On Unipotent Supports of Reductive Groups with a Disconnected Centre

Jay Taylor

Abstract. Let G be a simple algebraic group de�ned over a �nite �eld of good characteristic, with associated Frobenius endomorphism F. In this article we extend an observation of Lusztig, (which gives a numerical relationship between an ordinary character of G^{F} and its unipotent support), to the case where Z(G) is disconnected. We then use this observation in some applications to the ordinary character theory of G^{F}.

Throughout this article G will be a connected reductive algebraic group over \overline{}_{p}, an algebraic closure of the �nite �eld _{p}, where p is a good prime for G. Furthermore, we assume G is de�ned over _{q} ⊂ \overline{}_{p} (where q is a power of a p) and F : G → G is the associated Frobenius endomorphism. Throughout we will denote an algebraic group in bold and its corresponding rational structure in roman, for instance G := G^{F}.

1. Introduction

Let Irr(G) denote the set of ordinary irreducible characters of G. In [Lus92] Lusztig has associated to every χ ∈ Irr(G) a unique F-stable unipotent class of G called the unipotent support of χ, which we denote by O_{χ}. This was originally done under the assumption that p and q are su�ciently large, however the assumptions on p and q were later removed in [GM00]. A consequence of the existence of unipotent supports is that we have a surjective map

Φ_{G} : Irr(G) → {F-stable unipotent conjugacy classes of G}

given by Φ_{G}(χ) := O_{χ}.

Recall that Lusztig has shown for each χ ∈ Irr(G) there is a well de�ned integer n_{χ} such that n_{χ} · χ(1) is a polynomial in q with integer coe�cients. If x ∈ G then we write A_{G}(x) for the component group C_{G}(x)/C_{G}^{∘}(x). The following result was observed by Lusztig in [Lus84a, 13.4.4] and later veri�ed by H� ezard and Lusztig through a detailed case by case analysis, (see [Lus09] and [H� ez04]).

Theorem 1.1 (H�ezard, Lusztig). Assume G/Z(G) is simple, Z(G) is connected and O is an F-stable unipotent class of G. There exists a character χ ∈ Irr(G) such that Φ_{G}(χ) = O and n_{χ} = |A_{G}(u)| for any u ∈ O.

The usefulness of characters satisfying this technical condition can be seen in Kawanaka’s theory of generalised Gelfand�Graev representations (GGGRs). In particular, if χ satis�es this technical condition then it severely limits the appearance of χ as a constituent of the character of a GGGR associated to a unipotent element in O_{χ}^{F}. The main result of this article is an extension of Theorem 1.1 to include all simple groups, see Theorem 3.1.

\underline{2}

The layout of this paper is as follows. In Section 2 we will show there exist class representatives of each F-stable unipotent class such that a sensible generalisation of Theorem 1.1 can be made, (we call such representatives well chosen). In Section 3 we will state and prove our main result. Using this result we prove in Sections 4 and 6, (under the assumption p and q are large), that a conjecture of Kawanaka on unipotently supported class functions holds for simple groups which are not spin / half-spin groups. This is based on the work of Geck and H� ezard in [GH08] which uses Theorem 1.1. Finally, in Sections 7 and 8 we give an expression of a certain fourth root of unity that is associated to GGGRs when G is a special orthogonal or symplectic group. This is based on an argument used by Geck in [Gec99] which uses Theorem 1.1.

Acknowledgments: The work presented in this article was completed as part of the authors PhD quali�- cation under the supervision of Prof. Meinolf Geck. The author wishes to express his deepest thanks to Prof. Geck for proposing this problem and for his vital guidance. Part of this work was completed while the author spent six months at Lehrstuhl D f � ur Mathematik, RWTH Aachen University as an ERASMUS exchange student and also while the author spent two weeks at Universit� e Montpellier 2. The author warmly thanks both institutions for their kind hospitality during his visits. The author also thanks Dr. Frank L � ubeck for many essential discussions, (in particular for suggesting the approach taken in section 2 of this paper), and Prof. C� edric Bonnaf� e, (for assistance with some technical issues in type A).

2. Component Groups

For the remainder of this article, unless otherwise stated, we will assume G is simple. We �x a regular embedding ι : G → G̃, (as de�ned in [Lus88, §7]), where G̃ is a connected reductive group such that Z(G̃) is connected. We now �x a dual group of G, (resp. G̃), which we denote by G^{⋆}, (resp. G̃^{⋆}), and a Frobenius endomorphism F^{⋆} : G^{⋆} → G^{⋆} such that the pairs (G, F) and (G^{⋆}, F^{⋆}) are in duality. Similarly we denote again by F and F^{⋆} compatible Frobenius endomorphisms of G̃ and G̃^{⋆}. Recall that ι determines a corresponding surjective morphism ι^{⋆} : G̃^{⋆} → G^{⋆} between dual groups, which is de�ned over _{q}. For smoothness of the exposition if Z(G) is connected then we will take G̃ to be G and ι to be such that ι(g) = g for all g ∈ G.

The embedding ι de�nes a bijection between the unipotent conjugacy classes of G and the corresponding classes in G̃. If u is a unipotent element of G then we will implicitly identify u with its image ι(u) ∈ G̃. The purpose of this section is to show that in a simple algebraic group we can always pick representatives of an F-stable unipotent class such that they are well chosen. By this we mean the following.

De�nition 2.1. Let O be an F-stable unipotent class of G. We say a class representative u ∈ O^{F} is well chosen if |A_{G}(u)^{F}| = |Z_{G}(u)^{F}||AG̃(u)|, where Z_{G}(u) is the image of Z(G) in A_{G}(u).

Lemma 2.2. Let G be a connected reductive algebraic group and C an F-stable conjugacy class of G such that A_{G}(x) is abelian for some x ∈ C^{F}. Then the following are equivalent:

(i) the number of G-conjugacy classes contained in C^{F} is |A_{G}(x)|,

(ii) A_{G}(x)^{F} = A_{G}(x),

(iii) A_{G}(x′)^{F} = A_{G}(x′) for all x′∈ C^{F}.

Proof. By [Gec03, Theorem 4.3.5] we know the G-classes contained in C^{F} are in bijective correspondence with the F-conjugacy classes of A_{G}(x). Therefore the number of G-classes contained in C^{F} is the same as |A_{G}(x)| if and only if every F-conjugacy class of A_{G}(x) contains only one element. As A_{G}(x) is abelian this proves the equivalence of the �rst two statements. If (i)is true for x ∈ C^{F} then it is true for all x ∈ C^{F} as |A_{G}(x)| is independent of the choice of x. Therefore this together with the equivalence of (i) and (ii) gives the equivalence of (i)and (iii). ■

Lemma 2.3. Let G be a connected reductive algebraic group and C an F-stable conjugacy class of G. If the automorphism of A_{G}(x) induced by F, for some x ∈ C^{F}, is an inner automorphism then there exists x′ ∈ C^{F} such that A_{G}(x′)^{F} = A_{G}(x′).

Proof. Assume there exists an element y ∈ C_{G}(x) such that for all zC_{G}^{∘}(x) ∈ A_{G}(x) we have F(zC_{G}^{∘}(x)) = y^{−1} zyC_{G}^{∘}(x). As y ∈ G, by the Lang-Steinberg theorem, there exists g ∈ G such that y = g^{−1} F(g) ⇒ F(g) = gy. Clearly A_{G}(^{g}x) = ^{g}A_{G}(x) so gzC_{G}^{∘}(x)g^{−1} ∈ A_{G}(^{g}x). Furthermore we have

F(gzC_{G}^{∘}(x)g^{−1} ) = gy(y^{−1} zy)C_{G}^{∘}(x)y^{−1} g^{−1} = gzC_{G}^{∘}(x)g^{−1} ,

so F acts trivially on A_{G}(^{g}x). Taking x′ = ^{g}x ∈ C^{F} we have A_{G}(x′)^{F} = A_{G}(x′) as required. Note x′ is �xed by F because y ∈ C_{G}(x). ■

Proposition 2.4. Let G be a connected reductive algebraic group with connected centre such that G/Z(G) is simple, then for any F-stable unipotent class O ⊂ G we have A_{G}(u)^{F} = A_{G}(u) for some unipotent element u ∈ O^{F}.

Proof. The natural surjective morphism of algebraic groups π : G → G/Z(G) induces a bijection between the unipotent classes of G and G/Z(G). If u ∈ G is unipotent then the restriction of π to C_{G}(u) induces an isomorphism A_{G}(u) ≅ A_{G/Z(G)} (π(u)), which is de�ned over _{q}. Therefore we may prove this result using any group G which has a connected centre and simple quotient G/Z(G).

The structure of the component groups A_{G}(u) have been determined on a case by case basis. In the case where G is an adjoint exceptional group it is known that A_{G}(u) is either trivial or isomorphic to a symmetric group 𝔖_{2}, 𝔖_{3}, 𝔖_{4} or 𝔖_{5}, (see for example the tables in [Car93, §13.1]). Every automorphism of such a group is an inner automorphism, so the result holds by Lemma 2.3.

We now turn to the classical groups. The case of type A_{n} is trivial as all component groups are trivial. Assume Ḡ is either a symplectic or special orthogonal group and Ō ⊂ Ḡ is an F-stable unipotent class with representative ū ∈ Ō^{F}. The component group AḠ(ū) is an elementary abelian 2-group whose order is given in [SS70, IV - 2.26 and 2.27]. By inspecting the descriptions of |AḠ(ū)| we see by [Wal63, p. 38], (resp. [Wal63, p. 42]), for the symplectic groups, (resp. special orthogonal groups), that the number of Ḡclasses contained in Ō^{F} is the same as |AḠ(ū)|. Note the statement given in [Wal63] is for the orthogonal groups but the maximal number of Ḡ-classes contained in Ō^{F} is |AḠ(ū)| and there can only be less classes in the orthogonal group so this su�ces. As AḠ(ū) is abelian we have AḠ(ū)^{F} = AḠ(ū) by Lemma 2.2. The odd dimensional special orthogonal groups are adjoint groups of type B_{n}, so this case is covered.

Let G be an adjoint group of type C_{n} or D_{n} then there exists a symplectic or special orthogonal group Ḡ such that we have an isogeny π : Ḡ → G which is de�ned over _{q}. Let ū be a unipotent element of Ḡ and set u := π(ū). The restriction of π to CḠ(ū) induces a surjective morphism AḠ(ū) → A_{G}(u) de�ned over _{q}, therefore AḠ(ū)^{F} = AḠ(ū) implies A_{G}(u)^{F} = A_{G}(u) so we are done. ■

Remark 2.5. The statement of Proposition 2.4 is not a new result. It follows in most cases from a much stronger statement concerning the existence of so called split elements, see [Sho87, Remark 5.1]. However, we include this here as it gives a simpler argument for this statement and also circumvents the caveat that split elements do not always exist in type E_{8}.

