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Publicly Available Published by De Gruyter October 9, 2020

A sharp upper bound for the size of Lusztig series

Christine Bessenrodt EMAIL logo and Alexandre Zalesski
From the journal Journal of Group Theory

Abstract

The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group G of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group G in terms of its Lie rank and defining characteristic. When G is specified as G(q) and q is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of G.

1 Introduction

Let 𝐆 be a reductive connected algebraic group. Let F be a Frobenius (or Steinberg) endomorphism of 𝐆 and G:=𝐆F={gG:F(g)=g}. Then G is called a finite reductive group.

Let 𝐆* denote the dual group of 𝐆; see [3] or [5]. Then there is a Frobenius endomorphism F* of 𝐆* which defines a finite group

G*=𝐆*F*={gG*:F*(g)=g}with|G|=|G*|.

The group G* is called the dual group of G and plays an important role in the character theory of G. In particular, the Deligne–Lusztig theory partitions the set of irreducible characters of G as a disjoint union of the so called Lusztig (geometric) series s, where s runs through a set of representatives in G* of the geometric conjugacy classes of semisimple elements of 𝐆*; see [5, Definition 13.16]. The characters from 1 (that is, for s=1) are called unipotent.

One of the questions not yet answered in the framework of the character theory of finite reductive groups is how large a Lusztig series can be. This has already attracted some attention in the literature; in particular, one needs to have a uniform upper bound for |s|. Liebeck and Shalev [10, Lemma 2.1] obtained the bound |s|<|W|2, where W is the Weyl group of 𝐆, and used this to bound the number of irreducible character degrees of G, as well as for proving some asymptotic results. This bound has been later improved to |s||W| in [14, Theorem 8.2].

In this paper, we obtain a sharp upper bound for maxs|s| in terms of the rank of 𝐆, where 𝐆 is a simple algebraic group of classical type with trivial center (and s ranges over the semisimple elements of G*). In this case, |s| equals the number of unipotent characters of the group CG*(s) (see [5, Theorem 13.23]). In fact, we compute the maximum of the number of unipotent characters of CG*(s) when 𝐆* is a simple simply connected algebraic group of classical type. More precisely, we compute the maximum of |s| for G=G(q) with q large enough (where q is the well-known field parameter; usually, q>n-9 for q even and q>n-27 for q odd, where n is the rank of 𝐆).

To illustrate the nature of the problem, assume that G=GLn(q). Then GG*. If s=1, then CG*(s)G. The number of characters in 1 is well known to equal p(n), the number of partitions of n. One could expect that |s|p(n) for every s. However, such a conjecture is false, and the question of a sharp uniform upper bound for |s| does not have any obvious answer. One can refine this by asking for which s the number of unipotent characters of CG*(s) is maximal.

In this paper, we answer this question by determining the explicit value of the maximum for every classical group (for q large) and describe s for which the maximum is attained. Note that it is not a priori clear at all whether the above question is feasible and can have any precise answer. The content of this paper is in computing the maximum of certain combinatorial functions to which the original problem is reduced. It is interesting and somehow surprising that the formulae we obtain for maxs|s| are much simpler than those available for the number of unipotent characters of G and G* (see [11, § 3]).

We expect that our results have a certain conceptual significance and will be useful for applications, in particular, for improving known upper bounds for the sum of the character degrees of G (see [9, Chapter 5] and [14]).

Theorem 1.1.

Let G be a simple algebraic group of rank n of adjoint type in defining characteristic p and G=GF, a finite reductive group. Then the size of a Lusztig series of G does not exceed c5n/4 for some constant c bounded as follows (and specified explicitly in the detailed results below):

AnCnDnBnCnDn
p evenp evenp oddp oddp odd
c<1.5c<15c<6c<95c<209c<44

We do not deal with the groups 𝐆 of exceptional Lie type in defining characteristic p, and with G=D43(q), as in these cases, the sizes of Lusztig series are bounded by a constant which can be easily computed. For the other groups, the constant c depends on the defining characteristic p of 𝐆, on the congruence of n modulo 4 and, in case Dn, from the choice of the Frobenius endomorphism, which defines the groups Dn+(q) or Dn-(q). The fact that the above bound is sharp (with specified values of c) can be seen from Theorem 1.5 below, which provides an explicit value of the maximum size of a Lusztig series for q large enough and for each type of the group G. In addition, this highlights the nature of the constant c and reveals that the precise value of c in each case depends on the residue of n modulo 4, a phenomenon which could not be expected in advance.

Our starting point is a result of Bessenrodt and Ono [1]. Let β(n) be the maximal number of the form p(μ):=jp(μj), where μ=(μ1,,μj,) is a partition of n and p(μj) is the number of partitions of μj.

Theorem 1.2 ([1]).

For n=1,,7, we have β(n)=p(n)=1,2,3,5,7,11,15, respectively. Let π(n) denote a partition μ of n such that β(n)=p(μ). Then the partition π(n) is uniquely determined for n7, whereas β(7) is attained only at the two partitions (7) and (4,3).

For all n1,2,3,7, we have the following values for π(n) and β(n):

Table 1
nmod4π(n)β(n)
0(4n/4)β(4)n/4=5n/4
1(4(n-5)/4,5)β(5)β(n-5)=75(n-5)/4
2(4(n-6)/4,6)β(6)β(n-6)=115(n-6)/4
3(4(n-11)/4,5,6)β(5)β(6)β(n-11)=775(n-11)/4

In particular, we always have β(n)5n/4.

Using Theorem 1.2, we obtain the following statements.

Theorem 1.3.

If G=GLn(q) or Un(q), then the size of a Lusztig series does not exceed β(n), and the bound β(n)=5n/4 is attained if 4n and q>n4.

Our results for the other classical groups are more complex.

Let 𝐆 be a simple classical algebraic group of adjoint type and F a Frobenius endomorphism such that G=𝐆F is one of the groups SO2n+1(q), PCSp2n(q), (PSOo)2n±(q) in the notation of [12, Table 22.1]. Let G* be the dual group of G, so G* is Sp2n(q), Spin2n+1(q) and Spin2n±(q), respectively. For small n, our results, stated in Proposition 1.4, are obtained by straightforward computer computations. For large n, a sharp upper bound for the number of unipotent characters of CG*(s) and hence for the size of Lusztig series s of G when s ranges over the semisimple elements of G* is provided by Theorem 1.5 below.

Let α(m), α+(m) and α-(m) denote the number of unipotent characters of Sp2m(q), Spin2m+(q) and Spin2m-(q), respectively.

Proposition 1.4.

The following statements hold.

  1. Suppose that q is even and n<18. Then maxs||s=α(n),α±(n) for G of type Cn(q),Dn±(q), respectively.

  2. Suppose that q is odd and n<32 and n2,4,6 if G=Dn-(q). Then

    maxs|s|=maxa{α+(a)α(n-a)},maxa{α(a)α(n-a)},maxa{α+(a)α+(n-a)},maxa{α-(a)α+(n-a)}

    for G of type Bn(q), Cn(q), Dn+(q), Dn-(q), respectively, where a ranges between 0 and n. The explicit value of the maximum in each case is given by Tables 3 and 4.

Theorem 1.5.

Let G*{Spin2n+1(q);Sp2n(q);Spin2n±(q)}. For a semisimple element sG*, let Es denote the Lusztig series of irreducible characters of G.

  1. Let G*=Sp2n(q), q even, or Spin2n+1(q), q even. For n18, we have |s|f(n), where

    f(n)={α(16)β(n-16)=89885(n-16)/4𝑖𝑓n0mod4,α(15)β(n-15)=663965(n-21)/4𝑖𝑓n1mod4,α(14)β(n-14)=40205(n-14)/4𝑖𝑓n2mod4,α(15)β(n-15)=60365(n-15)/4𝑖𝑓n3mod4.
  2. Let G*=Spin2n±(q), q even. For n18, we have |s|f±(n), where

    f+(n)={α+(16)β(n-16)=41105(n-16)/4𝑖𝑓n0mod4,α+(17)β(n-17)=60075(n-17)/4𝑖𝑓n1mod4,α+(14)β(n-14)=18365(n-14)/4𝑖𝑓n2mod4,α+(15)β(n-15)=27305(n-15)/4𝑖𝑓n3mod4,
    f-(n)={α-(16)β(n-16)=40665(n-16)/4𝑖𝑓n0mod4,α-(17)β(n-17)=60075(n-17)/4𝑖𝑓n1mod4,α-(14)β(n-14)=18065(n-14)/4𝑖𝑓n2mod4,α-(15)β(n-15)=27305(n-15)/4𝑖𝑓n3mod4.

