Character bounds for regular semisimple elements and asymptotic results on Thompson’s conjecture

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Introduction
A conjecture of Thompson states that each finite simple group G contains a conjugacy class C ⊆ G such that C 2 = G.Inspired by this, we would like to study an asymptotic version of Thompson's conjecture when G is one of the finite classical groups SL n (q), SU n (q), Sp 2n (q), SO 2n+1 (q), and SO ± 2n (q), which are all closely related to simple groups.This asymptotic version treats target elements of sufficiently large support.We prove that regular semisimple conjugacy classes C = g G satisfy our asymptotic version of Thompson's conjecture whenever the characteristic polynomial of g is close to being irreducible.
If G = Cl(V ) is a finite classical group, with natural module V = F n q , we define the support supp(x) of an element x ∈ G to be the codimension of the largest eigenspace of x on V ⊗ Fq F q .The following is one of our main results and generalizes [LT,Theorem 7.8].
Theorem 1.1.For all integers k ∈ Z 1 there exists an explicit constant B = B(k) > 0 such that for all n ∈ Z 1 and all prime powers q the following statement holds.Suppose G is one of SL n (q), SU n (q), Sp 2n (q), SO 2n+1 (q), and SO ± 2n (q), and g ∈ G is a regular semisimple element whose characteristic polynomial on the natural module is a product of k pairwise distinct irreducible polynomials, of pairwise distinct degrees if G is of type Sp or SO.Then g G • g G contains every element x ∈ [G, G] with supp(x) B.
In fact, in the Sp and SO cases, we prove a slightly stronger result, see Theorem 10.2.We also note that the assumption x ∈ [G, G] is superfluous in the SL, SU, and Sp cases (since G = [G, G] in these cases, aside from known exceptions with n ≤ 3), but necessary in the SO case (since in this case [G, G] has index 2 in G and so In a special family of particularly favorable cases, Theorem 10.7 shows that all non-central elements of G lie in C 2 . If Irr(G) denotes the set of the complex irreducible characters of G, then the well-known formula of Frobenius states that x ∈ G is contained in g G • g G if and only if (1.1) To show that this is the case we need sufficiently good upper bounds on |χ(g)|.
To get these we realise our group as the fixed point subgroup G F of a Frobenius endomorphism F : G → G on a connected reductive algebraic group G and use the Deligne-Lusztig theory [DLu].
To illustrate our techniques suppose G F = Sp 2n (q) or SO 2n+1 (q).To each element w of the Weyl group W ∼ = C 2 ≀ S n of G, Deligne and Lusztig have associated a virtual character R w of G F whose irreducible constituents are called unipotent characters.The subspace Class 0 (G F ) ⊆ Class(G F ) of all C-valued class functions spanned by {R w | w ∈ W } is the space of uniform unipotent class functions.
If χ is a unipotent character then the (uniform) projection of χ onto Class 0 (G F ) is known to have the form for some class function f χ ∈ Class (W ), which is not irreducible in general.If g ∈ G F is semisimple then its characteristic function is uniform, which means χ(g) = R fχ (g) for all χ.
Our first step towards understanding R fχ (g) is to show that f χ satisfies a version of the recursive Murnaghan-Nakayama rule (or MN-rule), see Theorem 4.8.This is a consequence of a fundamental combinatorial result of Asai [As1], [As2] that relates the classical MN-rule for the irreducible characters of W and Lusztig's Fourier transform, whose proof we give in Section 3. If g ∈ G F is a regular semisimple element such that C • G (g) is a torus of type w then |χ(g)| = |R fχ (g)| = |f χ (w)| and we recover the MN-rule of Lübeck-Malle [LüMa,Thm. 3.3].
By working with uniform projections we may apply these results to non-unipotent characters, and we do so to obtain bounds on |χ(g)| whenever χ is a quadratic unipotent character and the cycle type of g ∈ G F is a product of k 1 pairwise distinct cycles (see §8 for precise definitions).In fact, following an argument of Larsen-Shalev [LSh] we obtain a bound on |χ(g)| that depends only on k, see Corollary 8.2 in the quadratic unipotent case.Bounds for arbitrary characters, involving k and n, are given in Corollary 7.6 and Corollary 7.8.Using the Mackey formula for tori we also give a bound in Theorem 6.2 that works for any irreducible character of any finite reductive algebraic group.
Now treating all characters χ in (1.1) involves a reduction to Levi subgroups using Deligne-Lusztig induction.The characters that contribute to the sum the most have a heavily restricted form.Our character bounds allow us to obtain sufficiently good bounds on the sum.Aside from these immediate applications, we believe our character bounds for regular semisimple elements will be useful in other situations as well.

Combinatorics
For any set X we will denote by Pow(X) the set of all subsets of X of finite cardinality.This is naturally an F 2 -vector space under symmetric difference, which we denote by A ⊖ B = (A ∪ B) − (A ∩ B) for any A, B ∈ Pow(X).Moreover, it is equipped with a nondegenerate symmetric bilinear form −, − : Pow(X) × Pow(X) → Z/2Z given by A, B = |A ∩ B| (mod 2).If e ∈ Z then we let Pow e (X) = {A ∈ Pow(X) | |A| ≡ e (mod 2)}.
We set X 2) .Elements of X (2) will be identified with their representatives in X × {0, 1}.We denote by δ : X (2) → Z/2Z the projection onto the second factor.
We denote by [A] the equivalence class containing A and B = Pow(N 0 )/∼ the set of all equivalence classes.These are called β-sets.The rank ρ([A]) = ρ(A) of [A] ∈ B is well defined.If n ∈ N 0 then B n ⊆ B denotes all β-sets of rank n.
2.1.Arrays.The elements of Pow(Z (2) ) will be called arrays.They will be identified with their images under the natural bijection Pow(Z (2) ) → Pow(Z) × Pow(Z) given by X → (X 0 , X 1 ), where We say X 0 is the top row of X and X 1 the bottom row of X.
Following Lusztig [Lu1], and modifying the notation of [W], we consider elements of P = Pow(N (2) 0 ) ⊆ Pow(Z (2) ).Recall the rank of X ∈ P is defined to be (2.1)rk(X) = x 0 ∈X 0 For d ∈ Z we let Pd ⊆ P be the set of arrays of defect d.We set Pod = d∈Z P2d+1 and Pev = d∈Z P2d so that P = Pod ⊔ Pev .For n ∈ N 0 we let Pn ⊆ P be the set of arrays of rank n.
Each X ∈ P gives rise to the following subsets of N 0 : X ∪ := X 0 ∪ X 1 , X ∩ = X 0 ∩ X 1 , X ⊖ = X 0 ⊖ X 1 .We let Sim(X) = {Y ∈ P | Y ∪ = X ∪ and Y ∩ = X ∩ } be the similarity class of X.All elements of Sim(X) have the same rank but different defects.
The defect def(X sp ) ∈ {0, 1} of this array satisfies def(X sp ) ≡ |X ⊖ | (mod 2).Thus we have an integer We have a map ♯ : P → Pow(N 0 ) given by Y ♯ = Y 1 ⊖ Y 1 sp ⊆ Y ⊖ .This restricts to a bijection ♯ : Sim(X) → Pow(X ⊖ ) for any X ∈ P. With this we define a C-linear map R : Here C[ P] denotes the free C-module with basis P and −, − is the symmetric F 2 -bilinear form defined above.Up to scaling this is the Fourier transform of the abelian group Pow(X ⊖ ).
Remark 2.1.We briefly make a few comments regarding the conventions and definitions in [Lu1].
x.This definite choice of M 0 over M ′ 0 , using X sp , is used in [Lu1,§4.18].Moreover, our definition of X sp agrees with the convention in [Lu2,17.2],that the smallest entry of X ⊖ occurs in the lower row of X sp .This is different to the definition of a distinguished symbol given in [GeMa,4.4.3].
We write this as X H when (d, i) is clear from the context or by X d,i λ = X λ when H = {λ} is a singleton.For brevity we write the map D 0,1 simply as (−) op .
If X ∈ P then the elements of the set Elsewhere in the literature (d, 0)-hooks and (d, 1)-hooks, with d > 0, are called d-hooks and d-cohooks respectively.Following [W] we define the leg parity of λ = (x, j) ∈ H d,i (X) to be If d > 0 and i = 0 then this has the same parity as the usual notion of the leg length of a hook.This is also easily seen to agree with the definitions in [GeMa,4.4.10].