We want to discuss how to relate the component group A_{G}(u) to the component group AG̃(u), (for this we follow [Gec96, p. 306]). The group G̃ is an almost direct product G̃ = GZ(G̃) so we get an almost direct product CG̃(u) = C_{G}(u)Z(G̃) and the restriction of ι to the centraliser C_{G}(u) induces a surjective map A_{G}(u) → AG̃(u). It is easy to see that the kernel of this map is the image of Z(G) in A_{G}(u) so we obtain a sequence of maps

Z(G) → A_{G}(u) → AG̃(u),  (2.1)

which is exact at A_{G}(u). As all morphisms are de�ned over _{q} this induces a sequence

Z(G)^{F} → A_{G}(u)^{F} → AG̃(u),  (2.2)

which is again exact at A_{G}(u)^{F}, (where here we assume u is chosen such that AG̃(u)^{F} = AG̃(u)). Although the map A_{G}(u) → AG̃(u) in (2.1)is surjective it is not necessarily the case that the map A_{G}(u)^{F} → AG̃(u) in (2.2) is surjective. If we can show that this is the case then the existence of well-chosen elements will be assured.

There are two cases where this is trivial to show. If |AG̃(u)| = 1 then it is obvious that such a map is surjective. The second case is when |A_{G}(u)| = |AG̃(u)| because then we must have A_{G}(u) ≅ AG̃(u), which implies A_{G}(u)^{F} ≅ AG̃(u)^{F}. In fact we are almost always in the situation of these two trivial cases, (see for example the information gathered in [Tay12, Chapter 2]). We deal with the remaining cases in exceptional groups in two lemmas. In the following lemmas we label the unipotent conjugacy classes as in [Car93, §13.1].

Lemma 2.6. Let G be the simply connected group of type E_{6} and O the unipotent class E_{6}(a_{3}). For any u ∈ O^{F} we have AG̃(u)^{F} = AG̃(u) and A_{G}(u)^{F} → AG̃(u) is surjective.

Proof. We see that A_{G}(u) is isomorphic to _{3} × _{2}, as it is a �nite group of order 6 with a non-trivial centre. The group AG̃(u) is abelian and AG̃(u)^{F} = AG̃(u) by Proposition 2.4 and Lemma 2.2. Let Z(G) = ⟨z⟩ and x ∈ A_{G}(u) be the element of order 2 such that A_{G}(u) = ⟨z, x⟩. As the action of F on AG̃(u) is trivial we must have F(x) ∈ {x, xz, xz^{2}}. However the element x is of order 2 but xz and xz^{2} are of order 6, so F(x) = x. ■

Lemma 2.7. Let G be the simply connected group of type E_{7} and O the unipotent class D_{4}(a_{1}) + A_{1}, E_{7}(a_{3}), E_{7}(a_{4}) or E_{7}(a_{5}). Then for some u ∈ O^{F} we have AG̃(u)^{F} = AG̃(u) and A_{G}(u)^{F} → AG̃(u) is surjective.

Proof. Let u ∈ O^{F} be an element such that its image u ∈ Õ^{F} has the property AG̃(u)^{F} = AG̃(u), which is possible because of Proposition 2.4. We use the work of Mizuno [Miz80] to con�rm that F only acts non-trivially on Z_{G}(u). By

(i) [Miz80, Lemma 30] for D_{4}(a_{1}) + A_{1} � Mizuno label D_{4}(a_{1}) + A_{1},

(ii) [Miz80, Lemma 13] for E_{7}(a_{3}) � Mizuno label D_{6} + A_{1},

(iii) [Miz80, Lemma 17] for E_{7}(a_{4}) � Mizuno label D_{6}(a_{1}) + A_{1}, (iv) and [Miz80, Lemma 21(2)] for E_{7}(a_{5}) � Mizuno label D_{6}(a_{2}) + A_{1},

we know the number of G-classes contained in O^{F} is |A_{G}(u)|. If O is not E_{7}(a_{5}) then the group A_{G}(u) is abelian so the result follows by Lemma 2.2.

Assume O is the class E_{7}(a_{5}), by [Miz80, Table 9] we see A_{G}(u) is isomorphic to 𝔖_{3} × _{2}. Let Z(G) = ⟨z⟩, then z is a generator for the _{2} component of A_{G}(u). We know the action of F on the component group AG̃(u) is trivial so we need only show F(x) ≠ xz for all x ∈ A_{G}(u) − Z_{G}(u). If x is a 3-cycle of A_{G}(u) then we cannot have F(x) = xz because x and xz are of di�erent orders. If x is a 2-cycle such that F(x) = xz then the number of F-conjugacy classes of A_{G}(u) would be less than 6 and this cannot possibly happen. ■

We now deal with the cases of classical type. Let G be a spin group, (in other words a simply connected group of type B_{n} or D_{n}), and let O be a unipotent class of G. Recall that to every unipotent class of G we have a corresponding partition λ of N, where N is the dimension of the spin group, such that any even number in λ occurs an even number of times. This partition is determined by the Jordan blocks of a corresponding class in a special orthogonal group. The non-trivial cases we need to consider occur when the partition λ contains at least one odd number and any odd number occurs at most once, (equivalently these are the classes such that Z(G) embeds into A_{G}(u)). We assume now that O is a class with such a partition. In [Lus84b, §14.3] Lusztig gives a description of the component group A_{G}(u), which we recall here.

First of all let Ḡ be a special orthogonal group such that we have an isogeny π : G → Ḡ. Recall that the kernel of this isogeny is a central subgroup of order 2, which we denote Ker(π) = {1, ϑ}. Every non-trivial element of Z(G) determines a non-trivial element of A_{G}(u), in particular ϑ determines a nontrivial element of A_{G}(u). Let I := {a_{1}, …, a_{k}} be the set of odd numbers occurring in λ. We let S be the group generated by the elements ϑ, x_{1}, …, x_{k}, which satisfy the relations

ϑ^{2} = 1  x_{i}^{2} = ϑ^{a_{i}(a_{i}−1)/2}  x_{i}x_{j} = x_{j}x_{i}ϑ  ϑx_{i} = x_{i}ϑ

for all i ≠ j. The group A_{G}(u) is then isomorphic to the subgroup of S which consists of all elements that can be expressed as a word in an even number of the generators x_{1}, …, x_{k}. For the remainder of this discussion we will identify A_{G}(u) with its image in S. For all 2 ⩽ i ⩽ k let y_{i} = x_{1}x_{i} then {ϑ, y_{2}, …, y_{k}} is a set of generators for A_{G}(u).

Consider the image ū = π(u) of u in the special orthogonal group Ḡ. The map π induces a surjective map A_{G}(u) → AḠ(ū) with kernel {1, ϑ}. By Proposition 2.4 and Lemma 2.2 we know the action of F on AḠ(ū) must be trivial for any class representative u ∈ O^{F}. Therefore given any element x ∈ A_{G}(u) we have F(x) ∈ {x, xϑ}, however we always have F(ϑ) = ϑ. Any automorphism of A_{G}(u) is uniquely determined by its action on the generators, so we break the study of F into two possible cases:

(a) F(y_{i}) = y_{i}ϑ for an even number of y_{i}’s,

(b) F(y_{i}) = y_{i}ϑ for an odd number of y_{i}’s.

In case (a)we claim that the automorphism is an inner automorphism. We start by noticing that if i_{1}, i_{2}, j_{1}, j_{2} are all distinct then x_{i_{1}}x_{i_{2}} and x_{j_{1}}x_{j_{2}} commute. Let σ_{ij} be the automorphism such that σ_{ij}(y_{i}) = y_{i}ϑ and σ_{ij}(y_{j}) = y_{j}ϑ but σ_{ij}(y_{ℓ}) = y_{ℓ} whenever i, j, ℓ are all distinct. It is easy to verify that σ_{ij} is conjugation by x_{i}x_{j}, so is inner. As F must be a composition of automorphisms of the form σ_{ij} we have F is also inner.

We now consider case (b). Assume G is of type D_{n} and by composing with a su�cient number of inner automorphisms let us assume that F acts non-trivially on precisely one generator y_{i}. As F acts nontrivially only on y_{i} it will be true that A_{G}(u)^{F} is the subgroup ⟨ϑ, y_{2}, …, y_{i−1} , y_{i+1} , …, y_{k}⟩, in particular |A_{G}(u)^{F}| = 2^{|I|−1} . As G is of type D_{n} we have

|A_{G}(u)^{F}|/|AG̃(u)| = 2^{|I|−1} /2^{|I|−2} = 2.  (2.3)

We claim that we also have |Z_{G}(u)^{F}| = 2. By the third paragraph of [Lus08, §3.7(a)] we know the centre of A_{G}(u) is given by {1, ϑ, z, zϑ} where z is the element x_{1}x_{2} ⋯x_{k−1} x_{k}. Note that this means the centre of A_{G}(u) coincides with the image of Z(G). We have the following expression of z in terms of our chosen generating set

z = x_{1}x_{2} ⋯x_{k} = y_{2}y_{3} ⋯y_{k}ϑ^{c}

for some c ∈ {0, 1}. Therefore we have F(z) = zϑ and F(zϑ) = z so Z_{G}(u)^{F} = {1, ϑ} as required.

Assume now G is of type B_{n}. Recall that λ partitions an odd number so k − 1 is even. By composing with a su�cient number of inner automorphisms let us assume that F acts on all but one generator nontrivially. Assume y_{i} is the generator such that F(y_{i}) = y_{i} then it is easy to check that y_{j}^{x_{1}x_{i}} = y_{j}θ whenever j ≠ i and clearly y_{i}^{x_{1}x_{i}} = y_{i}. In particular F is an inner automorphism. With this in hand we can prove the following proposition, which is vital for extending Theorem 1.1. Proposition 2.8. Assume G is a simple algebraic group then any F-stable unipotent class O of G contains a well-chosen representative. Proof. If G is adjoint then the existence of well-chosen representatives follows from Proposition 2.4, hence we may assume Z(G) is disconnected. We have already covered the two trivial cases where |A_{G}(u)| = |AG̃(u)| or |AG̃(u)| = 1, which covers the case of type A_{n}. If G is an exceptional group then the only cases left to consider are those covered by Lemmas Lemma 2.6 and Lemma 2.7. Assume G is a symplectic group or a special orthogonal group of type D_{n} then the statement was proved as part of the proof of Proposition 2.4. If G is a spin group then the result follows from the above discussion and Lemma 2.3. Finally assume G is a half spin group and π : G_{sc} → G is a simply connected cover of G. We need only comment that case (b)above cannot happen because the Frobenius cannot act non-trivially on the centre of G_{sc} as it must preserve Ker(π). In other words the map A_{G}(u)^{F} → AG̃(u) is always surjective. ■

From now on we assume for each F-stable unipotent class O of G that u ∈ O^{F} is a �xed well-chosen class representative of O.

