that φ(x) has only one Jordan block of size p for a long root element x and σ_{x}(ω(φ)) = p − 1 are given. It is also shown [12, Theorem 1.4] that the estimates in [12, Theorem 1.1] are asymptotically exact for the groups of type A, B, and D provided p > 2 in the last two cases.

In what follows ℂ is the complex field, G_{ℂ} is the simple algebraic group over ℂ of the same type and rank as G, Irr_{ℂ} is the set of irreducible rational representations of G_{ℂ} (considered up to the equivalence). For ρ ∈ Irr or Irr_{ℂ} and a unipotent element z ∈ G or G_{ℂ} denote by k_{ρ}(z) the degree of the minimal polynomial of ρ(z). It is well known that k_{ρ}(z) is equal to the maximal size of a Jordan block of ρ(z). If φ ∈ Irr, then φ_{ℂ} is the irreducible representation of G_{ℂ} with highest weight ω̄(φ). For unipotent x ∈ G put k_{φ_{ℂ}}(x) = k_{φ_{ℂ}}(y) where y ∈ G_{ℂ} is a unipotent element with the labelled Dynkin diagram Δ_{x} (this is correctly determined). Now let |x| = p. By the results [11, Theorem 1.1, Lemma 2.5, and Proposition 2.12], k_{φ_{ℂ}}(x) = σ_{x}(ω(φ)) + 1 and k_{φ}(x) = min{p, k_{φ_{ℂ}}(x)}. Hence if z is a long root element, then k_{φ}(z) = k_{φ_{ℂ}}(z) if and only if φ is not p-large. The results of [11] imply that for not very small p and r there exists a wide class of representations φ ∈ Irr such that k_{φ}(z) = k_{φ_{ℂ}}(z) < p for a long root element z ∈ G, but k_{φ}(x) = p < k_{φ_{ℂ}}(x) for many other elements x ∈ G of order p. In this connection in [13, Section 2] a notion of a p-large representation for a given element x of order p was introduced. A representation φ ∈ Irr was called p-large for x if σ_{x}(ω(φ)) ≥ p. It has been conjectured ([13, Conjecture 1] that if x ∈ G_{m}, r is large enough with respect to m and φ is p-large for x, then φ(x) has at least F (r) blocks of size p where F is an increasing function. Our Proposition 2 formally disproves this conjecture, but Theorem 1 actually proves a refined version of it with a stronger assumption on σ_{x}(ω(φ)). Thus for arbitrary elements x there is a gap between the class of representations φ ∈ Irr with k_{φ}(x) = k_{φ_{ℂ}}(x) and that of representations where asymptotic estimates for the number of Jordan blocks of size p in φ(x) hold. May be, for some classes of elements of order p stronger estimates than those of Theorem 1 are possible, but now it is not clear how to determine such classes.

The case p = 2 is not considered here, but in this situation φ ∈ Irr is 2-large if (ω(φ)) ≠ 2^{j}ω_{i}. For 2-large representations the estimates from [12, Theorem 1.1] are available. For remaining representations certain estimates could be obtain as well, but this article does not seem a proper place for this. We plan to handle this question in a subsequent paper which will be devoted to refining some estimates in [12].

The results of this article as well as those of [12] can be easily transferred to irreducible K-representations of finite classical groups in characteristic p.

Notation and preliminary comments

Throughout the article for a semisimple algebraic group S the symbols Irr(S), X(S), and X^{+}(S) mean the same as the similar ones for G introduced earlier; R(S) is the set of roots of S, ⟨S_{1}, …, S_{j}⟩ is the subgroup in S generated by subgroups S_{1}, …, S_{j}; Irr_{p}(S) ⊂ Irr(S) is the set of p-restricted representations, i.e. irreducible representations with p-restricted highest weights; X(φ) (X(M)) is the set of weights of a representation φ (a module M); dim M is the dimension of M; M(ω) is the irreducible S-module with highest weight ω; L is the Lie algebra of G; R = R(G), R^{+} ⊂ R is the set of positive roots; Irr_{p} = Irr_{p}(G); X_{β} ⊂ G and X_{β} ∈ L are the root subgroup and the root element associated with β ∈ R, X_{±i} = X_{±α_{i}}, and X_{±i} = X_{±α_{i}}. Set H(β_{1}, …, β_{j}) = ⟨X_{β_{1}}, X_{−β_{1}} …, X_{β_{j}}, X_{−β_{j}}⟩. For ω ∈ X(S) and α ∈ R(S) denote by ⟨ω, α⟩ the value of the weight ω on the root α. For an S-module M and a unipotent element x ∈ S define k_{M}(x) similarly to k_{φ}(x). If |x| = p, then n_{φ}(x) is the number of Jordan blocks of size p of the matrix φ(x) for a representation φ of S and n_{M}(x) denotes the same number for a module M affording φ.

