Enumeration of conjugacy classes in affine groups

We study the conjugacy classes of the classical affine groups. We derive generating functions for the number of classes analogous to formulas of Wall and the authors for the classical groups. We use these to get good upper bounds for the number of classes. These naturally come up as difficult cases in the study of the non-coprime k(GV) problem of Brauer.


Introduction
Let G be the group of affine transformations of a vector space V over a finite field.In this paper we derive generating functions for the number of conjugacy classes in this group and in the analogs for the other classical groups.For finite classical groups (not their affine versions), such generating functions were mostly obtained by Wall [1963] (see also [Fulman and Guralnick 2012] for orthogonal and symplectic groups in even characteristic).Besides the natural motivation for considering this, this is one of the most difficult cases in the noncoprime k(GV ) problem introduced by Brauer to obtain results about characters.This asks for bounds on the number of conjugacy classes k(H ), where H is a group with a normal abelian subgroup V .One of the major results in this area, based on work of many authors over a long period, is that k(H ) ≤ |V | if V is its own centralizer in H and gcd(|H/V |, |V |) = 1.In fact there is an entire book devoted to this topic [Schmid 2007].It turns out if we weaken this assumption, the result is no longer true but it still is close.One critical case is when L = H/V acts irreducibly on V (see [Guralnick and Tiep 2005] for reductions and for connections with representation theory).See [Guralnick and Maróti 2013;Guralnick and Tiep 2005;Keller 2006;Robinson 2004] for background and other results.One would like to prove that k(H ) < c|V | for some absolute constant c (under suitable hypotheses).Another motivation for studying this is the relationship with the conjugacy classes of the largest maximal parabolic subgroup of the classical groups.See [Nakada and Shinoda 1990] for the case of GL.
In [Guralnick and Tiep 2005], the focus was on the important case when L is close to simple and the same bound was proved in almost all cases studied.One of the main cases left open was the case that V is the natural module for a classical group L. It turns out that again aside from the case of AGL(n, q), the bound generally holds.We show that q n ≤ k(AGL(n, q)) < (q n+1 − 1)/(q − 1) < 2q n and obtain explicit and useful bounds in the analogs for other classical groups.
Variations on this theme and some other small families that were not considered in [Guralnick and Tiep 2005] will be studied in a sequel.
The paper is organized as follows.Section 2 gives some preliminaries which are fundamental to our two approaches for calculating exact generating functions for k(AGL), k(AGU) and k(ASp) and k(AO).The first approach writes k(AG) as a weighted sum over conjugacy classes of G.We work this out for all cases except for the famously difficult cases of characteristic two symplectic and orthogonal groups.Our second approach enumerates irreducible representations instead of conjugacy classes.This allows us to calculate k(AG) recursively, and has the additional benefit of working in both odd and even characteristic.
We dedicate this paper to Pham Huu Tiep, our friend and colleague, on the occasion of his 60th birthday.We note that he has done substantial work on the noncoprime k(GV ) problem; see [Guralnick and Tiep 2005].

Preliminaries
Let G be a finite group and let k be a finite field with A a finite dimensional kG-module.Then we consider the group H = AG, the semidirect product of the normal subgroup A and G.We say that H is the corresponding affine group.We will usually take A to be irreducible (and by replacing k by End G (A), we can assume that A is absolutely irreducible).
Our first approach, which we call the orbit approach, expresses k(AG) as a weighted sum over conjugacy classes of G. To describe this, let [g, A] denote (I − g)A, where I is the identity map.The number of orbits of the centralizer C G (g) on A/[g, A] depends only on the conjugacy class C of g, and we denote it by o(C).If g and x are elements of a group G, then we let x g = g −1 xg.
Lemma 2.1.Let G and A be as above.Then where the sum is over all conjugacy classes C of G.
Proof.Let g ∈ C with C a conjugacy class of G.We need to show that the number of conjugacy classes of elements h ∈ AG such that h is conjugate to some element of g A is the number of orbits of C G (g) on A/[g, A].