3. The Main Result

Let \overline{}_{ℓ} be an algebraic closure of the �eld of ℓ-adic numbers where ℓ > 0 is a prime distinct from p. We assume �xed once and for all an automorphism \overline{}_{ℓ} → \overline{}_{ℓ} of order two denoted by x ↦ \overline{x} such that \overline{ω} = ω^{−1} for all roots of unity ω ∈ \overline{}_{ℓ}^{×} . If H is a �nite group we denote by Cent(H) the space of all class functions f : H → \overline{}_{ℓ}. This is an inner product space with respect to the standard inner product ⟨−, −⟩_{H} : Cent(H) × Cent(H) → \overline{}_{ℓ} given by

⟨f, f′⟩_{H} = \frac{1}{|H|} ∑_{h∈H} f(h)\overline{f′(h)} ,

(we will write ⟨−, −⟩ when H is clear from the context). We have a set of irreducible characters Irr(H) ⊂ Cent(H) which forms an orthonormal basis of Cent(H). If K is a subgroup of H then we denote by Res_{K}^{H} : Cent(H) → Cent(K) and Ind_{K}^{H} : Cent(K) → Cent(H) the standard restriction and induction maps.

From now on representatives of F-stable unipotent classes will be assumed well chosen. Before we can state and prove the main result of this paper we must �rst recall some facts regarding the character theory of G. Let s ∈ G^{⋆} be a semisimple element then we denote by E(G, s) the rational Lusztig series determined by the G^{⋆}-conjugacy class of s. We denote by C_{G^{⋆}}^{∘}(s) the �xed point group C_{G^{⋆}}^{∘}(s)^{F^{⋆}} then this forms a normal subgroup of the full centraliser C_{G^{⋆}}(s). As such we have the quotient group A_{G^{⋆}}(s) := C_{G^{⋆}}(s)/C_{G^{⋆}}^{∘}(s) acts on Irr(C_{G^{⋆}}^{∘}(s)) by conjugation. This action stabilises each Lusztig series and we let E(C_{G^{⋆}}^{∘}(s), 1)/A_{G^{⋆}}(s) denote the orbits of the unipotent characters under the action of A_{G^{⋆}}(s).

We remark also that as G is a normal subgroup of G̃ we have a natural action of G̃/G · Z(G̃) on Irr(G) by conjugation. This action stabilises each Lusztig series of G. In [Lus88, Proposition 5.1] Lusztig shows that there exists a bijection

Ψ_{s} : E(G, s) → E(C_{G^{⋆}}^{∘}(s), 1)/A_{G^{⋆}}(s)  (3.1)

such that:

(i) the �bres of Ψ_{s} are the orbits of the action of G̃/G · Z(G̃) on E(G, s);

(ii) if Θ ∈ E(C_{G^{⋆}}^{∘}(s), 1)/A_{G^{⋆}}(s) is an orbit and Γ is the stabiliser of an element of Θ in A_{G^{⋆}}(s) then |Ψ_{s}^{−1} (Θ)| = |Γ|.

This is a generalisation of the Jordan decomposition of characters. To prove the above result Lusztig shows that the restriction of every irreducible character of G̃ to G is multiplicity free. With this framework in mind we can now state and prove the main result of this article.

Theorem 3.1. Assume G is a simple algebraic group and O is an F-stable unipotent class of G. There exists a character χ ∈ Irr(G) such that Φ_{G}(χ) = O and n_{χ} = |A_{G}(u)^{F}| = |Z_{G}(u)^{F}||AG̃(u)|, where u ∈ O^{F} is a well-chosen representative.

Proof. If G is an adjoint group then this is merely the statement of Theorem 1.1, so we may assume Z(G) is disconnected. We write (s̃, ψ) for a pair such that s̃∈ G̃^{⋆} is a semisimple element, (s := ι^{⋆}(s̃) ∈ G^{⋆}), and ψ ∈ E(C_{G^{⋆}}^{∘}(s), 1) is a unipotent character. We denote by ψ̃ ∈ E(G̃, s̃) the character uniquely determined by the sequence of bijections

E(G̃, s̃) → E(CG̃^{⋆}(s̃), 1) → E(C_{G^{⋆}}^{∘}(s), 1),  (3.2)

where the �rst bijection comes from the usual Jordan decomposition of characters and the last bijection comes from [DM91, Proposition 13.20].

Let us assume that (s̃, ψ) is a pair which satis�es the following properties:

(P1) n_{ψ} = |AG̃(u)|.

(P2) | Stab_{A_{G^{⋆}}(s)}(ψ)| = |Z_{G}(u)^{F}|.

(P3) ΦG̃(ψ̃) = O, where here we identify O with its image ι(O) in G̃.

By the Jordan decomposition of characters the character degree of ψ satis�es ψ̃(1) = ψ_{sc}(1)ψ(1), where ψ_{sc} ∈ E(G̃, s̃) is the unique semisimple character contained in the Lusztig series. By [Car93, Theorem 8.4.8] we have |G̃^{⋆}|_{p′} = |CG̃^{⋆}(s̃)|_{p′}ψ_{sc}(1) and by the order formula for �nite reductive groups, (see [Car93, pg. 75]), both |G̃^{⋆}|_{p′} and |CG̃^{⋆}(s̃)|_{p′} are monic polynomials in q. In particular ψ_{sc}(1) must also be a monic polynomial in q hence nψ̃ = n_{ψ} = |AG̃(u)| by (P1). Using the multiplicity free property mentioned above we have ResG̃ ; G(ψ̃) = χ_{1} + ⋯+ χ_{k}, where χ_{i} ∈ E(G, s) and χ_{i}(1) = | Stab_{A_{G^{⋆}}(s)}(ψ)|^{−1} ψ̃(1). For any 1 ⩽ i ⩽ k we have by (P2)that

n_{χ_{i}} = | Stab_{A_{G^{⋆}}(s)}(ψ)| · nψ̃ = |Z_{G}(u)^{F}||AG̃(u)| = |A_{G}(u)|.

By (P3)and [GM00, Theorem 3.7] we know O is the unipotent support of each χ_{i} hence any χ_{i} will satisfy the statement of the theorem. ■

Our proof of Theorem 3.1 will be complete once we have veri�ed the following result.

Theorem 3.2. For every simple algebraic group G with a disconnected centre and every F-stable unipotent class O of G there exists a pair (s̃, ψ), (as speci�ed in the proof of Theorem 3.1), satisfying (P1)to (P3). Furthermore s̃ can be chosen such that the image of s under an adjoint quotient of G^{⋆} is a quasi-isolated semisimple element. Let F ⊆ E(C_{G^{⋆}}^{∘}(s), 1) be the family of unipotent characters containing ψ then the following extra conditions hold, unless G is a spin / half spin group and A_{G}(u) is non-abelian:

(P4) the pair (s̃, W^{⋆}(F̃)) is d-good (for a de�nition see [Tay14, De�nition 2.4] and [Gec99, 4.4]).

(P5) 𝔛_{F} := {ψ ∈ F | | Stab_{A_{G^{⋆}}(s)}(ψ)| ≠ |Z_{G}(u)^{F}|} = ∅.

We will see that (P4)and (P5)will be used in applications below, in particular (P4)is required so that [Gec99, Theorem 4.5] is applicable. The detailed case by case veri�cation of Theorem 3.2 is the main result of [Tay14], which also forms the main content of the authors PhD thesis. The techniques used are those described in [GH08, §2] together with Cli�ord theory.

Remark 3.3. In [Tay14] the author works with geometric conjugacy classes of semisimple elements with representatives in a �xed maximal torus of G̃^{⋆}. One can do this if one modi�es the action of the Frobenius endomorphism on the centraliser of the semisimple appropriately, (i.e. by twisting with an element of the Weyl group � see [Tay14, §2]). It is an easy exercise for the reader that [Tay14, Theorem 2.5] gives a pair as in Theorem 3.2, (see also the proof of [Tay12, Theorem A]).

4. Generalised Gelfand�Graev Representations

Following [GH08] we will use the result of Theorem 3.1 to prove Kawanaka’s conjecture holds for most simple groups G with a disconnected centre. Firstly let us recall that in [Kaw86, §3.1] Kawanaka associates to every unipotent element u ∈ G a GGGR which we denote Γ_{u}. This is a representation of G whose construction depends only on the G-conjugacy class of u. Furthermore, these representations are such that Γ_{1} is the regular representation, (where 1 ∈ G is the identity), and Γ_{u} is a Gelfand�Graev representation if u is a regular unipotent element.

If Γ_{u} is a GGGR of G then we will write γ_{u} for the character of G admitted by Γ_{u} and we use the term GGGR to refer to both Γ_{u} and γ_{u}. We can now state Kawanaka’s conjecture.

Conjecture 4.1 (Kawanaka, [Kaw85, 3.3.1]). Let G be a connected reductive algebraic group and p a good prime for G. Let u_{1}, …, u_{r} be representatives for the unipotent conjugacy classes of G, then the set {γ_{u_{i}} | 1 ⩽ i ⩽ r} forms a -basis for the -module of all unipotently supported virtual characters of G.

If G is connected reductive with a connected centre and simple quotient G/Z(G) then, under the assumption that p, q are large enough, this was proved by Geck and H� ezard in [GH08, Theorem 4.5]. Here p, q large enough means that the results of [Lus92] are true. We will now assume that p, q are large enough so that we may use the results of [GH08].

For the following discussion of GGGRs to make sense we must �rst make some choices. We �x an F-stable Borel subgroup B̃ ⩽ G̃ which contains an F-stable maximal torus T̃ ⩽ G̃. We de�ne B := B̃∩ G and T := T̃∩ G, then these are similarly such groups for G. Let U be the common unipotent radical of both B and B̃ then B̃ = T̃⋉ U and B = T ⋉ U. We denote the �xed points of B̃, (resp. T̃, B, T, U), under the Frobenius endomorphism by B̃, (resp. T̃, B, T, U). Our GGGRs are then de�ned with respect to these choices.

