LOW-DIMENSIONAL REPRESENTATIONS OF SPECIAL UNITARY GROUPS

GERHARD HISS AND GUNTER MALLE

Abstract. We prove gap results for the low-dimensional representations of unitary groups in non-defining characteristics. More precisely, we give a lower bound for the third smallest degree of a non-trivial absolutely irreducible representation of the groups mentioned above, which is almost of the order of magnitude of the square of the two smallest such representations. As a corollary we obtain the uniqueness of Weil representations in all cross characteristics for the unitary groups. Our approach uses some results on ordinary representations which can be proved using Deligne-Lusztig theory and which might be of independent interest, as well as information on decomposition matrices.

1. Introduction

In this paper we investigate the low-dimensional representations of the special unitary groups G = SU_{n}(q) in characteristics ℓ not dividing q. We prove that the non-trivial representations of the two smallest degrees are (the ℓ-modular reductions of) the Weil representations, of degree approximately q^{n−1}. We also show that the degree of any other absolutely irreducible representation of G is at least in the order of magnitude of q^{2n−4}, thus giving a gap of roughly this size between the second and third lowest dimensions. It is easy to see that there do exist irreducible representations of dimension approximately q^{2n−4} so that the bound is best possible, up to lower order terms. A similar result for the special linear groups has recently been obtained by Guralnick and Tiep [18] and further improved by Brundan and Kleshchev [2]. But while the argument in [18] for the special linear groups requires only rather elementary ingredients, we have to make use of Deligne-Lusztig theory for the unitary case.

Results of this type seem to have many applications. We have at least two immediate applications in mind. The first is to the study of irreducible alternating and symmetric squares of representations of quasi-simple groups. The second is to the matrix recognition program in computational group theory.

The proof uses three main ingredients. First, we make use of the fact that the sets E_{ℓ}(G, s) are unions of ℓ-blocks, which was proved by Broué and Michel [1]. Secondly, Geck [12] has shown that the decomposition matrix of GU_{n}(q) is lower unitriangular. Finally, we are able to give upper bounds for the entries in a (small) part of the decomposition matrix of G modulo ℓ by constructing characters which are projective for all ℓ not dividing q, and which are not too big. Note that all of the above ingredients involve only assertions about ordinary characters of G.

Our general method to give lower bounds for the degrees of the irreducible modular characters of SU_{n}(q) is quite simple. Unfortunately, it only works well if n is not too small. For n ≤ 5 we have to resort to some unpleasant ad hoc arguments (see, for examples, Lemmas 12 or 13). These complications could have been avoided if the representation theory of these small rank groups had been studied in more detail before, as was done by Geck for SU_{3}(q), or by the first author for G_{2}(q). This shows once more that the investigation of series of groups of Lie type of small rank is of vital interest even for proving results for all groups of Lie type, very much like the “Atlas of Finite Groups” allows to deal with all kinds of degenerate situations in the theory of finite simple groups.

2. Results on ordinary characters of groups of Lie type

We first collect some general results on complex irreducible characters of finite groups of Lie type, which allow to restrict the blocks in which modular irreducible characters of small degrees may lie. Let G be a finite reductive group of characteristic p. In other words, G = G^{F} is the group of fixed points under a Frobenius morphism F of a connected reductive algebraic group G, where F is defined with respect to some 𝔽_{q}-structure with q a power of p. According to Lusztig, the set Irr(G) of complex irreducible characters of G may be partitioned into Lusztig series E(G, s), parametrized by conjugacy classes of semisimple F -stable elements s in a group G^{∗}, dual to G. Moreover, there is a bijection between E(G, s) and the unipotent characters E(C_{G^{∗}}(s), 1) of the centralizer of s in G^{∗} := G^{∗}^{F}. If C_{G^{∗}}(s) is not connected, the unipotent characters of C_{G^{∗}}(s) are, by definition, the irreducible constituents of the unipotent characters of C_{G^{∗}}(s)^{∘}^{F}, induced up to C_{G^{∗}}(s)^{F} (see [23, Corollary 6.1] and [6]). If ψ denotes this bijection then the degrees are related by

χ(1) = (\frac{|G|}{|C_{G^{∗}}(s)|})_{p′} ψ(χ)(1)  (1)

(see [5, Remark 13.24]). We write χ_{s,λ} for χ ∈ E(G, s) with ψ(χ) = λ.

We now describe the above bijection in a special case, which, however, always applies in the general unitary groups. For s ∈ G^{∗} semisimple, put G(s) := C_{G^{∗}}(s)^{∗}, and G(s) := G(s)^{F}. Suppose that C_{G^{∗}}(s) is an F -stable Levi subgroup (not necessarily 1-split) of G^{∗}. Then we may identify G(s) with an F -stable Levi subgroup of G, and Lusztig induction R_{G(s)}^{G} induces a bijection

E(G(s), 1) → E(G, s),  ϑ ↦ ε_{G}ε_{G(s)}R_{G(s)}^{G}(ŝϑ),  (2)

where ŝ is a suitable linear character of G(s), and ε_{G} = (−1)^{𝔽_{q}-rank of G} (see [5, Theorem 13.25(ii) and Proposition 13.30]).

Now let ℓ be a prime not dividing q. For a semisimple ℓ′-element s ∈ G^{∗} define

E_{ℓ}(G, s) := ⋃_{t} E(G, st)

where the union runs over all ℓ-elements t in C_{G^{∗}}(s). Then, by a result of Broué and Michel, E_{ℓ}(G, s) is a union of ℓ-blocks [1]. Since the degree of any ordinary character in E_{ℓ}(G, s) is divisible by (|G|/|C_{G^{∗}}(s)|)_{p′} by (1), and for all ℓ-elements t centralizing s, the centralizer of st is contained in the one of s, this implies the following.

Proposition 1. Let ρ be an absolutely irreducible ℓ-modular Brauer character of G. Assume that ρ lies in the union of ℓ-blocks E_{ℓ}(G, s) for some semisimple ℓ′- element s of G^{∗}. Then

(\frac{|G|}{|C_{G^{∗}}(s)|})_{p′} divides ρ(1).

In particular χ_{s,1}, the so-called semisimple character in E(G, s), is irreducible modulo ℓ.

This result already gives strong a priori information on the blocks which can possibly contain Brauer characters of low degree.

3. Harish-Chandra series

Keep the situation and notation of the preceding section. We have to recall some facts on the distribution of representations of a finite group of Lie type into Harish-Chandra series. For the results collected in this section see [19] and [7]. For each 1-split Levi subgroup L of G (where 1-split means that L is the Levi subgroup of a parabolic subgroup of G) we denote by ^{∗}R_{L}^{G} and R_{L}^{G} the maps “Harish-Chandra restriction” and “Harish-Chandra induction”, respectively, defined on generalized ℓ-modular characters of G, respectively L. Let ρ be an absolutely irreducible ℓmodular character of G. There is a minimal Levi subgroup L (unique up to conjugation) such that ^{∗}R_{L}^{G}(ρ) is non-zero. Any such minimal Levi subgroup L is called a Harish-Chandra vertex of ρ. The irreducible characters whose Harish-Chandra vertex equals G are called cuspidal. If L is a Harish-Chandra vertex of ρ, then ∗R_{L}^{G}(ρ) is a sum of irreducible, cuspidal ℓ-modular characters of L, all of which are conjugate in N_{G}(L). Any such constituent is called a Harish-Chandra source of ρ.

We shall also need the fact that Harish-Chandra restriction commutes with restriction to the commutator subgroup.

Proposition 2. Let G̃ be a closed, connected, F -stable subgroup of G such that G = TG̃ for a maximally split torus T of G. Put G̃ = G̃^{F} and T = T^{F}.

Let L be a Levi subgroup of G containing T. Then G̃ ∩ L is a Levi subgroup of G̃ and we have for all generalized ℓ-modular characters ρ of G:

Res_{G̃∩L}^{L}(^{∗}R_{L}^{G}(ρ)) = ^{∗}RG̃ ; G̃∩L(ResG ; G̃(ρ)).

A dual statement holds, of course, for Harish-Chandra induction.

Moreover,

∗RG̃ ; G̃∩L( ρ̃^{t}) = ^{t}(^{∗}RG̃ ; G̃∩L(ρ̃))

for all generalized ℓ-modular characters ρ̃ of G̃ and all t ∈ T.