Suppose that h = ga.Suppose that gc is conjugate to ga.Note that Thus if a, c ∈ A, ga and gc are conjugate in H if and only if a[g, A] and c[g, A] are in the same C G (g) orbit on A/[g, A], whence the result. □ In this paper we find (for all cases except even characteristic symplectic and orthogonal groups) exact formulas for o(C), which may be of independent interest.We then use these formulas, together with generating functions for k(G), to find exact generating functions for k(AG).
Our second approach, which we call the character approach, counts irreducible representations instead of conjugacy classes.This leads to recursive expressions for k(AG).Together with known generating functions for k(G), this enables us to obtain exact generating functions for k(AG).One nice feature of the character approach is that it works in both odd and even characteristic.
Crucial to the character approach is the next lemma, which is a well known elementary exercise.
Lemma 2.2.Let G be a finite group and V a finite G-module.Let J = V G be the semidirect product.Let be a set of G-orbit representatives on the set of irreducible characters of V .Then where G δ is the stabilizer of the character δ in G.
Proof.Let W be an irreducible ‫ރ‬J -module.Let δ be a character of V that occurs in W and set W δ to be the δ eigenspace of V .Note that δ is unique up to G-conjugacy and that the stabilizer of W δ in J is precisely J δ = V G δ .Thus, G δ acts irreducibly on W δ .Conversely given any irreducible G δ -module U , we can extend it to a J δ module by having V act via δ.Then inducing U from J δ to J gives an irreducible J -module.Thus, we see that k(J ) = δ∈ k(G δ ) as required.□ The following lemma is Euler's pentagonal number theorem (see for instance page 11 of [Andrews 1976]).
A few times in this paper quantities which can be easily re-expressed in terms of the infinite product ∞ i=1 (1 − 1/q i ) will arise, and Lemma 2.3 gives arbitrarily accurate upper and lower bounds on these products.Hence we will state bounds like without explicitly mentioning Euler's pentagonal number theorem on each occasion.We also use the following well-known lemma (see for instance [Odlyzko 1995]).
Lemma 2.4.Suppose that f (u) is analytic for |u| < R. Let M(r ) denote the maximum of | f | restricted to the circle |u| = r .Then for any 0 < r < R, the coefficient of u n in f (u) has absolute value at most M(r )/r n .
As a final bit of notation, we let |λ| denote the size of a partition λ.

AGL and related groups
Section 3A uses the orbit approach to calculate the generating function for k(AGL(n, q)).Section 3B uses the character approach to calculate the generating function for k(AGL(n, q)) and related groups.Section 3C uses these generating functions to obtain bounds on k(AGL(n, q)) and related groups.
3A. Orbit approach to k(AGL).We use Lemma 2.1 to determine a generating function for the numbers k(AGL(n, q)).
The following lemma calculates o(C) for a conjugacy class C of GL(n, q).This formula involves the number of distinct part sizes of a partition λ, which we denote by d(λ).For example if λ has 5 parts of size 4, 3 parts of size 2, and 4 parts of size 1, then d(λ) = 3.If λ is the empty partition, then d(λ) = 0. Lemma 3.1.Let C be a conjugacy class of GL(n, q), and let λ z−1 (C) be the partition corresponding to the eigenvalue 1 in the rational canonical form of an element of C. Then Proof.Let V be the natural module for GL(n, q).Let g ∈ C and let C(g) denote the centralizer of g in GL(V ).
] and the centralizer of g preserves this decomposition.Thus, we may assume that V = V 2 , i.e., we may assume that g is unipotent.
Now write where g | V i has all Jordan blocks of size i.We only consider the nonzero V i .So ] is the number of Jordan blocks of size i.It is well known that the centralizer of g induces the full GL(d i , q) and in particular any two nonzero elements of V i /[g, V i ] are in the same C(g) orbit. Consider . Thus, two elements in V which are in the same C(g)-orbit module [g, V ] must have the same highest nonzero (modulo [g, V ]) term.Conversely, we need to show that any two such vectors are in the same orbit and indeed are in the orbit of v j with v j ∈ V j \ [g, V j ].By induction, we may assume that j = m.Note that there exists h ∈ C(g) so that h is trivial on V / e<m V e and hv m − v m is an arbitrary element in e<m V e /[g, V e ].Thus, we see that v and v m are in the same orbit.Since C(g) induces GL(d m , q) on V m /[g, V m ] we see that orbit representatives for C(g) on V [g, V ] are 0 and one vector w i ∈ V i for each nonzero V i .The result follows.□ The following interesting identity will be helpful.