An element x ∈ G of order p can be embedded into a closed connected subgroup Γ of type A_{1} whose labelled diagram coincides with Δ_{x} (see [6, Theorem 4.2]). Set X_{1} = X(A_{1}(K)) (the simply connected group of this type) and identify X_{1} with ℤ mapping aω_{1} ∈ X_{1} into a ∈ ℤ. Then X(Γ) can be identified with a subset of ℤ. The canonical homomorphism τ_{x} can be obtained as the restriction of weights from a maximal torus T ⊂ G to a maximal torus T_{1} ⊂ Γ such that T_{1} ⊂ T. From now on we fix the tori T and T_{1} and all weights and roots of G and Γ are considered with respect to T and T_{1}. Throughout the text ε_{i} with 1 ≤ i ≤ r + 1 for G = A_{r}(K) and 1 ≤ i ≤ r otherwise are weights of the standard realization of G labelled as in [3, ch. VIII, §13]. Set e_{i} = τ_{x}(ε_{i}). One can choose Γ, T and T_{1} such that the restriction to Γ of the natural representation of G is a direct sum of irreducible components with p-restricted highest weights (see comments in [14, Section 3]); e_{i} ≥ e_{j} for i < j; e_{i} ≥ 0 if G = A_{r}(K) and i ≤ (r + 1)/2; and e_{i} ≥ 0 for all i ≤ r if G ≠ A_{r}(K). If H ⊂ G is a semisimple subgroup generated by some root subgroups, then T_{H} = T ∩H is a maximal torus in H. If T_{1} ⊂ T_{H}, we denote by the same symbol τ_{x} the homomorphism X(H) → ℤ determined by restricting weights from T_{H} to T_{1}. This causes no confusion. If an element v of some G-module is an eigenvector for T, we denote its weights with respect to T, T_{H}, and T_{1} by ω(v), ω_{H}(v), and ω_{Γ}(v). In what follows x is conjugate to an element of G_{m}, |x| = p, m and r − m are such as in the assertion of Theorem 1, and δ_{i} = τ_{x}(α_{i}), 1 ≤ i ≤ r.

Lemma 3. Set l = [(m + 2)/2] for G = A_{r}(K) and l = m otherwise. Then c_{x} = ∑_{i=1}^{l} δ_{i} and c_{x} ≤ p − 1.

Proof. Put k = l − 1 for G = A_{r}(K), m = 2t and k = l for G = A_{r}(K), m = 2t + 1. Our assumptions on e_{i}, m − r, and x imply that e_{i} = 0 for k < i < r + 2 − k if G = A_{r}(K) and e_{i} = 0 for i > m otherwise; notice that e_{k+1} = e_{k+2} = 0 for G = A_{r}(K). Now it follows from the definition of c_{x} and the formulae in [3, ch. VIII, §13] that c_{x} = ∑_{i=1}^{l} δ_{i} = e_{1} − e_{l+1} = e_{1}. As e_{1} is a weight of a p-restricted Γ-module, we have e_{1} < p. This yields the lemma.

Proof of Theorem 1. Set ω = ω(φ) and let ω = ∑_{i=1}^{r} a_{i}ω_{i}. It is clear that ω ≠ 0 as τ_{x}(ω) ≠ 0. Define subgroups H_{1} and H_{2} ⊂ G as follows. For G = A_{r}(K) set u = r − t + 2 if m = 2t and r − t + 1 if m = 2t + 1, β = ε_{t+1} − ε_{u},

H_{1} = H(α_{1}, …, α_{t}, β, α_{u}, …, α_{r}), H_{2} = H(α_{t+2}, …, α_{u−2})