Recall that δ : X (2) → Z/2Z is the second projection map.For a pair (d, i) ∈ Z (2) , with d = 0, and j ∈ {0, 1} we define a C-linear map Hj d,i : 2.4.Symbols.The map (−) →k : Pow(N 0 ) → Pow(N 0 ) defined above, for k ∈ N 0 , extends to map P → P given by (A, B) →k = (A →k , B →k ).As before this yields an equivalence relation on P. We denote by [X] the equivalence class containing X ∈ P and S the set of equivalence classes.The equivalence class [X] is called an (ordered) symbol.
Thus the maps R and Hj d,i preserve the kernel of the natural quotient map C[ P] → C[ S] and so factor through endomorphisms of C[ S] which we denote in the same way.
Recall that We take the rank and defect of X ∈ S to be rk( X ) = rk(X) and def( X ) = |def(X)|.We can then partition S with respect to the rank and defect as in Section 2.1.

More combinatorics
3.1.A combinatorial result of Asai.We will now prove the following fundamental combinatorial observation of Asai that relates the maps R and Hj d,i .This was first stated by Asai in [As1,Lem. 2.8.3] and [As2,Lem. 1.5.3]where it is left as a "direct calculation".However, as pointed out by §3.4] a sign is missing in the statement in [As1].
Aside from being an important ingredient in our work here, Asai's combinatorics form a core basis for the block theory of finite reductive groups and solutions of Lusztig's conjecture on almost characters for classical groups.In this second application the correctness of signs is crucial.In light of the importance of Asai's statements, we provide some details regarding the proof.
We note that some of the main ideas of the proof have recently appeared in [Ma,Prop. 6], where a weaker statement, leading essentially to Theorem 4.8, is proved.Unfortunately there are several errors in the proofs of [Ma] that are corrected by our arguments here.
Given X ∈ P we then have Before proving (i) of Theorem 3.1 we start with a lemma.
Lemma 3.2.Assume i = 0 and (H, λ) ∈ X (X, x) and (µ, G) ∈ Y (X, x) are two terms satisfying one of the following: Proof.Let Y = X 0,1 H, U = X d,0 µ, and V = U 0,1 G.We have We have to show the sum of these two expressions is 0.
For any z ∈ U ⊖ ∩ X ⊖ it follows from Eq. (2.3) that x − d} depending on whether δ(µ) = 0 or 1. Hence the statement clearly follows if (i) holds.So assume (ii) holds.As H ⊖ ⊆ X ⊖ we must have We will assume this is {x} as the case where it is {x − d} is identical.This means and the first term is equivalent to {x}, {0, . . ., x} ⊖ {0, . . ., x − d} .Adding this to Proof of Theorem 3.1(i).We can assume x ∈ X ∪ since otherwise X (X, x) and Y (X, x) are empty, and there is nothing to show.Similarly, this is the case if x − d ∈ X ∩ .The proof divides into several cases distinguished by the distribution of x − d and x amongst the rows of X. Case Hence, by Lemma 3.2 the coefficients of X H λ and X µ G in Eqs.(3.1) and (3.2) are the same.
If x−d and x occur in the same row of X then we must have X 0 (X, x) = {∅} and Y (X, x) = {∅}.Moreover, each pair in X 1 (X, x) gives rise to terms in Eq. (3.1) that cancel.If x − d and x occur in opposite rows of X then we must have X 1 (X, x) = {∅}.Moreover, we have a bijection Y (X, x) → X 0 (Λ, x) given by (µ, In this case (X µ) ⊖ = X ⊖ − {x, x − d} so 2s(X µ) = s(X) but X H λ and X H ′ λ op have the same coefficient in Eq. (3.1).Thus, the coefficients of X µ G = X G µ agree by Lemma 3.2 Case 4: x ∈ X ∩ and x − d ∈ X ⊖ .As in Case 1, we have a bijection X (X, x) → Y (X, x) given by Again, by Lemma 3.2 the coefficients of X H λ and X µ G in Eqs.(3.1) and (3.2) are the same.
We now consider the proof of (ii) of Theorem 3.1.The argument is exactly the same as (i), proceeding through the same cases.The bijection in Case (ii) is defined identically simply replacing D d,0 with D d,1 .Instead of providing a direct analogue of Lemma 3.2 we instead check directly in each case that the signs of the corresponding coefficients agree.As an example, we treat the analogue of Case 1, leaving the remaining cases to the reader.
Proof of Theorem 3.1(ii).Assume x ∈ X ⊖ and x − d ∈ X ∪ .We have a bijection X (X, x) → Y (X, x) given by As in the proof of (i) of Theorem 3.1, we need only check that the sign of the coefficient of Y λ in Eq. (3.1) and (−1) 1+def(V ) times the sign of the coefficient of V = U G in Eq. (3.2) agree, where Y = X H and U = X µ.We check this directly.
As µ is a (d, 1)-hook, we have We consider the two cases of the bijection above separately.
Suppose first that H ⊖ ∩ {x, x − d} = ∅.Clearly X 1 ⊖ U 1 , H ⊖ = 0, and as Summing the above terms it suffices to show that and this is straightforward.Finally we assume that As before it suffices to show that and again this is straightforward.
3.2.More symbols.We have a linear map − : for any X ∈ P. We then define a C-linear map ε : The following easy observations are stated in [W, §2].
Lemma 3.3.For any (d, i) ∈ Z (2) and j ∈ {0, 1} we have the following equalities of linear endomorphisms of C[ S]: It is straightforward to check that for any X ∈ P, we have H d,i (X op ) = H d,i (X) op and l d,i (λ, X) = l d,i (λ op , X op ), which gives the statement.
We define for each integer e ∈ Z the set P≡e = {X ∈ P | d(X) ≡ e (mod 2)}.
If X ∈ P then we let Sim e (X) = Sim(X) ∩ P≡e .Under the map ♯ : P → Pow(N 0 ), we have X ∈ P≡e if and only if |X ♯ | ≡ e (mod 2).Now we define Re : Lemma 3.4.For any X ∈ P and e ∈ Z we have: ( In particular, for any X ∈ P≡1 we have Re (X) = 0.
Proof.This follows by projecting (i) and (ii) of Lemma 3.3 onto each summand of the decomposition Here we use that Sim e (X) op = Sim e+def(X) (X), which follows from the above remarks.For the final statement note that (ii) shows that if By (i) of Lemma 3.4 we have R 0 factors through − to give an endomorphism of its image.We extend this to an endomorphism of C[S] by letting it fix pointwise the complement defined above (in other words, R 0 ( X ± ) = X ± for all degenerate X ∈ P).We also denote this by R 0 .We consider the subspaces C[S od ] and Proposition 3.5.For any (d, i) ∈ Z (2) , with d = 0, we have commutative diagrams If e ∈ 2Z is an even integer then we let A ev,e = U ev,e = C[S ev,0 ] ⊆ C[S ev ].The map R 0 restricts to an involution on the subspace C[S ev,0 ].As above we consider R 0 as a map A ev,e → U ev,e with inverse Q 0 : U ev,e → A ev,e .Let Pev,nd ⊆ Pev,0 be the subset of nondegenerate arrays.For any odd integer e ∈ 2Z + 1 we let U ev,e = C[S ev,1 ] and define the quotient space C .By Lemma 3.4 the map Re factors through a map A ev,e → U ev,e which we denote by R e .We define a right inverse Q e : U ev,e → A ev,e of this map by setting It follows easily from Lemma 3.3 that the endomorphism Hi d,0 of C[ Pev,nd ] factors through a well defined map H i d,0 : A ev,e → A ev,e+i for each e ∈ Z. Similarly we have H 0 d,i factors through a map U ev,e → U ev,e+i .Proposition 3.6.For any (d, i) ∈ Z (2) , with d = 0, and any e ∈ Z we have commutative diagrams A ev,e+i U ev,e+i Such a decomposition, which we call a cycle decomposition, is unique up to reordering.We call k 1 the cycle length.
There is a unique ordering of the orbits such that (−1) It determines the conjugacy class Cl W I (w) uniquely.If these inequalities are all strict then we say w has pairwise distinct cycles.
Lemma 4.3.Let w ∈ W I be as above and let If w has pairwise distinct cycles then the following hold: (i) Proof.Part (i) is given by the uniqueness of the cycle decomposition.Part (ii) follows from (i) and Lemma 4.2.Part (iii) follows from (i) because w, σ stabilises each I j , so they must be a union of the O i .