Proof. Let P be a parabolic subgroup of G with Levi decomposition P = LU. Then U ⊆ G̃ and G̃ ∩ P is a standard parabolic subgroup of G̃ with Levi decomposition G̃ ∩ P = (G̃ ∩ L)U. (See the first paragraph of the proof of [5, Proposition 13.20] for these statements.) Moreover, U is T -invariant, since T is contained in L. The results follow easily from these facts.

The above result will be applied to the case of G = GU_{n}(q) and G̃ = SU_{n}(q). We give one example of such an application.

Corollary 3. With the notation of Proposition 2, let ρ be an absolutely irreducible ℓ-modular character of G with Harish-Chandra vertex L. Then ResG ; G̃(ρ) has at most as many constituents as Res_{G̃∩L}^{L}(^{∗}R_{L}^{G}(ρ)).

Proof. Since T∩G̃, being a maximal torus of G̃, is connected, we have G = T G̃. Let ρ̃ be an irreducible constituent of ResG ; G̃(ρ). Then ResG ; G̃(ρ) = ∑_{t∈S} ρ̃^{t} for some

subset S ⊆ T. By Proposition 2 we have Res_{G̃∩L}^{L}(^{∗}R_{L}^{G}(ρ)) = ∑_{t∈S} ^{t}(^{∗}RG̃ ; G̃∩L(ρ̃)). This gives the desired result.

Under additional hypotheses we obtain a slightly stronger result.

Proposition 4. With the notation of Proposition 2, let ρ be an absolutely irreducible ℓ-modular character of G with Harish-Chandra vertex L and Harish-Chandra source σ. Suppose that σ̃ := Res_{G̃∩L}^{L}(σ) is irreducible. Suppose also that |W_{G}(L, σ)| = |WG̃(G̃ ∩ L, σ̃)|, where W_{G}(L, σ) denotes the ramification group of R_{L}^{G}(σ) (see e.g. [15, Section 3]). Then ResG ; G̃(ρ) is irreducible.

Proof. Let k be an algebraically closed field of characteristic ℓ and let Z be a kL-module with Brauer character σ. Put L̃ := G̃ ∩ L and Z̃ := ResL ; L̃(Z).

It follows from the first part of Proposition 2 that Z̃ is cuspidal and also that

ResG ; G̃(R_{L}^{G}(Z)) = RG̃ ; L̃(Z̃).

Our assumption on the ramification groups now implies that End_{kG}(R_{L}^{G}(Z)) = End_{kG̃}(RG̃ ; L̃(Z̃). It follows that every indecomposable direct summand of R_{L}^{G}(Z) restricts to an indecomposable direct summand of RG̃ ; L̃(Z̃). By [15, Theorem 2.4] each of these indecomposable direct summands has a simple head. Thus a simple head of an indecomposable summand of R_{L}^{G}(Z) restricts to a simple module. Since ρ is the character of such a head, the result follows.

By the comparison theorem [20, Theorem 5.9], the computation of Harish-Chandra induced unipotent characters can be reduced to the computation of induced characters in Weyl groups. In the cases we are interested in, Harish-Chandra induction of arbitrary characters can be reduced to Harish-Chandra induction of unipotent characters. Namely, let L be an F -stable Levi subgroup of G. Then L^{∗} is an F -stable Levi subgroup of G^{∗}. Let s be an F -stable semisimple element of L^{∗} and suppose that C_{G^{∗}}(s) is a Levi subgroup of G^{∗}. Then C_{L^{∗}}(s) = C_{G^{∗}}(s) ∩ L^{∗} is a Levi subgroup of L^{∗}. Using properties of Lusztig induction [5, 11.5, Proposition 12.6], we obtain the following commutative diagram.

[figure: E(L, s); E(G, s); E(L(s), 1); E(G(s), 1); R_{L}^{G}; R_{L(s)}^{G(s)}; ε_{G}ε_{G(s)}R_{G(s)}^{G}(ŝ–); ε_{L}ε_{L(s)}R_{L(s)}^{L}(ŝ–)]  (3)

This allows to compute R_{L}^{G} from R_{L(s)}^{G(s)}, applied to unipotent characters. Note that L(s) is a Levi subgroup of G(s).

4. Approximations to decomposition matrices

We now turn to the case of the general unitary group G = GU_{n}(q), where n ≥ 3. We will mainly be concerned with the set E(G, 1) of unipotent characters of G. The unipotent characters are parametrized by partitions λ ⊢ n [3, 13.8]. We denote the unipotent character corresponding to a partition λ by χ_{λ}. Since the set of unipotent characters forms a basic set for the unipotent blocks and the decomposition matrix is triangular [12], the unipotent Brauer characters can also be labelled canonically by partitions of n. We write φ_{λ} for the unipotent Brauer character labelled by λ.

We say that a matrix D′ approximates a decomposition matrix if the columns of D′ are non-negative integral linear combinations of the columns of the decomposition matrix. In other words, the columns of D′ describe the decomposition of some projective characters into ordinary characters. The same terminology is used for parts of decomposition matrices consisting of certain rows only.

The decomposition numbers of GU_{3}(q) were investigated by Geck, who showed the following.

Proposition 5. (Geck [11]) For all q and all ℓ |̸q the ℓ-modular decomposition matrix for the unipotent characters of GU_{3}(q) is approximated by

λ
(3) 1
(2, 1) . 1
(1^{3}) 1 q 1

Proposition 6. For all ℓ |̸q the ℓ-modular decomposition matrices for the unipotent characters of GU_{4}(q) and GU_{5}(q) are approximated by

\begin{matrix}λ \\ (4) & 1 \\ (3, 1) & 1 & 1 \\ (2^{2}) & 2 & 1 & 1 \\ (2, 1^{2}) & 1 & . & 1 & 1 \\ (1^{4}) & 1 & 1 & 1 & q & 1 \end{matrix}  \begin{matrix}λ \\ (5) & 1 \\ (4, 1) & . & 1 \\ (3, 2) & 1 & . & 1 \\ (3, 1^{2}) & 2 & q & 1 & 1 \\ (2^{2}, 1) & 1 & q & . & 1 & 1 \\ (2, 1^{3}) & . & 1 & . & . & q & 1 \\ (1^{5}) & 1 & q & 1 & 1 & q^{2} & q & 1 \end{matrix}

If n = 4 and ℓ = 2, then all the entries in the above matrix, except q, are decomposition numbers. In case n = 4 and odd ℓ dividing q + 1 one obtains the first three columns of the decomposition matrix of unipotent characters by subtracting the third column from the first.

Proof. First consider GU_{4}(q). The projective characters occurring in this and in subsequent tables come from two different sources. Firstly, we Harish-Chandra induce projective characters from proper Levi subgroups of GU_{4}(q). Thus, the first column is obtained as the permutation character on the unipotent radical of a Borel subgroup, the second is the Harish-Chandra induction of the projective cover of the Steinberg character of the Levi subgroup with semisimple part GU_{2}(q), and the third is the Harish-Chandra induction of the projective cover of the Steinberg character of the Levi subgroup with semisimple part GL_{2}(q^{2}). The last two projective characters are the generalized Gelfand-Graev characters corresponding to the partitions (3, 1) and (4) respectively (see [12]). The statements about decomposition numbers are proved in [27].

For GU_{5}(q) we just give the origins of the projective characters. The first, second and fourth are obtained by Harish-Chandra induction from the unipotent projective characters of the Levi subgroup with semisimple part GU_{3}(q) and the third by Harish-Chandra induction from the projective cover of the Steinberg character of the Levi subgroup with semisimple part GL_{2}(q^{2}). The last three projective characters are the generalized Gelfand-Graev characters corresponding to the partitions (3, 2), (4, 1), and (5) respectively (see [12]).

For n ≥ 6 the first seven rows of the unipotent part of the decomposition matrix can be approximated simultaneously as follows.