Differentiate this equation with respect to q and then set q = 1.The left hand side becomes By the product rule, the right hand side becomes (1) In what follows, for d ≥ 1, we let N (q; d) denote the number of monic irreducible polynomials φ(z) of degree d over F q for which φ(0) ̸ = 0, that is monic irreducible polynomials other than z.
The following well known identity (see for example Theorem 3.25 of [Lidl and Niederreiter 1994]) will be useful.
Theorem 3.4 derives a generating function for the number of conjugacy classes in AGL(n, q).Theorem 3.4.
where the sum is over all conjugacy classes C of GL(n, q).
Since conjugacy classes of GL(n, q) correspond to rational canonical forms, it follows from the previous equation and Lemma 3.1 that .
By Lemma 3.2 this is equal to Applying Lemma 3.3, this simplifies to Character approach to k(AGL) and related groups.We apply Lemma 2.2.Note that if δ is the trivial character, then G δ = G.We recall the case of G = GL(n, q) with V the natural module.The group J is usually denoted as AGL(n, q) the affine general linear group.Note that in this case | | = 2.Note that the stabilizer of a nontrivial linear character is isomorphic to AGL(n − 1, q) and so: As a corollary, we get another proof of Theorem 3.4.
Corollary 3.7.k(AGL(1, q)) = q and for n ≥ 2, Proof.By Theorem 3.4, the fact that q n ≤ k(AGL(n, q)) is equivalent to the statement that where the last step follows since (1 − u i )/(1 − qu i ) ≫ 1.In fact this argument shows that the strict inequality q n < k(AGL(n, q)) holds for n ≥ 2, since the coefficient of u i in (1 For a second proof that q n ≤ k(AGL(n, q)) with strict inequality if n ≥ 2, note that k(GL(n, q)) is at least q n − q n−1 and indeed is strictly greater for n > 1, since there are q n − q n−1 semisimple classes (i.e., different characteristic polynomials) and for n > 1, there are unipotent classes as well.Now use the fact (Lemma 3.5) that For the upper bound, we know from [Maslen and Rockmore 1997] that k(GL(m, q)) < q m for all m.So again by Lemma 3.5, Finally, we give a result for AH where H is between GL and SL.

AGU and related groups
Section 4A uses the orbit approach to calculate the generating function for k(AGU(n, q)).Section 4B uses the character approach to calculate the generating function for k(AGU(n, q)).Section 4C uses this generating function to obtain bounds on the number of conjugacy classes of AGU(n, q) and related groups.
4A. Orbit approach to k(AGU).This section uses the orbit approach to calculate the generating function for k(AGU(n, q)).
The following theorem calculates o(C) for a conjugacy class C of GU(n, q).This only involves λ z−1 (C), the partition corresponding to the eigenvalue 1 in the rational canonical form of the conjugacy class C. As in the GL case, let d(λ) be the number of distinct parts of the partition λ.In what follows we also let b(λ) denote the number of part sizes of λ which have multiplicity exactly 1.