Remark 4.2. By [Gec04, Remark 2.2] we know class representatives for all unipotent classes of G may be found in U hence we assume this to be the case from this point forward. This means we may also assume that our �xed well-chosen representatives lie in U, as any G-conjugate of a well-chosen representative is well chosen.

When Z(G) is disconnected we will want to relate the GGGRs of G to the GGGRs of G̃. If u ∈ G̃ is a unipotent element then u is also a unipotent element of G. We will denote the GGGR of G̃ associated to u by γ̃_{u} and the GGGR of G associated to u by γ_{u}. With this in mind we have the following observation, which follows immediately from the construction of GGGRs.

Lemma 4.3. Let Õ be a G̃-conjugacy class of unipotent elements and {u_{i}} ⊆ Õ a collection of class representatives for the G-conjugacy classes contained in Õ ∩ G, then γ̃_{u_{i}} = IndG̃ ; G(γ_{u_{i}}) for each i.

We will need another observation regarding the natural conjugation action of G̃ on a GGGR. The group G̃ is the product T̃· G, therefore we may assume that any left transversal of G in G̃ is contained in T̃. In particular, if χ is a class function of G then for any g ∈ G̃ there exists t ∈ T̃ such that χ^{g} = χ^{t}, (where χ^{g}(x) = χ(gxg^{−1} ) for all x ∈ G). With this notation we have the following result.

Proposition 4.4 (Geck, [Gec93, Proposition 2.2]). For any unipotent element u ∈ U and any t ∈ T̃ we have γ_{u}^{t} = γ_{tut^{−1}} .

5. The Case of Abelian Component Groups

To prove Kawanaka’s conjecture we will follow the same line of argument as in [GH08, §4]. In particular the focus will be on the following observation.

Lemma 5.1 (Geck�H�ezard, [GH08, Lemma 4.2]). Let u_{1}, …, u_{d} be representatives for the unipotent conjugacy classes of G. Assume that there exist virtual characters χ_{1}, …, χ_{d} of G such that the matrix of scalar products (⟨χ_{i}, γ_{u_{j}}⟩_{G})_{1⩽i,j⩽d} is invertible over , then Conjecture 4.1holds.

We will start by proving a result which is crucial in dealing with almost all cases. Let D_{G} : Cent(G) → Cent(G) denote the Alvis�Curtis duality map. It is known that D_{G} is an isometry of Cent(G) and D_{G}∘ D_{G} is the identity, (see for example [DM91, §8]). Hence, for any irreducible character χ ∈ Irr(G) there exists a sign ε_{χ} ∈ {1, −1} such that ε_{χ}D_{G}(χ) ∈ Irr(G). For any χ we denote ε_{χ}D_{G}(χ) by χ^{∗}. It will be useful for us to know the following relationship between the Alvis�Curtis duality maps DG̃ and D_{G}.

Lemma 5.2. For any connected reductive algebraic group G we have

ResG̃ ; G∘DG̃ = D_{G} ∘ ResG̃ ; G.

Proof. If H is a connected reductive algebraic group de�ned over _{q} then we denote by rk(H) its semisimple _{q}-rank, (see [DM91, De�nition 8.6]). Also if L ⩽ G is an F-stable Levi subgroup contained in an F-stable subgroup of G then we denote by R_{L}^{G} the corresponding Harish-Chandra induction map. By [DM91, Proposition 2.2] if H̃ is a Borel, parabolic or Levi subgroup of G̃ then H := H̃ ∩ G is also such a subgroup of G and the map H̃ → H gives a bijection between the sets of such subgroups. Identifying these groups in this way and using the statements in [Bon06, Proposition 10.10] we have

ResG̃ ; G∘DG̃ = ∑_{P̃⩾B̃}(−1)^{rk(P̃)}(ResG̃ ; G∘RG̃ L̃) ∘^{∗}RG̃ L̃,

= ∑_{P̃⩾B̃}(−1)^{rk(P̃)}R_{L}^{G} ∘ (ResL̃ ; L ∘^{∗}RG̃ L̃),

= ∑_{P⩾B}(−1)^{r(P)}(R_{L}^{G} ∘^{∗}R_{L}^{G}) ∘ ResG̃ ; G,

= D_{G} ∘ ResG̃ ; G.

To obtain the third equality we have used the fact that the inclusion morphism P → P̃ induces an isomorphism between the derived subgroups of P and P̃ which is de�ned over _{q}, hence rk(P) = rk(P̃). ■

Corollary 5.3. Let χ̃ ∈ Irr(G̃) and χ_{i} ∈ Irr(G) be such that

ResG̃ ; G(χ̃) = χ_{1} + ⋯+ χ_{k},

then εχ̃ = ε_{χ_{i}} for all 1 ⩽ i ⩽ k. In particular ResG̃ ; G(χ̃^{∗}) = χ_{1}^{∗} + ⋯+ χ_{k}^{∗}.

Proof. We know εχ̃DG̃(χ̃) ∈ Irr(G̃), in particular it is a character of G̃. This means

ResG̃ ; G(χ̃^{∗}) = εχ̃(ResG̃ ; G∘DG̃)(χ̃) = εχ̃(D_{G} ∘ ResG̃ ; G)(χ̃) = εχ̃D_{G}(χ_{1}) + ⋯+ εχ̃D_{G}(χ_{k}).

is a character of G, where the second equality is obtained by Lemma 5.2. As it is a character all coe�cients of irreducible constituents must be positive, therefore we cannot have ε_{χ_{i}} ≠ εχ̃ for any 1 ⩽ i ⩽ k. The �nal statement is clear by de�nition. ■

We temporarily �x the following notation. If O is an F-stable unipotent class of G then we denote by Õ_{i}, for 1 ⩽ i ⩽ d, the G̃-classes such that O^{F} = Õ_{1} ⊔ ⋯⊔ Õ_{d}. For each Õ_{i} we write O_{i,j}, for 1 ⩽ j ⩽ k_{i} (where k_{i} is a number depending upon the class O_{i}), for the G-classes such that Õ_{i} ∩ G = O_{i,1} ⊔ ⋯⊔ O_{i,k_{i}}. Finally we �x G-class representatives u_{i,j} ∈ O_{i,j} for each 1 ⩽ i ⩽ d and 1 ⩽ j ⩽ k_{i}.

Proposition 5.4. Let O be an F-stable unipotent class of G and assume A_{G}(u) is abelian then for every 1 ⩽ i ⩽ d we have k_{i} = |Z_{G}(u)^{F}|. Furthermore let (s̃, ψ) be the pair prescribed by Theorem 3.2, then there exist irreducible characters χ_{i,j} ∈ E(G, s) such that ⟨χ_{i,j}^{∗}, γ_{u_{x,y}}⟩ = δ_{i,x}δ_{j,y}, for 1 ⩽ i, x ⩽ d and 1 ⩽ j, y ⩽ |Z_{G}(u)^{F}|, (here δ_{∗,∗} denotes the Kronecker delta).

Proof. Recall that u ∈ O^{F} is a class representative such that AG̃(u)^{F} = AG̃(u). The G̃-classes in O^{F} are in bijection with the F-conjugacy classes in AG̃(u), therefore d = |AG̃(u)| by [Bon06, Exemple 1.1]. Let F̃ ⊆ E(G̃, s̃) be the family of characters which, under the map in (3.2) is in bijection with the family of characters containing ψ.

By [GH08, Proposition 4.3], (which we can use as Properties (P1), (P3) and (P4) hold), there exist irreducible characters χ̃_{1,1}, …, χ̃_{d,1} ∈ F̃ such that ⟨χ̃_{i,1}^{∗}, γ̃_{u_{x,1}} ⟩ = δ_{i,x} for all 1 ⩽ i, x ⩽ d. Using Lemma 4.3 followed by Frobenius reciprocity we see that

δ_{i,x} = ⟨χ̃_{i,1}^{∗}, γ̃_{u_{x,1}} ⟩G̃ = ⟨χ̃_{i,1}^{∗}, IndG̃ ; G(γ_{u_{x,1}})⟩G̃ = ⟨ResG̃ ; G(χ̃_{i,1}^{∗}), γ_{u_{x,1}}⟩_{G}.

As A_{G}(u) is abelian we know (P5)holds, in particular using Corollary 5.3 we know the restriction has the following decomposition into irreducible characters

ResG̃ ; G(χ̃_{i,1}^{∗}) = χ_{i,1}^{∗} + ⋯+ χ_{i,|Z_{G}(u)^{F}|}^{∗} ⇒ δ_{i,x} = ∑_{j=1}^{|Z_{G}(u)^{F}|}⟨χ_{i,j}^{∗}, γ_{u_{x,1}}⟩.

Without loss of generality we may assume the labelling to be such that ⟨χ_{i,j}^{∗}, γ_{u_{i,1}}⟩ = δ_{j,1}. We will write StabG̃(χ_{i,1}^{∗}) for the stabiliser of χ_{i,1}^{∗} in G̃ under the natural conjugation action of G̃ on Irr(G). By Cli�ord theory and the remark before Proposition 4.4 there exists a set {t_{i,1}, …, t_{i,|Z_{G}(u)^{F}|} } ⊆ T̃, (which can be completed to form a left transversal of StabG̃(χ_{i,1}^{∗}) in G̃), such that χ_{i,j}^{∗} := χ_{i,1}^{∗}^{t_{i,j}} satis�es the condition χ_{i,j}^{∗} = χ_{i,k}^{∗} if and only if j = k. We assume for convenience that t_{i,1} is the identity for all i.

Now as conjugation by elements of G̃ is an isometry of the space of all class functions we have

1 = ⟨χ_{i,1}^{∗}^{t_{i,j}}, γ_{u_{i,1}}^{t_{i,j}}⟩ = ⟨χ_{i,j}^{∗}, γ_{t_{i,j}(u_{i,1})t_{i,j}^{−1}} ⟩.

We claim that if j, k are distinct indices then we cannot have t_{i,j}(u_{i,1})t_{i,j}^{−1} and t_{i,k}(u_{i,1})t_{i,k}^{−1} are in the same G-class. If they were in the same G-class then we would have γ_{u_{i,1}}^{t_{i,j}} = γ_{u_{i,1}}^{t_{i,k}} by Proposition 4.4, which would mean

1 = ⟨χ_{i,1}^{∗}^{t_{i,k}}, γ_{u_{i,1}}^{t_{i,j}}⟩ = ⟨χ_{i,1}^{∗}^{t_{i,k}t_{i,j}^{−1}} , γ_{u_{i,1}}⟩.  (5.1)

Now χ_{i,1}^{∗}^{t_{i,k}t_{i,j}^{−1}} = χ_{i,ℓ}^{∗} for some 1 ⩽ ℓ ⩽ |Z_{G}(u)^{F}| but of all the components of ResG̃ ; G(χ̃_{i,1}^{∗}) only χ_{i,1}^{∗} satis�es the property in (5.1). In particular this implies that t_{i,k} and t_{i,j} have the same image in the quotient G̃/ StabG̃(χ_{i,1}^{∗}) but this is a contradiction.

