Green functions for groups of types E_{6} and F_{4}

in characteristic 2

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

In this paper we determine the Green functions for the connected groups E_{6}(2^{n}), ^{2}E_{6}(2^{n}) and F_{4}(2^{n}), and the outer Green functions for the disconnected groups E_{6}(2^{n}): 2 and^{2}E_{6}(2^{n}): 2. The Green functions of a finite Chevalley group G are a main ingredient for the determination of the character table. By the work of Deligne and Lusztig, the character values on semisimple elements may be calculated purely combinatorically. The values on arbitrary elements can be obtained if certain functions on the set of unipotent elements are known for all centralizers of semisimple elements. These functions consist of the Green functions, which were first introduced by Green [5] in the context of the general linear group, and a few more so called nonuniform functions, which are orthogonal to all Deligne-Lusztig characters.

The Green functions are known for connected classical groups [8], and for connected exceptional groups in good characteristic [1,14]. They were determined for disconnected groups of types A_{l}(2^{n}).2, l ≤ 5, D_{4}(2^{n}).2 and D_{4}(3^{n}).3 in [11]. Here we calculate them for connected and disconnected groups of type E_{6} and for groups of type F_{4} in characteristic 2. The Green functions for the twisted groups ^{2}F_{4}(2^{2n+1}) were already computed in [10].

Our computations rely on some properties known to be satisfied by Green functions, namely integrality, orthogonality relations and certain induction formulae. The determination of the Green functions in good characteristic by Beynon, Spaltenstein and Shoji made use of a recursive algorithm, the validity of which relies on the parametrization of unipotent classes by nilpotent orbits in the Lie algebra. But such a bijection does not exist in characteristic 2, and hence the algorithm cannot be applied here.

Our approach makes necessary calculations with matrices of polynomials in the indeterminate q. These were performed with the computer algebra system MAPLE. The results were put into TeX-format by MAPLE and then printed out to obtain the given tables. So the danger of misprints should be rather small.

Acknowledgement: The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft.

1 Properties of Green functions

Let G be a reductive algebraic group defined over the algebraic closure 𝔽̄_{p} of a finite field of characteristic p, and F : G → G a Frobenius map with fixed point set G := G^{F}, a finite group of Lie type. Denote by W the Weyl group of G. Then the G-classes of F -stable maximal tori of G are parametrized by the F -conjugacy classes of W. If T is F -stable and parametrized by the F -class of w ∈ W, we also write T = T_{w}, where w is determined up to F -conjugation.

If the algebraic group G is connected, Deligne and Lusztig defined for each G-class of maximal F -stable tori T = T_{w} and each (linear) character θ of T_{w}^{F} =: T_{w} a generalized character R_{T_{w},θ} of G, and showed that the ℂ-span of these characters exhausts almost all class functions on G. Class functions lying in this span are called uniform, the others nonuniform. It follows from the results of Deligne and Lusztig that the value of R_{T_{w},θ} is independent of θ on unipotent classes, so that on such classes R_{T_{w},θ} = R_{T_{w},1}, for which we will also write Q_{w}. If it is necessary to specify the group for which Q_{w} is defined, we indicate it with an upper index Q_{w}^{G}. The functions Q_{w} on the set of unipotent elements of G are called the Green functions of G, and they are the ones we are going to determine.

The Q_{w} are known to satisfy a number of nice properties, which we recall now. For this let P be an F -stable parabolic subgroup of G with F -stable Levi subgroup L, and denote by W_{L} the Weyl group of L.

Lemma 1: Let G, W and Q_{w} be as above. Then we have:

(a) The Q_{w} are integer valued.

(b) If T = T_{w} < L < P are F -stable, then Q_{w}^{G} is obtained by Harish-Chandra induction of Q_{w}^{L} from L to G.

(c) We have the orthogonality relations

\frac{1}{|G|} ∑_{{u∈G \atop unipotent}} Q_{v}(u)Q_{w}(u) = \{ \begin{matrix}\frac{|N_{G}(T_{w})|}{|T_{w}|^{2},} & if v ∼_{F} w \\ 0, & else \end{matrix} .

(d) The Green functions take value 1 on regular unipotent elements, and value ±(G : T_{w})_{p′} on the identity.

For proofs of these statements see for example [2, 7.4.4, 7.5.1, 7.6.2, 8.4.1]. Now assume that G is disconnected, obtained from its connected com-

ponent G^{o} by adjoining a graph automorphism σ of prime order. In this situation Digne and Michel have shown [4] that the construction of Deligne-Lusztig characters carries over from the connected case. Namely, if T^{o} ⊂ B^{o} denote a σ stable maximal torus contained in a σ stable Borel subgroup of G^{o}, then one sets B := N_{G}(B^{o}) and T := N_{B}(T^{o}), which are then called Borel subgroup and torus of G. The G^{o}-conjugacy classes of F, σ stable maximal tori T of G are parametrized by conjugacy classes of the Weyl group W^{σ} (an upper index σ denotes the fixed points under σ). For each such class and each (linear) character θ of T = T^{F} a generalized Deligne-Lusztig character may be defined on the coset G^{o}.σ.

