Disconnected groups of Lie type as Galois groups

Gunter Malle *, Mathematisches Institut und IWR, Im Neuenheimer Feld 368, D - 6900 Heidelberg, Germany

During the last ten years, big progress was made in the realization of finite simple groups as Galois groups of regular field extensions of ℚ^{ab}(t). For the alternating groups, this is an old result of Hilbert, for the classical simple groups of Lie type Belyi succeeded in finding Galois realizations [1,2], and the sporadic simple groups were investigated by a number of authors (see [24] for references). Most of the families of exceptional groups of Lie type, at least in good characteristic, could be realized as Galois groups over abelian number fields in [18]. But some cases, in particular a whole congruence class of q mod 3 for groups of types E_{6} and ^{2}E_{6} had to be left open. The aim of this paper, which may be thought of as a continuation of [18], is to study the aforementioned cases. We obtain:

Main Theorem: The following groups occur as Galois groups of regular field extensions of ℚ^{ab}(t):

(1) F_{4}(2^{n}) for all n ≥ 1,

(2) E_{6}(q) and E_{6}(q).2 for q = p^{n} ≢ 1 (mod 3), p > 3,
(3) E_{6}(q), E_{6}(q).2, 3^{.}E_{6}(q) and 3^{.}E_{6}(q).2 for q = p^{n} ≡ 1 (mod 3), p > 3,
(4) ^{2}E_{6}(q) and ^{2}E_{6}(q).2 for q = p^{n} ≢ −1 (mod 3), p > 3,
(5) ^{2}E_{6}(q), ^{2}E_{6}(q).2, 3^{.}^{2}E_{6}(q) and 3^{.}^{2}E_{6}(q).2 for q = p^{n} ≡ −1 (mod 3), p > 3,

Moreover, the following even occur as regular Galois groups over ℚ(t):

(6) E_{6}(p) and E_{6}(p).2 for primes p ≡ 4, 5, 6, 9, 16, 17 (mod 19), p ≡ −1 (mod 3),

(7) E_{6}(p), E_{6}(p).2, 3^{.}E_{6}(p) and 3^{.}E_{6}(p).2 for primes p ≡ 4, 5, 6, 9, 16, 17 (mod 19), p ≡ 1 (mod 3),

(8) ^{2}E_{6}(p) and ^{2}E_{6}(p).2 for primes p ≡ 2, 3, 10, 13, 14, 15 (mod 19), p ≡ 1 (mod 3),

(9) ^{2}E_{6}(p), ^{2}E_{6}(p).2, 3^{.}^{2}E_{6}(p) and 3^{.}^{2}E_{6}(p).2 for primes p ≡ 2, 3, 10, 13, 14, 15 (mod 19), p ≡ −1 (mod 3).

This result consists of Theorems 1.1, 2.1, 2.2, 2.3, 3.1, 3.2, 3.3 in sections 1–3. Precise abelian fields of definition for all extensions mentioned in (1) up to (5) are derived there, too.

Together with [18], [20] and the above mentioned results of the other authors we can therefore conclude, using the classification of finite simple groups:

* The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

Corollary: Let G be a finite simple group. If G is not an exceptional group of Lie type in bad characteristic, then G occurs as a Galois group over ℚ^{ab}.

More precisely there are at most the following exceptions: ^{2}B_{2}(2^{2n+1}), ^{2}F_{4}(2^{2n+1}), E_{6}(2^{2n}), ^{2}E_{6}(2^{2n−1}), groups of type E_{7} in characteristic 2 or 3, groups of type E_{8} in characteristic 2, 3, or 5.

The Galois realizations (2), (4), (6), and (8) in the Main Theorem for E_{6}(q).2 and 2E_{6}(q).2 for primes p = q are even GAR-realizations in the sense of [23]. Moreover with (3) and (5), all covering groups of E_{6}(q) and ^{2}E_{6}(q) for odd q are shown to occur as Galois groups over ℚ^{ab}, and for the primes p = q in (7) and (9) even over ℚ.

As in [18], the proof of the Main Theorem relies on the character theoretic version of the rationality criterion of Belyi, Matzat and Thompson for class structures C = (C_{1}, C_{2}, C_{3}). Namely, for a certain choice of conjugacy classes C_{i} of the given group G one has to check that the normalized structure constant n(C) takes the value one, and that secondly each triple (σ_{1}, σ_{2}, σ_{3}) with σ_{i} ∈ C_{i} and σ_{1}σ_{2}σ_{3} = ι generates G. The first step requires the knowledge of character values of G on the conjugacy classes C_{i}. In [18] these could be calculated from the Deligne-Lusztig theory. However, two problems arose. Firstly the character theory for groups of Lie type G(q), q = p^{n}, in bad characteristic p is presently not complete, hence in those cases not all character values may be computed from the theory. Secondly, for E_{6}(q) with q ≡ 1 (mod 3) no suitable class triple C with n(C) = 1 and satisfying the generation property could be found.

Here, for groups F_{4} in characteristic 2 we will show that the explicit knowledge of the Green functions describing the character values on unipotent elements is not necessary. Instead by using the Howlett-Lehrer comparison theorem and number theoretic properties of character values we are able to compute the normalized structure constants for suitable class structures and thus derive the existence of Galois realizations.

The second problem for E_{6} and ^{2}E_{6} may be circumvented by applying the rationality criterion not to G but to the extension with the graph automorphism Ĝ of G. The descent from a Galois extension with group Ĝ to one with group G is then achieved by a standard argument. Moreover this approach enables us to ‘mask’ the center of order three of the simply connected groups of that type by the action of the outer automorphism and hence to realize the covering groups as Galois groups too. This kind of argument was first used by Feit [11] for threefold coverings of sporadic groups, then in [19] in the case of the special unitary groups SU_{3}(p). The necessary character theory for the disconnected groups Ĝ was developed in [20] and [9], and the decomposition of generalized Deligne-Lusztig characters in the cases of interest for us in [22].

The methods explained so far allow the determination of n(C) = 1 for the groups given in the Main Theorem. To prove the generation we make use of properties of t.i.-tori, as in [18]. It turns out that the arguments of [18] easily carry over to our cases. Nevertheless it should be mentioned that all proofs of generation employ the classification of finite simple groups.

While it cannot be expected to obtain whole families G(q) for all prime powers q = p^{n} as Galois groups over the field of rational numbers ℚ with the existing rationality criteria, in some cases ℚ-realizations at least for q = p, i.e., for the group G(p) defined over the prime field may be obtained by a careful choice of the three conjugacy classes. This leads to the results (6)–(9) in the Main Theorem.

The calculations for induction and decomposition of principal series characters in section 1 were done using the computer algebra system CAS; for the computations with character values in G (as polynomials in q) a separate program was written.

Parts (1)–(3), (6) and (7) of the results in the Main Theorem were already announced in [20], mainly without proofs. This paper, together with [20] and parts of [22] constituted the authors Habilitationsschrift [21].

1 The groups F_{4}(2^{n}) as Galois groups

In [18], the groups F_{4}(q) for odd q were shown to occur as Galois groups over ℚ^{ab}. In characteristic 2 no rigid class structure consisting of three semisimple classes could be found. The determination of the normalized structure constant n(C) for other class triples requires the knowledge of values of unipotent characters on certain unipotent elements, hence of part of the Green functions, and the latter are not yet known in bad characteristic. Here we determine the values of some unipotent characters on one unipotent class of F_{4}(2^{n}), proving n(C) = 1 for a certain class structure. With this the groups F_{4}(2^{n}) are shown to occur as Galois groups over suitable abelian number fields. For n = 1 this is Theorem 5.3 in [18], for n ≥ 2 the proof is given in the third part of this section.

An important aid in the computation of the missing character values is given by the result of Howlett and Lehrer (comparison theorem, [13], Theorem 5.9, see also [25], Theorem 3.4). It describes the decomposition of Harish-Chandra induced unipotent characters from Levi subgroups. This yields conditions on the values of the noncuspidal unipotent characters of G, if the corresponding values for the Levi subgroups are known.

But it is well known that the information obtainable this way is not sufficient to calculate all values. Additional conditions on the missing values may be deduced from integrality properties and from the calculation of certain normalized structure constants which by definition have to be nonnegative integers. The inductive nature of the approach described above first makes necessary the determination of values of unipotent characters on two unipotent classes of the Levi factor C_{3}(2^{n}) ≅ B_{3}(2^{n}).

1.1 Descent to parabolic subgroups of F_{4}(2^{n}) From now on, let G be the simple group F_{4}(q), q = 2^{n}, n ≥ 2 (the case q = 2 was already treated in [18], the character table may be found in [6]), with BN-pair B = U.T, N = T.W. The notation for the root system of F_{4}(q) is adopted from [28]. The simple roots are

Δ = \{ϵ_{2} − ϵ_{3}, ϵ_{3} − ϵ_{4}, ϵ_{4}, \frac{1}{2}(ϵ_{1} − ϵ_{2} − ϵ_{3} − ϵ_{4})\}.

Hence we have the following positive roots

Φ^{+} = \{ϵ_{i} ± ϵ_{j} (1 ≤ i < j ≤ 4), ϵ_{i} (1 ≤ i ≤ 4), \frac{1}{2}(ϵ_{1} ± ϵ_{2} ± ϵ_{3} ± ϵ_{4})\} .

The roots \{ϵ_{3} −ϵ_{4}, ϵ_{4}, \frac{1}{2}(ϵ_{1} −ϵ_{2} −ϵ_{3} −ϵ_{4})\} =: Δ(C_{3}) generate the root system of the standard Levi subgroup C_{3}(q):

Φ^{+}(C_{3}) = \{ϵ_{3}, ϵ_{4}, ϵ_{3} ± ϵ_{4}, ϵ_{1} − ϵ_{2}, \frac{1}{2}(ϵ_{1} − ϵ_{2} ± ϵ_{3} ± ϵ_{4})\} .