Theorem 4.1.Let C be a conjugacy class of GU(n, q).Then Proof.It suffices to assume that C consists of unipotent elements and so corresponds to a partition λ.The proof is similar to the case of GL.Now write has all Jordan blocks of size i.We only consider the nonzero V i .So ] is the number of Jordan blocks of size i.It is well known that the centralizer of g induces the full GU(d i , q) and so there are q orbits of the form gv with 0 ̸ = v ∈ V i for d i > 1 and q − 1 orbits if d i = 1 (there are no nontrivial vectors of norm 0 if . Thus, two elements in V which are in the same C(g)-orbit module [g, V ] must have the same highest nonzero (modulo [g, V ]) term.Conversely, we need to show that any two such vectors are in the same orbit and indeed are in the orbit of v j with v j ∈ V j \ [g, V j ].By induction, we may assume that j = m.Note that there exists h ∈ C(g) so that h is trivial on V / e<m V e and hv m − v m is an arbitrary element in e<m V e /[g, V e ].Thus, we see that the v and v m are in the same orbit.The number of orbits for the nontrivial v m is q or q − 1 as above.The result follows. □ The following combinatorial lemma will also be helpful.

This is equal to
(2) The generating function Proof.The first part is just the well known generating function for the partition function.The second part is in the proof of Lemma 3.2.
For the third assertion, note that Differentiating with respect to x and setting x = 1 gives that Theorem 4.3 gives an exact generating function for k(AGU(n, q)).
Proof.By Lemma 2.1 and Theorem 4.1, k(AGU(n, q)) is equal to T 1 + T 2 − T 3 , where T 1 is k(GU(n, q)), and T 2 , T 3 are the following sums over conjugacy classes C of GU(n, q): To compute the generating function of T 2 , we take Wall's generating function for T 1 , divide it by the generating function for unipotent conjugacy classes in part (1) of Lemma 4.2, and multiply it by the weighted sum over unipotent classes in part (2) of Lemma 4.2.We conclude that T 2 is the coefficient of To compute the generating function of T 3 , we take Wall's generating function for T 1 , divide it by the generating function for unipotent conjugacy classes in part (1) of Lemma 4.2, and multiply it by the weighted sum over unipotent classes in part (3) of Lemma 4.2.We conclude that T 3 is the coefficient of Putting the pieces together, we conclude that k(AGU(n, q)) is the coefficient of ). Proof.We use the convention that GU(0, q) and AGU(0, q) are trivial groups and that GU(−1, q) and AGU(−1, q) are the empty set.We can identify the natural module and the character group of the module because the module is self dual viewed over the field of q-elements.
If n = 2, we note that GU(2, q) has precisely q nontrivial orbits on the natural module.The stabilizer of a nondegenerate vector is GU(1, q) and the stabilizer of a totally singular vector is elementary abelian of order q and again we see the result holds.Now suppose that n ≥ 3. Thus, we see that there are q − 1 orbits with stabilizer isomorphic to GU(n − 1, q) (corresponding to vectors with a given nonzero norm) and the stabilizer H of a singular vector.Note that H has a center Z of order q and H/Z ∼ = AGU(n − 2, q).Also note that any irreducible character of U = O p (H ) that is nontrivial on Z has dimension q n−2 and corresponds to one of the q − 1 nontrivial 1-dimensional characters on Z .Moreover each of these representations extends to a representation of H (this can be seen by considering the normalizer of U in the full linear group).Fix a nontrivial linear character of Z and an irreducible module W of H that affords this linear representation.It follows by Clifford theory [Curtis and Reiner 1962, 51.7] that any irreducible representation of H nontrivial on Z is of the form W ⊗ W ′ where W ′ is an irreducible H/U -module.Since there are q − 1 nontrivial central characters of U and there are k(GU(n − 2, q)) choices for W ′ , the result follows.□ We now give a second proof of Theorem 4.3.
Proof.Let k n = k(GU(n, q)) and let a n = k(AGU(n, q)).Then Lemma 4.4 gives (3) Let Multiplying (3) by u n and summing over n ≥ 1 gives that Solving for A(u), one obtains that From Wall [1963], and the theorem follows.□ 4C.Bounds for AGU and related groups.As a corollary, we obtain the following result.