Proposition 7. Let n = 2m ≥ 6 be even. Then part of the decomposition matrix for the unipotent characters of GU_{n}(q) is approximated by:

λ
(n)  1
(n − 1, 1)  1  1
(n − 2, 2)  m  1  1
(n − 2, 1^{2}) m − 1  .  1 1
(n − 3, 3)  m m − 1 1 . 1
(n − 3, 2, 1)  .  .  … 1
(n − 3, 1^{3}) m − 1 m − 1 1 q 1 q 1

Proof. It is sufficient to construct projective characters with the indicated decomposition for n = 6. Then the result for arbitrary even n ≥ 8 follows by induction. Indeed, we may assume that the table is correct for GU_{n−2}(q). The projective characters listed in the table are obtained by Harish-Chandra induction from the projective characters for GU_{n−2}(q), as can be checked by an easy calculation inside the Weyl group B_{m} of GU_{n}(q).

It remains to explain the origin of the projective characters for GU_{6}(q). The first four are Harish-Chandra induced from the first four projective characters of GU_{4}(q) in Proposition 6. The fifth is the Harish-Chandra induction of the projective cover of the Steinberg character of GU_{2}(q) × GL_{2}(q^{2}), and the sixth is the generalized Gelfand-Graev character corresponding to the partition (3, 2, 1).

Proposition 8. Let n = 2m+1 ≥ 7 be odd. Then part of the decomposition matrix for the unipotent characters of GU_{n}(q) is approximated by:

λ
(n)  1
(n − 1, 1)  .  1
(n − 2, 2) m − 1  .  1
(n − 2, 1^{2})  m  q  1 1
(n − 3, 3)  .  m − 2 . . 1
(n − 3, 2, 1)  1  q  . 1 . 1
(n − 3, 1^{3})  .  m − 1 . . 1 q 1

Proof. As in the previous Proposition we may use induction on m. The projective characters for GU_{2m+1}(q) given in the table are obtained via Harish-Chandra induction from those for GU_{2m−1}(q), as can be verified inside the Weyl group B_{m}. We now explain how to obtain the projective characters for GU_{7}(q). The first four and the sixth arise by Harish-Chandra induction from the first five projective characters of GU_{5}(q) in Proposition 6. For the fifth we consider GU_{3}(q)× GL_{2}(q^{2}). The second projective for GU_{3}(q) tensored with the projective cover of the Steinberg character of GL_{2}(q^{2}) is again projective. Harish-Chandra induction to GU_{7}(q) gives the fifth column in the table.

The generalized Gelfand-Graev characters used in the above proofs have been computed by Frank Lübeck. They are available in the CHEVIE-system [14], which was also used to compute the relevant scalar products.

For the convenience of the reader we collect, for later use, the degrees (multiplied by (q^{3} + 1)(q^{2} − 1)(q + 1)) of the unipotent characters occurring in the above tables (see [3, 13.8]).

\begin{matrix}λ & χ_{λ}(1) · (q^{3} + 1)(q^{2} − 1)(q + 1) \\ (n) & (q^{3} + 1)(q^{2} − 1)(q + 1) \\ (n − 1, 1) & q (q^{n−1} − (−1)^{n−1})(q^{3} + 1)(q^{2} − 1) \\ (n − 2, 2) & q^{2} (q^{n} − (−1)^{n}) (q^{n−3} − (−1)^{n−3})(q^{3} + 1) \\ (n − 2, 1^{2}) & q^{3} (q^{n−1} − (−1)^{n−1}) (q^{n−2} − (−1)^{n−2})(q^{3} + 1) \\ (n − 3, 3) & q^{3} (q^{n} − (−1)^{n}) (q^{n−1} − (−1)^{n−1}) (q^{n−5} − (−1)^{n−5}) \\ (n − 3, 2, 1) & q^{4} (q^{n} − (−1)^{n}) (q^{n−2} − (−1)^{n−2}) (q^{n−4} − (−1)^{n−4})(q − 1) \\ (n − 3, 1^{3}) & q^{6} (q^{n−1} − (−1)^{n−1}) (q^{n−2} − (−1)^{n−2}) (q^{n−3} − (−1)^{n−3}) \end{matrix}

5. A parabolic subgroup

In this section, let G_{n} = SU_{n}(q), n ≥ 3, where q is a power of the prime p. To make the exposition more transparent, we first introduce a convenient matrix representation for G_{n}. The (n×n)-identity matrix is denoted by I_{n}, and the (n×n)- matrix with ones along the anti-diagonal and zeroes otherwise, by J_{n}. If a is an element of an extension field of 𝔽_{q}, we put ā := a^{q}, and extend this definition to matrices by applying the bar-map to each entry of a matrix. The special unitary group G_{n} = SU_{n}(q) is embedded into SL_{n}(q^{2}) as

SU_{n}(q) = {a ∈ SL_{n}(q^{2}) | aJ_{n}ā^{t} = J_{n}}.

Let P_{n} be the stabilizer in G_{n} of an isotropic line in the natural module for G_{n}. Thus P_{n} is a maximal parabolic subgroup with unipotent radical Q_{n} and Levi subgroup L_{n}. Furthermore, Q_{n} is a special p-group of order q^{1+2(n−2)}, |Z(Q_{n})| = q, and Q_{n}/Z(Q_{n}) is isomorphic to the natural module for SU_{n−2}(q) (see for example [24]). In the above matrix representation, the Levi subgroup L_{n} may be taken to be

L_{n} = \{[ \begin{matrix}a & 0 & 0 \\ 0 & y & 0 \\ 0 & 0 & ā^{−1} \end{matrix} ] | a ∈ GL_{1}(q^{2}), y ∈ GU_{n−2}(q), a^{1−q} det(y) = 1\} ,

and the unipotent radical Q_{n} as

Q_{n} = \{[ \begin{matrix}1 & u & y \\ 0 & I_{n−2} & v \\ 0 & 0 & 1 \end{matrix} ]| y ∈ 𝔽_{q^{2}}, v ∈ 𝔽_{q^{2}}^{n−2}, u = −v̄^{t}J_{n−2}, y + v̄^{t}J_{n−2}v + ȳ = 0\} .

Let L̃_{n} and L′_{n} denote the subgroups of L_{n} consisting of those matrices with a^{q+1} = 1 and a = 1, respectively, and put P̃_{n} := Q_{n}L̃_{n}. Then L′_{n} ≅ SU_{n−2}(q) is contained in L̃_{n} and L̃_{n}/L′_{n} is cyclic of order q +1. Moreover, L′_{n} is the commutator subgroup of L_{n}, except in the case n = 4, q = 2.

6. The Weil characters of SU_{n}(q)

Assume that n ≥ 3 and let G = SU_{n}(q). We first show the existence of certain semi-simple elements in the dual group G^{∗} = PGU_{n}(q). These elements serve as labels for some of the Weil characters to be defined below. Let v be a non-isotropic vector in the natural module for GU_{n}(q), and let Ŝ be the subgroup of GU_{n}(q) which fixes the line ⟨v⟩ and induces the identity on its orthogonal complement. Then Ŝ, as well as its image S in PGU_{n}(q) are cyclic groups of order q + 1. The centralizer of each non-trivial element of Ŝ in GU_{n}(q) is a group of type GU_{1}(q) × GU_{n−1}(q). Since n > 2, the centralizer in PGU_{n}(q) of a non-trivial element of S is a group of type GU_{n−1}(q). Note that the centralizer of a non-trivial element of S in the algebraic group PGL_{n}(𝔽̄_{q}) is connected.

To define the Weil characters of G, we identify G with L′_{n+2} and put P := P_{n+2}, Q := Q_{n+2}, and P̃ := P̃_{n+2} (in the notation of the preceding section). Gérardin has shown that each of the q−1 ordinary irreducible characters of Q of degree q^{n} extends to QL′ (see [17, Theorem 3.3]). By multiplying with a suitable linear character of L′ (in case L′ is not perfect), we obtain an extension which is also invariant under the elements of L̃. Since P̃/QL′ is cyclic, it follows that the extension to QL′ extends further to P̃. Let ϕ denote the character of such an extension. The irreducible constituents of the restriction ResP̃ ; G(ϕ) are called Weil characters of G. (They are independent of the chosen extension ϕ.) Seitz [24] shows that ResP̃ ; G(ϕ) has q + 1 irreducible constituents, one of degree (q^{n} + (−1)^{n}q)/(q + 1) and q of degree (q^{n} − (−1)^{n})/(q + 1). The first of these is the unipotent character χ_{(n−1,1)}, the others are the semisimple characters χ_{s,(n−1)} in Lusztig series E(G, s) where s runs over the non-trivial elements of the subgroup S ≤ PGU_{n}(q) described above. Let x denote the smaller one of the two above-mentioned degrees, so that the other is equal to x + 1. Landazuri and Seitz have shown [22], that except in the cases (n, q) ∈ {(3, 2), (4, 2), (4, 3)}, any non-trivial irreducible character of G in characteristic ℓ not dividing q has degree at least x. In particular, the ordinary Weil characters of degree x are irreducible modulo ℓ, while those of degree x + 1 split off at most one trivial constituent. (Using the Modular Atlas [21] or the references [11] and [27], one can see that the same statement is true for the three exceptions to the Landazuri-Seitz bound.) The following lemma clarifies when the reducible case occurs.