Similarly the roots {ϵ_{2} − ϵ_{3}, ϵ_{3} − ϵ_{4}, ϵ_{4}} =: Δ(B_{3}) generate the root system of the Levi subgroup B_{3}(q)

Φ^{+}(B_{3}) = {ϵ_{2}, ϵ_{3}, ϵ_{4}, ϵ_{2} ± ϵ_{3}, ϵ_{2} ± ϵ_{4}, ϵ_{3} ± ϵ_{4}}.

The unipotent classes of G on which the character values shall be determined are the classes of the elements x_{1+2}(1) and x_{1}(1) conjugate under the graph automorphism, both with centralizer order |C_{G}(x_{1+2}(1))| = q^{24}(q^{2} − 1)(q^{4} − 1)(q^{6} − 1) ([28], Theorem 2.1).

First we consider the parabolic subgroup P_{1} := [q^{15}].C_{3}(q).(q − 1). For the induction of characters of P_{1} it has to be known, which classes of this parabolic subgroup fuse into the class of x_{1+2}(1) in G. Therefore representatives of the orbits of [x_{1+2}(1)]_{G} ∩ P_{1} under P_{1} have to be determined. Given the Bruhat normal form g = u_{1}hwu_{2} of a g ∈ G we obtain, using x_{1+2}(1) ∈ Z(U),

x_{1+2}(1)^{g} = x_{1+2}(1)^{u_{1}hwu_{2}} = x_{1+2}(1)^{hwu_{2}} = x_{1+2}(y)^{wu_{2}}.

As U is a 2-Sylow subgroup of P_{1}, we may assume x_{1+2}(1)^{g} ∈ U, hence the representatives of [x_{1+2}(1)]_{G} ∩ P_{1} in P_{1} are precisely the elements x_{1+2}(1)^{w} ∈ U. From the action of the Weyl group W = W (F_{4}) we obtain the following orbits (with the notation as in [28]):

{x_{1+2}(1)}, {x_{1±3}(1), x_{1±4}(1), x_{2±3}(1), x_{2±4}(1)}, {x_{3±4}(1), x_{1−2}(1)}. Only the third orbit has a nontrivial intersection with the Levi subgroup C_{3}(q).(q − 1) of P_{1}. Using the commmutator relations in [28], (2.1), the centralizers in P_{1} are determined to be

\begin{aligned}|C_{P_{1}}(x_{1+2}(1))| & = q^{24}(q^{2} − 1)(q^{4} − 1)(q^{6} − 1), \\ |C_{P_{1}}(x_{1+3}(1))| & = q^{23}(q − 1)(q^{2} − 1)(q^{3} − 1), \\ |C_{P_{1}}(x_{1−2}(1))| & = q^{19}(q − 1)(q^{2} − 1)(q^{4} − 1). \end{aligned}

The unipotent characters ρ of the Levi subgroup C_{3}(q).(q − 1) may be extended to the semidirect product P_{1} = [q^{15}].C_{3}(q).(q − 1) by ρ_{P_{1}}(u · s) = ρ(s) for s ∈ C_{3}(q)(q − 1), u ∈ [q^{15}]. The result of Howlett and Lehrer [13] describes the decomposition of the induced character ρ_{P}^{G} into irreducible characters of G in terms of some decomposition inside the Weyl group. By the above calculations we have shown the first part of the next Proposition, the proof of the second being entirely similar.

Proposition 1.1: (1) Let ρ be a unipotent character of C_{3}(q).(q − 1). Then the value of the Harish-Chandra induced character ρ_{P_{1}}^{G} on x_{1+2}(1) is given by

ρ_{P_{1}}^{G}(x_{1+2}(1)) = (1 + qΦ_{2}^{2}Φ_{4}Φ_{6})ρ(ι) + q^{5}Φ_{2}Φ_{3}Φ_{6}ρ(x_{1−2}(1)).

(2) Let ρ be a unipotent character of the Levi subgroup B_{3}(q).(q − 1) of the parabolic subgroup P_{2} = [q^{15}].B_{3}(q).(q − 1). Then ρ_{P_{2}}^{G} on x_{1+2}(1) is given by

ρ_{P_{2}}^{G}(x_{1+2}(1)) = Φ_{2}Φ_{3}Φ_{6}ρ(ι) + q^{4}Φ_{2}Φ_{3}Φ_{4}Φ_{6}ρ(x_{2+3}(1)).

(Here Φ_{i} denotes the value of the i-th cyclotomic polynomial at q.)

The degrees ρ(ι) of the unipotent characters of C_{3}(q) may be found in [4], p.467. But the values ρ(x_{1−2}(1)) are not yet known. In characteristic 2, the groups C_{3}(q) and B_{3}(q) are isomorphic. Under this isomorphism, the class of x_{3}(1) in C_{3}(q) corresponds to the class of x_{2+3}(1) in B_{3}(q). Hence to effectively evaluate the induction formulae for the parabolic subgroups P_{1} and P_{2}, we only have to know the values of the unipotent characters of C_{3}(q) on the two unipotent classes of x_{1−2}(1) and x_{3}(1). These may be calculated by a second descent to Levi subgroups of C_{3}(q).

The simple roots {ϵ_{3} − ϵ_{4}, ϵ_{4}} =: Δ(B_{2}) ⊂ Δ(C_{3}) generate the root system Φ^{+}(B_{2}) = {ϵ_{3}, ϵ_{4}, ϵ_{3}±ϵ_{4}} of a Levi subgroup B_{2}(q).(q−1) of C_{3}(q). To determine the induction formula from the parabolic subgroup P_{1,1} = [q^{5}].B_{2}(q).(q − 1), representatives for [x_{1−2}(1)]_{C_{3}} ∩ P_{1,1} and [x_{3}(1)]_{C_{3}} ∩ P_{1,1} and their respective centralizers have to be calculated. So one obtains

λ_{P_{1,1}}^{C_{3}}(x_{1−2}(1)) = λ(ι) + qΦ_{2}Φ_{4}λ(x_{3+4}(1)),
λ_{P_{1,1}}^{C_{3}}(x_{3}(1)) = Φ_{2}λ(ι) + q^{2}Φ_{2}λ(x_{3}(1)),

for the trivial extension to P_{1,1} of a unipotent character λ of B_{2}(q). The character table of the symplectic group B_{2}(2^{n}) = Sp_{4}(2^{n}) was calculated by Enomoto [10], Table IV-2. Hence the values of λ in the induction formulae are known. But in principle a further descent to parabolic subgroups of B_{2}(q) would also be possible: doing this, some misprints in Table IV-2 in [10] can be detected.

The theorem of Howlett-Lehrer will first be applied in the case of C_{3}(q). The character table of the Weyl group W (C_{3}) ≅ 2^{3}: S_{3} may easily be computed (see also [4], 11.4); it will not be included here. Finally, the character table of the dihedral group W (B_{2}) ≅ D_{8} is well known; the conjugacy classes of W (B_{2}) are in bijection with the classes of maximal tori in B_{2} and will be denoted according to this bijection in the following table ([3], Part G). Moreover the table also gives the bijection from characters of W (B_{2}) to unipotent characters of B_{2}(2^{n}) (see [4], pp.375 and 468), their degrees and their values on the elements u_{1} = x_{3+4}(1), u_{2} = x_{3}(1) according to [10] (after correction of the misprints). The last line of the table belongs to the cuspidal unipotent character of B_{2}, which does not correspond to any character of the Weyl group. The distinction between λ_{2} and λ_{3} is made by induction from the parabolic subgroup [q^{3}].A_{1}(q).(q − 1).

Table 1.1: Character table of W (B_{2}) and corresponding unipotent characters.

∅ A_{1} 2A_{1} Ã_{1} B_{2}  [ι]  [u_{1}]  [u_{2}]
λ_{1}  1 1 1 1 1  ϕ_{2,−}  1  1  1
λ_{2}  1 −1 1 1 −1 ϕ_{11,−}  \frac{1}{2}qΦ_{4} −\frac{1}{2}qΦ_{1} \frac{1}{2}qΦ_{2}
λ_{3}  1 1 1 −1 −1  ϕ_{−,2}  \frac{1}{2}qΦ_{4}  \frac{1}{2}qΦ_{2} −\frac{1}{2}qΦ_{1}
λ_{4}  1 −1 1 −1 1  ϕ_{−,11}  q^{4}  .  .
λ_{5}  2 . −2 . .  ϕ_{1,1}  \frac{1}{2}qΦ_{2}^{2} \frac{1}{2}qΦ_{2}  \frac{1}{2}qΦ_{2}
−  B_{2}, 1  \frac{1}{2}qΦ_{1}^{2} −\frac{1}{2}qΦ_{1} −\frac{1}{2}qΦ_{1}

The fusion of classes from W (B_{2}) into W (C_{3}) is deduced from the correspondence of classes to maximal tori. The irreducible characters of W (C_{3}) may be indexed by pairs of partitions of 3 ([4], 11.4.3). The decomposition of the induced λ_{i}^{W (C_{3})} is then given by

\begin{matrix}λ_{1}^{W} = & ϕ_{3,−} + ϕ_{21,−} + ϕ_{2,1}, \\ λ_{2}^{W} = & ϕ_{−,3} + ϕ_{−,21} + ϕ_{1,2}, \\ λ_{3}^{W} = & ϕ_{111,−} + ϕ_{21,−} + ϕ_{11,1}, \\ λ_{4}^{W} = & ϕ_{−,111} + ϕ_{−,21} + ϕ_{1,11}, \\ λ_{5}^{W} = & ϕ_{1,2} + ϕ_{1,11} + ϕ_{11,1} + ϕ_{2,1}. \end{matrix}

The second parabolic subgroup of C_{3}(q) we use is P_{1,2} = [q^{6}]: (A_{2}(q).(q − 1)) with the Levi factor A_{2}(q).(q − 1). In this case P_{1,2} acts transitively on [x_{1−2}(1)]_{C_{3}} ∩ P_{1,2} and we find |C_{P_{1,2}}(x_{1−2}(1))| = q^{9}Φ_{1}^{2}Φ_{2}. The induction formulae for a unipotent character μ of A_{2}(q) hence have the form

μ_{P_{1,2}}^{C_{3}}(x_{1−2}(1)) = Φ_{2}Φ_{4}μ(ι),
μ_{P_{1,2}}^{C_{3}}(x_{3}(1)) = Φ_{2}μ(ι) + q^{2}Φ_{2}^{2}μ(x_{4}(1)).