Now all coefficients of powers of u in
1−qu i is the generating function for the number of conjugacy classes of GL(n, q).By [Maslen and Rockmore 1997], k(GL(n, q)) is at most q n .Hence the coefficient of u n−m in it is at most q n−m .It follows that k(AGU(n, q)) is at most Since the coefficients of u m in are nonnegative, it follows that k(AGU(n, q)) is at most which (set u = 1/q) is equal to is visibly maximized among prime powers q when q = 2, when it is at most 20 (we used the remark after Lemma 2.3 to bound the infinite product).□ Corollary 4.6.k(AGU(n, q)) ≤ q 2n .Proof.By the preceding result, this holds if 20 ≤ q n .So we only need to check the cases n = 1, or n = 2, q = 2, 3, 4 or n = 3, q = 2 or n = 4, q = 2. From the generating function (Theorem 4.3), k(AGU(1, q)) = 2q, and the other finite number of cases are computed easily from the generating function and seen to be at most q 2n .□ We can also use the previous results to get bounds for the groups between ASU(n, q) and AGU(n, q).Since SL(2, q) ∼ = SU(2, q), we assume that n ≥ 3.With more effort one can get much better bounds as we did in the case of SL(n, q).We just obtain the bound required for the k(GV ) problem.
. This is at most q 2n unless q = 2 with n ≤ 5 or q = 3 or 4 and n = 3.These cases all follow using the exact values of k(G) (obtained from our generating function) in the bound k(H ) ≤ k(G)(q + 1), except for the cases q = 2, n = 3, 4. One computes (either using a recursion similar to Lemma 4.4 and exact values of k(SU) in [Macdonald 1981], or by Magma) that k(ASU(3, 2)) = 24 and k(ASU(4, 2)) = 49, completing the proof.□

ASp
Section 5A uses the orbit approach to calculate the generating function for k(ASp(2n, q)), assuming that the characteristic is odd.Section 5B uses the character approach to calculate the generating function for k(ASp(2n, q)) in both odd and even characteristic.Section 5C uses these generating functions to obtain bounds on k(ASp(2n, q)).
5A. Orbit approach to k(ASp), odd characteristic.This section treats the affine symplectic groups.We only work in odd characteristic.In this case the conjugacy class of a unipotent element is determined by its Jordan form (over the algebraic closure) and it is much more complicated to deal with the characteristic 2 case.Since our character approach works in characteristic 2, we will not pursue the direct approach in that case.So for this section, let q be odd.
The following theorem calculates o(C) for a conjugacy class C of Sp(2n, q).This only involves the unipotent part of the class C. Recall that the conjugacy class of a unipotent element is determined (over the algebraic closure) by a partition of 2n with a i parts of i.Moreover, a i is even if i is odd.Over a finite field, we attach a sign ϵ i for each even i with a i ̸ = 0 and this gives a description of all the unipotent conjugacy classes (see [Liebeck and Seitz 2012] for details).We let λ ± z−1 (C) denote this signed partition for the unipotent part of the class C.
Theorem 5.1.Suppose that the characteristic is odd.Let C be a conjugacy class of Sp(2n, q).Let a i be the number of parts of λ ± z−1 (C) of size i.Then o(C) is equal to where if a i = 2 and the sign is +, (q − 1) if a i = 2 and the sign is −, (q − 1)/2 if a i = 1 (independently of the sign). (4) Proof.The proof is similar to the case of GL and GU and reduces to the case of unipotent elements.So assume that C is a unipotent class.Let g ∈ C. Write V as an orthogonal direct sum of spaces V i where g has a i Jordan blocks of size i on V i .As in the previous cases, one can show that gv is either conjugate to g or for some i, g is conjugate to gv i where By [Liebeck and Seitz 2012], we see that there is a subgroup of C(g) acting as Sp(a i , q) for i odd or O ϵ i (a i , q) if i is even acting naturally on V i /[g, V i ].Thus, the number of classes of the form gv i with v I ∈ V i \ [g, V i ] is 1 if i is odd and f i as given above if i is even.□ The following combinatorial lemma will also be helpful.
Lemma 5.2.Suppose that the characteristic is odd.