Proposition 9. Let ℓ be a prime not dividing q.

(a) Let n ≥ 4 be even. The Weil character χ_{(n−1,1)} of degree x + 1 remains irreducible modulo ℓ if and only if ℓ |̸(q + 1).

(b) Let n ≥ 3 be odd. The Weil character χ_{s,(n−1)} of degree x + 1 remains irreducible modulo ℓ if and only if the order of s is not a power of ℓ.

Proof. The assertions are true for (n, q) ∈ {(3, 2), (4, 2), (4, 3)}, by the references given above. If n = 4, the result follows from [27].

Now assume that n ≥ 6 is even. The Harish-Chandra restriction of χ_{(n−1,1)} to the Levi subgroup L := L_{n} equals ^{∗}R_{L}^{G}(χ_{(n−1,1)}) = χ_{(n−3,1)}. Thus the result follows by induction.

In the case of odd n it follows from Proposition 1 that χ_{s,(n−1)} is irreducible if the order of s is not a power of ℓ. Otherwise, χ_{s,(n−1)} ∈ E_{ℓ}(G, 1) and we have χ_{s,(n−1)}(g) = 1 + χ_{(n−1,1)}(g) for all ℓ-regular elements g ∈ G. Hence the reduction modulo ℓ of χ_{s,(n−1)} contains the trivial Brauer character as claimed.

7. An induced character

Let n ≥ 3 and let G_{n} = SU_{n}(q). For every n ≥ 3 let ϕ_{n} denote an extension to P̃_{n} of an ordinary irreducible character of Q_{n} of degree q^{n−2}. We also write Φ_{n} := IndP_{n} ; P̃_{n}(ϕ_{n}), so that IndG_{n} ; P̃_{n}(ϕ_{n}) = Ind_{P_{n}}^{G_{n}}(Φ_{n}).

For the remainder of this section but Lemma 11 below, we omit the subscript n. Thus, in particular, G = SU_{n}(q). The next lemma shows why we are interested in the ordinary constituents of the induced character IndG ; P̃(ϕ).

Lemma 10. Let ℓ be a prime not dividing q. Let ϑ be a non-trivial absolutely irreducible ℓ-modular character of G satisfying ϑ(1) < q^{n−2}(q − 1)d_{n−2}, where d_{n−2} denotes the degree of the smallest non-trivial absolutely irreducible ℓ-modular character of L′. Then ϑ is a modular constituent of IndG ; P̃(ϕ).

Proof. Let k be an algebraically closed field of characteristic ℓ and let V be a kG-module affording ϑ. Since ℓ does not divide the order of Q, the restriction Res_{Q}^{G}(V) of V to Q is semi-simple. Since V is non-trivial and Z(Q) is not contained in any proper normal subgroup of G, this restriction has an irreducible direct summand of dimension q^{n−2}. Then all q − 1 irreducible kQ-modules M_{1}, …, M_{q−1} of dimension q^{n−2} are direct summands of Res_{Q}^{G}(V), since these are all conjugate in L.

Let M̃_{i} denote an extension of M_{i} to P̃ (see Section 6). Then, by Clifford theory, all constituents of IndP̃ ; Q(M_{i}) are of the form M̃_{i}⊗_{k}X_{i}, where X_{i} is an irreducible kL̃- module, inflated over Q. By Frobenius reciprocity, ResG ; P̃(V) has such a submodule M̃_{i} ⊗_{k} X_{i} for each i. Our assumption on dim(V) implies that the only X_{i} that can occur are 1-dimensional. Hence, for each 1 ≤ i ≤ q − 1, there is a submodule of ResG ; P̃(V) extending M_{i}. One of these lifts to an ordinary representation with character ϕ. Thus V is a modular constituent of the induced module IndG ; P̃(ϕ).

For the next lemma only we resume our original notation with the subscript n.

Lemma 11. Let n ≥ 5. Any irreducible constituent of ^{∗}R_{L_{n}}^{G_{n}}(Ind_{P_{n}}^{G_{n}}(Φ_{n})) is also a constituent of Ind_{P_{n−2}}^{L_{n}}(Φ_{n−2}). In particular, if χ_{λ} occurs in Ind_{P_{n}}^{G_{n}}(Φ_{n}) for some unipotent character χ_{λ} then ⟨Ind_{P_{n−2}}^{L_{n}}(Φ_{n−2}), χ_{μ}⟩ ≠ 0 for all partitions μ that can be obtained from λ by removing a 2-hook. (Here, χ_{μ} denotes the unique unipotent character of L_{n} extending the unipotent character with label μ of G_{n−2} = L′_{n}.)

Proof. We first consider Res_{P_{n}}^{G_{n}}(Ind_{P_{n}}^{G_{n}}(Φ_{n})). By the Mackey formula we have

Res_{P_{n}}^{G_{n}}(Ind_{P_{n}}^{G_{n}}(Φ_{n})) = ∑_{g∈P_{n}\G_{n}/P_{n}} Ind_{P_{n}∩^{g}P_{n}}^{P_{n}}(Res_{P_{n}∩^{g}P_{n}}^{^{g}P_{n}}(^{g}Φ_{n}))

where the sum runs over double coset representatives of P_{n} in G_{n}. It is well known (see for example [3, Proposition 2.8.1]) that for parabolic subgroups the double coset representatives can be chosen to be preimages of corresponding double coset representatives in the Weyl group. The Weyl group of G_{n} is of type B_{m}, with m = ⌊n/2⌋, generated by fundamental reflections s_{1}, …, s_{m}, say, where we choose notation such that s_{2}, …, s_{m} are conjugate, while s_{1} is not conjugate to s_{2}. Then it is easy to see that B_{m−1}\B_{m}/B_{m−1} = {1, s_{m}, s_{m} ⋯ s_{2}s_{1}s_{2} ⋯ s_{m}}. For g = 1 we get no contribution to ^{∗}R_{L_{n}}^{G_{n}}(Ind_{P_{n}}^{G_{n}}(Φ_{n})), since by construction Q_{n} has no fixed points on Φ_{n}.

Next consider g = s_{m}. Here, ^{g}Q_{n} ∩ L_{n} = Q_{n−2} is the unipotent radical of the parabolic subgroup P_{n−2} of L′_{n} (this follows from [3, Proposition 2.8.9]). Also, gZ(Q_{n}) is contained in L_{n} and hence equals the center of Q_{n−2}. Now ^{g}Φ_{n} restricted to ^{g}Q_{n} is the sum of the irreducible characters of ^{g}Q_{n} of degree q^{n−2}. None of these has ^{g}Z(Q_{n}) in its kernel. Hence the restriction of ^{g}Φ_{n} to Q_{n−2} only contains characters not having Z(Q_{n−2}) in their kernel. It follows that if ψ is an irreducible constituent of Res_{P_{n}∩^{g}P_{n}}^{^{g}P_{n}}(^{g}Φ_{n}), then ResP_{n}∩^{g}P_{n} ; P̃_{n−2}(ψ) contains ϕ_{n−2}. Thus any con-

stituent of Ind_{P_{n}∩^{g}P_{n}}^{P_{n}}(Res_{P_{n}∩^{g}P_{n}}^{^{g}P_{n}}(^{g}Φ_{n})) is also contained in IndP_{n} ; P̃_{n−2}(ϕ_{n−2}). Taking

Q_{n}-fixed points of the latter just gives Ind_{P_{n−2}}^{L_{n}}(Φ_{n−2}).