Now if μ_{1}, μ_{2} denote the trivial and the sign character, and μ_{3} the nonlinear irreducible character of W (A_{2}) ≅ S_{3}, then the induced characters decompose as follows for W (C_{3}):

\begin{matrix}μ_{1}^{W} = & ϕ_{3,−} + ϕ_{−,3} + ϕ_{1,2} + ϕ_{2,1}, \\ μ_{2}^{W} = & ϕ_{111,−} + ϕ_{−,111} + ϕ_{1,11} + ϕ_{11,1}, \\ μ_{3}^{W} = & ϕ_{21,−} + ϕ_{−,21} + ϕ_{1,2} + ϕ_{1,11} + ϕ_{11,1} + ϕ_{2,1}. \end{matrix}

For the application of the theorem of Howlett-Lehrer, the bijection between the characters of the Weyl group W (C_{3}) and the unipotent principal series characters of W (C_{3}) has to be specified. This, together with the degrees of these characters may be found in [4], p.467.

Proposition 1.2: The values of the twelve unipotent characters of C_{3}(q), q = 2^{n} ≥ 4, on the two classes [x_{1−2}(1)] and [x_{3}(1)] are as given in Table 2.

Before starting the proof we will recollect some well known properties of values of irreducible characters of finite groups.

Lemma 1.1: Let G be a finite group, χ an irreducible character, C a conjugacy class with representative element σ. Then we have

(i) |χ(σ)| ≤ χ(ι),

(ii) χ(σ) · |C|/χ(ι) is an algebraic integer.

Table 1.2: Values of the unipotent characters of C_{3}(2^{n}) on two unipotent classes.

[ι]  [x_{1−2}(1)]  [x_{3}(1)]
ϕ_{3,−}  1  1  1
ϕ_{1,2}  q^{2}Φ_{3}Φ_{6}  q^{2}  ?
ϕ_{11,1}  q^{3}Φ_{3}Φ_{6}  q^{3}Φ_{4}  ?
ϕ_{−,111}  q^{9}  .  .
ϕ_{21,−}  \frac{1}{2}qΦ_{2}^{2}Φ_{6}  \frac{1}{2}qΦ_{2}Φ_{4}  \frac{1}{2}qΦ_{2}
ϕ_{2,1}  \frac{1}{2}qΦ_{3}Φ_{4}  \frac{1}{2}qΦ_{2}Φ_{4}  \frac{1}{2}q(2q^{2} + q + 1)
ϕ_{−,3}  \frac{1}{2}qΦ_{4}Φ_{6}  −\frac{1}{2}qΦ_{1}Φ_{4}  ?
B_{2}, 1 \frac{1}{2}qΦ_{1}^{2}Φ_{3}  −\frac{1}{2}qΦ_{1}Φ_{4}  −\frac{1}{2}qΦ_{1}
ϕ_{−,21} \frac{1}{2}q^{4}Φ_{2}^{2}Φ_{6} −\frac{1}{2}q^{4}Φ_{1}Φ_{2}  \frac{1}{2}q^{4}Φ_{2}
ϕ_{1,11}  \frac{1}{2}q^{4}Φ_{3}Φ_{4}  \frac{1}{2}q^{4}Φ_{4}  \frac{1}{2}q^{4}Φ_{2}
ϕ_{111,−} \frac{1}{2}q^{4}Φ_{4}Φ_{6}  \frac{1}{2}q^{4}Φ_{4}  ?
B_{2}, ϵ \frac{1}{2}q^{4}Φ_{1}^{2}Φ_{3} −\frac{1}{2}q^{4}Φ_{1}Φ_{2}  −\frac{1}{2}q^{4}Φ_{1}

Proof of the Proposition: It was already mentioned that the conditions on the character values obtainable by induction from parabolic subgroups do not suffice for a complete determination. The missing information may in some cases be deduced from number theoretic considerations. The values of the trivial character ϕ_{3,−} and of the Steinberg character ϕ_{−,111} are already known. The theorem of Howlett-Lehrer and the decompositions of induced characters of the Weyl groups given above yield after a short calculation

\begin{aligned}(ϕ_{2,1} + ϕ_{1,11})(u_{1}) & = \frac{1}{2}qΦ_{4}(q^{3} + q + 1) \\ (ϕ_{−,3} + ϕ_{111,−})(u_{1}) & = \frac{1}{2}qΦ_{4}(q^{3} − q + 1). \end{aligned}

Here for brevity we have put u_{1} := x_{1−2}(1). As [u_{1}] is a rational class, using (C_{3}(q) : C_{C_{3}}(u_{1})) = Φ_{1}Φ_{2}Φ_{3}Φ_{6} we get from Lemma 1(ii) the divisibility properties

\frac{1}{2}q^{4}Φ_{4}|ϕ_{1,11}(u_{1}), \frac{1}{2}q^{4}Φ_{4}|ϕ_{111,−}(u_{1})

and from (i) the estimates

|ϕ_{2,1}(u_{1})| ≤ \frac{1}{2}qΦ_{3}Φ_{4}, |ϕ_{−,3}(u_{1})| ≤ \frac{1}{2}qΦ_{4}Φ_{6},

which together with the other equations for ρ(u_{1}) uniquely determine the values of the principal series characters on u_{1}.

By the general version of the theorem in [13], an induction of the cuspidal character of B_{2}(q) yields the condition (B_{2}, 1 + B_{2}, ϵ)(u_{1}) = −\frac{1}{2}qΦ_{1}(q^{4} + q^{3} + q^{2} + 1) on the values of the two subcuspidal characters B_{2}, 1 and B_{2}, ϵ on u_{1}. Again the assertions of Lemma 1 may be used to determine both values from this. This completes the proof of the second column of the table.

For the third column we first get y+z := (B_{2}, 1+B_{2}, ϵ)(u_{2}) = −\frac{1}{2}qΦ_{1}Φ_{2}Φ_{6}, \frac{1}{2}q^{4}Φ_{1}|z and |y| ≤ \frac{1}{2}qΦ_{1}^{2}Φ_{3}. This has only two possible solutions, namely y = −\frac{1}{2}qΦ_{1} or y = \frac{1}{2}qΦ_{1}^{2}Φ_{3}. In the second case we would have (B_{2}, 1)(ι) = (B_{2}, 1)(u_{2}), contradicting the fact that C_{3}(q) is a simple group. Hence y and z are uniquely determined. For ϕ_{21,−}(u_{2}), three cases have to be considered. One of them again leads to ϕ_{21,−}(ι) = ϕ_{21,−}(u_{2}), the second would imply a too big value for ϕ_{2,1}(u_{2}). Thus ϕ_{21,−}(u_{2}) is also known, and this finally allows the computation of the other values on u_{2} = x_{3}(1) in the table.

The values on u_{2} missing in the table may not be deduced from our calculations. But we will not need them for our further considerations.

Remark: Comparison with the character table of C_{3}(2) = S_{6}(2) in [6] shows that the results found above also hold for q = 2.

Finally, before returning to F_{4}(q), we also consider the parabolic subgroups of type [q^{20}]:(A_{2}(q).A_{1}(q).(q − 1)). The character values of the Levi subgroup A_{2}(q).A_{1}(q).(q − 1) are known by classical results of Green [12]. So we only have to give the relation with the Weyl group characters. The unipotent characters χ of groups of type A_{l} are parametrized by partitions of l + 1, as are the irreducible characters of the Weyl group. Hereby the following degrees χ(ι) correspond to the respective partitions:

\begin{matrix}(3) × (2) & (3) × (11) & (21) × (2) & (21) × (11) & (111) × (2) & (111) × (11) \\ 1 & q & qΦ_{2} & q^{2}Φ_{2} & q^{3} & q^{4} \end{matrix}

So all necessary values in the Levi subgroups are determined.

1.2 Character values in F_{4}(2^{n}) The character table of W (F_{4}) with a notation for classes and characters suitable for our purpose can be found in [4], p.413. The bijection from the characters to the unipotent principal series characters of G are also reproduced there on p.479. We now apply the theorem of Howlett-Lehrer to the groups G = F_{4}(q). The induction formulae for the unipotent characters of the Levi factors in P_{1} and P_{2} were already derived in Proposition 1. Analogously one obtains

ρ_{P_{3}}^{G}(x_{1+2}(1)) = Φ_{2}Φ_{3}Φ_{4}Φ_{6}(1 + q^{3}Φ_{2})ρ(ι) + q^{6}Φ_{2}^{2}Φ_{3}Φ_{4}Φ_{6}ρ(x_{2−3}(1))

for P_{3} = [q^{20}]: (A_{2}(q).Ã_{1}(q).(q − 1)) and

ρ_{P_{4}}^{G}(x_{1+2}(1)) = Φ_{2}Φ_{4}Φ_{6}(Φ_{2} + q^{2}Φ_{3} + q^{4}Φ_{2}Φ_{3})ρ(ι) + q^{7}Φ_{2}^{2}Φ_{4}Φ_{6}ρ(x_{2−3}(1))

for P_{4} = [q^{20}]: (Ã_{2}(q).A_{1}(q).(q − 1)).