  3. Let G*=Sp2n(q), q odd. For n32, we have |s|τ(n), where

    τ(n)={α(14)α(14)β(n-28)=161604005(n-28)/4𝑖𝑓n0mod4,α(15)α(14)β(n-29)=242647205(n-29)/4𝑖𝑓n1mod4,α(15)α(15)β(n-30)=364332965(n-30)/4𝑖𝑓n2mod4,α(16)α(15)β(n-31)=542515685(n-31)/4𝑖𝑓n3mod4.
  4. Let sG*=Spin2n+1(q), q odd. For n32, we have |s|θ(n), where

    θ(n)={α(16)α+(16)β(n-32)=369406805(n-32)/4𝑖𝑓n0mod4,α(15)α+(14)β(n-29)=110820965(n-29)/4𝑖𝑓n1mod4,α(14)α+(16)β(n-30)=165222005(n-30)/4𝑖𝑓n2mod4,α(15)α+(16)β(n-31)=248079605(n-31)/4𝑖𝑓n3mod4.
  5. Let G*=Spin2n±(q), q odd. For n32, we have |s|θ±(n), where

    θ+(n)={α+(16)α+(16)β(n-32)=168921005(n-32)/4𝑖𝑓n0mod4,α+(15)α+(14)β(n-29)=50122805(n-29)/4𝑖𝑓n1mod4,α+(16)α+(14)β(n-30)=75459605(n-30)/4𝑖𝑓n2mod4,α+(16)α+(15)β(n-31)=112203005(n-31)/4𝑖𝑓n3mod4,
    θ-(n)={α-(16)α+(16)β(n-32)=167112605(n-32)/4𝑖𝑓n0mod4,α-(15)α+(14)β(n-29)=50122805(n-29)/4𝑖𝑓n1mod4,α-(16)α+(14)β(n-30)=74651765(n-30)/4𝑖𝑓n2mod4,α-(15)α+(16)β(n-31)=112203005(n-31)/4𝑖𝑓n3mod4.

In addition, the detailed bounds given in (1)(5) are attained if q>n-9 if q is even, and q>n-27 if q is odd.

Note that Lusztig series were originally defined only for groups with connected center, but later, this notion has been extended to arbitrary connected reductive groups so that, again, the size of a Lusztig series equals the number of unipotent characters of CG*(s); see [5, Theorem 13.23]. Then the number of unipotent characters of CG*(s) does not exceed |CG*(s):CG*(s)0|ν(CG*(s)0), where CG*(s)0=(C𝐆*(s)0))F and ν(CG*(s)0) is the number of unipotent characters of CG*(s)0. The index |CG*(s):CG*(s)0| does not exceed r+1 for groups of type Ar and 4 for the other simple groups [15, Chapter II, Corollary 4.4]. So we can replace c by (r+1)c for the Ar-case and 4c for the other groups (in fact, the latter is needed only for q odd). However, these bounds may not be sharp.

Our strategy can be outlined as follows. The simplest case is where G=GLn(q) or Un(q); here we show (Section 3) that |s|β(n), and the bound is attained for q large enough. For the other classical groups, this bound is valid only if ±1 are not eigenvalues of s on the natural 𝔽qG*-module V for G* (Lemma 4.6). Suppose first that G*=Sp2n(q), q odd, and k,l are the multiplicities of the eigenvalues 1 and -1, respectively, of s on V. Then we show that

|s|maxα(k2)α(l2)β(m),wherek+l2+m=n.

This reduces the problem to computing the above maximum, and next we show that the bound is attained for some s if q is large enough. If G* is an orthogonal group, then we have a similar reduction with α(k2)α(l2) to be replaced by α±(k2)α±(l2) if dimV is even, and α(k2)α±(l2) if dimV is odd, with a certain choice of the signs. If q is even, then we argue similarly. The maximum of the products in question is computed in Section 5. The proof of Theorem 1.5 occupies Sections 6 and 7 for q even and odd, respectively.

Notation

The size of a finite set S is denoted by |S|. Also, we write |g| for the order of a group element, which does not lead to any confusion. For a group G, we denote by Z(G) the center of G, and by CG(S) the centralizer of a subset S of G in G. We use this notation also in the situation where V is a set on which G acts by permutations or a vector space on which G acts by linear transformations. So CV(S)={vV:sv=vfor allsS}. For SG, we denote by S the subgroup generated by S.

Idn is the identity (n×n)-matrix. By diag(x1,,xn), we denote the diagonal matrix with subsequent diagonal entries x1,,xn. A similar notation is used for a block-diagonal matrix.

We denote by the set of natural numbers. For n, p(n) denotes the number of partitions of n; for a partition μ=(μ1,,μt), we set p(μ)=j=1tp(μj). We then set β(n)=maxp(μ), where the maximum is taken over all partitions μ of n.

By 𝔽q, we denote the field of q elements. If K is a field, then K× denotes the multiplicative group of K and K¯ an algebraic closure of K.

All vector spaces considered in the paper are of finite dimension. By GL(V), we denote the group of all invertible linear transformations of a vector space V. If the ground field K is not algebraically closed and sGL(V) is a semisimple element, then the natural analog of eigenspaces are homogeneous components of s on V; these are the sum of all minimal non-zero Ks-submodules of V isomorphic to each other. If s has a single homogeneous component on V, we say that s is homogeneous.

For an algebraic group 𝐆, we denote by 𝐆0 the connected component of the identity of 𝐆. We use F to denote a Frobenius endomorphism of an algebraic group, and we often use it for different algebraic groups. We usually write G for 𝐆F={g𝐆:F(g)=g}. If 𝐆 is connected reductive, we call G=𝐆F a finite reductive group. For a finite reductive group G, we denote by ν(G) the number of unipotent characters of G. See Section 2.2 for more details. As mentioned in the introduction, α(m), α+(m), α-(m) stands for the number of unipotent characters of Sp2m(q), Spin2m+(q), Spin2m-(q), respectively. By 𝐆* and G*, we denote the dual groups of a reductive algebraic group 𝐆 and of a finite reductive group G, respectively.

Our notation for classical groups is standard, except for the special orthogonal groups of even characteristic; following [12], we denote by SO2n±(q) and SO2n(𝔽¯q) with q even the subgroup of index 2 in the full orthogonal group O2n±(q) and O2n(𝔽¯q), respectively. (The advantage is that certain results can be stated uniformly for q odd and even.) In addition, dealing with the groups SO2n+1(q), we assume that q is odd as SO2n+1(q)Sp2n(q) whenever q is even.

We expect a reader to be familiar with the geometry of classical groups; most necessary facts can be found in [8, Chapter 2]. In particular, for the notion of Witt defect of an orthogonal space, see [8, p. 28]. Nonetheless, we recall a few notions from this area.

An orthogonal space means a vector space V of finite dimension over a field K, say, endowed with a non-degenerate symmetric bilinear form f(v1,v2)K for v1,v2V, and if the characteristic of K equals 2, then the form is additionally assumed to be alternating (that is, f(v,v)=0 for vV) and non-defective [4, Chapter I, § 16]. The full orthogonal group is denoted by O(V). The spinor group of an orthogonal space V (we call it the full spinor group of V) is defined in terms of the Clifford algebra of V [4, Chapter II, § 7]; this yields the notion of spinor norm, which defines the subgroup Ω(V) of O(V) formed by elements of spinor norm 1. In particular, if the ground field is of characteristic 2, we have Ω(V)=SO(V) by convention. We use Spin(V) to denote the preimage of Ω(V) in the spinor group of V under the natural projection of it onto O(V); see [4, Chapter II, § 7].

2 Preliminaries

For later considerations, we will need the explicit formulae for β(n) from [1] which we now recall.

2.1 Some properties of the function β(n)

From Theorem 1.2, we deduce a number of properties of the numbers β(n).

Lemma 2.1.

Let 0<kn be integers. Then

β(k)<β(n)𝑓𝑜𝑟k<n𝑎𝑛𝑑β(k)β(n)β(k+n).

More precisely, if k1,2,3,7, then

β(k+n)β(k)β(n)={1𝑖𝑓k0𝑜𝑟n0mod4,5549𝑖𝑓k1𝑎𝑛𝑑n1mod4,1𝑖𝑓k1𝑎𝑛𝑑n2𝑜𝑟k2𝑎𝑛𝑑n1mod4,625539𝑖𝑓k1𝑎𝑛𝑑n3𝑜𝑟k3𝑎𝑛𝑑n1mod4,125121𝑖𝑓k2𝑎𝑛𝑑n2mod4,12577𝑖𝑓k2𝑎𝑛𝑑n3𝑜𝑟k3𝑎𝑛𝑑n2mod4,625539𝑖𝑓k3𝑎𝑛𝑑n3mod4.

For k=1,2,3,7 and n>7, the values of β(n+k)β(k)β(n) are as follows:

nk=1k=2k=3k=7
n0mod475111077757775
n1mod4115111025212521
n2mod475252235333533
n3mod412577252225212521

Proof.

Straightforward computations using Theorem 1.2. ∎

Lemma 2.2.

Let n>2 be an integer.

  1. If n is even, then β(n2)<β(3)β(n-3)=3β(n-3), and for n>4, we have β(n2)<β(5)β(n-5)=7β(n-5).

  2. We have 5(n-3)/4<β(n).

Proof.

(1) If n2n-5, then β(n2)β(n-5)<β(n-3) by Lemma 2.1. Otherwise, n<10, and the claim follows by inspection (Table 2).

(2) We have 5(n-3)/4<5(n-2)/4<5(n-1)/4<5n/4, and 5(n-i)/4=β(n-i) if 4(n-i) with i=0,1,2,3. If i>0, then β(n-i)<β(n) by Lemma 2.1. If i=0, then β(n)=5n/4>5(n-3)/4, whence the result. ∎

For use in later sections, we record the following lemma.

Lemma 2.3.

Let nN be even and n>6. Set β(n)=maxa𝑜𝑑𝑑β(a)β(n-a). Then we have β(n)=β(5)β(n-5). Explicitly, we have

β(n)={725(n-10)/4𝑖𝑓n2mod4,72115(n-16)/4𝑖𝑓n0mod4𝑎𝑛𝑑n16,

and β(8)=73, β(12)=715. In addition, β(6)=9, β(4)=3, β(2)=1.