For g = s_{m} ⋯ s_{2}s_{1}s_{2} ⋯ s_{m} we have P_{n} ∩ ^{g}P_{n} = L_{n}. By the results of Seitz cited in Section 6, the restriction of ^{g}ϕ_{n} to L′_{n} contains exactly the Weil characters of L′_{n} as constituents. But clearly induction from L_{n} to P_{n} and then taking Q_{n}- fixed points just gives back the same character, so we get all Weil characters as contribution to the Q_{n}-fixed points of Ind_{L_{n}}^{P_{n}}(Res_{L_{n}}^{^{g}P_{n}}(^{g}Φ_{n})), restricted to L′_{n}.

By Lemma 10 it follows that all Weil characters of L′_{n} occur in Ind_{P_{n−2}}^{L′_{n}}(Φ_{n−2}), thus the preceding discussion shows that all constituents of^{∗}R_{L_{n}}^{G_{n}}(Ind_{P_{n}}^{G_{n}}(Φ_{n})) occur in Ind_{P_{n−2}}^{L_{n}}(Φ_{n−2}), which is the first part of the assertion. For the second note that the constituents of ^{∗}R_{L_{n}}^{G_{n}}(χ_{λ}) are just those χ_{μ} where μ can be obtained from λ by removing a 2-hook.

In the next three Lemmas we investigate the ordinary constituents of IndG ; P̃(ϕ). The first result rules out certain unipotent characters as constituents if n is even. We will use it later on only in the case n = 4.

Lemma 12. Let n ≥ 4 be even. Let λ = (λ_{1}, λ_{2}) ⊢ n be a 2-part partition different from (n − 1, 1). Then ⟨IndG ; P̃(ϕ), χ_{λ}⟩ = 0.

Proof. Let R denote the stabilizer in G of a maximal totally isotropic subspace containing the line stabilized by P. Thus R is a maximal parabolic subgroup such that the derived subgroup of its Levi factor is isomorphic to SL_{m}(q^{2}), where n = 2m. Note that the constituents of the permutation character Ind_{R}^{G}(1) are precisely the unipotent characters χ_{λ} for 2-part partitions λ (see for example [16, 6.1.9]). We will prove the assertion by evaluating l := ⟨Ind_{P}^{G}(Φ), Ind_{R}^{G}(1)⟩. By the Mackey formula this equals

l = ∑_{g∈P \G/R}⟨Res_{P ∩^{g}R}^{P}(Φ), 1⟩,

where the sum ranges over a set of double coset representatives of P, R in G. In the Weyl group of G we have ⟨s_{1}, …, s_{m−1}⟩\B_{m}/⟨s_{2}, …, s_{m}⟩ = {1, s_{m} ⋯ s_{1}} (in the notation of the proof of Lemma 11).

Clearly the intersection P ∩ R of two standard parabolic subgroups is again a parabolic subgroup (of type A_{m−2}). Its unipotent radical contains the unipotent radical of P which (by the definition of ϕ) has no fixed points on ϕ. Hence the trivial character is not a constituent of Res_{P ∩R}^{P}(Φ) and thus the trivial double coset does not give a contribution to l.

Now consider g = s_{m} ⋯ s_{1}. Let T ≤ P be the maximally split maximal torus of G consisting of the diagonal matrices in G. Then clearly P = P̃T. Moreover T is contained in the intersection X := P ∩^{g}R. A short calculation using root subgroups shows that X has a normal unipotent subgroup Q_{1} which is an abelian subgroup of Q of order q^{n−2} intersecting Z(Q) trivially. Since Res_{Q}^{P}(Φ) vanishes outside Z(Q)

it follows that ResP̃ ; Q_{1}(ϕ) is the regular character of Q_{1}.

Since Q_{1} ≤ P̃ we may write P̃ ∩ Q_{1}T = Q_{1}T̃ for some torus T̃ ≤ T. We have Φ = IndP ; P̃(ϕ), hence

Res_{Q_{1}T}^{P} (Φ) = Res_{Q_{1}T}^{P} (IndP ; P̃(ϕ)) = Ind_{Q_{1}T̃}^{Q_{1}T}(ResP̃ ; Q_{1}T̃(ϕ))

by the Mackey formula, since P = P̃T. But this implies that

⟨Res_{X}^{P}(Φ), 1⟩ ≤ ⟨Res_{Q_{1}T}^{P} (Φ), 1⟩ = ⟨Ind_{Q_{1}T̃}^{Q_{1}T}(ResP̃ ; Q_{1}T̃(ϕ)), 1⟩ = ⟨ResP̃ ; Q_{1}T̃(ϕ), 1⟩

by Frobenius reciprocity. But the latter is certainly not larger than ⟨ResP̃ ; Q_{1}(ϕ), 1⟩,

which equals 1 since we saw above that ResP̃ ; Q_{1}(ϕ) is the regular character of Q_{1}.

Thus l ≤ 1. On the other hand, since χ_{(n−1,1)}(1) = (q^{n} + (−1)^{n}q)/(q + 1) is smaller than q^{n−2}d_{n−2}, Lemma 10 shows that χ_{(n−1,1)} is a constituent of Ind_{P}^{G}(Φ). Hence no other χ_{λ} occurs in Ind_{P}^{G}(Φ).

Lemma 13. Let n = 5 and denote by St the ordinary Steinberg character of G = SU_{5}(q), i.e., St = χ_{(1^{5})}. Then ⟨IndG ; P̃(ϕ), St⟩ = 0.

Proof. By Frobenius reciprocity it suffices to show that ⟨ϕ, ResG ; P̃(St)⟩ = 0. Since ϕ is the extension of an irreducible character of P′ = QL′ to P̃, this is the case if l := ⟨ResP̃ ; P′(ϕ), Res_{P′}^{G}(St)⟩ = 0. The values of the Steinberg character are wellknown (see for example [3, 6.5]); in particular St vanishes on all non-semisimple elements. Hence in order to evaluate l we have to determine the fusion of semisimple elements from L′ = SU_{3}(q) into G, and to find the values of ϕ on these elements.

By the result of Seitz the restriction of ϕ to L′ is known, hence its values can be computed from the known character table of SU_{3}(q) (see [25] or [14]). The fusion of semisimple elements of L′ into G is easily calculated from the explicit matrix representation given in Section 5. This allows the computation of l, which turns out to be equal to 0 in all cases (the values and the fusion depend on the congruence of q (mod 6)).

We are now ready to prove the crucial reduction on the possible characteristic 0 constituents of the induced character IndG ; P̃(ϕ).

Lemma 14. Let n ≥ 6. If ⟨IndG ; P̃(ϕ), χ_{λ}⟩ ≠ 0 then

λ ∈ { (n − 1, 1), (n − 3, 3), (n − 3, 2, 1), (n − 3, 1^{3}) } .

Let s be a semisimple element of PGU_{n}(q) with centralizer GU_{n−1}(q) (see Section 6). If ⟨IndG ; P̃(ϕ), χ_{s,λ}⟩ ≠ 0 then

λ ∈ { (n − 1), (n − 2, 1), (n − 3, 2), (n − 3, 1^{2}) }

for n ≥ 10. For 6 ≤ n ≤ 9, moreover λ ∈ {(n − 4, 3), (n − 4, 2, 1)} may occur, as well as λ = (2, 1^{3}) for n = 6.

Proof. First note that IndG ; P̃(ϕ) clearly does not contain the trivial character if n > 2. Thus by Lemma 11 it does neither contain χ_{(n−2,2)} nor χ_{(n−2,1^{2})} for n > 4. By Lemma 12 it does not contain χ_{(2^{2})} for n = 4.

From the definition of ϕ it follows that

IndG ; P̃(ϕ)(1) = q^{n−2} \frac{(q^{n} − (−1)^{n})(q^{n−1} + (−1)^{n})}{q + 1} .