We next define some semisimple classes of F_{4}(q). Let C_{T} be the class of an element of order Φ_{12}(q) = q^{4} − q^{2} + 1 (with representative h_{76} in Table III of [28]). As the centralizer T of such an element is a cyclic t.i.-Hall torus of G, the characters not vanishing on C_{T} may easily be described (see [18], §1 and §5, for a discussion). There occur twelve unipotent characters of G and a family of irreducible Deligne-Lusztig characters R_{T,θ} for the torus T. They are listed in Table 3.

Further let C_{q+1} be the class of an element of order q + 1 with centralizer of type A_{2} × A_{1} (representative h_{16} or h_{17} in Table III in [28]; the two are interchanged by the graph automorphism). According to Table II in [28] such elements exist for all q ≠ 2. Finally we choose for C_{q−1} a class of elements of order q − 1 with centralizer of type A_{2} × A_{1} (h_{4} or h_{5} in Table III in [28]). These exist for all q ≥ 8.

The values on the semisimple classes C_{T}, C_{q+1} and C_{q−1} of the twelve unipotent characters χ_{i} not vanishing on C_{T} may easily be computed using the formulae of Lusztig; for a description of the method see [4], 7.5, or [16], Theorem 1.5.

Proposition 1.3: The values of the unipotent characters of F_{4}(q), q = 2^{n} ≥ 4, not vanishing on C_{T}, on the classes C_{2} := [x_{1+2}(1)], C_{q+1} and C_{q−1} are as given in Table 3.

Table 1.3: Values of the characters of F_{4}(q) not vanishing on C_{T}.

[ι]  C_{2}  C_{T} C_{q+1} C_{q−1}
ϕ_{1,0}  1  1  1  1  1
ϕ_{1,24}  q^{24}  .  1  q^{4}  q^{4}
ϕ_{4,1}  \frac{1}{2}qΦ_{2}^{2}Φ_{6}^{2}Φ_{8}  \frac{1}{2}qΦ_{2}Φ_{6}(q^{4} + q^{3} + 1)  −1  .  Φ_{2}^{2}
B_{2}, 1  \frac{1}{2}qΦ_{1}^{2}Φ_{3}^{2}Φ_{8}  −\frac{1}{2}qΦ_{1}Φ_{3}(q^{4} − q^{3} + 1)  1  −Φ_{1}^{2}  .
ϕ_{4,13}  \frac{1}{2}q^{13}Φ_{2}^{2}Φ_{6}^{2}Φ_{8}  \frac{1}{2}q^{13}Φ_{2}Φ_{6}  −1  .  q^{2}Φ_{2}^{2}
B_{2}, ϵ  \frac{1}{2}q^{13}Φ_{1}^{2}Φ_{3}^{2}Φ_{8}  −\frac{1}{2}q^{13}Φ_{1}Φ_{3}  1 −q^{2}Φ_{1}^{2}  .
ϕ_{6,6′′}  \frac{1}{12}q^{4}Φ_{3}^{2}Φ_{4}^{2}Φ_{6}^{2}Φ_{8} \frac{1}{12}q^{4}Φ_{3}Φ_{4}Φ_{6}(3q^{4} + 2q^{2} + 1) 1 −2qΦ_{6} 2qΦ_{3}
B_{2}, r  \frac{1}{4}q^{4}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}^{2}Φ_{6}^{2}Φ_{8}  ?  −1  .  .
F_{4}[θ]  \frac{1}{3}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{8}  ?  1  .  .
F_{4}[θ^{2}]  \frac{1}{3}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{8}  ?  1  .  .
F_{4}[i]  \frac{1}{4}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{3}^{2}Φ_{6}^{2}  ?  1  .  .
F_{4}[−i]  \frac{1}{4}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{3}^{2}Φ_{6}^{2}  ?  1  .  .
R_{T,θ}  Φ_{1}^{4}Φ_{2}^{4}Φ_{3}^{2}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}  ?  {−1}  .  .

Remark: The characters in the family R_{T,θ} take constant values on all classes apart from those of nontrivial elements of T. Hence it suffices for the calculation of structure constants to know the sum of the values of the R_{T,θ} on C_{T}. This sum is equal to −1, which is indicated in the table by the symbol {−1}.

Proof : By the above considerations, only the values on the unipotent element u := x_{1+2}(1) remain to be verified. For this, first the theorem of Howlett-Lehrer is applied to the characters B_{2}, 1 and B_{2}, ϵ of the Levi factors C_{3}(q) and B_{3}(q). According to [4], p.479, in this case the quotient root system has type B_{2}, so that the decomposition of the induced B_{2}, 1 and B_{2}, ϵ in unipotent characters of F_{4}(q) is described by the multiplicities of the irreducible characters of W (B_{2}) in the permutation character of (nonnormal) subgroups of index two. From these equations it follows that

(B_{2}, ϵ)(u) − (B_{2}, 1)(u) = −\frac{1}{2}qΦ_{1}Φ_{3}(q^{12} − q^{4} + q^{3} − 1).

Together with |(B_{2}, 1)(u)| ≤ \frac{1}{2}qΦ_{1}^{2}Φ_{3}^{2}Φ_{8} and \frac{1}{2}q^{13}Φ_{1}Φ_{3}|(B_{2}, ϵ)(u) (using Lemma 1 as in the proof of Proposition 2) this has only the solution given in the table.

Next, induction of the principal series unipotent characters of the parabolic subgroups P_{1}, P_{2} and P_{3}, using the values calculated in Proposition 2 yields a linear system of equations for the values of the principal series unipotent characters. So one finds for instance

ϕ_{4,13}(u) − ϕ_{4,1}(u) = \frac{1}{2}qΦ_{2}Φ_{6}(q^{12} − q^{4} − q^{3} − 1).

As above, using Lemma 1, this uniquely determines both values. The Steinberg character ϕ_{1,24} vanishes on all non semisimple classes, hence only ϕ_{6,6′′}(u) remains to be calculated. For this, a further number theoretic property of character tables has to be cited. As the intersection of the centralizers of elements in C_{T} and C_{q+1} is always trivial, the normalized structure constant n(C_{2}, C_{q+1}, C_{T}) has to be a nonnegative integer. Moreover the irreducible characters R_{T,θ} for T vanish on C_{q+1} (for example because of the aforementioned property of the centralizers), hence all values necessary for the computation of the structure constant apart from ϕ_{6,6′′}(u) are known. Writing ϕ_{6,6′′}(u) = \frac{1}{12}q^{4}Φ_{3}Φ_{4}Φ_{6}(3q^{4} + 2q^{2} + 1) + x (i.e., we want to prove x = 0) we obtain

n(C_{2}, C_{q+1}, C_{T}) = \frac{|C_{2}| · |C_{q+1}| · |C_{T} |}{|G|^{2}} _{i}^{∑} \frac{χ_{i}(C_{2})χ_{i}(C_{q+1})χ_{i}(C_{T})}{χ_{i}(ι)} = 1 − \frac{24x}{q^{7}Φ_{1}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}} .

The same calculation is now also applied to the semisimple class C_{q−1}. Here we find

n(C_{2}, C_{q−1}, C_{T}) = 1 + \frac{24x}{q^{7}Φ_{1}^{3}Φ_{2}Φ_{3}Φ_{4}Φ_{6}} .

Denote the rational number 24x/q^{7}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6} by y. Then the two preceding equations imply 0 ≤ n(C_{2}, C_{q+1}, C_{T}) = 1 − Φ_{1}^{2}y ∈ ℤ, 0 ≤ n(C_{2}, C_{q−1}, C_{T}) = 1 + Φ_{2}^{2}y ∈ ℤ. Because of gcd(Φ_{1}, Φ_{2}) = 1 and Φ_{1} ≥ 3 this already forces y = x = 0, yielding the value of ϕ_{6,6′′}(u). These arguments hold for q ≥ 8, but for q = 4 there exists no class C_{q−1} as described above in G [28]. Here a similar argument with the class of the semisimple element h_{31} gives the desired conclusion.

Remark: Again a comparison with [6] shows that the values in Table 3 also remain correct for F_{4}(2).

1.3 The Galois realization The preceding observations clearly show that C = (C_{2}, C_{q+1}, C_{T}) is a suitable class structure (due to n(C) = 1, see the proof of Proposition 3). But there exist two classes of types C_{2}, C_{q+1} respectively, in G, in both cases interchanged by the graph automorphism σ, so a certain freedom of choice remains for C. This will be of importance for excluding ^{3}D_{4}(q) as possibility for the group generated by a tripel of elements from C. By [14], Prop. 5.2, for even q the group F_{4}(q) possesses exactly two classes of maximal subgroups ^{3}D_{4}(q): 3, which are interchanged by the graph automorphism. We now study the fusion of some classes of this maximal subgroup into the classes C_{2} and C_{q+1} of G.

By [30] there are two classes of involutions in ^{3}D_{4}(2^{n}), denoted there by A_{1} and 3A_{1}. Of these, A_{1} can only fuse in one of the classes C_{2}, σ(C_{2}) of G, as can be seen from the centralizer orders. Contrary to that, elements from 3A_{1} are contained in the normalizer of a maximal torus of order Φ_{12} in ^{3}D_{4}(q) (as follows by calculating n_{^{3}D_{4}}(3A_{1}, 3A_{1}, q^{4} − q^{2} + 1) from [30]), which is not the case for C_{2} (calculate n_{F_{4}}(C_{2}, C_{2}, C_{T}) in F_{4}(2) from [6]). (This may also be proved by determining the fusion of ^{3}D_{4}(2): 3 in F_{4}(2), both groups appearing in the Atlas, and now as all unipotent classes contain representatives defined already over 𝔽_{2}, the conclusion for all q follows.)