Proof.

For the additional statement, see Table 2. Suppose that i,j is odd with i+j=n, ij. Since n10, we have j5. If j3mod4 and j>7, then β(j-5)=β(6)β(j-11) by Theorem 1.2. Hence Theorem 1.2 and Lemma 2.1 imply for j7, β(i)β(j)=β(i)β(5)β(j-5)β(5)β(n-5). For j=7 and i{3,7}, we have

β(3)β(7)=315<72=β(5)2andβ(7)2=152<735=β(5)β(9).

So, in any case, β(n)=β(5)β(n-5). Hence, applying Theorem 1.2, we obtain the formulae for β(n) stated above. ∎

2.2 Unipotent characters

Let 𝐆 be a connected reductive algebraic group with Frobenius endomorphism F. For a precise definition of it, we refer to [3, p. 31] or [12, Section 2.1]. (Some authors use the terms “Frobenius map” or “Steinberg endomorphism”.) If 𝐆 is simple, then an algebraic group endomorphism F:𝐆𝐆 is Frobenius if and only if the subgroup 𝐆F={g𝐆:F(g)=g} is finite [12, Theorem 21.5]. Groups 𝐆F are called finite reductive groups (see [2, p. XIII] or [3, § 4.4]). (The term “finite groups of Lie type” is also in use in the literature; see [3, p. 31].) Thus, a finite reductive group is determined by the pair (𝐆,F), a connected reductive algebraic group 𝐆 and a Frobenius endomorphism F of it.

As briefly mentioned in Section 1, for every finite reductive group G=𝐆F, the Deligne–Lusztig theory partitions the set of irreducible characters of G as a disjoint union of the Lusztig (geometric) series s, where s runs through a set of representatives of the classes of semisimple elements of G* that are conjugate in 𝐆*; see [5, Definition 13.16]. The characters in 1 (that is, for s=1) are called unipotent. Note that the geometric series can be further refined to rational series parameterized by the conjugacy classes of semisimple elements in G*; if 𝐆 has connected center (assumed in this paper), then the geometric and rational series coincide [5, p. 107].

We emphasize that the Lusztig series (and hence unipotent characters) of a finite reductive group cannot be defined in terms of 𝐆F as an abstract group. One observes that a given finite group of Lie type can be obtained as 𝐆F from different pairs 𝐆,F. A typical example is as follows. Given a pair (𝐆,F), set 𝐇 to be the direct product 𝐆××𝐆 of m copies of 𝐆, and then define a Frobenius endomorphism F of 𝐇 as a mapping sending an element (g1,,gm) with g1,,gm𝐆 to (F(gm),g1,,gm-1). Then F(g1,,gm)=(g1,,gm) implies

F(gm)=g1=g2==gm,

so 𝐇F={(g,,g):g𝐆F}𝐆F. In fact, the general case reduces to the above example; see [3, p. 380], where it is stated that one can assume 𝐇 to be simple (if so is 𝐆), that is, m=1.

Lemma 2.4 ([5, p. 112]).

Let G=GF be a finite reductive group and sG* a semisimple element. Suppose that CG*(s) is connected. Then |Es|=ν(CG*(s)), the number of unipotent characters of CG*(s).

To be rigorous, we emphasize that CG*(s)=C𝐆*(s)F is a finite reductive group.

Lemma 2.4 reduces the computation of the sizes of Lusztig series to the computation of the number of unipotent characters, and our results in fact give sharp upper bounds for ν(CG*(s)) when s ranges over semisimple elements of G*. (Note that, if C𝐆(s) is not connected, one can extend the notion of a unipotent character so that Lemma 2.4 remains valid; see [5, p. 112]. However, in full generality, the problem of computing sharp upper bounds is more complex.)

For what follows, it is essential to decide whether or not C𝐆(s) is a connected reductive group if 𝐆 is and s𝐆 is a semisimple element. There are the following criteria for connectivity.

Lemma 2.5.

The group CG(s) is connected for all semisimple elements s of G if one of the following holds.

  1. The center of 𝐆* is connected.

  2. 𝐆 is semisimple and simply connected.

  3. 𝐆=SO(𝐕), where 𝐕 is an orthogonal space over 𝔽¯q, and the multiplicity of the eigenvalue 1 or the eigenvalue -1 of s on 𝐕 is at most 1.

Proof.

See [5, Remark 13.15] for (1), and [15, Chapter E, Remark 3.9] for (2). In fact, (2) is a particular case of (1) as the dual group of a simply connected semisimple algebraic group has trivial center; see [3, § 4.4].

(3) If dimV is even, then the multiplicity of the eigenvalue 1 as well as the eigenvalue -1 of s on 𝐕 is known to be even (see Lemma 4.2 below for a proof), so (3) follows from [16, Lemma 2.2] in this case. Now suppose that dimV is odd, so, by our convention, q is odd. Let 𝐕1 be the 1-eigenspace of s on 𝐕. Then dim𝐕1=1 (Lemma 4.2), so C𝐆(s) is contained in the stabilizer of 𝐕1 in 𝐆. With respect to a suitable basis of 𝐕, the latter can be written as

{diag(detg,g):gO(𝐕1)}.

Then C𝐆(s) is contained in the group diag(±1,CO(𝐕1)(s)), where s is the restriction of s to 𝐕1. As dim𝐕1 is even, CO(𝐕1)(s) is connected by the above and hence is contained in SO(𝐕1) (or see the proof of [16, Lemma 2.1]). Then C𝐆(s)=diag(1,CO(𝐕1)(s)), whence the result. ∎

Remark.

There is an inaccuracy in the statement of [16, Lemma 2.2], where “Let 𝐆=SO(𝐕)” is to be replaced by “Let 𝐆=SO(𝐕) if q is odd and Ω(𝐕) if q is even” with no change of the proof.

Thus, if Lemma 2.5 applies, then CG(s) is a finite reductive group. For the notion of a simply connected semisimple algebraic group, see for instance [12, Definition 9.14] or [3, p. 25]; if 𝐆 is of adjoint type, then G* is simply connected. Classical algebraic groups of adjoint and simply connected type can be described in terms of their traditional definition; see [12, Table 9.2] or [3, p. 40].

Lemma 2.6.

Let H=SO(V), where dimV is even, and let sH be a semisimple element.

  1. Suppose that either 1 or -1 is not an eigenvalue of s on V. Then CH(s) is a finite reductive group. In particular, this is the case if q is even.

  2. Suppose that neither 1 nor -1 is an eigenvalue of s on V. Then CO(V)(s)H.

Proof.

(1) Let 𝐕=V𝔽¯q be an orthogonal space defined with the same Gram matrix as V. It is well known that 𝐆=SO(𝐕) is an algebraic group and

SO(V)=SO(𝐕)Ffor some Frobenius morphismF:𝐆𝐆.

By [16, Lemma 2.2 (2)], the group C𝐆(s) is connected. As CG(s)=C𝐆(s)F, the claim follows.

(2) See [16, Lemma 2.1]. ∎

Let 𝐕 be an orthogonal space over 𝔽¯q. The group SO(𝐕) is a simple algebraic group; however, SO(𝐕) is not simply connected. Slightly abusing notation, we denote the simply connected covering of it by Spin(𝐕); this is the preimage of SO(𝐕) in the full spinor group of 𝐕. So Spin(𝐕) is a simply connected simple algebraic group, and there is a surjective algebraic group homomorphism η:Spin(𝐕)SO(𝐕) (see [2, p. 228]). If q is even, then η is an isomorphism of the underlying abstract groups.

Let h:𝐆𝐇 be a surjective homomorphism of connected algebraic groups with central kernel (that is, an isogeny), defined over 𝔽q, and let F be a Frobenius endomorphism of 𝐆. If kerh is F-stable, one defines the action of F on 𝐇 by F(h(g))=h(F(g)) for g𝐆. Set H:=𝐇F. With this notation, we have the following result.

Lemma 2.7 ([5, Proposition 13.20]).

Let ν(G), ν(H) be the number of unipotent characters of G,H, respectively. Then ν(G)=ν(H).

For instance, if

G=GLn(q)andH=PGLn(q)orG=Un(q)andH=PUn(q),

then the lemma applies. Moreover, ν(SLn(q))=ν(PGLn(q)) as (see [3, p. 39])

PSLn(𝔽¯q)F=PGLn(q)forn>1.

Lemma 2.7 allows us to ignore the case where 𝐆*=Spin2n+1(𝔽¯q), q even. Indeed, in this case, there exists an isogeny h:Spin2n+1(𝔽¯q)𝐇:=Sp2n(𝔽¯q), which also yields an isogeny C𝐆*(s)C𝐇(h(s)). Therefore, by Lemma 2.7, we have ν(CG*(s))=ν(CH(h(s))), where H=𝐇F=Sp2n(q). So it suffices to compute the maximum of ν(CH(s)) over semisimple elements sH.

For SO2n(𝔽¯q), there are Frobenius endomorphisms for which SO2n(𝔽¯q)F coincides with SO2n+(q) or SO2n-(q). (If n=4, there is one more type of Frobenius endomorphisms which yields the “triality group” D43(q); this is not considered in this paper.) Here SO2n+(q) and SO2n-(q) are special orthogonal groups SO(V), where V is an orthogonal space of Witt defect 0 and 1, respectively, dimV=2n.