Thus any unipotent constituent χ_{λ} of the induced character can have degree at most that large. If χ_{λ} is cuspidal, then either n = 6 and λ = (3, 2, 1), or n ≥ 10 and λ is a triangular partition. But then χ_{λ}(1) exceeds IndG ; P̃(ϕ)(1). We may hence assume that ^{∗}R_{L}^{G}(χ_{λ}) ≠ 0 so that Lemma 11 applies. It is straightforward to check (for example with MAPLE [4]) from the degree formula for χ_{λ}(1), that χ_{λ}(1) ≤ IndG ; P̃(ϕ)(1) implies that

λ ∈ {(6), (5, 1), (4, 2), (4, 1^{2}), (3^{2}), (3, 2, 1), (3, 1^{3}), (2^{3}), (2^{2}, 1^{2})}

for n = 6. We just argued that the third and fourth of these cannot occur in IndG ; P̃(ϕ). Furthermore, (3^{2}), (2^{3}) and (2^{2}, 1^{2}) can be obtained from (2^{2}) by adding a 2-hook, thus the corresponding characters do not occur in IndG ; P̃(ϕ) by Lemma 11. This proves the assertion for n = 6. For n = 7, the degree formula shows that only

λ ∈ {(7), (6, 1), (5, 2), (5, 1^{2}), (4, 3), (4, 2, 1), (4, 1^{3}), (3^{2}, 1), (3, 2^{2}), (3, 2, 1^{2})}

are possible. Again the third and fourth are out. The partitions (3^{2}, 1), (3, 2^{2}) and (3, 2, 1^{2}) can be obtained from either (3, 2) or (3, 1^{2}) by adding a 2-hook, but the two latter do not correspond to constituents for n = 5. Thus again by Lemma 11 we are done.

Now let n ≥ 8 and assume the result is true for n − 2. If χ_{λ} is a constituent of IndG ; P̃(ϕ)(1), Lemma 11 shows that λ can be obtained from an admissible partition of n − 2 by adding a 2-hook. It is easy to write down all such partitions: the ones which do not occur in the statement are precisely those of the form (n − 5, μ) with μ ⊢ 5, or (4^{2}). Except for

{(6, 5), (4, 3, 2), (4, 3, 1^{2}), (5, 4, 1), (3, 2, 1^{3}), (3, 2^{2}, 1)}

all of these possess another 2-hook whose removal leads to a non-admissible partition of n − 2. For the six partitions above, the corresponding character degree is larger than IndG ; P̃(ϕ)(1). Using Lemma 11 we are done by induction for the unipotent characters.

Now let’s consider χ_{s,λ}. By (1) its degree is given by (q^{n} −(−1)^{n})/(q +1)·χ_{λ}(1). If n = 10 or n = 11 then comparison with the degree of IndG ; P̃(ϕ) shows that only the λ in the statement can occur. For n ≥ 12 we may then proceed by induction as in the unipotent case above, using Lemma 11. The cases 6 ≤ n ≤ 9 can be checked from the degree formula.

Remark 15. It can be verified by explicit computations that χ_{λ} for λ = (2, 1^{3}) (respectively λ = (3, 1^{3})) is indeed a constituent of IndG ; P̃(ϕ) for G = SU_{5}(2) (respectively G = SU_{6}(2)). This shows that at least in these cases the unipotent character of largest degree among those listed in the preceding lemma does in fact occur.

On the other hand, if n is even, λ = (n − 3, 3) cannot occur by Lemma 12.

8. Lower bounds for dimensions of modules

We are now ready to prove the main result of our paper. In order to be able to state the theorem we introduce the quantity d(n, q, ℓ) (for an integer n ≥ 3, a prime power q and a prime ℓ not dividing q) as follows:

d(3, q, ℓ) := \begin{cases}\frac{1}{3}(q − 1)(q^{2} − q + 1), & if 3 | (q + 1), ℓ ≠ 3, \\ \frac{1}{6}(q − 1)(q^{2} + 3q + 2), & if 3 | (q + 1), ℓ = 3, \\ \frac{1}{3}(2q^{3} − q^{2} + 2q − 3), & if 3 |̸(q + 1), q > 3, \\ 14, & if q = 3, ℓ = 2, \end{cases}

d(4, q, ℓ) := \begin{cases}\frac{1}{2}(q^{2} + 1)(q^{2} − q + 1), & if q, ℓ odd, \\ \frac{1}{2}(q^{2} + 1)(q^{2} − q + 1) − 1, & if q odd, ℓ = 2, \\ (q^{2} + 1)(q^{2} − q + 1) − 1, & if q even, \end{cases}

d(n, q, ℓ) := \begin{cases}q^{n−2}(q − 1)(q^{n−2} − q)/(q + 1), & if n ≥ 5 odd, \\ q^{n−2}(q − 1)(q^{n−2} − 1)/(q + 1), & if n ≥ 6 even, \end{cases}

Theorem 16. Let G = SU_{n}(q), n ≥ 3, (n, q) ≠ (3, 2), (4, 2). Let k be an algebraically closed field of characteristic ℓ not dividing q, and let V be a non-trivial irreducible kG-module of dimension dim(V) < d(n, q, ℓ). Then

dim(V) ∈ \{ \frac{q^{n} − (−1)^{n}}{q + 1} , \frac{q^{n} + q(−1)^{n}}{q + 1} \} .

In particular, V is the ℓ-modular reduction of an ordinary Weil module of G.

If n is odd, there exists one such module of dimension (q^{n} − q)/(q + 1) and (q + 1)_{ℓ′} − 1 such modules of dimension (q^{n} + 1)/(q + 1), while if n is even, there exist (q +1)_{ℓ′} −ϵ(q, ℓ) inequivalent modules of dimension (q^{n} −1)/(q +1) and ϵ(q, ℓ) modules of dimension (q^{n} + q)/(q + 1). Here, ϵ(q, ℓ) = 0 if ℓ | (q + 1) and ϵ(q, ℓ) = 1 otherwise.

Proof. Let V be as in the statement and let ρ denote the Brauer character of V. First we determine the union of ℓ-blocks E(G, s) which possibly can contain ρ. If ρ is not unipotent, it can also lie in E(G, s) for a semisimple s with centralizer GU_{n−1}(q), s^{q+1} = 1. Since ρ(1) < d(n, q, ℓ) it follows from Proposition 1 and [26, Table V] that these are the only possibilities if n ≥ 5. There are further possibilities if n = 3 or 4. To find these, we used the explicit character table of SU_{3}(q) determined by Simpson and Frame [25], as well as the list of the semisimple class types of PGU_{4}(q) determined by Frank Lübeck with the help of CHEVIE.

For n = 3 and odd ℓ the result now follows from the tables of decomposition matrices computed by Geck in [11] (see also Proposition 5). For n = 3, ℓ = 2 and q ≡ 1 (mod 4), Erdmann has given the decomposition numbers of the principal block of SU_{3}(q) in [9, Lemma 11.4] (see also [8]). The decomposition matrices for the other relevant blocks are easy to determine. The result follows from these tables. For n = 3, ℓ = 2 and q ≡ 3 (mod 4), we have mimicked the argument of Geck ([11, Proof of Theorem 4.3(a)]) to bound the unknown decomposition number in the principal block. We obtain a bound similar to Geck’s and thus the result in this case as well. So from now on we may assume that n ≥ 4.

Step 1: Unipotent Constituents. Suppose first that ρ lies in a unipotent block. Then it must be a constituent of a unipotent character χ_{λ}, where λ is as in Lemma 14. Each of these χ_{λ}s has a unique extension to a unipotent character of GU_{n}(q), denoted by the same symbol. By the triangular shape of the decomposition matrix of GU_{n}(q), this implies that ρ is a constituent of the restriction to SU_{n}(q) of φ_{λ}.

By Clifford’s theorem, ρ(1) ≥ φ_{λ}(1)/(q + 1), if ρ is contained in the restriction of φ_{λ}. Of course, φ_{(n)} and φ_{(n−1,1)} restrict irreducibly to SU_{n}(q), the latter by [24]. If ℓ does not divide gcd(n, q + 1), then all unipotent Brauer characters of GU_{n}(q) restrict irreducibly to SU_{n}(q). This follows from [13] which asserts that the restrictions of the ordinary unipotent characters of GU_{n}(q) to SU_{n}(q) form a basic set for the unipotent Brauer characters.