Further, precisely the classes of ^{3}D_{4}(q) with representatives s_{14} in [8] fuse into classes of type C_{T}. By [30], ^{3}D_{4}(q) contains just one class of maximal tori of type (q + 1)(q^{3} + 1). In F_{4}(q), two such exist, namely those denoted by C_{3} + A_{1} and D_{4} in Table 1 of [28]. These again are interchanged by σ. Now if both these classes of tori would contain elements from [h_{16}], then both types would occur in C_{F_{4}}(h_{16}) ≅ ^{2}A_{2}(q).A_{1}(q).(q + 1). But in the latter group, all tori of the same order are conjugate, yielding a contradiction. Hence elements h_{16} lie in one of the two types of tori, elements h_{17} in the other one.

The above discussion has shown that the one class of maximal subgroups ^{3}D_{4}(q): 3 of G only contains elements from C_{2} and [h_{16}], while the other only contains elements from σ(C_{2}) and [h_{17}] = σ([h_{16}]). We may now choose C_{q+1} := [h_{17}], and C = (C_{2}, C_{q+1}, C_{T}), so that by construction no tripel from C lies in a subgroup ^{3}D_{4}(q): 3 of G.

Theorem 1.1: The groups F_{4}(q), q = 2^{n} ≥ 4, occur as Galois groups over ℚ^{ab} for the ramification structure C^{∗} = (C_{2}, C_{q+1}, C_{T})^{∗}. Here the respective classes contain elements of orders (2, q + 1, q^{4} − q^{2} + 1). More precisely the compositum of a field of index twelve in ℚ(ζ_{q^{4}−q^{2}+1}) with the totally real subfield of ℚ(ζ_{q+1}) is a proper field of definition.

Proof : From Proposition 3 we calculate n(C) = 1 for the relevant class structure. By the rationality criterion in [23], II, §4.2, Folgerung 3, it remains to show that any tripel (ρ, σ, τ) ∈ \overline{Σ}(C) generates the whole group G = F_{4}(q). Let H = ⟨ρ, σ, τ⟩. The torus T generated by τ is a self centralizing t.i.-Hall subgroup of G. Hence Lemma 1.3 of [18] may be applied. This was already partly carried out in [18], proof of Theorem 5.1. By that argument, the only candidates remaining for H are ^{3}D_{4}(q) and G itself. But the first two classes of C were carefully chosen so that they do not intersect the same subgroup ^{3}D_{4}(q). This excludes the last possibility, and we have H = G.

The remark about the precise field of definition also follows from [23], II, §4.2, Folgerung 3, and the description of normalizers of cyclic subgroups in G in [28].

Together with the results about F_{4}(q), 2 |̸ q, and q = 2, in [18] all groups of type F_{4} have now been realized as Galois groups over abelian number fields.

2 Groups E_{6}(q)_{sc}: 2 as Galois groups

To be able to apply rigidity even in the case where E_{6}(q)_{sc} has a nontrivial center, in this section we study the extension of E_{6}(q) with the diagram automorphism of order two. This extension exists for all q, independent of the congruence class of q modulo 3. Hence this yields a general approach to the whole family. Nevertheless it will turn out that the elements in the chosen class structure do not generate all of G for p = 2. So one of the conditions for rigidity is not satisfied, and no Galois realization in characteristic 2 may be obtained this way. Hence we may as well exclude p = 2 from the beginning on. Also, for p = 3, E_{6}(q)_{sc} always has trivial center, and for this case the existence of Galois realizations was shown in [20], Satz 2.4. Thus we need not consider p = 3 either, and may henceforth assume p to be a good prime for E_{6}(q).

2.1 Auxiliary results For the proof of generation we will have to calculate certain structure constants in A_{2}(q): 2; the necessary part of that character table will be computed first.

Let A_{2}(q) ≅ P SL_{3}(q) be the simple group of type A_{2} and A_{2}(q).⟨σ⟩ ≅ A_{2}(q): 2 the extension with the graph automorphism of order two. This acts on the Weyl group like the longest element w_{0} and the extended Weyl group is isomorphic to the direct product S_{3} × 2. Hence all three unipotent characters 1 = ϕ_{3}, ϕ_{21} and St = ϕ_{111} of A_{2}(q) extend to A_{2}(q): 2. Denote by σ a quasi-central outer element of A_{2}(q): 2 of order two [9]. Then its class is uniquely determined by this property. Moreover denote by τ a nontrivial element of the maximal torus of G of order Φ_{3}/δ, δ = gcd(q − 1, 3).

Proposition 2.1: The values of the extensions of the unipotent characters of A_{2}(q): 2 on the outer class [σ] and on the class of τ are as given in Table 1.

Table 2.1: The unipotent characters of A_{2}(q): 2.

[ι]  [τ]  [σ]
q^{3}Φ_{1}^{2}Φ_{2}Φ_{3}/δ Φ_{3}/δ qΦ_{1}Φ_{2}
ϕ_{3}  1  1  1
ϕ_{21}  qΦ_{2}  −1  .
ϕ_{111}  q^{3}  1  q

In this and the following tables, the second line gives the centralizer order in A_{2}(q), not in A_{2}(q): 2 (Atlas notation).

Proof : First note that all nontrivial elements of the torus T of order Φ_{3}/δ are regular, and hence each unipotent character takes the same fixed value on all such τ. That may easily be calculated from the Deligne-Lusztig theory. Next by [9], 4.13, the class of σ is uniform, hence the values of the extensions of unipotent characters on this class may be computed from the decomposition of the generalized Deligne-Lusztig characters given in [22], Theorem 4, and the formula for R_{T,1}(σ) in [9], 4.12. This proves the remaining entries of Table 1.

Let now G = E_{6}(q)_{sc}, q = p^{n}, p ≥ 5, be the simply connected group of type E_{6} over the field 𝔽_{q}. We want to realize this (not necessarily simple) group as Galois group over ℚ^{ab}(t). Denote by Ĝ the extension of G with the graph automorphism of order two. Then Ĝ is a disconnected group of Lie type. Various aspects of the character theory of such groups are studied in [9], [20] and [22]. Namely in [9], the notion of Deligne-Lusztig character is generalized to disconnected groups and several properties of these characters are proved. In [20] a generalization of the Howlett-Lehrer comparison theorem to disconnected groups is discussed, and finally in [22], using results from the other two references, the decomposition of generalized Deligne-Lusztig characters for disconnected groups of small rank, including E_{6}(q): 2, is determined. This will enable us to calculate enough character values in Ĝ to compute a structure constant, leading to rigidity.

Choose an outer element σ in Ĝ such that the centralizer in G has the structure F_{4}(q) (then σ is quasi-central in the terminology of [9]). Because F_{4}(q) has trivial center, σ is an involution, and the extension Ĝ = G.⟨σ⟩ splits. Moreover σ acts nontrivially on the center of G, if that has order three (i.e., if q ≡ 1 (mod 3)). So the usual rationality criteria of [23] are applicable to the group with trivial center Ĝ. This way of masking the center of a group was first used by Feit in the context of covering groups of sporadic groups [11].

Let u be a unipotent element in C_{G}(σ) = F_{4}(q) with |C_{F_{4}}(u)| = q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}, which hence lies central in a Sylow p-subgroup of F_{4}(q) (as p ≠ 2, the corresponding class of F_{4}(q) is uniquely determined by this property). Then uσ has order 2p and centralizer order 2q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6} in Ĝ. From the list of conjugacy classes of G in [26] it follows that u must have centralizer order q^{36}Φ_{1}^{5}Φ_{2}^{3}Φ_{3}^{2}Φ_{4}Φ_{5}Φ_{6} in G, because only that centralizer contains a subgroup C_{3}(q) (compare also with [4], p.402). Denote the class of uσ in Ĝ by C_{2p}.

Further let T be a maximal torus of G of order q^{6} + q^{3} + 1 = Φ_{9}. Choose τ ∈ T \Z(G), hence a regular element, and denote its conjugacy class by C_{T}. Then |CĜ(τ)| = Φ_{9} because the outer automorphism acts fixed point freely on T. Almost all irreducible characters of G vanish on the regular element τ ∈ C_{T}. More precisely, by Lusztigs Jordan decomposition of characters only unipotent characters and the Deligne-Lusztig characters R_{T,θ} for the torus T may take nonzero values on C_{T} (this principle was already used in [18] and [20]). Due to the fixed point free action of σ on T, the R_{T,θ} for θ ≠ 1 are pairwise interchanged under the action of σ. So they do not extend to Ĝ and will therefore not contribute to the structure constant we are going to calculate. This leaves the unipotent characters. They all extend to Ĝ, as was proved for example in [22].

2.2 Some unipotent characters of E_{6}(q)_{sc}: 2 The unipotent characters of G are parametrized by irreducible characters of W (E_{6}) if they lie in the principal series, otherwise by the minimal Levi subgroup from which they may be obtained by Harish-Chandra induction. We adopt the notation of [4], 13.9.

Proposition 2.2: The values on C_{T}, [σ] and C_{2p} of the extensions to Ĝ = E_{6}(q)_{sc}: 2, q = p^{n}, p > 3, of those unipotent characters not vanishing on T^{#} are as given in Table 2.

Table 2.2: Some unipotent characters of E_{6}(q)_{sc}: 2.