There exists a Frobenius endomorphism F, say, of Spin(𝐕) compatible with the natural mapping η:Spin(𝐕)SO(𝐕) in the sense that η(F(h))=F(η(h)). Then we set Spin(V)=Spin(𝐕)F. If q is odd, then η(Spin(V))=Ω(V)SO(V). Nonetheless, by Lemma 2.7, we have the following result.

Lemma 2.8.

If q is odd, then ν(Spin(V))=ν(SO(V)).

Lemma 2.9.

Let G=Spin(V) and sG a semisimple element. Let η:GSO(V) be the natural projection. Let W1, W2 be the 1- and -1-eigenspaces of η(s) on V, and W3=(W1+W2). Then η(CG(s))=SO(W1)×SO(W2)×CSO(W3)(s), where sSO(W3) is the restriction of η(s) to W3. (If W1=0 or W2=0, then the respective multiple is to be dropped.)

Proof.

Clearly, η(C𝐆(s)) stabilizes 𝐖1 and 𝐖2, and hence also 𝐖3. It follows that η(C𝐆(s))O(𝐖1)×O(𝐖2)×O(𝐖3). By Lemma 2.5 and the comments after Lemma 2.6, the group C𝐆(s) is connected, as well as η(C𝐆(s)).

Observe first that η(C𝐆(s)) has finite index in CO(𝐕)(η(s)). Indeed, let

M={g𝐆:[g,s]kerη},

which coincides with {g𝐆:[η(g),η(s)]=1}=η-1(CSO(𝐕)(η(s))). Then

η(M)=CSO(𝐕)(η(s)).

As kerηZ(𝐆), it follows that the mapping g[g,s] (gM) is a homomorphism MZ(𝐆) whose kernel is C𝐆(s). The group Z(𝐆) is finite, so C𝐆(s) has finite index in M. So η(C𝐆(s)) has finite index in η(M)=CSO(𝐕)(η(s)), and hence in CO(𝐕)(η(s)).

Choose a basis B, say, of 𝐕 such that B𝐖i is a basis of 𝐖i for i=1,2,3. Then, under this basis, the matrix t of η(s) on 𝐕 is diag(Id,-Id,s). Therefore, CO(𝐕)(t)O(𝐖1)×O(𝐖2)×CO(𝐖3)(s). Note that sSO(𝐖3) as dim𝐖2 is even (Lemma 4.2).

As ±1 are not eigenvalues of s, CO(𝐖3)(s) is connected (Lemma 2.5). In addition, SO(𝐖1)×SO(𝐖2)×CO(𝐖3)(s) is connected (as so is CO(𝐖3)(s)) and has finite index in O(𝐖1)×O(𝐖2)×CO(𝐖3)(s). So both

η(C𝐆(s))andSO(𝐖1)×SO(𝐖2)×CSO(𝐖3)(s)

are connected subgroups of finite index in O(𝐖1)×O(𝐖2)×CO(𝐖3)(s). As the connected component of the identity in an algebraic group is unique, these groups coincide, as stated. ∎

Lemma 2.9 implies the following result on unipotent characters which is essential in what follows.

Lemma 2.10.

Let G=Spin(V) or Sp(V), where V is an orthogonal or symplectic space over Fq. Let sG be a semisimple element, and let W1, W2 be the 1- and -1-eigenspaces of s on V. Let W3=(W1+W2). Then

ν(CG(s))=ν(SO(W1))ν(SO(W2))ν(CSO(W3)(s)),

where s is the restriction of s to W3.

Proof.

We omit the proof for Sp(V) as it is straightforward. Let G=Spin(V). Note that ν(CSO(W3)(s)) is meaningful as CSO(W3)(s) is a finite reductive group (Lemma 2.6). We use the notation of Lemma 2.9, assuming that 𝐕=V𝔽¯q and that the structure of an orthogonal space on 𝐕 is defined by the same Gram matrix as that of V. Then 𝐖i=Wi𝔽¯q for i=1,2,3. Let F be the Frobenius endomorphism of 𝐆 such that 𝐆F=G; we keep F for the Frobenius endomorphisms of SO(𝐕), SO(𝐖i) (i=1,2,3) inherited from that of 𝐆. By Lemma 2.9, we have η(C𝐆(s))=SO(𝐖1)×SO(𝐖2)×CSO(𝐖3)(s). By Lemma 2.7, we have ν((η(C𝐆(s)))F)=ν(C𝐆(s)F), and the left-hand side is equal to

ν(SO(𝐖1)F×SO(𝐖2)F×CSO(𝐖3)(s)F)=ν(SO(W1))ν(SO(W2))ν(CSO(W3)(s)),

as claimed. ∎

3 Proof of Theorem 1.3

Here G*GLn(q) or Un(q). To simplify notation, we deal below with G in place of G*, that is, we choose a semisimple element sG and show that the number of unipotent characters in CG(s) does not exceed β(n).

For our purpose, we quote the following well-known result; see [3, p. 465].

Lemma 3.1.

Let G=GLn(F¯q) and G=GFGLn(q) or Un(q) (depending on F). Then the number of unipotent characters of G equals p(n), the number of partitions of n.

Let G=GLn(q), V the natural 𝔽qG-module, and let sG be a semisimple element. We can write V=Vi, where Vi are the homogeneous components for s, that is, each Vi is a sum of isomorphic 𝔽qs-modules, and distinct Vi,Vj have no common irreducible constituents. Let siGL(Vi) be the restriction of s to Vi. Then CG(s)iGL(Vi), and CG(s)=iCGL(Vi)(si). Let di be the dimension of a minimal 𝔽qs-submodule of Vi. Then CGL(Vi)(si)GLdi(qmi), where mi=dimVi/di. One observes that the decomposition V=Vi is unique up to reordering the terms. Let k be the number of terms and ni=dimVi. Then s determines the string (n1,,nk) up to reordering of the n1,,nk, which is a partition of n, and we denote by π(s) the partition (n1,,nk). (We can assume n1nk, but we prefer to allow any ordering.) If sUn(q)GLn(q2), then π(s) is defined as the partition obtained for s in GLn(q2). The following lemma is well known.

Lemma 3.2.

Let G=GLn(F¯q), G=GFGLn(q), and let sG be a semisimple element. Then CG(s) is isomorphic to the direct product of groups GLdi(qmi), where idimi=n.

The following lemma is also well known, but we give a proof for the reader’s convenience and in order to make further discussions more transparent.

Lemma 3.3.

Let G=GLn(F¯q), G=GFUn(q), and let sG be a semisimple element. Then CG(s) is isomorphic to the direct product of groups GLdi(q2mi) and Uej(qlj), where li is odd and i2dimi+ejlj=n.

Proof.

Note that each of the sums i2dimi, ejlj can be absent. It is well known that there is an orthogonal decomposition V=(Vi)(Vj), where each Vj is a non-degenerate homogeneous component for s, and each Vi is the sum of two totally isotropic homogeneous components for s. Let H be the stabilizer in G of this decomposition, that is,

H={gG:gVi=Vi,gVj=Vjfor each termVi,Vj}.

Let ni=dimVi, nj=dimVj, and let Hi,Hj be the restriction of H to Vi,Vj, respectively. Then HjUnj(q) and HiGLni/2(q2). Therefore,

n=ni+njandH(iGLni/2(q2))×(jUnj(q)).

Let si,sj be the restriction of s to Vi,Vj, respectively. Then

siHi,sjHjandCG(s)=(iCHi(si))×(jCHj(sj)).

Using the isomorphism HiGLni/2(q2), we can view a homogeneous component Vi of Vi as a natural 𝔽q2GLni/2(q2)-module, and then si is a homogeneous element of GLni/2(q2), that is, Vi is a homogeneous 𝔽qsi-module. As in Lemma 3.2, CHi(si)GLdi(q2mi), where dimi=ni2. It is also known that CHj(sj)Uej(q2lj), where ejlj=nj and lj is odd. So the result follows. ∎

Lemma 3.4.

Let G=GLn(q) or Un(q), and let sG* be a semisimple element. Then

|s|=ν(CG(s))β(n).

Furthermore, suppose that equality holds. Then π(s)=π(n), where π(n) is defined in Theorem 1.2, and if G=GLn(q), then |s| divides q-1, if G=Un(q), then |s| divides q+1.

Proof.

If G=GLn(q), then, by Lemma 3.2,

CG(s)iGLdi(qmi),whereidimi=n.

Recall (Lemma 3.1) that the number of unipotent characters of GLn(q) equals p(n) and hence does not depend on q. So ν(CG*(s))=ip(di). Set n=di. Then ip(di)β(n). By Lemma 2.1, β(n)<β(n) for n<n; if equality holds above, then n=n, and hence mi=1 for every i. So the result follows from Lemma 3.1.

Let G=Un(q). Then CG(s) is a direct product of groups isomorphic to

GLmi(q2di),i=1,,k,andUlj(qfj),j=1,,k′′,

for some integers k,k′′0, and n=dimV=2i=1kmidi+j=1k′′ljfj. (Note that CG(s) may be a product of GLmi(q2di) or Ulj(qfj) only.) The number of unipotent characters of GLmi(q2di) equals p(mi) and that of Ulj(qfj) equals p(lj) (Lemma 3.1). Let n=mi, n′′=lj. Then

|s|=ν(CG(s))=p(mi)p(lj)β(n)β(n′′).