Recall from Section 2 that we have define the β-sets B n .After [GeP,§6.4.1] we have a bijection , defined as follows.First note that for any [X] ∈ Sδ n we have rk( X 0 ) + rk( X 1 ) = n by Eq. (2.1).Now choose a σ-stable partition I = I 0 ⊔ I 1 such that |I j | = 2 • rk(X j ) with j ∈ {0, 1} (note these subsets may be empty).We then have ρ where χ[X j ] is the inflation of χ [X j ] ∈ Irr(H I j ) under the map W I j → H I j .These characters satisfy the following MN-rule (or Murnaghan-Nakayama rule).
Proposition 4.4.Let O ∈ I/ w, N I be an orbit for some element w ∈ W I .Then we have a unique decomposition w = w 1 w 2 = w 2 w 1 with Proof.We refer the reader to [GeP,Thm. 10.3.1].For the correspondence between hooks of partitions and hooks of β-sets see [O, §I.1].
In [LSh,Theorem 7.2], a bound is given for the character values of the symmetric group at a given element in terms of its cycle length.The argument in [LSh] is a consequence of the MN-rule together with analogues of the following easy observations.Lemma 4.5.Let X ∈ P be an array with an (e, j)-hook λ ∈ H e,j (X) for some (e, j) ∈ N (2) is such a hook.Then X n,i λ is of rank 0 and so has no hooks.Hence, by the previous lemma the only possible (n, i)-hooks are λ and λ op .But it is easily seen that if λ op were a hook of X then it would also be one of X n,i λ, which is impossible.
n .We argue by induction on k.Suppose k = 1.If χ(w) = 0 then by Proposition 4.4, [X] has an (n, 0)-hook, but by Lemma 4.6, there is at most one such hook, so |χ(w)| 1.So assume k > 1.We have I/ w, σ = {O 1 , . . ., O k } and we set Clearly we may assume that χ(w) = 0.By Proposition 4.4 and the induction hypothesis we see that |χ(w)| Repeatedly applying Proposition 4.4, we see that there exist arrays X = X 0 , X 1 , . . ., X k ∈ P such that for 1 i k we have Now assume χ ∈ Irr(W 0 I ).If χ extends to W I then we are done so assume this is not the case.
n and is the sum If ∆(w) = 0 then we are done, so we may assume that ∆(w) = 0.A result of Stembridge [St,Theorem 7.5] shows that, in this case, |∆(w)| = 2 k |χ [A] (x)| for some character χ [A] ∈ Irr(S I + ) and some element x ∈ S I + which is a product of k disjoint cycles.Hence, by [LSh,Theorem 7.2], we have |∆(w)| 2 2k−1 • k!, which easily gives the bound.
For an ordered symbol [X] ∈ Sn of rank n we define a corresponding class function , where δ = def(X sp ).Somewhat remarkably these functions also satisfy a version of the MN-rule, which is the main point of Theorem 3.1.
Theorem 4.8.Let O ∈ I/ w, N I be an orbit for some element w ∈ W I .Then we have a unique Proof.By Proposition 4.4 we have )) and projecting the right hand side of this onto Q[ Sδ ] gives us that If def(X) is even then the same holds.but we must multiply through by (−1) j .Theorem 4.9.Fix an integer 1 k n.Then for each element w ∈ W I of cycle length k and each symbol [X] ∈ Sn of rank n, we have Proof.The proof is identical to that of Theorem 4.7.
We now associate functions to unordered symbols as follows.If X ∈ S od n has odd defect then we simply let φ X = φ [X] ∈ Class(W I ).Now fix e ∈ {0, 1}.Then for any X ∈ S ev,e n we let Note that Res ) so these functions are nonzero.
Remark 4.10.With some additional justifications, the statement in Theorem 4.8 may now equally be seen to hold for the class functions φ X .If X has odd defect then the same statement holds verbatim.
Assume now in the statement of Theorem 4.8 that δ(w) = e, so that w ∈ W e I .Then it makes sense to consider φ X (w) for any X ∈ S ev,e n .Clearly w 2 ∈ W e+j I , and if λ ∈ H d,j (X) then X λ ∈ S ev,e+j n , so the term φ X λ (w 2 ) makes sense.Hence, restricting the symbols to S ev,2e n , the statement in Theorem 4.8 continues to hold.

Lusztig series
In the next few sections we consider a general connected reductive group G defined over F = F p with Frobenius endomorphism F : G → G.We will follow the setup in [Lu1], see also the exposition in [GeMa,Chap. 2].This setting, whilst a little less frequently used, is more convenient as we wish to discuss character values of Deligne-Lusztig characters.In this section we just outline some notation.
Let T B G be a fixed F -stable maximal torus and Borel subgroup of G. Let X = X(T) = Hom(T, G m ), which we view as a Z-module.For any φ : T → T a morphism of algebraic groups we denote by φ * : X → X the map given by φ * (χ) = χ • φ for all χ ∈ X.
For each w ∈ W := N G (T)/T we fix an element n w ∈ N G (T) such that w = n w T. Let ι g : G → G, with g ∈ G, be the inner automorphism given by ι g (x) = gxg −1 .Then F w := F ι nw and wF := ι nw F are also Frobenius endomorphisms of G stabilising T. We write w * instead of (ι nw | T ) * and for brevity we let w λ = w * −1 (λ) for all w ∈ W and λ ∈ X.
If Z (p) is the localisation of Z at the prime ideal (p) ⊆ Z containing p then V = Z (p) ⊗ Z X is a free Z (p) -module of finite rank.We will identify X with its image in V under the canonical map x → 1 ⊗ x and similarly any homomorphism γ : X → X is identified with 1 ⊗ γ : V → V .Note the quotient V /X is a torsion Z-module.
Abusing terminology we call W aff = X ⋊ W the affine Weyl group.We have a natural action of W aff on V given by (χ, z) • λ = χ + z λ for any (χ, z) ∈ W aff and λ ∈ V .Let W aff (λ) be the stabiliser of λ and let If ZΦ ⊆ X is the Z-submodule generated by the roots then W • aff = ZΦ ⋊ W ⊳ W aff is the usual affine Weyl group.Again we let W • aff (λ) be the stabiliser of λ and let W • (λ) be the projection of We now fix an injective homomorphism κ : The following is straightforward; see [Lu1,Lem. 6.2] and [GeMa,Lem. 2.4.8].
Given w ∈ W and θ ∈ Irr(T wF ) we denote by R G w (θ) the virtual character of G F defined in [DLu,Def. 1.9].As usual we extend this by linearity to a map on all class functions.Moreover, for any (λ, w) ∈ C W (X, F ) we set R G w (λ) := R G w (λ wF ).We then define for any pair (λ, a) Suppose now that ι : G → G is a regular embedding.Then T = T • Z(G) is an F -stable maximal torus of G and if we let X = X( T) then we have a surjective Z-module homomorphism ι * : X → X given by ι * (χ) = χ • ι.Note this maps the roots of G in X bijectively onto the roots of G in X.Through ι we identify W with the Weyl group N G( T)/ T of G.

A character bound from the Mackey formula
We denote by W :F the semidirect product of W with the group F Aut(W ) such that F wF −1 = F (w) for all w ∈ W .The unique coset W F ⊆ W :F of W containing F is a W -set under conjugation and for w ∈ W we write C W (wF ) for the stabiliser of wF under this action.
For each w ∈ W we choose an element g w ∈ G such that g −1 w F (g w ) = n w ∈ N G (T).As usual the map wF → T w := gw T yields a bijection between the orbits of W acting on W F and the G F -classes of F -stable maximal tori.Note that t → gw t gives an isomorphism T wF → T F w .In particular, we have a bijection Irr(T F w ) → Irr(T wF ) given by θ → g −1 w θ = θ • ι gw .
Remark 6.1.For θ ∈ Irr(T F w ) we have a Deligne-Lusztig character R G Tw (θ) defined as in [DLu,1.20] We will implicitly use this equality in what follows.
We have an action of C W (wF ) on Irr(T wF ) by setting z θ = θ • ι −1 nz for any z ∈ C W (wF ) and θ ∈ Irr(T wF ).We denote by C W (wF, θ) C W (wF ) the stabiliser of θ ∈ Irr(T wF ).Now for any (λ, w) ∈ C W (X, F ) and z ∈ W it follows from (5.1) that z (λ wF ) = (F (z) * −1 λ) zwF z −1 .Therefore, (6.1) where the latter stabiliser is calculated with respect to the action of Of course, the type is only determined up to W -conjugacy.Moreover, an element g of type wF is then G F -conjugate to an element of the form gw t with t ∈ T wF .