This argument shows that there are at least |Z_{G}(u)^{F}| conjugacy classes contained in Õ_{i} ∩ G for each 1 ⩽ i ⩽ d, in other words |Z_{G}(u)^{F}| ⩽ k_{i}. Conversely this clearly gives us an inequality

|A_{G}(u)^{F}| = |Z_{G}(u)^{F}||AG̃(u)| = d|Z_{G}(u)^{F}| ⩽ ∑_{i=1}^{d} k_{i} = |A_{G}(u)^{F}|

so we must have k_{i} = |Z_{G}(u)^{F}| for all i. By this argument we may now rede�ne our class representatives to be such that, for all 1 ⩽ x ⩽ d and 1 ⩽ y ⩽ |Z_{G}(u)^{F}|, we have u_{x,y} := t_{x,y}u_{x,1}t_{x,y}^{−1}. With all this in mind the statement of the proposition is now simple. Indeed we �rst see that

⟨χ_{i,j}^{∗}, γ_{u_{x,y}}⟩ = ⟨χ_{i,1}^{∗}^{t_{i,j}}, γ_{u_{x,1}}^{t_{x,y}}⟩ = ⟨χ_{i,1}^{∗}^{t_{i,j}t_{x,y}^{−1}}, γ_{u_{x,1}}⟩

but it is clear that ⟨χ_{i,1}^{∗}^{t_{i,j}t_{x,y}^{−1}}, ResG̃ ; G(χ̃_{i,1}^{∗})⟩ = 1, which tells us the above inner product is 0 unless i = x. Now assume i = x then ⟨χ_{i,1}^{∗}^{t_{i,j}t_{i,y}^{−1}}, γ_{u_{i,1}}⟩ = 1 if and only if χ_{i,1}^{∗}^{t_{i,j}t_{i,y}^{−1}} = χ_{i,1}^{∗}, which is true if and only if j = y so we are done. ■

For convenience we restate the above proposition with slightly less cumbersome notation.

Corollary 5.5. Let O be an F-stable unipotent class of G and assume A_{G}(u) is abelian. Let d′:= |A_{G}(u)^{F}| and denote by O_{i} the G-classes such that O^{F} = O_{1} ⊔ ⋯⊔ O_{d′}, furthermore let u_{i} ∈ O_{i} denote a class representative. Consider the pair (s̃, ψ) prescribed by Theorem 3.2then there exist irreducible characters χ_{i} ∈ E(G, s) such that ⟨χ_{i}^{∗}, γ_{u_{j}}⟩ = δ_{i,j}, for all 1 ⩽ i, j ⩽ d′.

Remark 5.6. Let us relax the condition that A_{G}(u) is abelian in the above result but instead assume AG̃(u) is abelian. In particular let G be a spin / half spin group and let u be such that Z_{G}(u) = Z(G) then we are in the following situation. The family F̃ described in the proof of Proposition 5.4 has order m^{2}, where m is a power of 2. The group AG̃(u) is abelian so we may use [GH08, Proposition 4.3], as in the above proof, to obtain m characters in F̃ satisfying the inner product condition. In this situation (P5)fails and in fact there are only m characters in F̃ whose restriction to G contains the correct number of irreducible constituents, (see [Tay14, Propositions 9.3 and 11.10]). The problem is that it is not clear that the characters provided by the result of Geck and H� ezard coincide with the characters which have the correct restriction from G̃ to G. This is the key obstruction to showing Kawanaka’s conjecture holds for spin / half spin groups.

One way to �x this problem would be to make sure that the character sheaves used in the proof of [GH08, Corollary 3.5] can be chosen to have the same labelling as the irreducible characters in F̃ whose restriction to G is not irreducible. The author considered this in the case where G is a spin group of type B_{n} by trying to adapt the explicit results obtained in [Lus86]. However it seems that trying to understand the correspondence between the two labellings is signi�cantly complicated.

6. Kawanaka’s Conjecture

The result in the previous section will be crucial in determining Kawanaka’s conjecture when G is of classical type. If G is of exceptional type and Z(G) is disconnected then we must have G is of type E_{6} or E_{7}. In these groups there are only three classes such that A_{G}(u) is non-abelian and they are all such that AG̃(u) ≅ 𝔖_{3}. Assume O is one of these three classes and (s̃, ψ) is the pair prescribed by Theorem 3.2.

Write Õ_{1}, Õ_{2} and Õ_{3} for the G̃-classes such that O^{F} = Õ_{1} ⊔ Õ_{2} ⊔ Õ_{3} and �x representatives u_{i} ∈ Õ_{i} ∩ G. The matrix of multiplicities between the irreducible characters in E(G̃, s̃) and the Alvis�Curtis duals of the characters of the GGGRs is given in [Gec99, Proposition 6.7]. Using this table we see that there exist three characters χ̃_{1}, χ̃_{2}, χ̃_{3} ∈ Irr(G̃) such that ⟨DG̃(χ̃_{i}), γ̃_{u_{j}} ⟩ = ⟨χ̃_{i}, DG̃(γ̃_{u_{j}})⟩ = δ_{i,j}. In particular εχ̃ = 1 for all χ̃ ∈ E(G̃, s̃), which means ⟨χ̃_{i}^{∗}, γ̃_{u_{j}} ⟩ = δ_{i,j}. Two out of three of the classes satisfy AG̃(u) ≅ A_{G}(u). If this is the case then O_{i} := Õ_{i} ∩ G is a single G-class and O^{F} = O_{1} ⊔ O_{2} ⊔ O_{3}. Let u_{i} ∈ O_{i} denote a G-class representative then as (P5)holds we know χ_{i} := ResG̃ ; G(χ̃_{i}) is irreducible. In particular for each 1 ⩽ i, j ⩽ 3 we have ⟨χ_{i}^{∗}, γ_{u_{j}}⟩ = δ_{i,j}.

For the remaining class we have |A_{G}(u)| = 2|AG̃(u)|. The number of F-conjugacy classes in A_{G}(u) is equal to 6 = |A_{G}(u)^{F}|. We know (P5)holds for this class, so for each 1 ⩽ i ⩽ 3 we have ResG̃ ; G(χ̃_{i}) = χ_{i,1} + χ_{i,2} where χ_{i,j} ∈ Irr(G). Using the techniques in the proof of Proposition 5.4 it is easy to see that each of the three G̃-classes Õ_{i} is such that Õ_{i}^{F} ∩ G = O_{i,1} ⊔ O_{i,2}, where O_{i,j} is a G-class for 1 ⩽ j ⩽ 2. Let us choose class representatives u_{x,y} ∈ O_{x,y} for each 1 ⩽ x ⩽ 3 and 1 ⩽ y ⩽ 2 then the following relation holds ⟨χ_{i,j}^{∗}, γ_{u_{x,y}}⟩ = δ_{i,x}δ_{j,y}. In particular we have the following corollary.

Corollary 6.1. Let O be an F-stable unipotent class of G and assume AG̃(u) ≅ 𝔖_{3}. Let us write O_{i}, for 1 ⩽ i ⩽ d′ for the G-classes such that O^{F} = O_{1} ⊔ ⋯⊔ O_{d′} and let u_{i} ∈ O_{i} denote a class representative. Consider the pair (s̃, ψ) prescribed by Theorem 3.2then there exist irreducible characters χ_{i} ∈ E(G, s) such that ⟨χ_{i}^{∗}, γ_{u_{j}}⟩ = δ_{i,j} for all 1 ⩽ i, j ⩽ d′.

We have now gathered all the preliminary information that we need to prove the following theorem.

Theorem 6.2. Assume p, q are large enough and G is a simple algebraic group, which is not a spin or half-spin group, then Conjecture 4.1holds.

Proof. If G is adjoint then this is just the result obtained by Geck and H� ezard so we may assume that G has a disconnected centre. Let O_{1}, …, O_{d} be the distinct F-stable unipotent classes of G and for each 1 ⩽ i ⩽ d let (s̃_{i}, ψ_{i}) be the pair prescribed by Theorem 3.2 for O_{i}. For each 1 ⩽ i ⩽ d we write O_{i,1}, …, O_{i,k_{i}} for the G-conjugacy classes such that O_{i}^{F} = O_{i,1} ⊔ ⋯⊔ O_{i,k_{i}}. By Corollary 5.5 and Corollary 6.1 we can �nd irreducible characters {χ_{i,1}, …, χ_{i,k_{i}}} ⊆ E(G, s_{i}) such that ⟨χ_{i,j}^{∗}, γ_{u_{i,y}}⟩ = δ_{j,y}. In particular the matrix of multiplicities (⟨χ_{i,j}^{∗}, γ_{u_{x,y}}⟩), (where we have 1 ⩽ i, x ⩽ d, 1 ⩽ j ⩽ k_{i}, 1 ⩽ y ⩽ k_{x}), is a square block matrix with identity matrices on the diagonal. We may now argue as in the proof of [GH08, Theorem 4.5] to show that all blocks in the lower triangular part of this matrix are zero. In particular this matrix of multiplicities is invertible over so by Lemma 5.1 Kawanaka’s conjecture holds. ■

Remark 6.3. In [GH08, Proposition 4.6] Geck and H� ezard give a characterisation of GGGRs in terms of character values. This characterisation follows as a formal consequence of Kawanaka’s conjecture, so this now also holds for all simple groups which are not a spin / half spin group.

7. GGGRs and Fourth Roots of Unity

In this section we would like to use Theorem 3.1 together with the techniques of [Gec99, §3] to compute certain fourth roots of unity that arise in connection to GGGRs. We begin by recalling the setup of [Gec99]. Let N_{G} denote the set of all pairs ι = (O_{ι}, ψ_{ι}), where O_{ι}is a unipotent conjugacy class of G and ψ_{ι}∈ Irr(A_{G}(u)) for u ∈ O_{ι}. We say a pair ι is F-stable if F(O_{ι}) = O_{ι}and ψ_{ι}∘ F = ψ_{ι}. Note that we assume u is well chosen, hence u ∈ O^{F} so F induces an automorphism of A_{G}(u). We denote the subset of F-stable pairs by N_{G}^{F} ⊆ N_{G}.