By Lemma 2.1, β(n)β(n′′)β(n+n′′)β(n). If the equality holds, then n=n+n′′, whence n=0, n=n′′ and fj=1 for all j=1,,k′′. It follows that |s| divides q+1, and π(s)=π(n) again follows from Lemma 3.1. ∎

We now show that the bound is attained for every n for q large enough.

Lemma 3.5.

Let n,iN, i{0,1,2,3} with inmod4. Assume that n>3 if i=0,1,2, and n>10 for i=3. Let G=GLn(q), respectively, Un(q). If n4(q-1)+i, respectively, n4(q+1)+i, then ν(CG(s))=β(n) for a suitable semisimple element sG.

Proof.

Let n=4k+i. Then kq-1, respectively, q+1. Therefore, there exist distinct elements a1,,akGL1(q), respectively, U1(q). If i=0, then we set s=diag(a1Id4,a2Id4,,akId4). If i=1, then we take the last scalar to be akId5; if i=2, then we take the last scalar to be akId6. If i=3, then we take the last two scalars to be ak-1Id5 and akId6. If G=Un(q), then we choose an orthogonal basis of the underlying space in order to get sUn(q). Then CG(s) is the direct product of groups GL4(q) (respectively U4(q)) if n0mod4, with obvious adjustments in the other cases. Then ν(CG*(s))=β(n). So the bound β(n) is attained. ∎

Proof of Theorem 1.3.

This follows from Lemmas 3.5 and 3.4. ∎

Lemma 3.6.

Let C be a cyclic group, |C|>2. Set l=|C|-22 if |C| is even, and l=|C|-12 if |C| is odd. Then there are l distinct elements a1,,alC such that aiaj1 for all 1i,jl.

Proof.

Let C=a. Then set ai=ai. As the elements ai (1i|C|-1) are all distinct and al+1 is of order 2 if |C| is even, it follows that {aj:1jl} satisfies the conclusion of the lemma. ∎

For application to other classical groups, we need a slightly different version of Lemma 3.5. We view GLn(q) as a matrix group over 𝔽q and Un(q) as a matrix group over 𝔽q2 whose subgroup of diagonal matrices is diag(U1(q),,U1(q)). In Lemma 3.7 below, D denotes the group of diagonal matrices in G. For dD, the set of distinct diagonal entries of d is denoted by Spec(d).

Lemma 3.7.

Let G=GLn(q) or Un(q), and let G2 be the subgroup of G of index 2 if q is odd, and G2=G if q is even. Suppose that qn+5. Then there exists a semisimple element sDG2 such that

Spec(s)Spec(s-1)=𝑎𝑛𝑑ν(CG(s))=β(n).

Proof.

Let C=GL1(q) or U1(q) if q is even, and let C be the subgroup of index 2 in these groups if q is odd. Let l be as in Lemma 3.6. Then l=q-22 if q is even, q-32 if q3mod4 and q-52 if q1mod4. By Lemma 3.6, for every kl, there are distinct elements a1,,akC such that aiaj1 for all 1i,jk.

Then we take k=n-r4, where 0r<4 and nrmod4. As qn+5 by assumption, we have k=n-r4q-54l. Let us choose these elements a1,,ak for a similar reasoning as in the proof of Lemma 3.5 to construct suitable elements sD.

Then ν(CG(s))=β(n) by Lemma 3.1. In addition, as s is a diagonal matrix with entries a1,,ak (with certain multiplicities), the condition aiaj1 for all 1i,jk implies Spec(s)Spec(s-1)=. As each diagonal entry of s lies in C, it follows that sG2. ∎

4 Other classical groups

4.1 Remarks on classical groups

We start with observations on the centralizers of semisimple elements of classical groups. Let

H{O2n+1(q),qodd,Sp2n(q),O2n±(q)},

and let V be the underlying space for H. Recall that Ω2n±(q) denotes the subgroup of O2n±(q) formed by elements of spinor norm 1, and in even characteristic Ω2n±(q)=SO2n±(q) by convention.

The following two lemmas are well known.

Lemma 4.1 ([8, Proposition 2.5.13]).

For q odd, set ε(n)=(-1)(q-1)n/2. The group Ω2n+(q), respectively, Ω2n-(q) contains -Id if and only if ε(n)=1, respectively, ε(n)=-1. In particular, Ω2n+(q) contains -Id if n is even or q is a square.

Lemma 4.2.

Let G{SO2n+1(q),q𝑜𝑑𝑑,SO2n±(q),Sp2n(q)}, and let V be the natural module for G. Let gG be a semisimple element, and let V1 and V2 be the 1- and -1-eigenspaces of g on V. (If q is even, then V2=0 by convention.) Then

  1. V1 and V2 are non-degenerate and orthogonal to each other;

  2. dimV2 and dim(V1+V2) are even;

  3. dimV1 is even unless G=SO2n+1(q), in which case dimV1 is odd.

Proof.

Let i{1,2}. (1) If Vi is degenerate, then U:=ViVi0 is totally isotropic. Let 0uU, so dimu=dimV-1 [8, Lemma 2.1.5]. As g is semisimple, u has a g-invariant complement U, say. Let vU, and let f be the form on V defining G. Then we have 0f(u,v)=f(gu,gv)=af(u,gv), where a=1 or -1. It follows that gv=av, which is a contradiction as such a v must be in Vi.

If V1,V20, then q is odd; choose 0viVi; then

f(v1,v2)=f(gv1,gv2)=-f(v1,v2),

whence f(v1,v2)=0.

(2) It suffices to prove this statement for the respective groups over 𝔽¯q; in this case, V is the sum of the eigenspaces of g, and ±1 are not eigenvalues of g on W:=(V1+V2). Let e be an eigenvalue of g on W, so e±1, and let We be the respective eigenspace. Then, for 0wWe, we have

f(w,w)=f(gw,gw)=e2f(w,w)=0

as e21. One easily observes that w=w+W, where W is a g-stable non-degenerate subspace of w. By induction, dimW is even, and hence so is dimW.

Moreover, if vw, then, as in the proof of (1), gv=e-1v+x for xw. This implies by induction that the determinant of gW, the restriction of g to W, equals 1. As detg=1 and g acts on V2 as -Id, it follows that dimV2 is even, as claimed.

(3) is obvious as dimV1=dimV-dimV2-dimW. ∎

Next we describe the structure of centralizers of semisimple elements in H. This is treated in [6, § 1] and elsewhere, but we choose to briefly recall the main facts in a form compatible with what follows.

Let hH be a semisimple element. Viewing V as h-space, we can write V=V1VkVk+1Vk+l, where V1,,Vk+l are homogeneous components of V for h. (In other words, V1,,Vk+l are h-stable; for every i{1,,k+l}, all irreducible constituents of Vi|h are isomorphic to each other and not isomorphic to those of Vj|h for every ji.) Furthermore, each Vi is either non-degenerate or totally isotropic; see for instance [13, Lemma 3.3]. By reordering the terms, we assume that V1,,Vk are totally isotropic (unless k=0), whereas Vk+1,,Vk+l are non-degenerate (unless l=0). In the former case, for every ik, there is another totally isotropic homogeneous component Vj, say, such that Vi|h and Vj|h are dual to each other and Vi+Vj is non-degenerate [13, Lemma 3.3]. It follows that k=2m is even. We can reorder V1,,Vk so that Vi,Vk-i+1 are dual as h-modules, i=1,,m. Set hi=h|Vi and ni=dimVi for i=1,,k+l. If hi=±Id, then Vi is non-degenerate (Lemma 4.2), and hence i>k in this case.

For i{1,,k}, set Hi=GL(Vi), and for i{k+1,,l}, set

Hi={gH:g|Vj=Idwheneverji}I(Vi).

(For uniformity, we use I(Vi) to denote the classical groups defined by the relevant form on Vi.) Then

CH(h)CH1(h1)××CHm(hm)×CHk+1(hk+1)××CHk+l(hk+l).

Let di be the dimension of each irreducible constituent of hi, i=1,,k+l. As Vi is homogeneous, ni is a multiple of di. Write ni=diei. If ik, then CHi(hi)=CGL(Vi)(hi).

(a) Suppose that H is symplectic. Then

CHi(hi){Spni(q)ifhi=±Id,GLei(qdi)ifik,Uei(qdi/2)ifi>kandhi±Id.

Here we write Uei(qdi/2) due to our notation for unitary groups, that is,

Uei(qdi/2)GLei(qdi).

(b) Suppose that H is orthogonal. Then

CHi(hi){O(Vi)ifhi=±Id,GLei(qdi)ifhi±Idandik,Uei(qdi/2)ifhi±Idandi>k,whereeiis oddif and only if the Witt defect ofViis 1.

In case (b), fix some Vi of Witt defect 1 (assuming the existence of it). Then Vi is a direct sum of ei irreducible non-degenerate hi-modules isomorphic to each other. Denote by D one of them, so hi acts irreducibly on D. Here dimD>1 as hi±1. Therefore, the Witt defect of D is 1 because otherwise O(D) has no irreducible element [7, Satz 3 (c)]. So the assertion on the parity of ei follows from [8, Proposition 2.5.11 (ii)]. (Note that di=12dimD can be even.)

We state the above information in a uniform way as follows.

Proposition 4.3.