As in [DiM], if H is a finite group then we denote by π H h ∈ Class(H) the function taking the value |C H (h)| on Cl H (h) and the value 0 on H − Cl H (h). For any f ∈ Class(H) we then have f, π H h = f (h).We also write [g, h] = g −1 h −1 gh for the commutator of g, h ∈ H. Theorem 6.2.Assume g ∈ G F is a regular semisimple element of type wF .Fix a pair (λ, a) ∈ D W (X, F ) and let . By [DLu,Prop. 9.18], see also [DiM,Prop. 10.3.6],we have For any θ ∈ Irr(T wF ) we have by the Mackey formula for tori, see [DLu,Thm. 6.8] or [DiM,Cor. 9.3.1],that Applying this to χ, π G F g we get Let V (w) be as in Lemma 5.1 so that V (w)/X ∼ = Irr(T wF ).If µ ∈ V (w) is such that χ, R G w (µ) = 0 then we must have (µ, wW • (µ)) and (λ, a) are in the same W aff -orbit by the disjointness of Lusztig series.This happens if and only if there exists a z ∈ W such that µ − z λ ∈ X and This last condition is equivalent to z ∈ X w (λ, a).
If z ∈ X w (λ, a) and v ∈ C W (F x, λ) then one checks easily that zv ∈ X w (λ, a), and clearly zv λ = z λ.Hence, the sum in (6.2) can be taken over X w (λ, a)/C W (F x, λ) with θ replaced by ( z λ) wF .The group C W (F w) = F (C W (wF )) acts on X w (λ, a) by left multiplication, and the term in the sum is constant on orbits.Hence, we can sum as in the statement once we multiply through by the size of the orbit |C W (F w)/C W ( z λ) (F w)|.We now cancel terms using (6.1).
Remark 6.3.The above follows the same argument as [GMa,Thm. 5.4] where one finds the bound |C W (F w)|.The above yields this bound when the sum contains only one term.However, this does not hold in general so this should be corrected as above.In the extreme cases where W (λ) = W or W • (λ) = {1} then this does give the bound |C W (F w)|.

Further bounds from Deligne-Lusztig characters
The following result giving the value of a Deligne-Lusztig character at a regular semisimple element is well known.We simply translate this into the setup we utilise here, see [Ge,Prop. 4.5.8]for an equivalent formulation.
Proposition 7.1.Assume g ∈ G F is a regular semisimple element of type wF and let t ∈ T wF be such that gw t ∈ Cl G F (g). Then for any (λ, x) ∈ C W (X, F ) we have In particular, we have Proof.As noted in the proof of Theorem 6.2 we simply have to calculate ) .The statement now follows immediately from the inner product formula for Deligne-Lusztig characters and the identification V (w)/X ∼ = Irr(T wF ).
To make invoking Lusztig's classification results for the irreducible characters of G F simpler it will be beneficial to assume that Z(G) is connected.As usual we invoke a regular embedding to achieve this.We note that for our purposes we do not need the significantly more difficult multiplicity freeness results obtained by Lusztig.
where GF χ is the stabiliser of χ in GF .
Proof.Let T G be an F -stable maximal torus and θ ∈ Irr(T F ).We then have T = T •Z( G) is an F -stable maximal torus of G. Let θ ∈ Irr( TF ) be an irreducible character such that Res TF T F ( θ) = θ.By [B,Prop. 10.10] we have Res This shows that for any c ∈ GF we have c χ ∈ Irr(G F ) and χ have the same uniform projection.As for some integer e 1. Hence χ(g) = e| GF / GF χ |χ(g) giving the bound.
Fix a pair (λ, a) ∈ D W (X, F ).For a class function f ∈ Class(F a) on the coset of W • (λ) we define a corresponding class function We denote by Irr(F w.W • (λ)) the set of restrictions Res , where φ ∈ Irr(W • (λ):F w) restricts irreducibly to W • (λ).These functions on the coset F a depend on our choice of representative w, so we include a period to indicate this choice.There is, however, a natural choice w a ∈ a which is the element of minimal length (determined by our choice of Borel subgroup B).
Lemma 7.3.Assume g ∈ G F is a regular semisimple element of type wF .For any irreducible character φ ∈ Irr(F x.W • (λ)), with x ∈ a, we have but by [I,Lem. 8.14(c)], the sum on the right hand side is equal to 1.
Lemma 7.4.Assume g ∈ G F is a regular semisimple element of type wF .Then for any class function f ∈ Class(F a) we have Proof.By Proposition 7.1 we may restrict the sum over x ∈ a, found in the definition of R G λ,a (f ), to those elements satisfying F x ∈ Cl W (F w).Alternatively, via the bijection C W (F w)\W → Cl W (F w), given by C W (F w)z → z −1 F wz, we can sum over all cosets C W (F w)z ∈ C W (F w)\W such that z −1 F wz ∈ F a. Grouping together elements in the same W • (λ)-orbit and bringing |W • (λ)| into the sum gives the statement.
Observe that, if W • (λ) = W and w ∈ a then R G λ,a (f )(g) = f (F w)λ wF (t), see [LsMa,Prop. 3.3].Together with Theorem 4.7 and Theorem 4.9, this implies Corollary 7.5.Suppose G is a quasisimple group of classical type In this case we have W is isomorphic to S n and F induces an inner automorphism on W , which is either trivial or conjugation by the longest element.Hence, the coset W F can be identified with W so it makes sense to speak of the cycle type of an element of W F .
Corollary 7.6.Assume all the quasi-simple components of G are of type A. If g ∈ G F is a regular semisimple element of type wF then Proof.By Lemma 7.2, we can assume that Z(G) is connected.In that case every irreducible character is, up to sign, of the form R G λ,a (φ) with φ ∈ Irr(F w.W (λ)) so this is just Lemma 7.3.For the final statement we need only show that . Now an arbitrary w may be written as a pairwise commuting product w = w 1 • • • w r such that for each 1 i r we have w i is a product of k i 1 disjoint cycles of length m i 1 and the lengths m 1 , . . ., m r are pairwise distinct.We then have and by the previous calculation For the next statement we wish to define an integer r(W, F ) 0 as follows.Let S ⊆ W be the set of Coxeter generators determined by our choice of Borel B. Write W = W 1 • • • W m as a direct product of its irreducible components, all of which are assumed to be of type A through D. We then have a corresponding decomposition S = S 1 ⊔ • • • ⊔ S m .The Frobenius F permutes the W i .Suppose first that it does so transitively.Then we define Here we consider the trivial group as being of type A 0 .Now grouping together the W i we can write where each W (i) is an F -stable subgroup such that F permutes transitively its irreducible components.Hence, we are in the previous situation and we define r(W, F Theorem 7.7.Assume g ∈ G F is a regular semisimple element of type wF and every quasisimple component of G is of classical type A to D. If F is a Frobenius endomorphism then for any irreducible character χ ∈ E(G F , λ, a) we have where w a ∈ a is the unique element of minimal length and r = r(W • (λ), F w a ) is defined as above.
Proof.Again, by Lemma 7.2 we can assume Z(G) is connected.By Lemma 5.2 this implies that [Lu1,Chp. 4] Lusztig has defined a partition of Irr(W (λ)) into families.
Denote by w a ∈ a = Z W (λ, F ) the unique element of minimal length.The automorphism γ := F w a of W (λ) permutes the families.Suppose F ⊆ Irr(W (λ)) is a γ-stable family.For each γ-fixed character φ ∈ F γ we fix an extension φ ∈ Irr(W (λ):γ) that is realisable over Q.
As all factors are of classical type, we have G F is a (possibly trivial) elementary abelian 2-group.We also have two sets M(G F , γ) and M(G F , γ) and a pairing From the formula for this pairing, we see that By [Lu1,Thm. 4.23], we have a bijection E(G F , λ, F) → M(G F , γ), which we denote by χ → x χ , and an injection F γ → M(G F , γ), denoted by φ → x φ.This latter map depends on our choice of extension.Now, by [Lu1,4.26 The group of roots of unity acts on M(G F , γ), and the number of orbits is the same as for all χ ∈ Irr(G F ).