Assume now that ι is an F-stable pair. As ψ_{ι}is invariant under the action of F we may extend ψ_{ι}to a character of the semidirect product A_{G}(u) ⋊ ⟨F⟩, where F acts as a cyclic group of automorphisms. Let ψ̃_{ι} be a �xed choice of such an extension. For each x ∈ A_{G}(u) we write u_{x} for an element of O_{ι}^{F} obtained by twisting u with the element x ∈ A_{G}(u). We then de�ne

Y_{ι}(g) = \{ \begin{matrix}ψ̃_{ι}(xF) & if g = u_{x} for some x ∈ A_{G}(u), \\ 0 & otherwise \end{matrix}

for all g ∈ G. The function Y_{ι} is a G-class function and the set Y = {Y_{ι} | ι ∈ N_{G}^{F}} forms a basis for the space of all unipotently supported class functions of G.

Let ι ∈ N_{G}^{F} be an F-stable pair and let u_{1}, …, u_{d} ∈ O_{ι}^{F} be class representatives for the G-classes contained in O_{ι}^{F}. We de�ne, as in [Lus92, 7.5], for any pair ι ∈ N_{G}^{F} the G-class function

γ_{ι}= ∑_{r=1}^{d} [A_{G}(u_{r}) : A_{G}(u_{r})^{F}]Y_{ι}(u_{r})γ_{u_{r}}.

The set X = {γ_{ι} | ι ∈ N_{G}^{F}} is then also a basis for the space of all unipotently supported class functions of G.

Recall that in [Lus84b] Lusztig has associated to every pair ι ∈ N_{G} a unique pair (L_{ι}, υ_{ι}), up to G conjugacy, where L_{ι} is a Levi subgroup of G and υ_{ι} ∈ N_{L_{ι}} is a cuspidal pair. This is known as the generalised Springer correspondence. Lusztig has shown that we have a disjoint union

N_{G} = ⨆_{(L,υ)} ℐ (L, υ)  where  ℐ (L, υ) = {ι ∈ N_{G} | (L_{ι}, υ_{ι}) = (L, υ)}.

We call ℐ (L, υ) a block of N_{G}. A pair ι ∈ N_{G} is a cuspidal pair if and only if it lies in a block of cardinality

1. To each pair ι ∈ N_{G} we also assign the following value

b_{ι}= \frac{1}{2}(dim G − dim O_{ι}− dim Z^{∘}(L_{ι})).

In [Lus92, Theorem 7.3] Lusztig explicitly constructs the change of basis from X to Y. The expression of the function γ_{ι} ∈ X in terms of elements of Y involves an unknown fourth root of unity ζ_{ι}. This root of unity is de�ned in [Lus92, Proposition 7.2] and it is shown there that ζ_{ι} = ζ_{ι′}whenever ι, ι′ belong to the same block. Following [Lus92, 8.4] to each ι we de�ne δ_{ι}= (−1)^{rank(L_{ι}/Z(L_{ι}))} and ζ′_{ι}= δ_{ι}ζ_{ι}^{−1} . As δ_{ι}and ζ_{ι} only depend on the block containing ι the same must be true for ζ′_{ι}. Moreover we have the following more precise statement.

Lemma 7.1 (Lusztig, [Lus92, Proposition 7.2]). Let ℐ (L, υ) be a block of N_{G} and assume that υ ∈ N_{L}^{F}. If ζ′_{υ} is the fourth root of unity associated to υ in L^{F} then ζ′_{ι}= ζ′_{υ} for all ι ∈ ℐ (L, υ).

There is one extreme case where we always know the value ζ′_{ι}. If L_{ι}is a maximal torus then from the de�nitions it is clear that ζ′_{ι}= 1. The results of [Gec99] are based on the following two axioms, (these are known to be true when p and q are su�ciently large).

(A1) Let ι, ι′ ∈ N_{G}^{F} be such that O_{ι} = O_{ι′} then ⟨D_{G}(γ_{ι}), Y_{ι′}⟩ = |A_{G}(u)|ζ′_{ι}q^{−b_{ι}}δ_{ι,ι′}, where δ_{ι,ι′}denotes the Kronecker delta.

(A2) If D_{G}(γ_{ι})(x) ≠ 0 for some x ∈ G then ⟨x⟩ ⩽ O_{ι}where ⟨x⟩ is the G-conjugacy class containing x.

As we will use the work of Geck we will also take these as axioms upon which our arguments are built. Our main focus in this section is to show that using Theorem 3.1 statements similar to [Gec99, Theorem 3.8] can be made for symplectic and special orthogonal groups. To do this we need the following easy extensions of results of Geck.

Proposition 7.2. Let O be an F-stable unipotent class of G then there exists a character χ ∈ Irr(G) with O_{χ} = O and an element x ∈ A_{G}(u) such that u_{x} ∈ O^{F} is well chosen and

⟨χ^{∗}, γ_{ι}⟩ = \frac{|A_{G}(u)|}{|A_{G}(u)^{F}|} · \overline{Y_{ι}(u_{x})},

for all ι ∈ N_{G}^{F} satisfying O_{ι} = O. Furthermore we have ⟨χ^{∗}, γ_{u_{x}}⟩ = 1 and ⟨χ^{∗}, γ_{u_{y}}⟩ ≠ 0 implies x and y are F-conjugate, (i.e. u_{x} and u_{y} are G-conjugate).

Proof. The proof of this result is similar in nature to that of Proposition 5.4 and with this in mind we will assume that the G-class representatives of O^{F} are enumerated as before Proposition 5.4. Let (s̃, ψ) be the pair prescribed by Theorem 3.2 for the class O and denote by ψ̃ ∈ E(G̃, s̃) the character corresponding to ψ under the bijection in (3.2). By the proof of Theorem 3.2 we have nψ̃ = |AG̃(u)| so by [Gec99, Proposition 3.1] there exists an index 1 ⩽ i ⩽ d such that ⟨ψ̃^{∗}, γ̃_{u_{x,1}} ⟩G̃ = δ_{i,x} for all 1 ⩽ x ⩽ d. Using the same argument as in Proposition 5.4 we see that there is a unique irreducible constituent χ in ResG̃ ; G(ψ̃) satisfying ⟨χ^{∗}, γ_{u_{x,y}}⟩_{G} = δ_{i,x}δ_{1,y} for all 1 ⩽ x ⩽ d and 1 ⩽ y ⩽ k_{i}, (note that we use here that (P2)holds for ψ). Recall our assumption that u ∈ O^{F} is a well-chosen element then by [Gec99, 2.8(b)] and the proof of Theorem 3.2 we have

⟨χ^{∗}, γ_{(O,1)}⟩ = \frac{|A_{G}(u)|}{n_{χ}} = \frac{|A_{G}(u)|}{|A_{G}(u)^{F}|} .

From the de�nition of γ_{ι}we conclude that

⟨χ^{∗}, γ_{ι}⟩ = ⟨ χ^{∗}, ∑_{x=1}^{d} ∑_{y=1}^{k_{i}} \frac{|A_{G}(u_{x,y})|}{|A_{G}(u_{x,y})^{F}|} Y_{ι}(u_{x,y})γ_{u_{x,y}}⟩ = \frac{|A_{G}(u_{i,1})|}{|A_{G}(u_{i,1})^{F}|} · \overline{Y_{ι}(u_{i,1})}

and taking ι = (O, 1) we see that |A_{G}(u_{i,1})^{F}| = |A_{G}(u)^{F}|. As |Z_{G}(u)^{F}| = |Z_{G}(u_{i,1})^{F}| this forces u_{i,1} to be a well-chosen element, as required. ■

Corollary 7.3. Let O, χ and x be as in Proposition 7.2and assume that either A_{G}(u) is abelian or A_{G}(u) is non-abelian and A_{G}(u)^{F} = A_{G}(u) then

ε_{χ} · χ(u_{x}) = \frac{1}{|A_{G}(u)^{F}|} ∑_{ι} ζ′_{ι}q^{b_{ι}}ψ_{ι}(1)^{2},  (7.1)

where the sum is taken over all ι ∈ N_{G}^{F} with O_{ι} = O. In particular the expression on the right hand side is an algebraic integer.

Proof. From [Gec99, Corollary 2.6] we get

χ(u_{x}) = \frac{1}{|A_{G}(u)|} ∑_{ι} ζ′_{ι}q^{b_{ι}}Y_{ι}(u_{x})⟨χ, D_{G}(γ_{ι})⟩,

= ε_{χ} · \frac{1}{|A_{G}(u)|} ∑_{ι} ζ′_{ι}q^{b_{ι}}Y_{ι}(u_{x})⟨χ^{∗}, γ_{ι}⟩,

= ε_{χ} · \frac{1}{|A_{G}(u)^{F}|} ∑_{ι} ζ′_{ι}q^{b_{ι}}Y_{ι}(u_{x})\overline{Y_{ι}(u_{x})}.

Assume �rst that A_{G}(u) is abelian and recall Y_{ι}(u_{x}) = ψ̃_{ι}(xF). As ψ̃_{ι} will be a linear character every character value will be a root of unity, in particular Y_{ι}(u_{x})\overline{Y_{ι}(u_{x})} = 1 = ψ_{ι}(1)^{2}. We assume now that A_{G}(u) is non-abelian and A_{G}(u)^{F} = A_{G}(u) then as u_{x} is a well-chosen element we must also have A_{G}(u_{x})^{F} = A_{G}(u_{x}). Let g ∈ G be such that u_{x} = gug^{−1} , (i.e. g^{−1} F(g) ↦ x ∈ A_{G}(u)), then the component group has the form A_{G}(u_{x}) = {gyC_{G}^{∘}(u)g^{−1} | y ∈ A_{G}(u)}. If every element of A_{G}(u_{x}) is �xed by F then

F(gy)C_{G}^{∘}(u)F(g)^{−1} = gyC_{G}^{∘}(u)g^{−1} ⇒ (g^{−1} F(g))yC_{G}^{∘}(u)(g^{−1} F(g))^{−1} = yC_{G}^{∘}(u)

Group  Condition  Partition  |A_{G}(u)|
Sp_{2n}  n = 1 + ⋯+ k  (2, 4, 6, …, 2k)  2^{k}
SO_{2n} n = 2k^{2}  (1, 3, 5, …, 4k − 1) 2^{2k−1}

Table 8.1: Conditions for the existence of cuspidal pairs.

for all y ∈ A_{G}(u). As the image of g^{−1} F(g) in A_{G}(u) is x this implies that x is in the centre of A_{G}(u). From the description of the component groups we see that we must have x^{2} = 1 hence ψ_{ι}(x) = ±ψ_{ι}(1) which ensures Y_{ι}(u_{x})\overline{Y_{ι}(u_{x})} = ψ_{ι}(1)^{2}. ■

This corollary is the key ingredient for our argument. It provides a divisibility criterion for the fourth root of unity ζ′_{ι}in the ring of algebraic integers, which will lead to a divisibility criterion in .