Let hH be a semisimple element, and let V1,V2 be the 1- and -1-eigenspace of h on V. Then

CH(h)I(V1)×I(V2)×iGLdi(qli)×jUej(qmj),

where 12(dimV1+dimV2)+idili+ejmj=n.

Corollary 4.4.

Let G{SO2n+1(q),q𝑜𝑑𝑑,SO2n±(q),Sp2n(q)}, and let V be the natural module for G. Let sG be a semisimple element. Suppose that s does not have eigenvalues -1 on V and the multiplicity of the eigenvalue 1 is at most 1. Then CG(s)iGLdi(qli)×jUej(qmj), where idili+ejmj=n.

Proof.

Let H=I(V), so GH. Suppose that dimV is even. Then, under these assumptions, CG(s)=CH(s) by Lemma 2.6 (2), so the result follows from Proposition 4.3. If dimV is odd, then

O(V)=SO(V)×{±Id},

so CH(s)=CG(s)×{±Id}. ∎

Lemma 4.5.

Let sG=SO2n-(q) be a homogeneous semisimple element, and s±Id. Then CG(s)Ue(qd), where ed=n, e is odd, and ν(CG(s))p(n), where n is the greatest odd divisor of n. In addition, if (n,q)(n,3), there exists a (homogeneous) semisimple element sΩ2n-(q) such that CG(s)Un(qn/n).

Proof.

By the comment prior to Proposition 4.3 and Lemma 2.6 (2), we have CG(s)Ue(qd), where e is odd and n=de. By Lemma 3.1, ν(Ue(qd))=p(e), and p(e)p(n). For the additional claim, decompose the natural 𝔽qG-module V as a direct sum of n non-degenerate subspaces of dimension 2nn and of Witt defect 1. Let D be one of them. Then SO(D)SO2n/n-(q), so SO(D) contains an irreducible element t, say, of order qn/n+1 (see [7]). Then t2 is still irreducible on D unless n=n and q=3. Choose s to be an element of G stabilizing each direct summand (which is isomorphic to D) and acting on each of them as t2 does. Then s is homogeneous and CG(s)Un(qn/n) by the above. So the claim follows. ∎

4.2 Subgroups of classical groups and their unipotent characters

We assume the group 𝐆* to be simply connected, which in turn guarantees C𝐆*(s) to be connected for every semisimple element s𝐆*; see Lemma 2.5 (2). In view of Lemma 2.4, our task is to obtain a sharp upper bound for ν(CG*(s)). The information on the number of unipotent characters of G is given in [3, Section 13.8].

If G*=Spin2n+1(q), q odd, or Spin2n±(q), then the natural module V, say, for O2n+1(q) or O2n±(q) can be viewed as 𝔽qG*-module under the natural homomorphism of G* into the respective classical group. So we refer to V as the natural module for G*.

We have seen that the function β(n) plays a significant role in this paper. It is not true that |s|β(n), but the following lemma singles out an important special case where this is true.

Lemma 4.6.

Let G*{Spin2n+1(q) for q odd, Spin2n±(q), Sp2n(q)}, and let V be the natural module for G*. Let sG* be a semisimple element such that the multiplicity of eigenvalues 1 and -1 of s on V does not exceed 1.

  1. |s|=ν(CG*(s))β(n).

  2. If V=VV′′ is an orthogonal decomposition such that

    sV=V,sV′′=V′′𝑎𝑛𝑑Homs(V,V′′)=0

    (equivalently, s has no common eigenvalue on 𝐕,𝐕′′), then

    ν(CG*(s))=ν(CSO(V)(s1))ν(CSO(V′′)(s2)),

    where s1,s2 are the restriction of s to V,V′′, respectively.

Proof.

By Lemma 4.2, the multiplicity of the eigenvalue -1 is always even, as is that of the eigenvalue 1 unless G*Spin2n+1(q), where the multiplicity of the eigenvalue 1 is always odd. Therefore, the assumption implies that -1 is not an eigenvalue of s, as well as 1, provided G*Spin2n+1(q). By Lemma 2.5, C𝐆*(s) is connected, so by Lemma 2.4, |s|=ν(CG*(s)).

(1) Let 𝐇{SO2n+1(𝔽¯q),qodd,SO2n(𝔽¯q),Sp2n(𝔽¯q)}, and η:𝐆*𝐇 the natural homomorphism. Keep F to denote the Frobenius endomorphism of 𝐇 inherited from that of 𝐆*, and set H=𝐇F. Then η is surjective, and H is one of the groups SO2n+1(q), q odd, SO2n±(q), Sp2n(q) (depending on G* and F). As -1 is not an eigenvalue of η(s), by Lemma 2.5 (3), the group C𝐇(η(s)) is connected and, by Lemma 2.6, ν(CG*(s))=ν(CH(η(s))). By Corollary 4.4, we have CH(s)iGLdi(qli)×jUej(qmj), where idili+ejmj=n. Furthermore, the number of unipotent characters of each factor is equal to p(di) or p(ej) (Lemma 3.1), so the total is ip(di)jp(ej). By [1], this number is not greater than β(n), whence (1).

(2) By Lemma 2.5 (3), the group CO(𝐕i)(si)=CSO(𝐕i)(si) is connected for i=1,2, so CO(𝐕)(s)=CSO(𝐕)(s)=CSO(𝐕1)(s1)×CSO(𝐕2)(s2). In addition,

CSO(𝐕i)(si)F=CSO(Vi)(si),

so CSO(V)(s)=CSO(V1)(s1)×CSO(V2)(s2). This implies (2). ∎

The following lemma tells us that the bound in Lemma 4.6 (1) is attained if q is large enough and G*Spin2n-(q).

Lemma 4.7.

Let G*{Spin2n+1(q),q𝑜𝑑𝑑,Sp2n(q),Spin2n+(q)}, and let V be the natural FqG*-module. Suppose that nq-5. Then there exists tG* such that V is the sum of the eigenspaces of t, the multiplicity of the eigenvalues 1 and -1 of t is at most 1 and ν(CG*(t))=β(n). In addition, if q is odd and G* is orthogonal, then t can be chosen in a subgroup of index 2 of G*.

Proof.

It is well known that there exist totally singular subspaces V1,V2 of V such that V1V2=0, dimV1=dimV2=n, V1+V2 is non-degenerate, and there are dual bases in V1,V2 in the sense that if gG with gVi=Vi (i=1,2) and gi is the matrix of g on Vi, then g2=g1-1T, where g1T is the transpose of g1. Moreover, for H=GL(V1)GLn(q), there is an embedding λ:HG such that λ(h)=diag(h,h-1T) or diag(h,1,h-1T) for hH. Let W be the natural module for H. It follows that V1|HW and V2|H is dual to W.

Let sH be as in Lemma 3.7 and t=λ(s). Then the statement on the eigenvalues of t on V is obvious. Since W is the sum of the eigenspaces of s, it follows that V is the sum of the eigenspaces of t. In addition, the choice of s in Lemma 3.7 implies every eigenspace of λ(s) to lie in V1 or V2. It easily follows that CG*(t)=λ(CH(s))CH(s). Therefore, the number of unipotent characters of CG*(t) and CH(s) is the same. By Lemma 3.7, the latter is equal to β(n), whence the result.

However, to be precise, the isomorphism CH(s)CG*(t) should be accompanied with an isomorphism of algebraic groups C𝐇(s)C𝐆*(t) such that

C𝐇(s)F=CH(s)andC𝐆*(t)F=CG*(t).

(As above, we use the same letter F for the Frobenius endomorphism of different groups C𝐇(s) and C𝐆*(t)).

Let 𝐆*=Spin2n+1(𝔽¯q), Sp2n(𝔽¯q), or Spin2n(𝔽¯q) and 𝐇=GLn(𝔽¯q). In each case, we choose for F the standard Frobenius endomorphism arising from raising matrix entries of elements of the above groups to the q-power (see [5, p. 37]). (For this, we choose a basis B in V as above, such that BV1 and BV2 are dual bases, and view it as a basis of the underlying space 𝐕 of 𝐆.) Then G*=𝐆*F and H=𝐇F. The latter holds true when we consider 𝐇 as GL(𝐕1) or as a subgroup of 𝐆* stabilizing 𝐕1 and 𝐕2. Then C𝐇(s) and C𝐆*(t) are isomorphic as the eigenvalues of s on 𝐕2 are the inverses of those on 𝐕1 (see Lemma 3.7). In addition, we have C𝐇(s)F=CH(s) and C𝐆*(t)F=CG*(t).

For the additional statement for q odd, let G2 be the subgroup of index 2 in G*. Then |λ(H):(λ(H)G2)|2. By Lemma 3.7, s can be chosen in the subgroup of index 2 in H, whence the claim. ∎

Now we consider the case where G*=Spin2n-(q). We shall see that the statement of Lemma 4.7 remains true for n odd but fails otherwise. Recall that

β(n)=maxaoddβ(a)β(n-a);

by Lemma 2.3, β(n)=β(5)β(n-5) for n>6.

Lemma 4.8.

Let G*=Spin2n-(q), and let V be the natural module for G*. Let sG* be a semisimple element such that 1 and -1 are not eigenvalues of s on V.

  1. ν(CG*(s))β(n) for n odd, and ν(CG*(s))β(5)β(n-5) for n>6 even.

  2. Suppose that qn+5. Then the bounds in (1) are attained for some s.

  3. If n=6,4,2, then the maximum of ν(CG*(s)) equals 9,3,2, respectively.

Proof.