Clearly, if the defining characteristic p of G divides the order of the graph automorphism, there exist unipotent elements in the coset G^{o}.σ. Then the values of the generalized Deligne-Lusztig characters on these elements turn out to be independent of θ, and hence as in the connected case we obtain outer Green functions Q_{w} on the unipotent classes of G^{o}.σ, this time indexed by the classes of W^{σ}. The rôle of the identity element in the connected case is here played by the quasi-central elements, which are the outer unipotent elements characterized by the fact that they lie inside a torus T of G (see [4, 1.28]).

Lemma 2: Let G be a disconnected group as defined above and Q_{w} the corresponding outer Green functions. Then we have:

(a) The Q_{w} are integer valued.

(b) If T = T_{w} is contained in the F, σ-stable Levi subgroup L < G of an F, σ-stable parabolic subgroup P, then Q_{w}^{G} is obtained by ordinary Harish-Chandra induction of Q_{w}^{L} from L^{o}.σ to G^{o}.σ.

(c) We have the orthogonality relations

\frac{1}{|G^{o}|} ∑_{u∈Go.σ_{unipotent}} Q_{v}(u)Q_{w}(u) = \{ \begin{matrix}\frac{|N_{(Gσ)o}(T^{σ})|}{|T^{σ}^{o}|^{2}}, & if v ∼ w in W^{σ} \\ 0, & else \end{matrix} .

(d) The Green functions take value 1 on outer regular unipotent elements, and value ±(G^{σ} : T_{w}^{σ})_{p′} on quasi-central elements.

Proofs for the above statements may be found in [4] and [11].

In the case of disconnected groups, the Green functions are at present only known for (A_{l}, 2), l ≤ 5, (A_{l}(2^{n}) extended by the graph automorphism of order two), (D_{4}, 2) and the exceptional case (D_{4}, 3) [11]. The other exceptional case (E_{6}, 2) is treated in this paper. The infinite families (A_{l}, 2), l ≥ 6, and (D_{l}, 2), l ≥ 5, are still open. The corresponding twisted cases were not considered in [11] since they were expected to agree with the untwisted cases by a Shintani argument. This is proved for the cases in [11] and for (E_{6}, 2) in Section 5.

2 The connected groups of type E_{6}

The easiest case to handle is the connected group of type E_{6} in characteristic 2, because there twenty out of twentyfive Green functions can be obtained by Harish-Chandra induction. The unipotent classes of groups of type E_{6} were determined by Mizuno [12]. It turns out that the adjoint group is the one with the least number of unipotent classes, so let G := E_{6}(2^{n})_{ad}. By the results of Lusztig, the Green functions are the same for all isogeny types, so we may well make this choice. Before we can determine the Q_{w} for G, the ones for groups of types D_{4} and D_{5} have to be calculated. It is usual in the literature not to print out the Green functions themselves, but the almost characters formed from them by twisting with Weyl group characters (see [1] for example). We give lists of this equivalent information. The notation for the Weyl group characters is the same as in [2].

Proposition 1: Representatives of the unipotent classes of D_{4}(q), q = 2^{n}, and their centralizer orders are as given in Table 1. The values of the almost characters R_{χ}, χ ∈ Irr(W (D_{4})), on these classes are given by Table 2.

Proof : Due to the small rank of D_{4}, it is rather easy to compute the unipotent classes in the Borel subgroup of D_{4}(2^{n}), using the Steinberg relations. From that the fusion into D_{4} and the centralizer orders may be determined. Also, the corresponding classes in the algebraic group can be identified, using Spaltenstein’s description [15, I.2.6]. The details are omitted.

For the proof of Table 2 we use Lemma 1(b). All proper Levi subgroups of D_{4} have type A, and for linear groups all Green functions in all characteristics were described already by Green [5]. To apply Lemma 1 we still have to know the fusion of unipotent elements from the parabolic subgroups into D_{4}(q). But this follows easily from the previous calculation of the unipotent classes in the Borel subgroup. Thus ten of the thirteen Green functions of D_{4}(q) may be obtained. One of the missing functions, namely the one corresponding to the Coxeter torus of D_{4}, was determined by Lusztig in [6]. This leaves two unknown functions on fourteen unipotent classes. Lemma 1(d) gives the values on the two regular classes and on the identity. The orthogonality relations in Lemma 1(c) yield eleven linear equations for the remaining eleven unknown values of both Green functions, thereby determining them.

Proposition 2: Representatives of the unipotent classes of D_{5}(q), q = 2^{n}, and their centralizer orders are as given in Table 3. The values of the almost characters R_{χ}, χ ∈ Irr(W (D_{5})), on these classes are given by Table 4.