8. The Special Orthogonal and Symplectic Groups

We want to adapt the argument of [Gec99, Theorem 3.8] to symplectic groups and even dimensional special orthogonal groups. Recall that we assume p is a good prime, in particular p ≠ 2. Our computation of the fourth root of unity will use a result of Digne�Lehrer�Michel using Gauss sums which depends upon two choices, the �rst being a choice of primitive fourth root of unity in \overline{}_{ℓ}. Following [Bon06, §1.B] we �x an injective homomorphism of groups φ : / → \overline{}_{ℓ}^{×} and denote by φ̃ : → \overline{}_{ℓ}^{×} the composition of φ with the natural quotient map → / ; we have Ker(φ̃) = . We now de�ne j to be φ̃(1/4), which is a primitive fourth root of unity in \overline{}_{ℓ}.

The second choice we need to make is of a square root of p in \overline{}_{ℓ}, which we will now do following [Bon06, §36]. We �x a non-trivial additive character χ_{1} : _{p} → \overline{}_{ℓ}^{×} and denote by χ_{s} : _{p^{s}} → \overline{}_{ℓ}^{×} the additive character χ_{s} = χ_{1} ∘ Tr_{s} where Tr_{s} : _{p^{s}} → _{p} is the �eld trace. We denote by θ_{s} : _{p^{s}}^{×} → \overline{}_{ℓ}^{×} the unique linear character of degree 2 and de�ne the associated Gauss sum to be

G_{s}(θ_{s}) = ∑_{x∈ _{p^{s}}^{×}} θ_{s}(x)χ_{s}(x)

We denote by p^{\frac{1}{2}} our �xed square root of p in \overline{}_{ℓ}, which is chosen in the following way

p^{\frac{1}{2}} = \{ \begin{matrix}G_{1}(θ_{1}) & if p ≡ 1 (mod 4), \\ j^{−1} G_{1}(θ_{1}) & if p ≡ 3 (mod 4). \end{matrix}

Now given any a ∈ we denote by p^{\frac{a}{2}} the term (p^{\frac{1}{2}})^{a}.

In Table 8.1we have listed the conditions for the existence of a cuspidal pair in special orthogonal and symplectic groups. This information has been adapted from [Lus84b] and we see that in any given case there is only one cuspidal pair. The argument we will employ is based on induction and for this to work we will need to know ζ′_{ι_{0}} where ι_{0} is the unique cuspidal pair of G = SL_{2}(\overline{}_{p}) and G = SL_{2}(q). However this can be deduced from a result of Digne, Lehrer and Michel.

Lemma 8.1 (Digne�Lehrer�Michel). Assume G = SL_{2}(\overline{}_{p}) where p ≠ 2 and G = SL_{2}(q) with q = p^{a}. Let ι_{0} be the unique cuspidal pair of G then

ζ′_{ι_{0}} = \{ \begin{matrix}(−1)^{a} & if p ≡ 1 (mod 4), \\ (−j)^{a} & if p ≡ −1 (mod 4). \end{matrix}

Proof. Taking n = e = 2 in [DLM97, Proposition 2.8] and using [Bon06, 36.3] for the computation of the Gauss sum G_{s}(θ_{s}) we have

ζ_{ι_{0}} = \{ \begin{matrix}(−1)^{a−1} & if p ≡ 1 (mod 4), \\ (−1)^{a−1} j^{a} & if p ≡ −1 (mod 4). \end{matrix}

As δ_{ι_{0}} = −1 the statement clearly follows by the de�nition of ζ′_{ι_{0}}. ■

Remark 8.2. It should be noted that the statement of [DLM97, Proposition 2.8] depends upon the validity of (A1). However as the full character table of SL_{2}(q) is known, this could be computed without this result.

We now turn to the �nal piece of information we will need regarding the symplectic groups, which involves a simple counting argument. Let ι_{0} ∈ N_{G} be a cuspidal pair then we de�ne X_{ι_{0}} = {ι ∈ N_{G}^{F} | O_{ι}= O_{ι_{0}}}.

Lemma 8.3. Let G = Sp_{2n}(\overline{}_{p}) and assume n = 1 + 2 + ⋯+ k, for some k ⩾ 1. Under this assumption there exists a unique cuspidal pair ι_{0} in N_{G} and we have

|{ι ∈ X_{ι_{0}} | m_{ι}is even}| = |{ι ∈ X_{ι_{0}} | m_{ι}is odd}| = 2^{k−1} ,

where m_{ι}= rank(L_{ι}/Z(L_{ι})) = rank(L_{ι}) − dim(Z^{∘}(L_{ι})).

Proof. By [Lus84b, 11.6.1] the elements of X_{ι_{0}} are parametrised by certain unordered pairs (A, B), where A is a �nite subset of _{⩾0} and B is a �nite subset of _{⩾1} such that |A| + |B| is odd. Speci�cally we have the following, which follows as a consequence of [Lus84b, Corollary 12.4].

Let d_{ι_{0}} = −k if k is odd and k + 1 if k is even then d_{ι_{0}} is an odd number satisfying n = \frac{1}{2}d_{ι_{0}}(d_{ι_{0}}− 1). If k is even we have a bijection ι ↦ (A_{ι}, B_{ι}) between X_{ι_{0}} and the set of all symbols (A_{ι}, B_{ι}) satisfying

(i) A_{ι}⊆ {0, 2, 4, …, 2d_{ι_{0}}− 2},

(ii) B_{ι}⊆ {2, 4, …, 2d_{ι_{0}}− 2},

(iii) A_{ι}∩ B_{ι}= ∅ and A_{ι}∪ B_{ι}= {0, 2, 4, …, 2d_{ι_{0}}− 2}.

In particular this bijection is such that ι_{0} ↦ ({0, 2, 4 …, 2d_{ι_{0}}− 2}, ∅). If k is odd we again have a bijection ι ↦ (A_{ι}, B_{ι}) between X_{ι_{0}} and the set of all symbols (A_{ι}, B_{ι}) satisfying

(i) A_{ι}, B_{ι}⊆ {1, 3, 5, …, 1 − 2(d_{ι_{0}}+ 1)},

(ii) A_{ι}∩ B_{ι}= ∅ and A_{ι}∪ B_{ι}= {1, 3, 5, …, 1 − 2(d_{ι_{0}}+ 1)}.

In particular this bijection is such ι_{0} ↦ (∅, {1, 3, 5, …, 1 − 2(d_{ι_{0}}+ 1)}). We de�ne the defect of a pair (A_{ι}, B_{ι}) to be the value d_{ι}:= |A_{ι}| − |B_{ι}|. In both cases we have (A_{ι_{0}}, B_{ι_{0}}) is the unique symbol of defect d_{ι_{0}}.

The reason for introducing the defect of a symbol is that it is related to the value m_{ι}via the formula

m_{ι}= \frac{1}{2}d_{ι}(d_{ι}− 1).

We can now consider the number of pairs ι ∈ X_{ι_{0}} such that m_{ι}is even and the number of pairs such that m_{ι}is odd. Note that d_{ι}is always an odd number, say 2y_{ι}+ 1, and we have m_{ι}≡ y_{ι} (mod 2).

Assume k = 2y is even then for each 0 ⩽ x ⩽ k there are precisely \binom{k}{x} subsets B_{ι}⊆ \{2, 4, …, 2d_{ι_{0}}− 2\} such that |B_{ι}| = x. The defect of the symbols (A_{ι}, B_{ι}) associated to these subsets is d_{ι} = k + 1 − 2x = 2(y − x) + 1. By the above we have m_{ι}≡ y − x (mod 2). The result then follows from the following basic fact of binomial coe�cients

∑_{x even} \binom{k}{x} = ∑_{x odd} \binom{k}{x} = ∑_{z=0}^{k−1} \binom{k − 1}{z} = 2^{k−1} .  (8.1)

Assume k = 2y + 1 is odd then for each 0 ⩽ x ⩽ k there are precisely \binom{k}{x} subsets A_{ι} ⊆ \{1, 3, …, 1 − 2(d_{ι_{0}}+ 1)} such that |A_{ι}| = x. The symbols (A_{ι}, B_{ι}) associated to these subsets have defect d_{ι}= 2x − k = 2(x − y − 1) + 1. Again by the above remark we have m_{ι} ≡ y − x + 1 (mod 2). The result then follows from (8.1). ■

Remark 8.4. It should be noted that some of the statements of [Lus84b, Corollary 12.4] are not quite correct. In particular the description of the symbol corresponding to the cuspidal pair in the case k is odd. This was corrected by Shoji in [Sho97, Remark 5.8] and we have used the corrected statement above. Finally we know [Lus84b, Corollary 12.4(c)] is correct by the remark at the end of [GM00, §2.B], where the work of Shoji is also referenced. This is all we have used from this reference.

We are now in a state where we may prove the �nal result of this paper. The following fourth roots of unity were computed by Waldspurger for symplectic / special orthogonal groups in [Wal01, §V.8 - Proposition]. Here we merely give an alternative proof for these values, (note that one easily checks that the expressions coincide).

Theorem 8.5. Assume G_{n} is Sp_{2n}(\overline{}_{p}) or SO_{2n}(\overline{}_{p}) and, if such a pair exists, let ι_{0} be the unique cuspidal pair in N_{G_{n}}. Let ε ∈ {±1} be such that p ≡ ε (mod 4), then we have

ζ′_{ι_{0}} = \{ \begin{matrix}ε^{\frac{an}{2}} & if n is even, \\ (−1)^{an} & if n is odd andε = 1, \\ (−j)^{an} & if n is odd andε = −1. \end{matrix}

Proof. The proof of this statement is an adaptation of the proof of [Gec99, Theorem 3.8], which is a proof by induction on n. The proof is identical for the case of special orthogonal groups but requires slightly more work for the case of symplectic groups. Note we will let k be the appropriate value as described in Table 8.1. If n = 0 then G_{n} is a torus and the formula is trivially true. If n = 1 then G_{n} = Sp_{2}(\overline{}_{p}) ≅ SL_{2}(\overline{}_{p}) and we can see that the formula coincides with that given by Lemma 8.1. If n = 2 then G_{n} = SO_{4}(\overline{}_{p}) and the simply connected covering G_{sc} of G_{n} is isomorphic to SL_{2}(\overline{}_{p}) × SL_{2}(\overline{}_{p}). By [Lus84b, §10.1] ι_{0} is the image of the direct product υ_{0} × υ_{0}, where υ_{0} is the unique cuspidal pair of SL_{2}(\overline{}_{p}). There are two possible rational structures on G_{sc}, either G_{sc}^{F} = SL_{2}(q) × SL_{2}(q) or G_{sc}^{F} = SL_{2}(q^{2}).