Let η:G*SO(V) be the natural homomorphism and

t=η(s)Ω(V)Ω2n-(q).

As observed in Lemma 4.6, ν(CG*(s))=ν(CSO(V)(t)).

If n7, the claims follow by inspection, whence (3). Suppose that n>7.

(2) Let V=V1V2, where V1,V2 are non-degenerate subspaces of V, the Witt defect of V1 is 1, the Witt defect of V2 is 0, and dimV1=10, dimV2=2n-10. By Lemma 4.7, there is an element s2Ω(V2)Ω2n-10+(q) such that V2 is the sum of eigenspaces of s2 (whence s2q-1=1) and ν(CSO(V2)(s2))=β(n-5).

Furthermore, there is a homogeneous element s1Ω(V1)Ω10-(q) such that |s1|>2 divides q+1 and CSO(V1)(s1)U5(q) (see Lemma 4.5 and its proof). As s2q-1=1, it follows that s1, s2 have no common eigenvalues over 𝔽¯q. Let t=diag(s1,s2), sG* be such that η(s)=t (such s exists as η(G*)=Ω(V)). Then CO(V)(t)=CO(V1)(s1)×CO(V2)(s2); it follows from Lemma 2.5 (3) that CSO(𝐕)(t)=CSO(𝐕1)(s1)×CSO(𝐕2)(s2), and also that CSO(V)(t), CSO(V1)(s1) and CSO(V2)(s2) are finite reductive groups. So

ν(CSO(V)(t))=ν(CSO(V1)(s1))ν(CSO(V2)(s2))=β(5)β(n-5).

By Lemma 2.10, ν(CG*(s))=β(5)β(n-5), so we are done if n is even and n>6. If n is odd, then n-50 or 2mod4; in both cases, β(5)β(n-5)=β(n) by Lemma 2.1, provided n-54, whence the result.

(1) If n is odd, this is already proven in Lemma 4.6. Suppose that n is even. Suppose the contrary, and let sG* be such that ν(CG*(s))>β(5)β(n-5).

Choose a decomposition

V=(i=1kVi)(j=1lVj)

described after Lemma 4.2; in particular, each term is a minimal non-degenerate s-stable subspace of V, each Vj (j=1,,l) is minimal and each Vi (i=1,,k) is the sum of two minimal s-stable subspaces of V. By Corollary 4.4,

CG(s)i=1kGLdi(qli)×j=1lUej(qmj),

where dimVi=2dili, dimVj=2ejmj, so idili+ejmj=n. Note that each Vi has Witt defect 0. By [8, Proposition 2.5.11], at least one Vj has Witt defect 1; in particular, l0.

Observe first that the case k=0 and l=1 does not hold. Indeed, otherwise, n=ejmj and CG*(s)Uej(qmj). By Lemma 4.5, ν(CG*(s))p(n), where n is the odd part of n. Then p(n)p(n2) as n is even, and p(n2)β(n2). By Lemma 2.2, we have β(n2)<7β(n-5) and β(n2)<3β(n-3). In the latter case, if n-31mod4, then 3β(n-3)=375(n-8)/4, and this is less than

β(5)β(n-5)=7775(n-16)/4

(as n-53mod4). This is a contradiction.

Choose j so that the Witt defect of Vj is 1. Set W=Vj, so W0 is the sum of all terms in the above decomposition but Vj. Then the Witt defect of W equals 0. Let sj,s be the restriction of s to Vj, W, respectively. We show that nj:=ejmj is odd. Indeed, by Lemma 4.6, ν(CSO(V)(s))=ν(CSO(W)(s))ν(CSO(W)(s)), and ν(CSO(W)(s))β(n-nj) by Lemma 4.7. If nj is even, then, by Lemma 4.5, ν(CSO(Vj)(sj))=p(nj), where nj is the odd part of nj. By Theorem 1.2, we have p(nj)β(nj2). By Lemma 2.2, β(nj2)3β(nj-3). Then

3β(nj-3)β(n-nj)3β(n-3)

by Lemma 2.1. By the above, this is less than β(5)β(n-5).

So nj must be odd, and hence ν(CG*(s))β(n)=maxaoddβ(a)β(n-a). If n>6, then β(n)=β(5)β(n-5) by Lemma 2.3, as required. ∎

Recall that α(n), α+(n), α-(n) denote the number of unipotent characters of the group Sp2n(q), Spin2n+(q), Spin2n-(q), respectively, and note as well that ν(Spin2n+1(q))=α(n).

An essential role in what follows is played by Lemmas 4.9 and 4.10 which generalize Lemma 3.5 to other classical groups.

Lemma 4.9.

Let G*{Spin2n+(q),Spin2n-(q),Sp2n(q)}, q even. Let a,c0 be integers such that a+c=n and if G*=Spin2n-(q), then c<n. If qn-a+5, then there exists a semisimple element sG* such that

ν(CG*(s))={α(a)β(c)𝑖𝑓G*=Sp2n(q),α+(a)β(c)𝑖𝑓G*=Spin2n+(q),α-(a)β(c)𝑖𝑓G*=Spin2n-(q).

Proof.

Let V be the natural module for G*. (Note that Spin(V)Ω(V).) Then V contains a non-degenerate subspace W, say, of dimension 2c and of Witt defect 0. Set H={gG*:gx=xfor everyxW}. Then HSpin2c+(q) or Sp2c(q). By Lemma 4.7 and its proof, there is an element hH such that ν(CH(h))=β(c) and h does not have eigenvalue 1 on W. Then

CG*(h)CH(h)×X,whereXSp2a(q)orSpin2a±(q).

By Lemma 4.6,

ν(CG(h))=ν(CH(h))ν(X)=β(c)x,

where x=α+(a), α-(a) or α(a) when G*Spin2n+(q), Spin2n-(q), Sp2n(q), respectively. This is recorded in the statement. ∎

Lemma 4.10.

Let G*{Spin2n+1(q),Spin2n+(q),Spin2n-(q),Sp2n(q)}, q odd, and let a,b,c0 be integers such that a+b+c=n, a1, b1, and if G*=Spin2n-(q), then b+c<n. If G*Sp2n(q), then suppose that b(q-1)2 is even. Suppose that qn-a-b+5. Then there exists a semisimple element sG* such that

ν(CG*(s))={α(a)α(b)β(c)𝑖𝑓G*=Sp2n(q),α(a)α+(b)β(c)𝑖𝑓G*=Spin2n+1(q),α+(a)α+(b)β(c)𝑖𝑓G*=Spin2n+(q),α-(a)α+(b)β(c)𝑖𝑓G*=Spin2n-(q).

Proof.

Let V be the natural module for G* and the respective classical group. Consider an orthogonal decomposition V=W1W2W3, where dimW1=2a or 2a+1, dimW2=2b and W3=(W1+W2). If V is orthogonal, choose both W2,W3 to be of Witt defect 0. The condition b+c<n makes this possible if G*=Spin2n-(q); in the other cases, this is well known to be possible.

Choose a basis

B={b0,b1,,b2n}inV,

where b0 is dropped unless G*=Spin2n+1(q). We can assume that

b2n-2c+1,,b2nW3,b2n-2c-2b+1,,b2n-2cW2

and the remaining elements of B are in W1. With respect to this basis, consider the matrix t=diag(Id,-Id2b,s), where s is in Ω2c+(q) or Sp2c(q). (Note that -Id2bΩ(W2) by Lemma 4.1.) By Lemma 4.7 and its proof, we can choose s to be such that ±1 are not eigenvalues of s and the number of unipotent characters of CSO(W3)(s) or CSp(W3)(s) equals β(c).

If G*=Sp2n(q), then CG*(t)=Sp(W1)×Sp(W2)×CSp(W3)(s), and the result follows as ν(Sp(W1))=α(a) and ν(Sp(W2))=α(b).

Suppose that G* is orthogonal. Then ν(SO(W2))=α+(b) as W2 is of Witt defect 0, whereas ν(SO(W1))=α(a), α+(a) or α-(a) depending on whether G*=Spin2n+1(q), Spin2n+(q), Spin2n-(q), respectively. So again, the result follows from Lemma 2.10. ∎

5 Some relations between α(n), α+(n), α-(n) and β(n)

For x, let [x] denote the maximum integer that does not exceed x.

The enumeration of unipotent characters in our context has a nice combinatorial description (see [11, Theorem 8.2] or [3, Section 13.8]); for computing α(n), α+(n) and α-(n) for small n (Table 2), we use Lusztig’s formulae [11, § 3] expressing these functions in terms of p(m) with mn.

Lemma 5.1.

For nN odd, α-(n)=α+(n), and for all nN,

α-(n)α+(n)α(n).

Proof.

From Lusztig’s generating function [11, (3.4.2)], α+(n)-α-(n)=0 for n odd, and α+(n)-α-(n)=2p(n2) for n even, so always α+(n)α-(n).

Let p2(n) denote the number of pairs of partitions that sum up to n; hence p2(n)=m=0np(m)p(n-m). Again from [11], we have

α+(n)=12p2(n)+34(1+(-1)n)p(n2)+j>1evenp2(n-j2),
α(n)=j0p2(n-(j2+j)).

For n13, the claim is easily checked directly (Table 2). For n14, the easy inequality 32p(n2)<p(n-6)+p(n-7)<p2(n-6) and a comparison of the summands in the sums above gives the claim. ∎

Proposition 5.2.