Quadratic unipotent characters
Recall that a bound for the values of unipotent characters at regular semisimple elements was obtained in Corollary 7.5.In this section, we establish a bound for the more general class of quadratic unipotent characters.Consider a connected reductive group G whose center Z(G) is connected, of dimension 1, and whose derived subgroup G der = [G, G] G is a symplectic or special orthogonal group.We specify G by its root datum (X, Φ, q X, q Φ).Firstly we have X = n i=0 Ze i and q X = n i=0 Zq e i with perfect pairing −, − : X × q X → Z given by e i , q e j = δ ij (the Kronecker delta).We assume n 2.
One easily calculates that ( α i , q α j ) is a Cartan matrix and X/ZΦ ∼ = Z is generated by e 0 + ZΦ.Let X der = X/(Ze 0 ) and q X der = n i=1 q e i .Denote by : X → X der the natural quotient map.We then have (X der , Φ, q X der , q Φ) is the root datum of G der .Fix a prime power q = p a .We describe F by defining F * as an endomorphism of X.For any 0 i n with i = 1 we have F * e i = qe i .We then have F * e 1 is either qe 1 or q(e 0 − e 1 ) with this latter case occurring only when G F is of type 2 D n (q).
If V der = Z (p) ⊗ Z X der then we have der , λ, a), see Lemma 5.2.Consider the totally ordered set Letting F act on X via q −1 F * we get an action of W :F on X der that gives an injective homomorphism W :F → S I .We implicitly identify W :F , hence also W , with its image which is contained in W I .Via this identification we can speak of the signed cycle type of any element of W :F .Theorem 8.1.Let (λ, a) ∈ D W (X, F ) be such that 2λ ∈ X der and let g ∈ G F be a regular semisimple element of type wF .If wF has cycle length k 1 and pairwise distinct cycles then Proof.Recall that we have an isomorphism R G λ,a : Class(F a) → Class 0 (G F , λ, a).Thus, if pr : Class(G F ) → Class(G F ) is the projection onto the subspace of uniform functions then there exists a unique class function f χ ∈ Class(F a) such that R G λ,a (f χ ) = pr(χ).Now pr(χ) and χ have the same value at g, so it suffices to bound the value of R G λ,a (f χ ) at g.By Proposition 7.1 and Lemma 7.4, we have We can assume R G λ,a (f χ )(g) = 0 and hence assume that F x = z −1 F wz ∈ F a is a conjugate of F w. We now bound: |C W (F x)|/|C W (λ) (F x)|, the number of terms in the sum, and finally |f χ (F x)|.Let us note that the number of terms in the sum is precisely the number of W (λ)-orbits on F a that meet the centraliser C W (F w).
We will take this case by case.First let us note that as wF has pairwise distinct cycles, so does its conjugate F x = zF (wF )F −1 z −1 .Now, by replacing λ with an element in the same W aff -orbit, we can assume that λ = 1 2 (e 1 + • • • + e m ) for some 0 m n where λ = 0 when m = 0. We set For convenience, we let π i = (e i , −e i ) ∈ W I for any 1 i n.
Type B n .We have W (λ) = F a = H.Lusztig has shown that there is an isomorphism m and Y ∈ S od n−m } maps onto the series E(G F , λ).The images of the Fourier transforms R 0 ( X ) ⊗ R 0 ( Y ) are Lusztig's almost characters.By [Lu1,4.23],this bijection may be chosen such that if Using Theorem 4.7, we thus get the following bound on the character value If α = (α 1 , . . ., α k 0 ) and β = (β 1 , . . ., β k 1 ) are the signed cycle types of x 0 and x 1 respectively then, up to reordering the entries, α ∪ β = (α 1 , . . ., α k 0 , β 1 , . . ., β k 1 ) is the signed cycle type of F x. Thus, there can certainly be at most k c=0 k c = 2 k terms in the above sum.Putting things together we get the bound 2 2k−2 • k! in this case.
Type C n .We have W (λ) = W 0 I 0 W I 1 and F a = W e I 0 W I 1 for some e ∈ {0, 1}.In this case, we have an isomorphism and Y ∈ S od n−m } maps onto the series E(G F , λ).By [Lu1,4.23],this bijection may be chosen such that for any Appealing to Theorem 4.7, when X is degenerate, we find, as above, that |f χ (F x)| 2 2k−1 • k!.As we have |C W (F w)\W/W (λ)| 2|C W (F w)\W/H|, there are at most 2 k+1 terms in the above sum.Putting things together gives the bound 2 3k+1 • k! in this case.
Type D n .We have . These cases are similar to the above.We have Arguing similarly, we get that the above sum has at most 2 k+2 terms, and |f χ (F x)| 2 2k • k!.Putting things together gives the bound 2 3k+4 • k!.
We end with a comment about the final coset π 0 π 1 W (λ). In this case we have an isomorphism n−m → Class(G F , λ, a).Lusztig's Theorem in this case says that this isomorphism can be chosen such that for any Corollary 8.2.Assume G is a symplectic or special orthogonal group and χ ∈ Irr(G F ) is a quadratic unipotent character.Furthermore, let wF ∈ W F have cycle length k 1 and pairwise distinct cycles.Then |χ(g)| 2 3k+4 • k! for any regular semisimple element g ∈ G F of type wF .
Proof.Recall that being quadratic unipotent means that χ lies in a series E(G F , λ, a) with 2λ ∈ X.The statement is thus an immediate consequence of Theorem 8.1 and Lemmas 5.2 and 7.2.

Character degrees
In this section, we prove the following theorem.
Theorem 9.1.There exists an absolute constant C > 0 such that for every finite quasisimple group G of Lie type of rank r and every positive integer D, the number of irreducible characters of G of degree ≤ D is at most D C/r .Taking C large enough, we can ignore any finite number of quasisimple groups G, and thus we may assume that G = G F for a simple, simply connected algebraic group G of rank r and a Frobenius endomorphism F : G → G.The Landazuri-Seitz bound [LaSe] implies that the minimal non-trivial character of G has degree at least |G| ǫ , where ǫ depends only on the rank of G. Therefore, we are justified in assuming that r is as large as we wish, so, in particular, G is of classical type and F is an endomorphism of Steinberg type.
Our proof closely follows the character degree estimates of Liebeck and Shalev [LiSh].Liebeck and Shalev prove a more precise result [LiSh,Theorem 1.1] than Theorem 9.1 when q is sufficiently large in terms of r and a weaker result [LiSh,Theorem 1.2] for general q.
What is needed to obtain good bounds in high rank for small q is an estimate for the number of unipotent characters of G and certain related groups of bounded degree.This follows in principle from the degree formulas for unipotent characters of classical groups in [Lu1].We begin with these computations.
Proposition 9.2.There exists an absolute constant C ′ such that for every finite quasisimple group G of classical Lie type of rank r and every positive integer D, the number of unipotent characters of G of degree ≤ D is at most D C ′ /r .Proof.For every prime power q, we have by [LMaT,Lemma 4.1(i) (i) If G is of type A r or 2 A r , then the unipotent characters of G are indexed by sets A of positive integers such that ρ(A) = r + 1 in the notation of §2.Denoting the elements of A by λ The terms µ i := λ i + 1 − i in this sum give a partition of r + 1, so in particular, m ≤ r + 1.
Here, G = SL ε n (q) with n = r + 1.The degree d A of the character with given set A is the absolute value of (9.3) 1≤j<i≤m ((εq) .
For any fixed j, we have from (9.1) that Treating the other factors of (9.3) in the same way (and using (9.2) for the products in the denominator), we obtain . As is well-known (see e.g. the proof of [GLT,Lemma 5.3], the exponent of the second factor on the right-hand side is 1 2 and so where p denotes the partition function.As p(i) is sub-exponential in i, when D ≥ q n/3 this number is e O(log D/ log q n ) = D O(1)/n , yielding a uniform upper bound of the form D C ′ /r for the number of unipotent characters of degree ≤ D. However, by [LaSe], for D ≤ q n/3 , there are no non-trivial irreducible characters of degree ≤ D, and in particular no such unipotent characters.
(ii) The proof for the remaining classical groups follows the same pattern.Let [X] denote an equivalence class of ordered symbols, and let X = (X 0 , X 1 ) be the representative such that 0 ∈ X ∩ = X 0 ∩ X 1 .Let the elements of X 0 and X 1 respectively form the increasing sequences of non-negative integers λ 0 , so assuming X 0 and X 1 are both nonempty, we have λ 0 1 + λ 1 1 > 0. We note that X 0 and X 1 determine (possibly improper) partitions {λ 0 i + 1 − i} i and {λ 1 j + 1 − j} i of the ranks ρ(X 0 ) and ρ(X 1 ) respectively.More precisely, if the sequence obtained by first merging X 0 and X 1 and then sorting, without eliminating repetitions.Thus (9.4) By (2.1), the rank r of the symbol X is given by We have ν k ≥ ⌊(k − 1)/2⌋ for all k, with strict equality when k is even, so r ≥ ⌊n/2⌋.Thus, (9.7) ≥ ρ(X 0 ) + ρ(X 1 ).