However, in both cases we have ζ′_{ι_{0}} = (ζ′_{υ_{0}})^{2} and by Lemma 8.1 we can see the formula is valid.

Now we assume that n ⩾ 3 and the statement is true for all G_{m} with m < n. If ι ∈ X_{ι_{0}} then by [Lus84b, §10.4 and §10.6] we have L_{ι}/Z^{∘}(L_{ι}) is isomorphic to G_{m_{ι}} for some m_{ι}⩽ n. Let N be the number of positive roots of G then as rank G = rank L_{ι}for all ι we can express b_{ι}in the following way

b_{ι}= \frac{1}{2}(dim G − dim O_{ι}− dim Z^{∘}(L_{ι})),

= \frac{1}{2}(2N + rank G − dim O_{ι}− dim Z^{∘}(L_{ι})),

= \frac{1}{2}(2N − dim O_{ι}) + \frac{1}{2}(rank L_{ι}− dim Z^{∘}(L_{ι})),

= dim 𝔅_{u} + \frac{m_{ι}}{2},

where we have used the dimension formula given in [Spa82, II - 2.8] to obtain the last equality.

We now consider the criterion given to us from Corollary 7.3. Recall that u is a well-chosen element of a symplectic or special orthogonal group, so in particular A_{G}(u)^{F} = A_{G}(u). Furthermore the component group A_{G}(u) is abelian which means ψ_{ι}(1) = 1 for all ι. Using the information we have gathered we may rewrite the sum in Corollary 7.3 in the following way

∑_{ι} ζ′_{ι}q^{b_{ι}}ψ_{ι}(1)^{2} = ζ′_{ι_{0}}q^{dim 𝔅_{u}+ \frac{n}{2}} + ∑_{ι≠ι_{0}}ζ′_{ι}q^{dim 𝔅_{u}+ \frac{mι}{2}} = q^{dim 𝔅_{u}} ( ζ′_{ι_{0}}q^{\frac{n}{2}} + ∑_{ι≠ι_{0}}ζ′_{ι}q^{\frac{mι}{2}}) .

The statement of Corollary 7.3 says that |A_{G}(u)| divides the expression on the right hand side in the ring of algebraic integers. The order of the component group is a power of 2, therefore as q is a power of an odd prime we must have

|A_{G}(u)| divides ζ′_{ι_{0}}p^{\frac{an}{2}} + ∑_{ι∈X_{ι_{0}}\\{ι_{0}\}}ζ′_{ι}p^{\frac{amι}{2}}  (8.2)

in the ring of algebraic integers. As n ⩾ 3 we have |A_{G}(u)| ⩾ 4 so 4 divides the expression on the right which means we may consider the expression modulo 4. However we must be careful only to reduce integer powers of p modulo 4 and never rational powers.

We wish to consider the possibilities for the terms in the sum modulo 4. Let us assume ι ∈ X_{ι_{0}}\ {ι_{0}} then reducing only integer powers of p modulo 4 we see

ζ′_{ι}p^{\frac{amι}{2}} ≡ \{ \begin{matrix}1 (mod 4) & if m_{ι}is even, \\ (−1)^{a}p^{\frac{a}{2}} (mod 4) & if m_{ι}is odd and ε = 1, \\ (−j)^{a}p^{\frac{a}{2}} (mod 4) & if m_{ι}is odd and ε = −1. \end{matrix}

Let us assume G_{n} = SO_{2n}(\overline{}_{p}). From Table 8.1we see m_{ι}is even for all ι ∈ X_{ι_{0}} then (8.2)tells us

4 divides ζ′_{ι_{0}}p^{\frac{an}{2}} − 1  (8.3)

in the ring of algebraic integers. Multiplying the term on the right by the same term with − exchanged by + we have

4 divides (ζ′_{ι_{0}})^{2}p^{an} − 1.

All the terms on the right are integers so we must have 4 divides this term in . In other words (ζ′_{ι_{0}})^{2}p^{an} − 1 ≡ 0 (mod 4) ⇒ (ζ′_{ι_{0}})^{2} ≡ 1 (mod 4) as n is even, hence ζ′_{ι_{0}} = ±1. Returning to (8.3) we see that all the values in the expression are integers, so it must be true that 4 divides this term in . In particular ζ′_{ι_{0}}ε^{\frac{an}{2}} ≡ 1 (mod 4), which provides the desired formula.

Now assume G_{n} = Sp_{2n}(\overline{}_{p}). Using Lemma 8.3 we have the sum in (8.2)can be expressed as

ζ′_{ι_{0}}p^{\frac{an}{2}} + ∑_{ι∈X_{ι_{0}}\\{ι_{0}\}}ζ′_{ι}p^{\frac{amι}{2}} = \{ \begin{matrix}ζ′_{ι_{0}}p^{\frac{an}{2}} + (2^{k−1} − 1) + 2^{k−1} η^{a}p^{\frac{a}{2}} & if n is even, \\ ζ′_{ι_{0}}p^{\frac{an}{2}} + 2^{k−1} + (2^{k−1} − 1)η^{a}p^{\frac{a}{2}} & if n is odd. \end{matrix}  (8.4)

where η = −1 or −j depending upon the congruence of p modulo 4. We would like to eliminate the rational powers of p. Assume for the moment that k > 2 then 4 divides 2^{k−1} , so we may simplify extensively the expressions in (8.4). Let us consider the following products

(ζ′_{ι_{0}}p^{\frac{an}{2}} − 1)(ζ′_{ι_{0}}p^{\frac{an}{2}} + 1)  and  (ζ′_{ι_{0}}p^{\frac{an}{2}} − η^{a}p^{\frac{a}{2}})(ζ′_{ι_{0}}p^{\frac{an}{2}} + η^{a}p^{\frac{a}{2}})

where the left comes from the case n is even and the right comes from the case n is odd. By (8.2) and (8.4)we have 4 divides these products in the ring of algebraic integers. Expanding the brackets we see all terms in the expression are in , hence 4 must divide these expressions in . Reducing modulo 4 we get (ζ′_{ι_{0}})^{2}p^{an} − 1 ≡ 0 (mod 4), (because η^{2} = ε and ε^{a}p^{a} ≡ 1 (mod 4)), in particular (ζ′_{ι_{0}})^{2} ≡ ε^{an} (mod 4).

If n is even then the argument is identical to the above argument for SO_{2n}(\overline{}_{p}). Therefore let us assume n is odd, then (ζ′_{ι_{0}})^{2} ≡ ε^{a} (mod 4) hence ζ_{ι_{0}} = ±η^{a} where η is as in (8.4). Returning to (8.2)and (8.4)we see

4 divides ζ′_{ι_{0}}p^{\frac{an}{2}} − η^{a}p^{\frac{a}{2}}

in the ring of algebraic integers. Assume n = 2x + 1 then we have p^{\frac{an}{2}} = p^{ax}p^{\frac{a}{2}} ≡ ε^{ax}p^{\frac{a}{2}} (mod 4). This implies

4 divides (ζ′_{ι_{0}}ε^{ax} − η^{a})p^{\frac{a}{2}}

in the ring of algebraic integers. Assume for a contradiction that ζ′_{ι_{0}} = −ε^{ax}η^{a} then this says 4 divides −2η^{a}p^{\frac{a}{2}} in the ring of algebraic integers. In particular there exists an algebraic integer y such that 2η^{a}p^{\frac{a}{2}} = 4y ⇒ η^{a}p^{\frac{a}{2}} = 2y. Squaring this expression we see η^{2a}p^{a} = 4y^{2} but η^{2a}p^{a} ∈ and as 2 divides the right hand side we must have 2 divides p^{a} in but this is impossible as p is odd. Therefore ζ′_{ι_{0}} = ε^{ax}η^{a} and the result follows by noticing η^{an} = η^{2ax}η^{a} = ε^{ax}η^{a}.

To �nish the proof we must deal with the case where k = 2, in other words n = 3 and G_{n} = Sp_{6}(\overline{}_{p}). In this case 2^{k−1} = 2, so returning to (8.4)we see

ζ′_{ι_{0}}p^{\frac{an}{2}} + ∑_{ι∈X_{ι_{0}}\\{ι_{0}\}}ζ′_{ι}p^{\frac{amι}{2}} = ζ′_{ι_{0}}p^{\frac{an}{2}} + 2 + η^{a}p^{\frac{a}{2}}.  (8.5)

Again we consider the product

(ζ′_{ι_{0}}p^{\frac{an}{2}} + 2 + η^{a}p^{\frac{a}{2}})(ζ′_{ι_{0}}p^{\frac{an}{2}} − 2 − η^{a}p^{\frac{a}{2}}) = (ζ′_{ι_{0}})^{2}p^{an} − 4 − 4η^{a}p^{\frac{a}{2}} − η^{2a}p^{a}.

Reducing this modulo 4 we see (ζ′_{ι_{0}})^{2} ≡ ε^{a} (mod 4), hence ζ_{ι_{0}} = ±η^{a} as before. Again we write n = 2x + 1 then returning to (8.5)and reducing modulo 4 we have

4  divides  2 + (ζ′_{ι_{0}}ε^{ax} + η^{a})p^{\frac{a}{2}}

in the ring of algebraic integers. If ζ′_{ι_{0}} = −ε^{ax}η^{a} then the above statement says 4 divides 2 in the ring of integers, a contradiction. Hence we must have ζ′_{ι_{0}} = ε^{ax}η^{a} and as above this is η^{an} as required. ■

One would hope that the above argument may work for other simple groups with a disconnected centre, unfortunately this is not the case. For example when G is a spin group the degree of ψ_{ι_{0}} is a power of 2, where ι_{0} is the cuspidal pair not coming from the generalised Springer correspondence of the corresponding special orthogonal group. In particular when we reduce modulo 4 in the above argument the term containing the unknown fourth root of unity will become zero.

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FB Mathematik, TU Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

Email: taylor@mathematik.uni-kl.de