For n43, α(n)>β(n), and for all n>43, α(n)<β(n).

Proof.

For n43, the stated inequalities for α(n) hold by computation (Table 2); these also show that α(n)<β(n) for 44n300.

For n>43, we use very rough estimates to give an upper bound for α(n). First we have p(n)<2[n/2]+1 for all n (for example, use [1]). Hence

p2(m)=i=0mp(i)p(m-i)(m+1)2[m/2]+2for allm.

Applying this, for any n2, we have

α(n)=j0p2(n-(j2+j))(n2-1)2[n/2]+2.

One easily checks that α(n)(n2-1)2[n/2]+2<5(n-3)/4 for n244. Using Theorem 1.2, we conclude that α(n)<β(n) for all n>43. ∎

Proposition 5.3.

For 2<n38, we have β(n)<α-(n)α+(n). For all n39, we have α-(n)α+(n)<β(n).

Proof.

For 2<n43, the stated inequalities hold by Table 2; for n>43, these follow from Lemma 5.1 and Proposition 5.2. ∎

Corollary 5.4.

Let n>43. Then, for fixed n but varying a, the maximum of each function α(a)β(n-a), α+(a)β(n-a) and α-(a)β(n-a) is attained for a43.

Proof.

Suppose on the contrary that the maximum is attained at some a>43. Then α-(a)β(n-a)α+(a)β(n-a)α(a)β(n-a)<β(a)β(n-a)β(n). As n>43, by Theorem 1.2, we have

β(n)=5β(n-4)<10β(n-4)=α-(4)β(n-4)<α+(4)β(n-4)<α(4)β(n-4),

a contradiction. ∎

5.1 The products α(a)β(n-a), α+(a)β(n-a) and α-(a)β(n-a)

Lemma 5.5.

The following statements hold.

  1. If 13<a43, then

    α(a+4)<5α(a),
    α+(a+4)<5α+(a),
    α-(a+4)<5α-(a).

  2. If 0<a13, then

    α(a+4)>5α(a),
    α+(a+4)>5α+(a),
    α-(a+4)>5α-(a).

  3. If 13<a43, then

    α(a+4)β(m)<α(a)β(m+4),
    α+(a+4)β(m)<α(a)+β(m+4),
    α-(a+4)β(m)<α-(a)β(m+4)

    for every integer m0.

  4. If a13, m>3, m5,6,11, then

    α(a)β(m)<α(a+4)β(m-4),
    α+(a)β(m)<α+(a+4)β(m-4),
    α-(a)β(m)<α-(a+4)β(m-4).

    More precisely,

    α(a)β(m)<α(a+4)β(m-4)

    if m=11, a<13 or m=6, a<12 or m=5, a<8.

  5. If a<n, then the maximum of

    α(a)β(n-a),α+(a)β(n-a)𝑎𝑛𝑑α-(a)β(n-a)

    is attained for a17. If n>24, then, additionally, a>13.

Proof.

(1) and (1a) follow directly by Table 2.

(2) By Theorem 1.2 and Table 1, we have 5β(m)β(m+4), so the claim follows from (1). (Note that 5β(m)=β(m+4) for m1,2,7.)

(3) If m>3, m5,6,11, then 5β(m-4)=β(m). Therefore, by (1a),

α(a+4)β(m-4)>5α(a)β(m-4)=α(a)β(m),

similarly for α+(a),α-(a) in place of α(a).

Let m=11. Then β(11)=77, β(7)=15, so β(11)=7715β(7). So the result follows if α(a+4)>7715α(a). This is true if a<13.

Let m=6. Then β(6)=11, β(2)=2, so β(6)=112β(2). So the result follows if α(a+4)>112α(a). This is true for a<12.

Let m=5. Then β(5)=7, β(1)=1, so β(5)=7β(1). So the result follows if α(a+4)>7α(a). This is true for a<8.

(4) By Corollary 5.4, we may assume that a43. Suppose that a>17. Then, by (2), α(a)β(n-a)<α(a-4)β(n-a+4), a contradiction. If a13, then n>24 implies n-a>11, so α(a)β(n-a)<α(a+4)β(n-a-4) by (3), similarly for α+(a), α-(a) in place of α(a). ∎

Proposition 5.6.

The following statements hold.

  1. For n<18, the maximum of α(a)β(n-a), α+(a)β(n-a), α-(a)β(n-a) is attained for a=n.

  2. Let n18. Then the maximum of α(a)β(n-a) is attained for a=16, 15, 14, 15 when n0,1,2,3mod4.

  3. The maximum of α+(a)β(n-a) and of α-(a)β(n-a) is attained for a=16, 17, 14, 15 when n0,1,2,3mod4, respectively (in particular, namod4).

Proof.

By computer calculation, the claim is easily checked up to n=29. Let n>29. By Lemma 5.5 (4), the maximum of each of these functions is attained for a with 13<a17. Then n-a>7. Write n-a=r+4k, where 7<r11 and k0 is an integer. By Theorem 1.2, β(n-a)=5kβ(r). So

α(a)β(n-a)=5kα(a)β(r),wherea+r<29.

By the above, the maximum of α(a)β(r) is attained for a=16,15,14,15 if a+r is congruent to 0,1,2,3mod4, respectively. Say, if 4(a+r), then a=16, and

α(a)β(n)=5kα(a)β(r)5kα(16)β(r)=α(16)β(n),

whence the result. The other cases are similar, as well as the cases with α+(a), α-(a) in place of α(a). ∎

5.2 The products α(a)α(b)β(n-a-b) and α(a)α+(b)β(n-a-b)

Lemma 5.7.

Let n,a,b0 be integers such that a+bn. For n fixed, the maximum of

α(a)α(b)β(n-a-b),α(a)α+(b)β(n-a-b),
α+(a)α+(b)β(n-a-b),α+(a)α-(b)β(n-a-b)

is attained for a17, b17.

Furthermore,

  1. if n-a-b>11, then a>13, b>13;

  2. if n>45, then a>13, b>13.

Proof.

The first statement follows from Corollary 5.4 and Lemma 5.5 (4). Furthermore, n-a>11 and n-b>11. Suppose that a13. Then, by Lemma 5.5 (2), α(a)α(b)β(n-a-b)<α(a+4)α(b)β(n-a-b-4), a contradiction, similarly for the other three functions, as well as for b13, whence (1). In addition, if n>45, then n-a-b>11 as a+b34, whence (2). ∎

Proposition 5.8.

Let n,a,b0 be integers, na+b.

In the table below, we record for each of the functions α(a)α(b)β(n-a-b), α(a)α+(b)β(n-a-b), α(a)+α+(b)β(n-a-b) and α(a)α-(b)β(n-a-b) the pairs (a,b) where the functions attain their maximum for n28 or n29; for the first and third function, we list the pairs with ab. (Here we write 4 for the congruence modulo 4.)

functionboundn40n41n42n43
α(a)α(b)β(n-a-b)n28(14,14)(15,14)(15,15)(16,15)
α(a)α+(b)β(n-a-b)n29(16,16)(15,14)(14,16)(15,16)
α+(a)α+(b)β(n-a-b)n29(16,16)(15,14)(16,14)(16,15)
α+(a)α-(b)β(n-a-b)n29(16,16)(14,15)(14,16)(16,15)

Proof.

The assertion was checked to hold for n50 by computer, so we may assume n>50. Lemma 5.7 shows that the values a,b at which all four products in question attain their maximum satisfy 13<a,b<18. Write n=4k+r, with a+b+7ra+b+10, and some integer k0. Then r44. Let γn(a,b) stand for any of the functions above. As r>7, by Theorem 1.2, we have β(n-a-b)=β(r-a-b)β(4)k, and hence γn(a,b)=β(4)kγr(a,b). Since 35r44, the claim holds for r, and the result follows. ∎

Remarks.

(1) For all n<32, the maximum of α(a)α(b)β(n-a-b) is attained for pairs (a,b) such that a+b=n.

(2) For n33, the maxima of the functions γn(a,b) defined in the proof of Proposition 5.8 have been calculated by computer and are shown in Tables 3 and 4 at the end of the paper.

6 Proof of the main results for q even

In this section, q is even and G*{Sp2n(q),Spin2n±(q)Ω2n±(q)}. For q large enough, we determine the maximum of ν(CG*(s)) when s runs over the semisimple elements of G*.

Let V be a vector space of dimension 2n over 𝔽q viewed as the natural module for G*, so V is endowed with a suitable form defining G*. Denote by V1 the 1-eigenspace of s on V. By Lemma 4.2, V1 is non-degenerate and dimV1=2a is even. Set W=V1, so V=V1W. Let s denote the restriction of s to W. We keep this notation until the end of this section.

Lemma 6.1.

Proposition 1.4 is true for q even.

Proof.

Suppose that G*Spin2n±(q) (the proof for G*=Sp2n(q) is similar and hence omitted). By Lemma 2.10,

ν(CG*(s))=ν(SO(V1))ν(CSO(W)(s))=α±(a)ν(CSO(W)(s)).

By Lemma 4.6 (1), ν(CSO(W)(s))β(n-a). If n<18, then the maximum of α±(a)β(n-a) is attained for a=n (Proposition 5.6 (1)), which is realized for s=1. ∎

6.1 Symplectic groups in even characteristic

Let G*=Sp2n(q). Then CG*(s)Sp2a(q)×