For every prime power q and every symbol X, there is at least one associated unipotent character of at least one classical group G of rank r over the field F q .If def(X) is odd, we obtain characters of G = Sp 2r (F q ) and of G = Spin 2r+1 (F q ) in this way.If it is divisible by 4, there is a character of G = Spin + 2r (F q ); otherwise, there is a character of G = Spin − 2r (F q ).If X 0 = X 1 , then there are two unipotent characters for Spin + 2r (F q ) associated to X; otherwise, there is only one for each possible G.Moreover, all unipotent characters for groups of type B, C, and D arise in this way for a unique equivalence class of unordered symbols.
The degree d X of the unipotent character of G associated to the symbol X is (at least) with n j := |X j |.In terms of the sequence ν i , this takes the form .

Note that
Reasoning as in case A, for cases B and C (i.e., n = 2m + 1 is odd) and using the fact that n ≤ 2r + 1, we have We define By (9.6), the exponent of the second term on the right-hand side is Here we have used the identity Thus, Since |Irr(G)| ≤ q C 1 r for some absolute constant C 1 by [FuG], by enlarging C ′ (which then covers all small ranks), we may assume log d X / log q r ≤ r/2 − 8.5, and so (9.9) max By (9.4) and (9.5) and the integrality of the ν i , we have and so by (9.9), and this violates (9.6).Hence, ν i − ⌊ i−1 2 ⌋ achieves its maximum at only i = n, and ν n > ν n−1 , again by (9.9); also, Applying the Landazuri-Seitz bound as before, we may assume that d X ≥ q r .The rank r ′ of the symbol, obtained from X by deleting the largest single term ν n , call it X ′ , is bounded above by log d X / log q r + 7.5, by (9.6) and (9.8).Applying (9.7) to X ′ = ((X ′ ) 0 , (X ′ ) 1 ), we see that (9.10) ρ((X ′ ) 0 ) + ρ((X ′ ) 1 ) ≤ log d X / log q r + 7.5 ≤ x + 7.5 if d X ≤ q rx with x ≥ 0. In fact, we can show that such symbols X satisfy (9.11) Indeed, without loss we may assume that µ 1 ≥ 1, so the sequence {µ j + 1 − j} of |T ′ | integers is a proper partition of ρ((X ′ ) 1 ), and so |(X ′ ) 1 | ≤ ρ((X ′ ) 1 ) ≤ x + 7.5 by (9.10).Applying (9.7) to Hence |(X ′ ) 0 | < 2x + 14, and .5, as stated.By (9.11), even when λ j 1 = 0, the number of zero entries in the sequence {λ j i + 1 − i} is at most |(X ′ ) j | < 3x + 22.5.Now, counting the number of (possibly improper) partitions {λ j i + 1 − i} of ρ((X ′ ) j ) and using (9.10), we see that the number of possibilities for the symbol X with d X < q rx is bounded above by (3x + 22.5) an exponential in x for x ≥ 1, proving the proposition for types B r and C r .
(iii) For types D r and 2 D r , that is, when n = 2m is even, we have n ≤ 2r, and The exponent of the second term on the right hand side is therefore Here we have used the inequality and the identity The argument finishes as before.
Proof of Theorem 9.1.Let G * = (G * ) F * denote the dual group of G.We partition the irreducible characters of G into rational Lusztig series E(s), indexed by conjugacy classes of semisimple conjugacy classes (s) of G * .There is a bijection between the elements of E(s) and unipotent characters of C G * (s); this correspondence multiplies degrees by We restate [LiSh,Lemma 3.2] in a form more convenient for our purposes.Since the ratio n/r between the dimension n of the natural module and the rank of G is bounded between 1 and 3, and since the constants d and d ′ in [LiSh] are absolute, if δ is greater than some absolute constant δ 0 , then the number of semisimple conjugacy classes (s) with |G * | p ′ /|C G * (s)| p ′ ≤ q δr is less than q δA for some absolute constant A. Moreover, C G * (s) contains a factor, the large factor, which is classical of rank r ′ ≥ r − Bδ for some absolute constant B, and this large factor is A r ′ (q) or 2 A r ′ (q) when G is of type A.
For D < q r/3 , there is only one irreducible character, by [LaSe].Hence we may assume D ≥ q r/3 .Enlarging C if necessary, we may assume that D = q δr with δ ≥ max(δ 0 , 1/2).By [FuG], |Irr(G)| ≤ q C 1 r for some absolute constant C 1 , so the result follows if δC ≥ C 1 r.Again enlarging C if necessary, we may assume without loss of generality that δ < r/2B.
If χ is an irreducible character of degree ≤ D, then it belongs to the Lusztig series E(s) for some s with |G| p ′ /|C G * (s)| p ′ ≤ D. The number of such semisimple classes s is bounded above by q δA .Following the proof of [LiSh,Lemma 3.4], note that for each s, C G * (s) contains a subgroup C G * (s) • which is the group of F * -fixed points of the connected reductive algebraic group s).Lifting s to an element ŝ of GL ε n (q), we see that every eigenvalue of ŝ has multiplicity ≤ n/2, but this contradicts the existence of the large factor of C G * (s) which is of type A r ′ (q) or 2 A r ′ (q) with r ′ > r/2.So we have C G * (s) = C G * (s) • for type A. Thus, there are at most 4 unipotent characters of C G * (s) of degree ≤ D for each unipotent character of C G * (s) • of degree ≤ D. Taking F * -fixed points of the derived group of C G * (s) • we obtain a subgroup whose unipotent characters correspond to those of C G * (s), and this subgroup is a product of classical groups whose ranks sum up to at most r and at least one of which, the large factor, has rank r ′ at least r − Bδ ≥ r/2.The total number of unipotent characters of the product of all the factors other than the large factor can therefore be bounded above by q C 1 Bδ by [FuG].The number of unipotent characters of the large factor of degree ≤ D = q δr is bounded above by D C ′ /r ′ ≤ q 2δC ′ , by Proposition 9.2.Hence, the number of unipotent characters of degree ≤ D of C G * (s) • is bounded by q (BC 1 +2C ′ )δ , and the number for C G * (s) is likewise bounded by an exponential in δ.Thus the number of characters of degree at most D is bounded by q C 2 δ for some absolute constant C 2 , and the theorem follows by taking C ≥ C 2 .
For later use, we prove the following related statement: Proof.(a) Embed G in G := GL ε n (q).Since the support of an element of G is at most n, by enlarging B, we are free to make n ≥ k as large as we wish.
(c) Now let j denote the level of χ ∈ Irr( G), as defined in [GLT].Assuming χ(1) > 1, we have j > 0. If j ≥ n/2, then χ(1) ≥ q n 2 /4−2 by [GLT,Theorem 1.2(ii)], and, as shown in (b), the contribution of all such characters to the left hand side of (10.1) is o(1).Hence it remains to consider the characters χ with j < n/2; any such character is irreducible over G, see [GLT,Corollary 8.6].Up to a linear factor, we may assume that χ has true level j.By [GLT, Theorem 3.9], any such character χ is of the form is a (possibly non-proper) Levi subgroup of G with 0 ≤ m < n, ϕ = ϕ λ is the unipotent character of GL ε n−m (q) labeled by a partition λ ⊢ (n − m) with largest part λ 1 = n − j, so, in particular, (10.2) m ≤ j, and ψ ∈ Irr(GL ε m (q)) when m > 0.Moreover, the total number of characters of G of true level j is |Irr(GL ε j (q))|, which is shown in [FuG,Propositions 3.5,3.9]to be at most 9q j .
As g is regular semisimple, the Steinberg character St G of G takes value ±1 at x. Applying [DiM,Cor. 10.2.10] we have (10.4) q denotes the natural module of G (endowed with a Hermitian form when ε = −), then the L-module V is a direct (orthogonal when ε = −) sum of two non-isomorphic irreducible modules V 1 := F n−m q and V 2 := F m q , with m ≤ j < n/2, see (10.2).In particular, if y ∈ N G (L), then y preserves each of V 1 and V 2 , and thus N G (L) = L.