1a. If n = 4, we use the decomposition numbers of the unipotent blocks of GU_{4}(q) determined by Wings (see [27] and also Proposition 6). Since the restriction of φ_{λ} to SU_{4}(q) is irreducible if ℓ is odd, and has at most 4 irreducible constituents otherwise, the result follows except for ℓ = 2 and ρ a constituent of the restriction of φ_{(2^{2})}. In the latter case the Harish-Chandra vertex of φ_{(2^{2})} is the Levi subgroup isomorphic to GL_{2}(q^{2}) and the Harish-Chandra source is the modular Steinberg character. Its restriction to SL_{2}(q^{2}) has at most two irreducible constituents, giving the result with the help of Corollary 3.

1b. Assume that n ≥ 5. Then, by [24], d(n, q, ℓ) ≤ q^{n−2}(q − 1)d_{n−2}, where d_{n−2} denotes the degree of the smallest non-trivial irreducible Brauer character of SU_{n−2}(q). By Lemma 10, ρ is a modular constituent of IndG ; P̃(ϕ) (in the notation of that Lemma). This is no restriction if n = 5. In this case we use the approximation to the decomposition matrix of the unipotent characters of GU_{5}(q) given in Proposition 6. As in the case for n = 4 this gives the desired result except for λ = (1^{5}). But χ_{(1^{5})} is not contained in IndG ; P̃(ϕ) by Lemma 13, so we are done in this case.

If n > 5, then ρ is a constituent of the restriction to SU_{n}(q) of φ_{λ} where λ is larger, in the lexicographic ordering of partitions, than (n − 3, 1^{3}). From the approximations to the decomposition matrices in Propositions 7 and 8 and the formulas for the degrees χ_{λ}(1) given in Section 4 we obtain lower bounds for the degrees of the Brauer characters φ_{λ} of GU_{n}(q).

Step 2: Restriction to SU_{n}(q). We next investigate the possible splittings of these φ_{λ}s on restriction to SU_{n}(q). It is easy to check, using MAPLE, that φ_{λ}(1) > d(n, q, ℓ) for all n and q, if λ is not one of (n) or (n − 1, 1). Thus the result follows if ℓ does not divide q + 1.

Suppose now that ℓ | q + 1. Then φ_{λ}(1) > d(n, q, ℓ)(q + 1), if λ ∈ {(n − 3, 1^{3}), (n − 3, 2, 1)}, or if n ≥ 6 is even and λ = (n − 3, 3). We illustrate the proof of this claim with the example of φ_{(n−3,3)} and n ≥ 6 even. First we have φ_{(n−3,3)}(1) ≥ χ_{(n−3,3)}(1) − χ_{(n−2,2)}(1) − (m − 1)χ_{(n−1,1)}(1) − m, by Proposition 7. We then find, using MAPLE once again:

[φ_{(n−3,3)}(1) − (q + 1)d(n, q, ℓ)] (q^{3} + 1)(q^{2} − 1)(q + 1) ≥
q^{3n−3} − q^{2n+3} + 2 q^{2n+1} − q^{2 n} − 2 q^{2n−1} + 3 q^{2n−2} − q^{2n−3} − q^{2n−4} +
q^{n+5} − 2 q^{n+3} + q^{n+2} + q^{n+1} − 3 q^{n} + q^{n−1} + q^{6} + q^{5} − q^{4} + q^{2} − q +
m [−q^{n+5} + q^{n+3} − q^{n+2} + q^{n} − 2 q^{6} − q^{5} + 2 q^{4} − q^{3} − q^{2} + 2 q + 1] ≥
q^{3n−3} − q^{2n+3} + q^{2n+1} − q^{2n−1} +
q^{2n+1} − q^{2 n} − q^{2n−1} − q^{2n−3} − q^{2n−4} +
3q^{2n−2} − mq^{n+5}
> 0,

for n ≥ 8 (for n > 8 use m ≤ n/2 < q^{n/2}). For n = 6 the claim can be proved directly using the first of the above estimates.

It remains to consider the three characters φ_{(n−2,2)}, φ_{(n−2,1^{2})} and φ_{(n−3,3)}, the latter only for odd n ≥ 7. By our construction of projective characters in Section 4, it follows that the projective cover of φ_{(n−2,2)} is a summand of a projective character Harish-Chandra induced from the Levi subgroup M ≅ GL_{2}(q^{2}) × T, where T is the maximally split torus of GU_{n−4}(q). By Frobenius reciprocity (for Harish-Chandra induction), the Harish-Chandra vertex of φ_{(n−2,2)} is contained in M. Moreover, if M is the Harish-Chandra vertex of φ_{(n−2,2)}, then ϑ := ^{∗}R_{M}^{G}(φ_{(n−2,2)}) is irreducible and equals the Harish-Chandra source of φ_{(n−2,2)}, a fact which can easily be checked by counting multiplicities. Since φ_{(n−2,2)} is unipotent, T is in the kernel K of ϑ. Also, T (M ∩ SU_{n}(q)) = M since n ≥ 5. Hence K(M ∩ SU_{n}(q)) = M, and so the restriction of ϑ to M ∩ SU_{n}(q) is irreducible. It follows from Corollary 3 that the restriction of φ_{(n−2,2)} to SU_{n}(q) is irreducible. On the other hand, if the Harish-Chandra vertex of φ_{(n−2,2)} is properly contained in M we are done by Proposition 4.

Now suppose that n is even and consider φ_{(n−2,1^{2})}. Here we find that its Harish-Chandra vertex is contained in GU_{4}(q)×T, where T is the maximally split torus of GL_{(n/2−2)}(q^{2}). Moreover, if the Harish-Chandra vertex of φ_{(n−2,1^{2})} is not equal to the maximally split torus of GU_{n}(q), then its Harish-Chandra source is irreducible. As above one shows that the restriction of φ_{(n−2,1^{2})} to SU_{n}(q) is irreducible.

Finally, let n be odd. The Harish-Chandra vertex of φ_{(n−2,1^{2})} is contained in M_{1} ≅ GU_{3}(q) × T_{1} and that of φ_{(n−3,3)} in M_{2} ≅ (GU_{3}(q) × GL_{2}(q^{2})) × T_{2} where T_{1} and T_{2} are appropriate tori (and T_{2} = {1} in case n = 7). We may assume that the Harish-Chandra vertices are larger than the maximally split torus of GU_{n}(q). Thus M_{1} is the Harish-Chandra vertex of φ_{(n−2,1^{2})} and its Harish-Chandra source is the modular Steinberg character of GU_{3}(q), inflated to M. Moreover, ∗R_{M_{1}}^{G}(φ_{(n−2,1^{2})}) is irreducible and restricts irreducibly to M_{1} ∩ SU_{n}(q). We are done in this case.

Similarly we can prove the desired result if the Harish-Chandra vertex of φ_{(n−3,3)} equals M_{2} (with some extra care if n = 7). It remains to consider the case that the Harish-Chandra vertex of φ_{(n−3,3)} equals M_{1}. In this case its Harish-Chandra source equals φ_{(2,1)}, which restricts irreducibly to SU_{n}(q) ∩ M_{1}. Proposition 4 implies that φ_{(n−3,3)} restricts irreducibly to SU_{n}(q).

Step 3: Non-unipotent Constituents. If ρ is not unipotent, then we have to study the decomposition matrix for E(G, s) with s as above. If n ≥ 6 we may apply Lemma 14. It shows that if χ_{s,λ} is a constituent of IndG ; P̃(ϕ), for λ ⊢ (n − 1), then the first part of λ is at least n − 4, but λ ≠ (n − 4, 1^{3}) (except possibly for n = 6, where λ = (2, 1^{3}) may also occur).

Let Ĝ = GU_{n}(q). We may and will identify Ĝ with its dual group, so that we have s ∈ Ĝ. Every column in one of the tables of Propositions 6, 7 and 8 which was obtained by Harish-Chandra induction, also gives a column in an approximation to the decomposition matrix of E(Ĝ, s). This follows from the fact that s may be chosen to lie in a maximally split torus, if n is odd, and in the Levi subgroup with simple component GU_{2}(q), if n is even. Therefore, every Levi subgroup of CĜ(s) is of the form M ∩ CĜ(s) where M is a Levi subgroup of Ĝ containing s. Now apply the commutative diagram (3) to induce the projective characters of M contained in E(M, s) to Ĝ.

Thus the first five columns of the matrix in Proposition 7 approximate the first five columns of the decomposition matrix for E(Ĝ, s), for odd n = 2m + 1 ≥ 7. Similarly, the first five columns of the matrix in Proposition 8 approximate the first five columns of the decomposition matrix for E(Ĝ, s), for even n = 2(m + 1) ≥ 8.