(1) The generating function for the number of unipotent classes of the groups Sp(2n, q) is

This is equal to
(2) The generating function (3) Let f j be as in Theorem 5.1.The generating function Proof.For the first part, the unipotent conjugacy classes of Sp(2n, q) correspond to signed partitions λ ± of size 2n.Clearly the generating function for such partitions is equal to For the second part, first note that arguing as in the first part, one has that For the third part, This is equal to

Now clearly
and the third part of the lemma follows.□ Theorem 5.3.In odd characteristic, k(ASp(2n, q)) is equal to the coefficient of u n in i (1 Proof.By Lemma 2.1 and Theorem 5.1, k(ASp(2n, q)) is equal to T 1 + T 2 + T 3 , where T 1 is k(Sp(2n, q)), and T 2 , T 3 are the following sums over conjugacy classes C of Sp(2n, q): To compute the generating function of T 2 , we take Wall's generating function for T 1 , divide it by the generating function for unipotent conjugacy classes in part (1) of Lemma 5.2, and multiply it by the generating function for the weighted sum over unipotent classes in part (2) of Lemma 5.2.We conclude To compute the generating function of T 3 , we take Wall's generating function for T 1 , divide it by the generating function for unipotent conjugacy classes in part (1) of Lemma 5.2, and multiply it by the generating function for the weighted sum over unipotent classes in part (3) of Lemma 5.2.We conclude that T 3 is the coefficient of u n in the proof of the theorem is complete.□ 5B.Character approach to k(ASp(2n, q)), any characteristic.We apply Lemma 2.2, as in the other cases.
To begin we treat the case of odd characteristic.
Proof.We take ASp(0, q) and Sp(0, q) to be the trivial group.If n = 1, then G = SL(2, q).It is straightforward to see that k(SL(2, q)) = q + 4 and that k(ASL(2, q)) = 2q + 4 and so the formula holds.So suppose that n ≥ 2. Let V be the natural module for G.Note that in this case G acts transitively on the nontrivial characters of V and the stabilizer of such a character is the stabilizer H of a vector in Sp(2n, q).Let U = O p (H ) and let Z = Z (H ).Then H/Z ∼ = ASp(2n − 2, q).If an irreducible character of H does not vanish on Z , then there are q − 1 possibilities (depending on the restriction to Z ) and arguing as in the unitary case, we see that the number of such characters of H is (q − 1)k(Sp(2n − 2, q)).This gives k(ASp(2n, q)) = k(Sp(2n, q)) + k(ASp(2n − 2, q)) + (q − 1)k(Sp(2n − 2, q)) as desired.□ We use this recursion to give another proof of the generating function for k(Sp(2n, q)) in odd characteristic.
Second proof of Theorem 5.3.Let k n = k(Sp(2n, q)) and let a n = k(ASp(2n, q)).Lemma 5.4 gives that (5) Let Multiplying ( 5) by u n and summing over n ≥ 1 gives that Solving for A(u) gives From Wall [1963], and the result follows.□ In even characteristic, the unipotent radical is abelian but not irreducible.So let G = Sp(2n, q) with q even.Let BG denote the semidirect product W G, where W is the 2n + 1 dimensional indecomposable module with G having a one dimensional fixed space W 0 and W/W 0 ∼ = V .
Note that the G-orbits of characters of B consist of the trivial character, one orbit of nontrivial characters with W 0 contained in the kernel and 2(q −1) orbits of characters which are nontrivial on W 0 .The stabilizer of a character in the second orbit is isomorphic to B Sp(2n − 2, q) while in the final case the stabilizers are O ± (2n, q) (with q − 1 of each type).This gives the following: Lemma 5.5.Let q be even. ( The next lemma follows immediately from the previous lemma.We use the convention that ASp(0, q) and O + (0, q) are the trivial groups and that O − (0, q) is the empty set.So Now we obtain the generating function for k(ASp(2n, q)) in even characteristic.
Theorem 5.7.In even characteristic, k(ASp(2n, q)) is equal to the coefficient of u n in Proof.We define three generating functions: Multiplying the recursion from Lemma 5.6 by u n and summing over n ≥ 1 gives that From Theorems 3.13 and Theorem 3.21 of [Fulman and Guralnick 2012], elementary manipulations, give that and the result follows.□ 5C.Bounds on k(ASp(2n, q)).As a corollary, we obtain the following results.