Now we count the number N of elements y ∈ G such that y −1 gy ∈ L, i.e. g ∈ yLy −1 .Then g acts on each of the subspaces yV 1 and yV 2 .On the other hand, the decomposition p V (g) = k i=1 f i (X) leads to a decomposition V = ⊕ k i=1 U i , where p U i (g) = f i (X), and each U i is a minimal g -invariant, non-degenerate if ε = −, subspace.Moreover, the g -modules U i are pairwise non-isomorphic.Hence (yV 1 , yV 2 ) is uniquely determined by choosing a subset of {U 1 , . . ., U k } (so that yV 1 is the sum over this subset and yV 2 is the sum over the complement).Thus the total number of possibilities for yLy −1 = N G (yV 1 , yV 2 ) is at most 2 k .On the other hand, yLy −1 = y ′ Ly ′−1 if and only if Suppose y −1 gy = diag(g 1 , g 2 ) ∈ L, with g 1 ∈ L 1 := GL ε n−m (q) and g 2 ∈ L 2 := GL ε m (q).Let k i denote the number of irreducible factors of the characteristic polynomial of g i on the natural module for L i .Then k 1 + k 2 ≤ k.Since ϕ is unipotent and g 1 is regular semisimple, we have |ϕ(g 1 )| ≤ 2 k 1 −1 • k 1 !by Corollary 7.5.On the other hand, when m > 0, (10.2) and Corollary 7.6 show that It now follows from (10.4) and (10.5) that With (10.3), this shows that the total contribution of characters of a fixed true level 1 ≤ j < n/2 to (10.1) is at most 9q and q 1 := q σB/3−k (here we use the estimate q j ≥ 2 j ≥ j 2 ).Note that (10.6) is o(1) when B is large enough.Hence, the total contribution of characters χ, of level at least 1 and less than n/2, to (10.1), is less than (q − ε)A(k)/(q 1 − 1) = o(q − ε), and the theorem follows.
10.2.Types BCD.Recall the involution f → f on the set F * q of monic irreducible polynomial f ∈ F q [t] with f (0) = 0: if deg(f ) = m then f (t) = t m f (1/t)/f (0) (equivalently, λ ∈ F q is a root of f if and only if 1/λ is a root of f ).In the following theorem, the condition that the regular semisimple element g ∈ G has pairwise distinct cycles implies that the characteristic polynomial of g on the natural F q G-module is of the form Theorem 10.2.For all k ∈ Z ≥1 , there exists B > 0 such that the following statement holds for all n ∈ Z ≥1 and all prime powers q.If G = Sp 2n (q), SO 2n+1 (q), or SO ± 2n (q), and g ∈ G is a regular semisimple element with cycle length k and pairwise distinct cycles, then g Proof.(a) Enlarging B, we are free to make n ≥ max(k, 5) as large as we wish.Write G = G F for a corresponding simple algebraic group of type Sp or SO.Then C G (g) is a maximal torus, so, using the well-known structure of centralizers of semisimple elements in the finite group G, we see that T := C G (g) has order at most 2(q + 1) n .
Next we bound |χ(g)|, again using (10.4).Let V = F d q denote the natural module of G endowed with a symplectic or quadratic form, d = 2n or 2n + 1, and let L := L F .Then the L-module V is an orthogonal sum of two non-degenerate L-invariant subspaces 1 , with L 1 of the same type as of G, and L 1 acts trivially on V 2 .Next, L 2 := L F 2 ∼ = GL m (q) or GU m (q), with V 1 a minimal L 2 -invariant non-degenerate subspace, and L 2 acts trivially on V 1 .In particular, if y ∈ N G (L), then y preserves each of V 1 and V 2 , and thus N G (L) = L. Now we count the number N of elements y ∈ G such that y −1 gy ∈ L, i.e. g ∈ yLy −1 .Then g acts on each of the subspaces yV 1 and yV 2 .On the other hand, since g has cycle length k with pairwise distinct cycles, V admits an orthogonal decomposition V = ⊕ k i=1 U i , where each U i is a minimal g -invariant non-degenerate subspace.Moreover, the g -modules U i are pairwise nonisomorphic.Hence (yV 1 , yV 2 ) is uniquely determined by choosing a subset of {U 1 , . . ., U k } (so that yV 1 is the sum over this subset and yV 2 is the sum over the complement).Thus the total number of possibilities for yLy Suppose y −1 gy = diag(g 1 , g 2 ) ∈ L, with g 1 ∈ L 1 and g 2 ∈ L 2 = GL ± m (q).Let k i denote the cycle length of g i .Then k 1 + k 2 ≤ k.Since ϕ 1 is quadratic unipotent and g 1 is regular semisimple, we have by Corollary 8.2.On the other hand, when m > 0, the statement (β) and Corollary 7.6 show that It now follows from (10.4) and (10.10) that With (10.9), this shows that the total contribution of characters of degree satisfying q n(j−1) ≤ χ(1) < q nj to (10.7) is at most 1 , where A(k) = (2 4k+4 • k!) 2 and q 1 := q σB/2−C−k+1 (here we again use q j ≥ j 2 ).Recalling (10.6), we conclude that the total contribution to (10.7) of characters χ, of degree at least 2 and less than q (n 2 −4n)/4 , is less than A(k)/(q 1 − 1) = o(1), and hence the theorem follows.10.3.Another result for SL.For any positive interger k, let a denote a fixed increasing sequence a 1 < • • • < a k of positive integers.By an a-flag in an F q -vector space V , we mean a flag , where we define a 0 := 0 and a by [LMaT,Lemma 4.1], we have (10.11) where In particular, as N goes to infinity, (10.12) where the implicit constant depends on a.Moreover, (10.13) Lemma 10.3.Let k and m be positive integers, and let a be an increasing sequence of k positive integers.If N is sufficiently large in terms of m and a, then for all g ∈ GL N (q) with supp(g) = m, the number of g-stable a-flags in F N q can be written Proof.We may assume N > 3m, so the eigenvalue λ of multiplicity N − m is unique and therefore lies in F q .Let W λ ⊂ F N q denote the generalized λ-eigenspace of g and W λ the direct sum of the generalized eigenspaces of g for all eigenvalues other than λ.Thus dim If dim V k ∩ W λ < a k , then by applying (10.11) to sequences of length 1, we see that the number of possibilities for V k is less than so the total number of possibilities for the whole flag is less than q (a k −1)N − O(1) .
If dim V k ∩ W λ = a k , then V k ⊂ W λ .Let I λ and K λ denote the image of λ − g on W λ and the kernel of λ − g respectively.Because V k is g-stable, either V k ⊂ K λ or V k ∩ I λ = {0}.In the latter case, V k is spanned by a non-zero vector in I λ and a subspace of W λ of dimension a k − 1.As dim I λ ≤ k < N/3, the number of possibilities for the whole flag is less than q (a k −1)N − O(1) .
Finally, we consider the number of possibilities when V k ⊂ K λ .As g acts on K λ as scalar multiplication, all a-flags with V k ⊂ K λ are g-stable.The total number is F a (dim K λ ) = F a (N − m) = q a k m (1 − q −N +O(1) )F a (N ), by (10.13).The lemma follows.
The unipotent characters of GL N (q) are indexed by partitions λ ⊢ N , and we say χ = χ λ has level N − λ 1 , where the parts of λ are arranged from largest to smallest, see [GLT,§3].Also recall that, for λ, µ ⊢ N , the Kostka number K λµ is the number of semistandard Young tableaux of shape λ and weight µ.
Proof.In any semistandard Young tableaux of shape λ and weight µ, the first µ 1 entries of the first row must have filled with value 1, and the remaining boxes in the first row are all to the right of every box in the remaining rows.Therefore, such a tableau is determined by choosing from the µ 2 values 2, the µ 3 values 3, and so on, an arbitrary weakly increasing sequence for the λ 1 − µ 1 remaining boxes in the first row, and from the values that remain, a semistandard Young tableau of shape (λ 2 , λ 3 , . ..).The number of such choices depends only on (λ 2 , λ 3 , . ..) and (µ 2 , µ 3 , . ..), but not on N .
Proposition 10.5.Let m and n be fixed positive integers.If N is a positive integer sufficiently large in terms of m and n, χ is a unipotent character of GL N (q) of level n, and g ∈ GL N (q) has support m, then (10.15) q mn χ(g) χ(1) − 1 < q −N/3 .