(we have H_{1} = H(α_{1}, ε_{2} − ε_{r+1}) for m = 2 and H_{1} = H(ε_{1} − ε_{r+1}) for m = 1). For G = B_{r}(K), C_{r}(K), or D_{r}(K) put β = ε_{m}, 2ε_{m}, or ε_{m−1} + ε_{m}, respectively,

H_{1} = H(α_{1}, …, α_{m−1}, β), H_{2} = H(α_{m+1}, …, α_{r})

(here H_{1} = H(β) for G = C_{r}(K) and m = 1). One easily observes that the sets of roots in brackets used to define H_{1} and H_{2} yield bases of the systems R(H_{1}) and R(H_{2}), respectively. Denote these bases by B_{i}. In all cases H_{1} is conjugate to G_{m} in G. We have H_{2} ≅ A_{r−m−1}(K), B_{r−m}(K), C_{r−m}(K), or D_{r−m}(K) for G = A_{r}(K), B_{r}(K), C_{r}(K), or D_{r}(K), respectively. It is clear that the subgroups H_{1} and H_{2} commute. Set H = H_{1}H_{2}. Let U_{i} = ⟨X_{γ} | γ ∈ R^{+}, X_{γ} ⊂ H_{i}⟩, i = 1, 2, and U = U_{1}U_{2}. It is not difficult to conclude that U_{i} is a maximal unipotent subgroup in H_{i} and U is such subgroup in H. We can assume that x ∈ U_{1}, Γ ⊂ H_{1} and T_{1} ⊂ T_{H_{1}}. We shall write a weight μ ∈ X(H) in the form (μ_{1}, μ_{2}) where μ_{i} ∈ X(H_{i}) is the restriction of μ to T_{H_{i}}. Set M = M(ω).

It is clear that n_{V} (x) = dim(x − 1)^{p−1}V for each H-module V. Taking this into account, it is not difficult to conclude the following. If 0 ⊂ W_{1} ⊂ … ⊂ W_{t} = V is a filtration of V, F_{i} = W_{i}/W_{i−1}, 1 ≤ i ≤ t and n_{F_{i}}(x) = n_{i}, then

n_{V} (x) ≥ ∑_{i=1}^{t} n_{i}. (1)

First suppose that φ ∈ Irr_{p}. Since passing to the dual representation does not influence the Jordan form of φ(x), one can assume that a_{i} ≠ 0 for some i ≤ (r+1)/2 if G = A_{r}(K). As for p-large representations the estimates of [12, Theorem 1.1] hold, we also assume that φ is not p-large. Hence ⟨μ, α⟩ < p for all μ ∈ X(φ) and long roots α (for all α if G = A_{r}(K) or D_{r}(K)). By the formulae for the maximal roots of the classical groups in [2, Tables 1-4], this forces that

a_{1} + … + a_{r} < p for G = A_{r}(K) or C_{r}(K),
a_{1} + 2a_{2} + … + 2a_{r−1} + a_{r} < p for G = B_{r}(K),
a_{1} + 2a_{2} + … + 2a_{r−2} + a_{r−1} + a_{r} < p for G = D_{r}(K).

(2)

Now we proceed to construct two composition factors M_{1} and M_{2} of the restriction M|H such that n_{M_{1}}(x) ≥ d(r − m) and n_{M_{2}}(x) > 0. This will be done for almost all ω. In exceptional cases we shall find one factor M_{1} such that n_{M_{1}}(x) > d(r−m). By (1), this would yield the assertion of the theorem.