Proof.From Theorem 5.3, k(ASp(2n, q)) is the coefficient of u n in ) is the generating function for the number of conjugacy classes in GL(n, q).By [Maslen and Rockmore 1997], k(GL(n, q)) is at most q n .Hence the coefficient of u n−m in it is at most q n−m .It follows that k(ASp(2n, q)) is at most which is equal to The term is visibly maximized among odd prime powers q when q = 3, when it is at most 27 we bounded the infinite product i (1 + 1/q i ) 4 /(1 − 1/q i ) using the remark after Lemma 2.3 .□ Corollary 5.9.In odd characteristic, k(ASp(2n, q)) ≤ q 2n , except for k(ASp(2, 3)) = 10.
Proof.We rewrite the generating function for k(ASp(2n, q)) in Theorem 5.7 as Now arguing exactly as in the odd characteristic case (Corollary 5.8), one sees that k(ASp(2n, q)) is at most and the result follows.□ Next we classify when k(ASp(2n, q)) ≤ q 2n .

Orthogonal Groups
Section 6A uses the orbit approach to calculate the generating function for k(AO) when the characteristic is odd.Section 6B uses the character approach to calculate the generating function of k(AO) in any characteristic.To be more precise, we actually derive two generating functions, one for k(AO + )+k(AO − ) and one for k(AO + ) − k(AO − ).Clearly this is equivalent to deriving generating functions for k(AO + ) and k(AO − ).
Section 6C derives some bounds on k(AO).
6A. Orbit approach for k(AO), odd characteristic.For the orbit approach we assume the characteristic is odd.It is somewhat more convenient to work in orthogonal groups than the special orthogonal group (there is essentially no difference in the result below for SO).The conjugacy class of a unipotent element in O ϵ (m, q) gives rise to a partition of m with a i pieces of size i.Moreover, a i is even for i even.This determines the conjugacy class over the algebraic closure.Over the finite field, we attach a sign ϵ i for each odd i with a i nonzero and this determines the class (see [Liebeck and Seitz 2012]).We let λ ± z−1 (C) denote this signed partition corresponding to the unipotent part of a conjugacy class C.
The proof of the next result is essentially identical to the case of symplectic groups and so we omit the details (and we can also use the character theory approach below).Theorem 6.1.Suppose that the characteristic is odd.Let C be a conjugacy class of O ϵ (n, q).Let a i be the number of parts of λ ± z−1 (C) of size i.Then o(C) is equal to where if a i = 2 and the sign is +, (q − 1) if a i = 2 and the sign is −, (q − 1)/2 if a i = 1 (independently of the sign). (6) The following combinatorial lemma will also be helpful.Lemma 6.2.Suppose that the characteristic is odd.
(1) The generating function for the number of unipotent classes of the groups O(n, q) is This is equal to (2) The generating function (3) Let f i be as in Theorem 6.1.Then Proof.For the first part, the unipotent conjugacy classes of the groups O(n, q) correspond to signed partitions λ ± of size n.The generating function for such partitions is clearly equal to i odd For the second part, first note that arguing as in the first part, one has that For the third part, This is equal to and the result follows.□ As a corollary, we derive a generating function for k(AO + ) + k(AO − ).
Theorem 6.3.In odd characteristic, Proof.By Lemma 2.1 and Theorem 6.1, ), and T 2 , T 3 are the following sums over conjugacy classes C of O + (n, q) and O − (n, q): To compute the generating function for T 2 , we take Wall's generating function for T 1 , divide it by the generating function for unipotent conjugacy classes in part (1) of Lemma 6.2, and multiply it by the generating function for the weighted sum over unipotent classes in part (2) of Lemma 6.2.We conclude that T 2 is the coefficient of u n in u To compute the generating function for T 3 , we take Wall's generating function for T 1 , divide it by the generating function for unipotent conjugacy classes in part (1) of Lemma 6.2 and multiply it by the generating function for the weighted sum over unipotent classes in part (3) of Lemma 6.2.We conclude that T 3 is the coefficient of u n in the result follows.□ Next, we derive a generating function for k(AO + ) − k(AO − ).