For each µ ⊢ N , we define the increasing sequence a µ of positive integers such that the sequence a 1 = a 1 − a 0 , . . ., a k+1 − a k = N − a k gives the parts of µ in increasing order.
As every Kostka matrix K (for partitions of N ) is unitriangular, we can invert and write χ = χ λ as a linear combination of permutation characters associated to φ µ , where µ λ.We can therefore express each unipotent character of level n, including χ λ , as a linear combination of permutation characters χ b,N associated to flags b with maximal dimension ≤ n, with coefficients which are entries in the inverse Kostka matrix K −1 .
Note that, for any fixed n, the set of partitions λ ⊢ N with λ 1 ≥ N − n depends only on n, but not on N , m, or q.The unitriangularity of K implies that the submatrix of K −1 , truncated to only partitions of N with the first part ≥ N − n, is the inverse of the submatrix of K, truncated to the same set of partitions.Applying Lemma 10.4, we see that all entries of this truncated submatrix of K −1 are bounded by some constant O(1) that depends only on n: Define ǫ µ so that (10.18) φ µ (g) = q (µ 1 −N )m (1 + ǫ µ )φ µ (1).
By Lemma 10.3, for fixed m and µ, |ǫ µ | < q −N/2 if N is sufficiently large and g is of support m.
It is clear that q − 1 divides both factors.If a prime ℓ divides gcd(c(q − 1), |T |) = gcd((q a + q b )(q − 1), q p − 1), then it divides |T |, so it cannot divide q.It must also divide either q a + q b or q − 1 or both.If it divides q − 1 but not q a + q b , then the highest power of ℓ dividing (q a + q b )(q − 1) is the same as the Theorem 10.7.For all but finitely many ordered pairs (p, q) where p ≥ 3 is prime and q is a prime power, the following statement holds.If t is a generator of the norm-1 subgroup T 1 ∼ = C (q p −1)/(q−1) of F × q p then every non-central element of SL p (q) is a product of two conjugates of t.Proof.Fixing an F q -basis of F q p we can identity F × q p with the centralizer T in G := GL p (q) of any generator of F q p .As p is prime, every non-central element of T generates F q p as F q -algebra, so no such element is contained in a proper parabolic subgroup of G. Therefore, every Harish-Chandra induced character of G vanishes on every element of T Z(G); in particular at our element t.By [Gr,(12)], a primary (i.e.not Harish-Chandra induced) character of G can be non-zero at t if and only if it is of the form I k 1 [p] or of the form I k p [1].In the first case, it belongs to the set X of Proposition 10.6.In the second case, it is the product of a unipotent character and a linear character.Since C G (t) = T , the conjugate classes of t in G and in SL p (q) are the same.
By Theorem 10.1, we may assume that our target element g has bounded support.By the Frobenius formula, (q −1) −1 times the number of representations of g as a product of two conjugates of t in G is χ(t) 2 χ(g) χ( 1) .
We divide this sum into a sum over χ which are unipotent characters times linear characters and a sum over χ ∈ X .
For the first, we note that t and g are both in SL p (F q ), and all linear characters of G are trivial on this subgroup.So we can simply sum over unipotent characters and omit the factor (q − 1) −1 .The contribution of the trivial character to the sum χ∈Irr(G) χ(t) 2 χ(g) χ( 1) is 1.For the other unipotent characters, by [Gr,Theorem 12], χ λ (t) is given by the value at a p-cycle of the character of the symmetric group S p associated to the partition λ ⊢ p, and by the Murnaghan-Nakayama rule, this value is ±1 if λ is of the form 1 n (p − n) 1 and 0 otherwise.By the main theorem of [LT], since g / ∈ Z(G), there exists an absolute constant ǫ > 0 such that (10.22) |χ(g)| ≤ χ(1) 1−ǫ/p for all χ ∈ Irr(G).By the dimension formula for primary characters of G [Gr,Lemma 7.4], χ λ (1) ≥ q n(p−n/2−1/2) ≥ q np/3 .
On the other hand, if p is large enough compared to A, then by Proposition 10.5, for 1 ≤ n < A we have χ 1 n (p−n) 1 (g) > 0, and is simply the Fourier transform on the abelian group Pow 0 (X ⊖ ).As such R 2 0 is the identity on C[S od ].To have a compatible notation we consider R 0 as a map U od → A od and denote by Q 0 : A od → U od its inverse.Assume (d, i) ∈ Z (2) with d = 0.It follows from Lemma 3.3 that the restriction of the map H0 d,i to C[ Sod ] factors through a well-defined endomorphism of U od and A od which we denote by H 0 d,i .Similarly H1 d,i • ε factors through an endomorphism which we denote by H 1 d,i .The following is now simply a consequence of Theorem 3.1.
≺) is a finite totally ordered set of cardinality |I| = 2n.Denote by † : I → I the unique order reversing bijection on I.We say a ∈ I is positive or negative if a ≻ a † or a ≺ a † respectively.This gives a decomposition I = I + ⊔ I − into subsets of cardinality n.If O ⊆ I is a subset then (O, ≺) is also a totally ordered set.Example 4.1.We could take I = {−n ≺ • • • ≺ −1 ≺ 1 ≺ • • • ≺ n} then for any a ∈ I we have a † = −a so I + = {1, . . ., n} and I − = {−1, . . ., −n}.If S I is the symmetric group on I then we define W I = C S I (σ) to be the centraliser of the involution σ = a∈I + (a, a † ).Let δ I : S I → Z/2Z be the unique non-trivial homomorphism.Given e ∈ {0, 1} we let W e I = {w ∈ W I | δ I (w) = e} so that we have a decomposition W I = W 0 I ⊔ W 1 I into the cosets of W 0 I ⊳ W I .Note we have a semidirect product decomposition W I = N I ⋊ H I where N I = (a, a † ) | a ∈ I + and H I = {w ∈ W I | w I + = I + } ∼ = S I + .For any σ-stable subset O ⊆ I, equivalently O = O † , we have a natural injective homomorphism W O → W I whose image is the pointwise stabiliser of I O.We identify W O with its image in W I .We say w ∈ W I is an I-cycle if the subgroup w, σ W I acts transitively on I. Thus w = nh, with n ∈ N I and h ∈ H I acting on I + as cycle of length n.The following is an elementary calculation.

Theorem 4. 7 .
Fix an integer 1 k n.Then for each element w ∈ W I of cycle length k and each irreducible character χ ∈ Irr(W I ) we have |χ Y op ] ) = (−1) e Res W I W e I (ρ [Y ] Clearly this is contained in the subspace of all C-class functions Class(G F , λ, a) spanned by the Lusztig series E(G F , λ, a).In fact, R G λ,a gives an isomorphism Class(F a) → Class 0 (G F , λ, a) onto the subspace spanned by {R G x (λ) | x ∈ a}; see the arguments in [DiM, §11.6].Suppose we choose a representative a = wW • (λ) of the coset.We may then form the semidirect product W • (λ):F w by the group F w Aut(W • (λ)) as above.The coset of W • (λ) in W • (λ):F w containing F w can be identified, W • (λ)-equivariantly, with the same coset in W :F .
It follows from[Lu1, 4.21.6] that |F γ | |G F | 2 .We now use Lemma 7.3.We assume F is a Frobenius endomorphism.If [G, G] is quasisimple of type B n or C n then W ∼ = W I is a hyperoctahedral group, and F induces the identity on W .If [G, G] is quasisimple of type D n then W ∼ = W 0 I and either F , F 2 , or F 3 induces the identity on W .When F 2 induces the identity on W , we have an embedding W :F → W I .Thus, it makes sense to speak of the cycles of an element of the coset W F .The following is now just a simple application of Lemma 4.3.Corollary 7.8.Assume [G, G] is quasisimple of type B n (n 2), C n (n 2), or D n (n 4), and F is a Frobenius endomorphism with F 2 inducing the identity on W .If wF ∈ W F has cycle length k 1 and pairwise distinct cycles, then for any regular semisimple element g ∈ G F of type wF we have |χ(g)| 2 n+k • n k and n 1 > . . .> n d (and c + d ≤ k ≤ c + d + 2 for the cycle length k).Note that if the irreducible factors of the characteristic polynomial of g have pairwise distinct degrees, then g has pairwise distinct cycles.
This clearly factors through a map C[ S] → C[S].The image of − has a natural complement in C[S], namely X + − X − | X ∈ P is degenerate C .