Let v ∈ M be a nonzero highest weight vector. Put μ_{i} = ω_{H_{i}}(v). The vector v generates an indecomposable H-module V_{1} with highest weight μ = (μ_{1}, μ_{2}). Using (2), one can deduce that ⟨μ_{1}, β⟩ < p for all β ∈ B_{1}. Here for G = B_{r}(K) we take into account that m > 1. Hence μ_{1} is p-restricted. Now assume that either G ≠ B_{r}(K), or a_{i} ≠ 0 for some i < r. For such representations we construct another weight vector w ∈ M that is fixed by U. Take l as in Lemma 3. First suppose that a_{j} ≠ 0 for some j ≤ l (Case 1). Choose maximal such j and put w = X_{−l} … X_{−(j+1)}X_{−j}v. Now let a_{j} = 0 for all j ≤ l (Case 2). Our assumptions on a_{i} imply that a_{i} ≠ 0 for some i with α_{i} ∈ B_{2}; furthermore, one can take i ≤ (r + 1)/2 for G = A_{r}(K) and i < r for G = B_{r}(K). Choose minimal such i and set w = X_{−l} … X_{−(i−1)}X_{−i}v if G ≠ D_{r}(K) or i < r and w = X_{−l} … X_{−(r−3)}X_{−(r−2)}X_{−r}v for G = D_{r}(K) and i = r. It follows from [12, Lemma 2.1(iii) and Lemma 2.9] that in all cases w ≠ 0. Using [10, Lemma 72] and analyzing the roots in B_{1} and B_{2} and the weight system X(φ), we get that U fixes w in all situations. Here it is essential that the case G = B_{r}(K) with ω = a_{r}ω_{r} is excluded. In the latter case we cannot assert that X_{β} fixes w. Set λ_{i} = ω_{H_{i}}(w), i = 1, 2. Now it is clear that w generates an indecomposable H-module V_{2} with highest weight λ = (λ_{1}, λ_{2}). We claim that λ_{1} is p-restricted. Write down all the situations where ⟨λ_{1}, γ⟩ ≠ ⟨μ_{1}, γ⟩ for some γ ∈ B_{1}. We have ⟨λ_{1}, β⟩ = ⟨μ_{1}, β⟩−1 in Case 1 if j = l and G ≠ B_{r}(K) or j = l −1 and G = D_{r}(K) and in Case 2 for G ≠ B_{r}(K) and all i; and ⟨λ_{1}, β⟩ = ⟨μ_{1}, β⟩ − 2 for G = B_{r}(K) both in Case 1 with j = l and in Case 2. In Case 1 we also have ⟨λ_{1}, α_{j−1}⟩ = ⟨μ_{1}, α_{j−1}⟩ + 1 if j > 1 and ⟨λ_{1}, α_{j}⟩ = ⟨μ_{1}, α_{j}⟩ − 1 if j < l. In Case 2 one gets ⟨λ_{1}, α_{l−1}⟩ = ⟨μ_{1}, α_{l−1}⟩ + 1 if l > 1. In all other situations we have ⟨λ_{1}, γ⟩ = ⟨μ_{1}, γ⟩. Now apply (2) to conclude that λ_{1} is p-restricted.

Set M_{1} = M(μ), M_{2} = M(λ), M_{1}^{j} = M(μ_{j}), and M_{2}^{j} = M(λ_{j}), j = 1, 2. Obviously, M_{i} is a composition factor of V_{i}. It is well known that M_{i} = M_{i}^{1} ⊗ M_{i}^{2}. It is clear that τ_{x}(μ_{1}) = τ_{x}(ω) ≥ p. Since x ∈ H_{1}, we have δ_{t} = 0 if α_{t} ∈ B_{2}. So by Lemma 3,

τ_{x}(λ_{1}) = τ_{x}(ω(w)) ≥ τ_{x}(ω) − ∑_{i=1}^{l} δ_{i} = τ_{x}(ω) − c_{x} ≥ p − 1.

It follows from [11, Theorem 1.1, Lemma 2.5, and Proposition 2.12] that k_{M_{i}^{1}}(x) = p. Hence n_{M_{i}}(x) ≥ dim M_{i}^{2}. One easily observes that M_{1}^{2} and M_{2}^{2} cannot both be trivial H_{2}-modules. Our assumptions on r − m and [5, Proposition 5.4.13] imply that the dimension of a nontrivial irreducible H_{2}-module is at least d(r − m). In the exceptional case where G = B_{r}(K) and ω = a_{r}ω_{r} we need to evaluate dim M_{1}^{2}. It is clear that this dimension is not less than the size of the orbit of ω under the action of the Weyl group of G. The latter is equal to 2^{r−m} > d(r − m) for our values of r − m. By (1), this yields the assertion of the theorem for φ ∈ Irr_{p}.