Theorem 6.4.In odd characteristic, Proof.By Lemma 2.1 and Theorem 6.1, Here C + ranges over conjugacy classes of O + (n, q), and C − ranges over conjugacy classes of O − (n, q).From To compute the generating function of T 2 , we take the generating function for T 1 , multiply it by i (1 − u 4i ) (which corresponds to removing the unipotent part).Then to add in the weighted unipotent part, one multiplies by j even To compute the generating function of T 3 , we take the generating function for T 1 , multiply it by i (1 − u 4i ) (which corresponds to removing the unipotent part).Then to add in the weighted unipotent part, one multiplies by Note that the terms involving f i canceled out (except for the a i = 2 case).The upshot is that the generating function for Character approach for k(AO), any characteristic.Next we consider orthogonal groups.In this case, the natural module V can be identified with its character group and the nontrivial G-orbits correspond to nonzero vectors of V of a given norm.First consider the case G = O ϵ (n, q) with q odd.The stabilizers are thus AO ϵ (m − 2, q) (for an isotropic vector) and (q − 1)/2 copies each of O + (n − 2, q) and O − (n − 2, q).Note that we use the convention that O ϵ (0, q) and AO ϵ (0, q) are empty if ϵ = − and are the trivial group if ϵ = +.Similarly, AO ϵ (−1, q) is the empty set.And as in earlier cases, the trivial group has one conjugacy class and the empty set has zero conjugacy classes.This yields the following result.Lemma 6.5.Let q be odd and n ≥ 1.Then k(AO ϵ (n, q)) = k(O ϵ (n, q)) + k(AO ϵ (n − 2, q)) + (q − 1) k(O + (n − 1, q)) + k(O − (n − 1, q)) /2.
As a corollary, we obtain a second proof of Theorems 6.3 and 6.4.
Finally we turn to characteristic 2. In this case odd dimensional orthogonal groups are isomorphic to symplectic groups, so we need only consider the even dimensional case.So consider G = O ϵ (n, q) with q and n both even.The argument is similar.The only difference is that the stabilizer of a vector of nonzero norm in AO ϵ (n, q) is Sp(n − 2, q) × ‫2/ޚ‬ and so: Lemma 6.6.Let q be even and n ≥ 2 be even.Then k(AO ϵ (n, q)) is equal to k(O ϵ (n, q)) + k(AO ϵ (n − 2, q)) + 2(q − 1) k(Sp(n − 2, q)) .
Combining the results of the previous two paragraphs proves the corollary, as (53 + 3.3)/2 ≤ 29.□ Corollary 6.10.Let q be odd.Then k(AO ± (2n, q)) ≤ q 2n .Proof.The result follows from the previous corollary whenever 29q n ≤ q 2n .So we only need to check the cases n = 1, or n = 2, q = 3, 5, or n = 3, q = 3.These cases are easily checked from our generating function for k(AO ± (2n, q)).□ Next we treat the case of odd dimensional groups in odd characteristic.In this case, the upper bound is not of the form constant times q rank .This is because every element in the classical group has eigenvalue 1. Corollary 6.11.Let q be odd.Then k(AO(2n + 1, q)) ≤ 20q n+1 .
Lemma 4.2.(1) The generating function for the number of unipotent classes of GU(n, q) is λ u |λ| .

K
Sp (u) = 1 + n≥1 k(Sp(2n, q))u n , K O (u) = 1 + n≥1 [k(O + (2n, q)) + k(O − (2n, q))]u n , A O (u) = 1 + n≥1 which simplifies to the desired result.□ 4B.Character approach to k(AGU).We use Lemma 2.2 to find a recursion for k(AGU).Then we use this to compute the generating function for k(AGU), giving another proof of Theorem 4.3.Recall that if H is a finite group and p is a prime, then O p (H ) is the (unique) maximal normal p-subgroup of H .