For the cases n = 4, 5 and 6 (the latter only for λ = (2, 1^{3})), we need approximations to the other decomposition numbers of E(Ĝ, s) as well. We obtain these by computing the restrictions to the unipotent elements of the characters in E(Ĝ, s) (for 4 ≤ n ≤ 6), and tensoring them with the generalized Gelfand Graev characters. Again this was done with the help of CHEVIE [14], which contains tables of Green functions for the relevant groups. In all cases we obtain approximations as in Proposition 5 (for n = 4) and Proposition 6 (for n = 5 and 6).

Similar, but easier calculations and considerations as in the case of unipotent characters now give the desired bounds on the degrees of the characters.

Remarks 17. (1) Under the restrictions on n and q made in the theorem, the group SU_{n}(q) is the Schur covering group of the simple group U_{n}(q), except when (n, q) ∈ {(4, 3), (6, 2)}. Thus our result can also be viewed as a result on dimensions of projective representations (in the sense of Schur) of U_{n}(q). The three smallest nontrivial degrees of absolutely irreducible projective representations in the exceptional cases may be found in the Modular Atlas [21] or in GAP [10]. We collect the result in Table 1.

(2) The bounds given in the theorem are asymptotically correct. For example, if n ≥ 5 and ℓ > 3, then φ_{(n−2,2)} remains irreducible on restriction to SU_{n}(q), and φ_{(n−2,2)}(1) has leading term q^{2n−4} (as polynomial in q).

Table 1. First three modular degrees for some small groups

G  d_{1} d_{2} d_{3}  char.
U_{4}(2) 4 5  6 ℓ ≠ 2, 3
4 5 10 ℓ = 3
U_{4}(3) 6 15 20 ℓ ≠ 3
U_{6}(2) 21 22 56 ℓ ≠ 2, 3
21 56 120 ℓ = 3

For odd q and ℓ = 2 we give, without proof, the degree of φ_{(n−2,2)}. Write n = 2m + 1 − ε with ε ∈ {0, 1} and m ≥ 2, and let ε_{m} denote the parity of m. Then

φ_{(n−2,2)}(1) = \frac{q^{2}(q^{n} − (−1)^{n})(q^{n−3} − (−1)^{n−3})}{(q^{2} − 1)(q + 1)} − ε\frac{q(q^{n−1} − (−1)^{n−1})}{q + 1} − (1 − ε_{m}).

(3) The bounds for n = 4 are best possible. For odd q and ℓ there exist ordinary irreducible characters of degree d(4, q, ℓ) (see [26, Table V]), which are irreducible modulo ℓ by Proposition 1. If q is odd and ℓ = 2, then φ_{(2^{2})} has degree (q^{2} + 1)(q^{2} − q + 1) − 2, and the restriction of φ_{(2^{2})} to SU_{4}(q) splits into the sum of two irreducibles. Finally, if q is even and ℓ divides q + 1, then φ_{(2^{2})} has degree (q^{2} + 1)(q^{2} − q + 1) − 1. These statements can be derived from the decomposition matrices given in [27].

(4) The bound for n = 3, ℓ ≠ 3 and 3 dividing q + 1 is best possible. The reason is the same as for n = 4 and q, ℓ odd.

Geck [11] has determined the decomposition numbers for odd primes ℓ of SU_{3}(q) up to one unknown parameter and conjectured that this parameter should take the smallest possible value. An analogous conjecture can be made for ℓ = 2. If these conjectures are true, then the smallest degree of an irreducible representation of SU_{3}(q) (with q > 2) larger than q^{2} − q + 1 equals (q − 1)(q^{2} − q + 1)/gcd(3, q + 1), exactly as in characteristic 0. Okuyama and Waki have announced a proof of Geck’s conjecture.

Acknowledgment: We thank Frank Lübeck for the computation of the generalized Gelfand-Graev characters as well as for the enumeration of the semisimple class types of PGU_{4}(q).

References

[1] M. Broué and J. Michel, Blocs et séries de Lusztig dans un groupe réductif fini, J. reine angew. Math. 395 (1989), 56–67.

[2] J. Brundan and A. Kleshchev, Lower bounds for degrees of irreducible Brauer characters of finite general linear groups, J. Algebra 223 (2000), 615–629.

[3] R.W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, Wiley, Chichester, 1985.

[4] B. W. Char, K. O. Geddes, G. H. Gonnet, B. L. Leong, M. B. Monagan, and S. M. Watt, Maple V, language reference manual, Springer-Verlag, 1991.

[5] F. Digne and J. Michel, Representations of finite groups of Lie type, London Math. Soc. Students Texts 21, Cambridge University Press, 1991.

[6] F. Digne and J. Michel, Groupes réductifs non connexes, Ann. Scient. Éc. Norm. Sup., IV. Ser. 27 3 (1994), 345–406.

[7] R. Dipper and J. Du, Harish-Chandra vertices, J. reine angew. Math. 437 (1993), 101–130.

[8] K. Erdmann, On 2-blocks with semidihedral defect groups, Trans. Amer. Math. Soc. 256 (1979), 267–287.

[9] K. Erdmann, Algebras and semidihedral defect groups II, Proc. London Math. Soc. 60 (1990), 123–165.

[10] The GAP Group, GAP – Groups, algorithms, and programming, Version 4, Aachen, St An-

drews, 1998, (http://www-gap.dcs.st-and.ac.uk/~gap).

[11] M. Geck, Irreducible Brauer characters of the 3-dimensional special unitary groups in nondefining characteristic, Comm. Algebra 18 (1990), 563–584.

[12] M. Geck, On the decomposition numbers of the finite unitary groups in non-defining characteristic, Math. Z. 207 (1991), 83–89.

[13] M. Geck, Basic sets of Brauer characters of finite groups of Lie type, II, J. London Math. Soc. 47 (1993), 255–268.

[14] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE—A system for computing and processing generic character tables, AAECC 7 (1996), 175–210.

[15] M. Geck, G. Hiss, and G. Malle, Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353–386.

[16] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, Oxford University Press, Oxford, 2000.

[17] P. Gérardin, Weil representations associated to finite fields, J. Algebra 46 (1977), 54–101.

[18] R. M. Guralnick and P. H. Tiep, Low-dimensional representations of special linear groups in cross characteristics, Proc. London Math. Soc. 78 (1999), 116–138.

[19] G. Hiss, Harish-Chandra series of Brauer characters in a finite group with a split BN-pair, J. London Math. Soc. 48 (1993), 219–228.

[20] R. B. Howlett and G. I. Lehrer, Representations of generic algebras and finite groups of Lie type, Trans. Amer. Math. Soc. 280 (1983), 753–779.

[21] C. Jansen, K. Lux, R. A. Parker, and R. A. Wilson, An atlas of Brauer characters, Clarendon Press, Oxford, 1995.

[22] V. Landazuri and G. M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418–443.

[23] G. Lusztig, On the representations of reductive groups with disconnected centre, in: Orbites unipotentes et représentations, I. Groupes finis et algèbres de Hecke, Astérisque 168, 1988, pp. 157–166.

[24] G. M. Seitz, Some representations of classical groups, J. London Math. Soc. 10 (1975), 115–120.

[25] W. A. Simpson and J. S. Frame, The character tables for SL(3, q), SU(3, q^{2}), P SL(3, q), P SU(3, q^{2}), Can. J. Math. 25 (1973), 486–494.

[26] P. H. Tiep and A. E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), 2093–2167.

[27] E. Wings, Die Zerlegungszahlen der Hauptblöcke der unitären Gruppen GU_{4}(q^{2}) in nichtdefinierender Charakteristik, Diplomarbeit, RWTH Aachen, Germany, 1990.

G.H.: Lehrstuhl D für Mathematik, RWTH Aachen, Templergraben 64, D-52062 Aachen, Germany

G.M.: FB Mathematik/Informatik, Universität Kassel, Heinrich-Plett-Straße 40, D-34132 Kassel, Germany

E-mail address: G.H.:  Gerhard.Hiss@Math.RWTH-Aachen.DE
E-mail address: G.M.:  Malle@Mathematik.Uni-Kassel.DE