Now suppose that φ ∈ Irr \ Irr_{p}. By the Steinberg tensor product theorem [9, Theorem 1.1], φ can be represented in the form ⊗_{j=1}^{s}φ_{j} Fr^{j} where Fr is the Frobenius morphism of G associated with raising elements of K to the pth power and all φ_{j} ∈ Irr_{p}. It is clear that the morphism Fr does not influence the Jordan form of φ(x). Hence one can assume that φ = ψ ⊗ θ where θ = φ_{j} Fr^{j} for some j and both ψ and θ are nontrivial. Set a = σ_{x}(ω(ψ)), ν = ω(φ_{j}), b = τ_{x}(ν) and define by μ the restriction of ν to T_{H}. Now it follows from the definitions of σ_{x} and τ_{x} that σ_{x}(ω) = a + b. By [11, Theorem 1.1, Lemma 2.5 and Proposition 2.12], k_{ψ}(x) = min{a + 1, p} and k_{θ}(x) = min{b + 1, p}. First suppose that a or b ≥ p − 1. Set ρ = ψ if a ≥ p − 1 and ρ = θ otherwise and denote by π the remaining representation from the pair (ψ, θ). Then k_{ρ}(x) = p and [4, ch. VIII, Lemma 2.2] implies that n_{φ}(x) ≥ dim π. Let d(r) be the value of d(r − m) if one formally sets m = 0. Then by [5, Proposition 5.4.13], dim π ≥ d(r) > d(r − m) which settles the case under consideration.

Now assume that both a and b < p − 1. Then k_{ψ}(x) = a + 1 and k_{θ}(x) = b + 1. Since σ_{x}(ω) ≥ p − 1 + c_{x}, we have b > c_{x}. Arguing as for p-restricted φ, we can and shall suppose that ⟨ν, α_{i}⟩ ≠ 0 for some i ≤ (r+1)/2 if G = A_{r}(K). Put M′ = M(ν) and construct the composition factors M_{i}, i = 1, 2, of the restriction M′|H as for p-restricted M before. Transfer the notation μ_{1}, λ_{1}, and M_{j}^{i}, i, j = 1, 2, to M′.

Again we have the exceptional case G = B_{r}(K) and ν = a_{r}ω_{r} where we do not construct M_{2} and consider M_{1} only. Obviously, τ_{x}(μ_{1}) = b. As before, we deduce that τ_{x}(λ_{1}) ≥ b − c_{x}. By [11, Theorem 1.1, Lemma 2.5, and Proposition 2.12], k_{M_{1}}(x) = b+1 and k_{M_{2}}(x) ≥ b+1−c_{x}. Looking at the realizations of M_{i} as tensor products, one easily observes that x contains at least dim M_{1}^{2} Jordan blocks of size b + 1 in its action on M_{1} and at least dim M_{2}^{2} blocks of size ≥ b + 1 − c_{x} in its action on M_{2}. Set F_{1} = M(ω(ψ)) ⊗ M_{1}, F_{2} = M(ω(ψ)) ⊗ M_{2} and consider F_{i} as H-modules in the natural way. In the general case the H-module M has a filtration two of whose quotients are isomorphic to F_{1} and F_{2}, respectively. In the exceptional case F_{1} is a quotient of a submodule in M. Observe that a + k_{M_{2}}(x) ≥ p. Using [4, ch. VIII, Theorem 2.7] that describes the canonical Jordan form of a tensor product of unipotent blocks, we obtain that k_{M_{i}}(x) = p and n_{F_{i}}(x) ≥ dim M_{i}^{2}. As for p-restricted M, we have dim M_{1}^{2} + dim M_{2}^{2} > d(r − m) in the general case and dim M_{1}^{2} > d(r − m) in the exceptional one. Now (1) completes the proof.