[ι]  C_{T}  [σ]  C_{2p}
|E_{6}(q)_{sc}|  Φ_{9}  |F_{4}(q)|  q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}
ϕ_{1,0}  1  1  1  1
ϕ_{20,2}  q^{2}Φ_{4}Φ_{5}Φ_{8}Φ_{12}  −1  qΦ_{4}Φ_{8}Φ_{12}  q(q^{6} + q^{4} + 1)
ϕ_{64,4}  q^{4}Φ_{2}^{3}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}Φ_{12}  1  .  .
ϕ_{64,13}  q^{13}Φ_{2}^{3}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}Φ_{12}  1  .  .
ϕ_{20,20}  q^{20}Φ_{4}Φ_{5}Φ_{8}Φ_{12}  −1  q^{13}Φ_{4}Φ_{8}Φ_{12}  q^{13}
ϕ_{1,36}  q^{36}  1  q^{24}  .
ϕ_{90,8}  \frac{1}{3}q^{7}Φ_{3}^{3}Φ_{5}Φ_{6}^{2}Φ_{8}Φ_{12}  −1  \frac{1}{3}q^{4}Φ_{3}^{2}Φ_{6}^{2}Φ_{8}Φ_{12}  \frac{1}{3}q^{4}Φ_{3}Φ_{6}(2q^{4} + 1)
E_{6}[θ]  \frac{1}{3}q^{7}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}  −1  \frac{1}{3}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{8}  −\frac{1}{3}q^{4}Φ_{1}^{3}Φ_{2}^{3}Φ_{4}
E_{6}[θ^{2}]  \frac{1}{3}q^{7}Φ_{1}^{6}Φ_{2}^{4}Φ_{4}^{2}Φ_{5}Φ_{8}  −1  \frac{1}{3}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{8}  −\frac{1}{3}q^{4}Φ_{1}^{3}Φ_{2}^{3}Φ_{4}

Proof : It is readily seen that the unipotent characters figuring in the table are precisely those not vanishing on τ. (Namely they are those with degree not divisible by Φ_{9}.) Their values on the semisimple inner element τ follow from [7] and the known decompositions of the ordinary Deligne-Lusztig characters (see also [18], §6). The outer element σ is quasicentral, so its characteristic function is uniform by [9], 4.13. Hence the value of a unipotent character ρ on the element σ is given by the formula

ρ(σ) = \frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑}⟨ρ, R_{T_{w},1}⟩_{σ} · R_{T_{w},1}(σ),

where W^{σ} = W (F_{4}) is the Weyl group of the centralizer of σ in G, and the R_{T_{w},1} denote the generalized Deligne-Lusztig characters for the disconnected group Ĝ, as defined and studied in [9] and [22]. The R_{T_{w},1}(σ) are equal to the degrees of the corresponding Deligne-Lusztig characters for T_{w}^{σo} in the centralizer G^{σ} = F_{4}(q) by [9]. Moreover the decomposition of the R_{T_{w},1} and hence the multiplicities ⟨ρ, R_{T_{w},1}⟩_{σ} were determined in [22], Theorem 8(1). From this, the values on σ follow.

The class C_{2p} causes some more problems. Still, by the result on the decomposition of the generalized Deligne-Lusztig characters in [22], the extensions of ϕ_{20,2}, ϕ_{20,20} and the Steinberg character ϕ_{1,36} to Ĝ coincide with certain almost characters, hence are uniform functions on the coset G.σ. So their values are given by the same formula as above, and for its application we only need to know R_{T_{w},1}(uσ). As uσ is the commuting product of the quasi-central element σ and the inner unipotent element u, by [9], Proposition 2.9, we have

RĜ ; T̂,1(uσ) = R_{T^{σ},1}^{G^{σ}}(u).

The R_{T^{σ},1}^{G^{σ}}(u) are just the ordinary Green functions of G^{σ} = F_{4}(q) on the unipotent element u central in a Sylow p-subgroup. These Green functions were determined by Shoji for p good and q large [29] and this was later shown by Lusztig [17] to be correct under the single assumption that p be good. As we have assumed p ≥ 5 the tables of Shoji yield the values of ϕ_{20,2}, ϕ_{20,20} and ϕ_{1,36} on uσ. Clearly we would be done if we knew that the class uσ is uniform; then all values of unipotent characters could be calculated in the above way. By 2.6, 4.10 in [9] the class of uσ in Ĝ is uniform if the class of the unipotent element u in G^{σ} is uniform. But G^{σ} = F_{4}(q), and we have assumed p ≥ 5, so the Green functions of G^{σ} on the class of u are given by the results of Shoji [29]. By the criterion of Lusztig for a unipotent class to be uniform it may now easily be checked that indeed the class of u in F_{4}(q) (which was identified after the definition of u) is uniform. So the values of the unipotent characters on uσ follow by the result of [9].

In [20], proof of Bemerkung 3.1, the determination of character values on C_{2p} required a rather long and technical computation because the theory of generalized Deligne-Lusztig characters developed by Digne and Michel [9] was not yet available.

2.3 The Galois realization for E_{6}(q)_{sc}: 2 From Table 2 it already follows that the class structure C = (C_{2p}, C_{2p}, C_{T}) of E_{6}(q)_{sc}: 2 has structure constant n(C) = 1. We now prove:

Theorem 2.1: The groups E_{6}(q)_{sc}: 2, q = p^{n}, p > 3, occur as Galois groups over ℚ^{ab} for the ramification structure C^{∗} = (C_{2p}, C_{2p}, C_{T})^{∗}. Here the respective classes contain elements of orders (2p, 2p, q^{6} + q^{3} + 1). More precisely a subfield of index 18 in ℚ(ζ_{q^{6}+q^{3}+1}) is a proper field of definition.

Proof : Table 2 yields n(C) = 1. For an application of [23], II, §4.2, Folgerung 3, to the group Ĝ = E_{6}(q)_{sc}: 2 with trivial center it remains to show that a tripel (λ_{1}, λ_{2}, λ_{3}) ∈ \overline{Σ}(C) generates all of Ĝ. For this it suffices to consider \overline{G} := Ĝ/Z(E_{6}(q)_{sc}) (which coincides with Ĝ for q ≢ 1 (mod 3)). Clearly the images λ̄_{1}, λ̄_{2}, λ̄_{3} under the canonical epimorphism Ĝ → \overline{G} generate \overline{G} precisely when the preimages generate Ĝ. Now λ̄_{3} generates a self centralizing cyclic t.i.-Hall subgroup \overline{T} in \overline{G}. Hence Lemma 1.3 of [18], which gives strong restrictions on the possibilities for \overline{H} := ⟨λ̄_{1}, λ̄_{2}, λ̄_{3}⟩ is applicable. Indeed, one reaches the situation of [18], Theorem 6.1. The possibility \overline{H} ≤ N\overline{G}(λ̄_{3}) is immediately excluded because \overline{H} contains elements of order p and we assumed p > 3. Alternatively, λ̄_{2}^{p} = σ̄ does not lie in N\overline{G}(λ̄_{3}) = \overline{T}: 18, because from Table 2 we may calculate n([σ], [σ], C_{T}) = 0. As in the proof of Theorem 6.1 in [18] it is seen that \overline{H} is not local. That argument may be simplified using [5].

Hence we have R ≤ \overline{H} ≤ Aut(R) for one of the nonabelian simple groups R in [18], Lemma 6.1. Now \overline{H} contains an element of order o(τ) ≥ Φ_{9}(5) = 15751, so only the groups of Lie type in the same characteristic remain as candidates for R. Of these, only R = A_{2}(q^{3}) and R = E_{6}(q) have order divisible by Φ_{9}, as is shown in [18]. If the first of these can be ruled out, we are done.

We cannot have \overline{H} = A_{2}(q^{3}), since then \overline{H} ∩ E_{6}(q) would be a subgroup of A_{2}(q^{3}) of index 2, and such a subgroup does not exist. So A_{2}(q^{3}) < \overline{H}, and because of o(σ̄) = 2, A := ⟨A_{2}(q^{3}), λ̄_{2}^{p}⟩ is an extension of A_{2}(q^{3}) of degree two. More precisely, this has to be the extension by the graph automorphism: if q is not a square, then A_{2}(q^{3}) has only this outer automorphism of order two. If on the other hand q = r^{2} is a square, then the corresponding field automorphism centralizes a A_{2}(r^{3}), the graph-field automorphism a ^{2}A_{2}(r^{3}). But both are not contained in F_{4}(q) × 2 = C_{G}(λ̄_{2}^{p}) = C_{G}(σ), because a primitive divisor of Φ_{9}(r) (resp. Φ_{18}(r)) divides |A_{2}(r^{3})| (resp. |^{2}A_{2}(r^{3})|), but not |F_{4}(q)|. Thus we see that A must be the extension with the graph automorphism. This particular extension was studied in Proposition 1, and from Table 1 we calculate the structure constant n_{A}([σ], [σ], C_{T}) = 1, contrary to nĜ([σ], [σ], C_{T}) = 0 (see above). So R = A_{2}(q^{3}) is not possible, and we conclude H = E_{6}(q)_{sc}: 2.

Theorem 2.2: Let E_{6}(q) be the simple group of type E_{6} and let q = p^{n} with p > 3. Then E_{6}(q), E_{6}(q): 2, and for q ≡ 1 (mod 3) also the covering group 3^{.}E_{6}(q) occur as Galois groups over ℚ^{ab}.

Proof : This follows by descent from E_{6}(q)_{sc}: 2 to E_{6}(q)_{sc}. By construction the regular Galois extension for E_{6}(q)_{sc}: 2 over ℚ^{ab}(t) is ramified at precisely three points. As E_{6}(q)_{sc} has index two in E_{6}(q)_{sc}: 2, the fixed field of E_{6}(q)_{sc} in this extension is again a rational function field over ℚ^{ab}(t), with the same field of definition as the original Galois extension. It is well known that E_{6}(q)_{sc} is the full covering group of E_{6}(q). Being a factor group of the latter, the simple group E_{6}(q) clearly also occurs as Galois group over the same field of definition. The same argument holds for E_{6}(q): 2 as factor group of E_{6}(q)_{sc}: 2.