Proof of Proposition 2. Let a, x, m, and c be such as in the assertion of the proposition. Assume that p < ac ≤ p + c − 1. So we have (a − 1)c ≤ p − 1. Set M = M(ω) and denote by M_{t} the weight subspace of weight t ∈ ℤ in the Γ-module M. It is clear that the Weyl group of Γ interchanges M_{t} and M_{−t}, hence dim M_{t} = dim M_{−t}. Put e = (a − 1)c, V_{1} = ⊕_{t>e}M_{t}, V_{2} = ⊕_{t>e}M_{−t}, and V = M_{e}. Set f = [(m + 1)/2] for G = A_{r}(K), f = m for G = B_{r}(K) or C_{r}(K), and f = m − 1 for G = D_{r}(K). Let v ∈ M be a nonzero highest weight vector and put w = X_{−f} … X_{−2}X_{−1}v. By [12, Lemma 2.9], w ≠ 0. We need a subgroup S which can be defined as follows. Put I = {i | 1 ≤ i ≤ r, δ_{i} = 0} and S = ⟨X_{i}, X_{−i} | i ∈ I⟩. The canonical Jordan forms of x in the standard realizations of G_{m} and G are well known. We have |x| = p since the dimension of the first realization is at most p due to our assumptions. Taking into account these Jordan forms, one easily obtains the values of δ_{i}, 1 ≤ i ≤ r, and using Lemma 3, deduces the following facts: I = {i | f + 1 ≤ i ≤ r − f} for G = A_{r}(K) and m = 2f, I = {i | f + 1 ≤ i ≤ r} for G = D_{r}(K), and S = H_{2} in all other cases where H_{2} is the subgroup defined in the proof of Theorem 1; c_{x} = ∑_{i=1}^{f} δ_{i} = c, τ_{x}(ω) = ac; and w ∈ V. Next, observe that S ≅ A_{r−m} for G = A_{r}(K) and m = 2f and S ≅ D_{r−m+1} for G = D_{r}(K). Our construction of the vector w shows that X_{i} fixes w if i ∈ I. This forces that w generates an indecomposable S-module M_{S} with highest weight ω_{S}(w). Then one immediately concludes that M_{S} ≅ M(ω_{1}). This yields that dim M_{S} = r − m + 1 = d(r − m) + 1 for G = A_{r}(K) and m = 2f, dim M_{S} = 2(r −m+1) = d(r −m)+2 for G = D_{r}(K), and dim M_{S} = d(r − m) otherwise. It is clear that M_{S} ⊂ V. Denote by X_{f} ⊂ X(M) the subset of weights of the form ω − ∑_{i=1}^{f} b_{i}α_{i} and by M_{A} the irreducible A_{f}(K)-module with highest weight aω_{1}. By Smith’ theorem [8], for each μ ∈ X_{f} the dimension of the weight subspace M_{μ} ⊂ M coincides with that of the weight subspace in M_{A} whose weight differs from aω_{1} by the same linear combination of the simple roots. Hence dim M_{μ} does not depend upon r. Set W = ⊕_{μ∈X_{f}} M_{μ}. Since M is an irreducible L-module and p > 2, observe that M is a linear span of vectors of the form X_{−i_{s}} … X_{−i_{2}}X_{−i_{1}}v. Now, analyzing the weight structure of M, we conclude that V_{1} ⊂ W and V = (V ∩ W) ⊕ M_{S}. This implies that dim V_{1} (= dim V_{2}) and dim(V ∩ W) do not depend upon r.

It follows from [10, Lemma 72] that

(x − 1)^{p−1}M_{t} ⊂ ⨁_{i≥t+2p−2} M_{i}. (3)

Let M_{t} ⊄ V_{2}. Then t ≥ −e. Obviously, e < p − 1 if ac < p − 1 + c_{x} and e = p − 1 for ac = p − 1 + c_{x}. Thus (3) implies that

(x − 1)^{p−1}M_{t} ⊂ ⨁_{t>p−1} M_{t} ⊂ V_{1}

in the first case and

(x − 1)^{p−1}M_{t} ⊂ ⨁_{t≥p−1} M_{t} ⊂ V_{1} ⊕ V

in the second one. This forces that n_{M}(x) ≤ dim V_{2} + dim V_{1} = 2 dim V_{1} in the first case and n_{M}(x) ≤ 2 dim V_{1} + dim(V ∩ W) + dim M_{S} in the second one. We have seen before that dim M_{S} = d(r − m) + u with u = 0, 1, or 2. Hence one can take N_{G}(a, m, p) = 2 dim V_{1} and Q_{G}(a, m, p) = 2 dim V_{1} + dim(V ∩ W) + u to complete the proof.

Remark 4. For G = A_{r}(K) or C_{r}(K) we could give a shorter proof of Proposition 2 using the realization of φ in the ath symmetric power of the standard module (see [7, 1.14 and 8.13]), but we need the proof above for B_{r}(K) and D_{r}(K).

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Institute of Mathematics, National Academy of Sciences of Belarus, Surganov str. 11, Minsk, 220072, Belarus

E-mail address: suprunenko@im.bas-net.by