We even get some Galois realizations over the rationals:

Theorem 2.3: For primes p ≡ 4, 5, 6, 9, 16, 17 (mod 19) the groups E_{6}(p)_{sc}: 2, E_{6}(p): 2, E_{6}(p)_{sc} and E_{6}(p) occur as Galois groups over ℚ, where the ramification structure may be deduced from the one for E_{6}(p)_{sc}: 2, C^{∗} = (C_{2p}, C_{2p}, C_{19})^{∗}.

Proof : The primes p satisfying the congruences in the Theorem have multiplicative order 9 mod 19. Hence in this case 19 divides Φ_{9}(p), but no other Φ_{i}(p) with i < 9, i.e., the prime 19 is a primitive divisor of Φ_{9}(p) in the terminology of Zsigmondy. So the maximal torus T of G := E_{6}(q)_{sc} contains elements of order 19, and these are rational in Ĝ = G.2 due to |N_{G}(T)/T | = 9, and because the outer automorphism acts fixed point freely on T. The values of the unipotent characters on the corresponding class C_{19} coincide with those on C_{T} given in Table 2, because unipotent characters take constant values on T \Z(G). Thus we have n(C_{2p}, C_{2p}, C_{19}) = 1, and it remains to prove generation.

As in the previous proof we immediately see that the group H generated by a tripel from the corresponding class structure cannot be local (using that 2 and 3 are primitive roots mod 19 and that 19 does not divide |W (E_{6})|). By [18], Lemma 1.3, applied to \overline{G} := Ĝ/Z(Ĝ) we conclude that R ≤ \overline{H} ≤ Aut(R) where \overline{H} := H/(H ∩ Z(Ĝ)), with a non abelian simple subgroup R of \overline{G} as in Lemma 6.1 of [18]. Of those only L_{2}(19), L_{2}(37), HN, T h, B, M, J_{1}, ON, J_{3} and some of the groups of Lie type in the same characteristic p have order divisible by 19. But T h, B, M and ON possess Frobenius subgroups of type 31:15, which is possible inside \overline{G} only for p = 31 ([18], Lemma 1.7(2)). Now 31 ≡ 12 (mod 19), which is excluded in the Theorem. Similarly HN and J_{1} contain subgroups 11:10, whence they may be ruled out as candidates for R. Further we know p||H|, but |L_{2}(37)| is not divisible by any prime p admitted in the Theorem. The only possible case for the occurrence of L_{2}(19) is for p = 5. As H contains outer elements of G.2, \overline{H ∩ G} is a subgroup of \overline{H} of index two. If we had L_{2}(19) ≤ \overline{H} ≤ Aut(L_{2}(19)) = L_{2}(19): 2, then consequently \overline{H} = L_{2}(19): 2. But this is impossible since L_{2}(19): 2 does not possess outer elements of order 10 = 2p. The remaining case R = J_{3} might only happen for p ∈ {5, 17}, as we have |J_{3}| = 2^{7} · 3^{5} · 5 · 17 · 19. The case p = 5 is excluded as above. Finally by [6] the (34, 34, 19)-structure constants of J_{3}: 2 are too large to come from (C_{34}, C_{34}, C_{19}) in Ĝ.

Of the groups of Lie type remain, as in [18], Theorem 6.1, apart from E_{6}(p) only A_{2}(p^{3}). Now the end of the proof of Theorem 1 may be used to exclude R = A_{2}(p^{3}), and the assertion follows from [23], II, §4.2, Folgerung 3.

Remark: The above Theorems realize E_{6}(q)_{sc}: 2 as a Galois group for the class structure C = (C_{2p}, C_{2p}, C_{T}). By the ”Spiegelungssatz” [23], III, §2, Satz 4, we may obtain a further Galois realization for a class structure C′ := (C_{2}, C_{2p}, C_{T}) of Ĝ, with C_{2} some class of outer involutions different from [σ]. The systems of generators contained in Σ(C′) arise because the generating triple in C is a fixed point under the topological automorphism interchanging the first two (identical) classes. Indeed, the structure constant for C′ may be calculated as in Proposition 2, and turns out to be equal to 1. This gives a second proof for the Galois realizations. Analogously one finds further rational extensions for E_{6}(p)_{sc}: 2 with p as in Theorem 3, for a class structure (2, 2p, 19).

Theorems 1, 2 and 3 were first proved in [21], using numerous complicated calculations in Levi subgroups. These are replaced here by the results of [9] and of [22] about the decomposition of generalized Deligne-Lusztig characters.

3 Groups ^{2}E_{6}(q)_{sc}: 2 as Galois groups

The approach of the last section may be transferred to the case^{2}E_{6}(q). So let G := ^{2}E_{6}(q)_{sc}, q = p^{n}, p ≥ 5. We consider the extension Ĝ := ^{2}E_{6}(q)_{sc}: 2 of G with the graph field automorphism of order two. The resulting group has trivial center, so that the rationality criteria are applicable.

3.1 Auxiliary results Before treating Ĝ, as in the last section we will first calculate some character values in 2A_{2}(q): 2. Let ^{2}A_{2}(q) ≅ U_{3}(q) be the simple group of type ^{2}A_{2} and consider the extension 2A_{2}(q): 2 with the graph field automorphism of order two. The unitary group possesses three unipotent characters, indexed by partitions of three, as in the case A_{2}. Denote by σ an outer quasi-central involution, and by τ some nontrivial element in the maximal torus of order Φ_{6}/δ, δ := gcd(q + 1, 3), which is then necessarily regular.

Proposition 3.1: The values of the extensions of the unipotent characters of ^{2}A_{2}(q): 2 on the outer class [σ] and on the class of τ are as given in Table 1.

Proof : This is completely analogous to the proof of Proposition 2.1, again using the fact that σ is quasi-central and hence uniform, and the explicit decomposition of generalized Deligne-Lusztig characters in [22].

Table 3.1: The unipotent characters of ^{2}A_{2}(q): 2.

[ι]  [τ]  [σ]
q^{3}Φ_{1}Φ_{2}^{2}Φ_{6}/δ Φ_{6}/δ qΦ_{1}Φ_{2}
ϕ_{3}  1  1  1
ϕ_{21}  qΦ_{1}  −1  .
ϕ_{111}  q^{3}  −1  q

The next result, the proof of which was shown to the author by R. Gow, will be used to deduce the values of the pair of subcuspidal characters of Ĝ coming from ^{2}A_{5}(q).

Proposition 3.2 (Gow) : The cuspidal unipotent character ϕ_{321} of ^{2}A_{5}(q) is extendable to the extension with the graph field automorphism. The extensions have Frobenius-Schur indicator 0.

Proof : The extendability easily follows from the known degrees of the irreducible characters of ^{2}A_{5}(q). In Proposition 7.6 of [15] Lusztig shows that the representation belonging to the cuspidal character ϕ_{321} possesses a ^{2}A_{5}(q)-invariant antisymmetric bilinear form, hence has the Frobenius-Schur indicator ν(ϕ) = −1. We want to conclude that the extensions ϕ̂ have indicator ν(ϕ̂) = 0. Obviously the indicator of the extension is either equal to 0 or to −1. Assume ν(ϕ̂) = −1. Then ϕ̂ would have symplectic type. By [27], Theorem C, and the Ennola duality (proved by Lusztig) there exists in ^{2}A_{5}(q) a unipotent element u such that |ϕ(u)| = |ϕ(ι)|_{p}. In particular ϕ takes an odd value on this element. The unipotent elements are inverted by the outer involution σ, so ⟨u, σ⟩ is a dihedral group in ^{2}A_{5}(q): 2. As all irreducible characters of dihedral groups have indicator 1, ϕ̂(u) = ϕ(u) would have to be twice an algebraic integer. This yields a contradiction, so ν(ϕ̂) = 0 follows.

3.2 Some unipotent characters of ^{2}E_{6}(q)_{sc}: 2 We again choose a unipotent element u ∈ C_{G}(σ) ≅ F_{4}(q) in the centralizer of the graph field automorphism σ with |C_{F_{4}}(u)| = q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}. The class of the element uσ with centralizer order 2q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6} in Ĝ will be denoted by C_{2p}. Further C_{T} is defined to be the conjugacy class of a regular element of the maximal torus T of order Φ_{18}(q). Thus we have set up the analogue of the situation for E_{6}(q) in the last section.

The notation for unipotent characters of G = ^{2}E_{6}(q) follows [4]. Those unipotent characters not vanishing on a regular element of the torus T of order Φ_{18}(q), hence on C_{T}, are immediately recognized to be those of degree not divisible by Φ_{18}(q).

Proposition 3.3: The values on C_{T}, [σ] and C_{2p} of the extensions of those unipotent characters of Ĝ = ^{2}E_{6}(q)_{sc}: 2, q = p^{n}, p > 3, not vanishing on T^{#} are as given in Table 2.

Table 3.2: Some unipotent characters of ^{2}E_{6}(q)_{sc}: 2.

[ι]  C_{T}  [σ]  C_{2p}
|^{2}E_{6}(q)_{sc}|  Φ_{18}  |F_{4}(q)|  q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}
ϕ_{1,0}  1  1  1  1
ϕ_{4,1}  q^{2}Φ_{4}Φ_{8}Φ_{10}Φ_{12}  −1  qΦ_{4}Φ_{8}Φ_{12}  q(q^{6} + q^{4} + 1)
2A_{5}, 1  q^{4}Φ_{1}^{3}Φ_{3}^{2}Φ_{4}^{2}Φ_{8}Φ_{12}  −1  .  .
2A_{5}, ϵ  q^{13}Φ_{1}^{3}Φ_{3}^{2}Φ_{4}^{2}Φ_{8}Φ_{12}  1  .  .
ϕ_{4,13}  q^{20}Φ_{4}Φ_{8}Φ_{10}Φ_{12}  −1  q^{13}Φ_{4}Φ_{8}Φ_{12}  q^{13}
ϕ_{1,24}  q^{36}  1  q^{24}  .
ϕ′′_{6,6}  \frac{1}{3}q^{7}Φ_{3}^{2}Φ_{6}^{3}Φ_{8}Φ_{10}Φ_{12}  1  \frac{1}{3}q^{4}Φ_{3}^{2}Φ_{6}^{2}Φ_{8}Φ_{12}  \frac{1}{3}q^{4}Φ_{3}Φ_{6}(2q^{4} + 1)
2E_{6}[θ]  \frac{1}{3}q^{7}Φ_{1}^{4}Φ_{2}^{6}Φ_{4}^{2}Φ_{8}Φ_{10}  1  \frac{1}{3}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{8}  −\frac{1}{3}q^{4}Φ_{1}^{3}Φ_{2}^{3}Φ_{4}
2E_{6}[θ^{2}]  \frac{1}{3}q^{7}Φ_{1}^{4}Φ_{2}^{6}Φ_{4}^{2}Φ_{8}Φ_{10}  1  \frac{1}{3}q^{4}Φ_{1}^{4}Φ_{2}^{4}Φ_{4}^{2}Φ_{8}  −\frac{1}{3}q^{4}Φ_{1}^{3}Φ_{2}^{3}Φ_{4}

Proof : By the Howlett-Lehrer comparison theorem, the Harish-Chandra induced character from the cuspidal character ϕ_{321} of the Levi subgroup of type ^{2}A_{5} decomposes into the sum of the two subcuspidal characters ^{2}A_{5}, 1 and ^{2}A_{5}, ϵ. Both of the latter possess extensions to Ĝ for degree reasons, so correspondingly the Harish-Chandra induced characters from the extensions of ϕ_{123} decomposes into the sum of extensions of those subcuspidal characters. By Proposition 2 the extension of ϕ_{321} is not real valued, so the same applies to the four extensions of ^{2}A_{5}, 1 and ^{2}A_{5}, ϵ. In particular these extensions must vanish on the rational outer classes [σ] and C_{2p}. This proves the third and fourth row of the table.

Next we determine the remaining values on the class [σ]. This is done by the same argument as in the proof for the untwisted case E_{6}(q). Namely σ is quasi-central, and the decomposition of the generalized Deligne-Lusztig characters R_{T,1} on G.σ is known by [22], Theorem 8(2). From the formulae in [9] the values may now be calculated.

Also the characters ϕ_{4,1}, ϕ_{4,13} and ϕ_{1,24} are uniform by the result of [22], so their values on C_{2p} are again determined from the knowledge of the Green functions for F_{4} in good characteristic [29, 17].

By Lemma 1.1 we may now put ϕ′′_{6,6}(uσ) = \frac{1}{3}Φ_{3}Φ_{6}c, ^{2}E_{6}[θ](uσ) = ^{2}E_{6}[θ^{2}](uσ) = \frac{1}{3}Φ_{1}^{3}Φ_{2}^{3}Φ_{4}d, with integers c and d. From this, the structure constant n_{1} := n(C_{T} , [σ], C_{2p}) becomes

n_{1} = −\frac{3q^{4}(2q^{5} + q^{2} − q + 1) − Φ_{3}c − 2Φ_{1}^{3}Φ_{2}Φ_{4}d}{3q^{15}Φ_{1}^{3}Φ_{2}Φ_{3}Φ_{4}}.

This has to be a nonnegative integer. Estimates using Lemma 1.1 now show n_{1} = 0, and putting c = 2q^{8} + q^{4} + c_{0}, d = −q^{4} + d_{0} we hence derive

Φ_{3}c_{0} = −2Φ_{1}^{3}Φ_{2}Φ_{4}d_{0}.

Secondly, the structure constant n_{2} := n(C_{T} , C_{2p}, C_{2p}) is equal to

n_{2} = 1 − 2d_{0} \frac{2q^{12}Φ_{3} − (q^{10} − q^{9} + q^{8} − q^{6} + 3q^{5} − q^{4} + q^{2} − q + 1)d_{0}}{q^{19}Φ_{3}^{2}} .

This again has to be an integer, whence q^{9}Φ_{3}|d_{0}. If d_{0} ≠ 0, then we get a value for^{2}E_{6}[θ](uσ) which is larger than the square root of |C_{G}(uσ)|, which is impossible. So d_{0} = c_{0} = 0, and the table is proved.

Alternatively, arguments as in the proof of Proposition 2.2 might have been used to derive the character values on C_{2p}.

3.3 The Galois realization for ^{2}E_{6}(q)_{sc}: 2 From Proposition 3 it is already known that the class structure C = (C_{2p}, C_{2p}, C_{T}) of 2E_{6}(q)_{sc}: 2 has the structure constant n(C) = 1. We now prove:

Theorem 3.1: The groups ^{2}E_{6}(q)_{sc}: 2, q = p^{n}, p > 3, occur as Galois groups over ℚ^{ab} for the ramification structure C^{∗} = (C_{2p}, C_{2p}, C_{T})^{∗}. Here the respective classes contain elements of orders (2p, 2p, q^{6} − q^{3} + 1). More precisely a subfield of index 18 in ℚ(ζ_{q^{6}−q^{3}+1}) is a proper field of definition.

Proof : In the proof of Proposition 3 we already saw that n(C) = 1. For an application of the rationality criterion to the group Ĝ = ^{2}E_{6}(q)_{sc}: 2 with trivial center it remains to check the generation. As for E_{6}(q) it obviously suffices to prove generation in the factor group \overline{G} := Ĝ/Z(^{2}E_{6}(q)_{sc}). Due to the t.i.-Hall property of T Lemma 1.3 of [18] can be applied. The possibility of local subgroups is excluded as usually (use for example [5]). This leaves the case ^{2}A_{2}(q^{3}). This last group may again be excluded as in the previous section using the part of the character table of ^{2}A_{2}(q): 2 calculated in Proposition 1. The argument becomes even simpler, as groups of type ^{2}A_{2} possess only field automorphisms. Hence a triple from C must generate all of Ĝ. The assertion about the proper field of definition follows from the rationality properties of the classes in C.

As an immediate consequence we have Theorem 3.2: Let ^{2}E_{6}(q) be the simple group of type ^{2}E_{6} and let q = p^{n} with p > 3. Then 2E_{6}(q), ^{2}E_{6}(q): 2, and for q ≡ −1 (mod 3) also the covering group 3^{.}^{2}E_{6}(q) occur as Galois groups over ℚ^{ab}.

Proof : This follows like Theorem 2.2 for E_{6}(q) because by construction the regular Galois extension of ℚ^{ab}(t) for ^{2}E_{6}(q)_{sc}: 2 is ramified at precisely three points.

With a little bit more work we again obtain some Galois realizations over ℚ:

Theorem 3.3: For primes p ≡ 2, 3, 10, 13, 14, 15 (mod 19), p > 3, the groups ^{2}E_{6}(p)_{sc}: 2, 2E_{6}(p): 2, ^{2}E_{6}(p)_{sc} and^{2}E_{6}(p) occur as Galois groups over ℚ, where the ramification structure may be deduced from the one for ^{2}E_{6}(p)_{sc}: 2, C^{∗} = (C_{2p}, C_{2p}, C_{19})^{∗}.

Proof : A prime satisfying one of the congruences of the Theorem has multiplicative order 18 mod 19. Hence in this case 19 divides Φ_{18}(p), but no other Φ_{i}(p) with i < 18. So the prime 19 is a primitive divisor of Φ_{18}(p) in the sense of Zsigmondy.

Thus there exist elements of order 19 in the maximal torus T of G := ^{2}E_{6}(q)_{sc}. Because of |N_{G}(T)/T | = 9 these are semirational in G, and as σ acts fixed point freely on T, they become rational in Ĝ. The values of the unipotent characters on the corresponding class C_{19} coincide with those on C_{T} in Table 2, because the unipotent characters take constant value on T^{#}. This proves n(C_{2p}, C_{2p}, C_{19}) = 1, and leaves only the generation to be checked.

The local subgroups may be excluded as in the previous cases, using the element of order 19. By Lemmas 1.3 and 6.1 in [18] and the considerations in the proof of Theorem 2.3, only the simple groups L_{2}(19), L_{2}(37), HN, T h, B, M, J_{1}, ON, J_{3} and groups of Lie type in the same characteristic remain as candidates. Of these, T h, B, M and ON contain Frobenius groups of type 31:15, which is possible in Ĝ only for p = 31. But p = 31 ≡ 12 (mod 19) is excluded in the Theorem. Next, HN and J_{1} contain Frobenius groups 11:10 and thus do not lie in Ĝ. This leaves L_{2}(19), L_{2}(37) and J_{3}. Now the group generated by a triple from C clearly has order divisible by p, hence p has to be one of p = 2, 3, 5, 19 for L_{2}(19), p = 2, 3, 19, 37 for L_{2}(37) or p = 2, 3, 5, 17, 19 for J_{3}. None of these primes is allowed in the Theorem (note that P GL_{2}(19) possesses a (6, 6, 19) generating system, so p = 3 would have to be excluded at this point).

Of the groups of Lie type in the same characteristic, arguments as in [18], Theorem 6.1, or in the proof of Theorem 2.3 show that apart from ^{2}E_{6}(p) only ^{2}A_{2}(p^{3}) might possibly occur. Now use the end of the proof of Theorem 2.2 to exclude this last group. The result then follows from [23], II, §4.2, Folgerung 3.

As with E_{6}(q), we obtain a second Galois realization for the groups in Theorems 1–3 from the Spiegelungssatz [23], belonging to a ramification structure (C_{2}, C_{2p}, C_{T}).

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Heidelberg June 1991