Proof : We first sketch the determination of the unipotent classes. The calculation of all unipotent classes in the Borel subgroup of D_{5} is already quite tedious. But in fact we just need the classes in the maximal parabolic subgroups for Harish-Chandra induction. Let P = UL be a parabolic subgroup with Levi complement L of type D_{4}(q).(q − 1) and unipotent radical U. The unipotent classes in P all contain representatives of the form v_{1}v_{2} with v_{1} lying in the unipotent radical U, and v_{2} being a representative of a unipotent conjugacy class of D_{4}(q). The latter are already known, so we just have to split up the U-cosets of suitable representatives of the fourteen unipotent D_{4}(2^{n})-classes, using the Steinberg relations. One finds 47 unipotent classes in P, fusing to twenty unipotent classes in D_{5}(2^{n}). The details are omitted.

Thus the fourteen previously determined Green functions of D_{4} may be Harish-Chandra induced to D_{5}. The same procedure applies to the maximal parabolic subgroup with Levi complement of type A_{4}. Here the parabolic subgroup contains 39 unipotent classes. From these two steps thirteen Green functions are obtained.

Two more types of maximal tori of D_{5} lie in maximal parabolic subgroups, but instead of performing the tedious computation of unipotent classes in these parabolics, we interpolate the induction formulae. Namely, let Q = V M be a maximal parabolic subgroup with Levi subgroup M and unipotent radical V. Denote by v_{1} = 1, …, v_{r} a system of representatives for the unipotent classes of M, and similarly by u_{1}, …, u_{s} one for the unipotent classes of G. Then there exist non-negative integers a_{ij}, i = 1, …, r, j = 1, …, s, (depending on q) such that for any class function χ of M we have

χ̂^{G}(u_{j}) = r ; ∑ ; i=1 a_{ij} χ(v_{i}), for 1 ≤ j ≤ s,

where χ̂ denotes the inflation of χ to Q, and χ̂^{G} is the induction to G of χ̂. If χ_{k}, k = 1, …, t, denote the Green functions of M whose induction to G is already known, then we obtain the s matrix equations

\underline{χ}(u_{j}) = (χ_{k}(v_{i}))_{k,i} (a_{ij})_{i}, for 1 ≤ j ≤ s,

where \underline{χ} = (χ_{1}, …, χ_{k})^{t}. This gives s · t linear equations in the unknowns a_{ij}. Now if M is of type A, then it has the same number of unipotent classes and of Green functions. Assume that the Harish-Chandra induction of t = r − 1 of these functions is already known. If we can compute the a_{1j}, j = 1, …, s, i.e., the unipotent Q-classes of V = V.[v_{1}] itself, and hence the fusion formula for V into G, this yields another s conditions on the unknowns a_{ij}. Also, the non-negativity of the a_{ij} and the knowledge of χ_{k}(u_{s}) = 1 for the class [u_{s}] of regular unipotent elements forces a_{rs} = 1, a_{is} = 0 for 1 ≤ i ≤ r −1. In general these equations suffices to uniquely determine the a_{ij}. This finally allows to compute the Harish-Chandra induction of the last Green function.

In our case, for parabolic subgroups with Levi complement of type A_{3}+A_{1} or A_{2} + 2A_{1}, the assumptions for the application of the interpolation are satisfied, since all but one of the induced Green functions are already known by induction from the first two parabolic subgroups. Thus we obtain two more Green functions, and just three are missing on the twenty unipotent classes. The one corresponding to the Coxeter torus again follows from Lusztig’s paper [6]. The remaining two functions may now be calculated from the orthogonality relations, using the known values on regular unipotent elements.

The conjugacy classes of simply connected groups of type E_{6} were determined by Mizuno [12]. He also gives explicit expressions as products of root elements for representatives of unipotent classes. From his result it is easy to find the unipotent conjugacy classes in the groups of adjoint type. In Table 5 we give the correspondence between his notation and ours for representatives of the unipotent classes.

The calculations needed to obtain the unipotent classes are briefly described in the proof of the next Theorem.

Theorem 1: For G = E_{6}(2^{n}), the values of the almost characters R_{χ}, χ ∈ Irr(W (E_{6})), on the unipotent classes in G are given by Table 6.

Proof : As was already mentioned above, most of the Green functions may be obtained by usual Harish-Chandra induction, applying Lemma 1(b). The group G has a maximal parabolic subgroup P_{1} ≅ [q^{16}].D_{5}(q).(q − 1) with Levi subgroup of type D_{5}. The eighteen Green functions of that group were determined in Proposition 2. For the induction the fusion of unipotent classes from P_{1} to G has to be found. For this we employ the method described in the proof of Proposition 2, using the knowledge of the classes in D_{5}(q). The calculations are rather lengthy but straightforward, and we omit them. Comparison with the list of classes in [12] constitutes a good test for the results obtained.

Induction of the eighteen Green functions of D_{5}(q) as given in Proposition 2 yields sixteen different functions for G (two pairs of P_{1}-classes of maximal tori fuse in G). The fusion pattern from the maximal parabolic subgroup P_{2} ≅ [q^{21}].A_{5}(q).(q − 1) may now be obtained by interpolation, as described in the proof of Proposition 2: All but two of the Green functions induced from A_{5} are already known. We just have to compute the fusion for cosets with respect to the unipotent radical for two of the unipotent classes of A_{5}, then the remaining fusion can be interpolated. With that we obtain the induction of the two new Green functions. Similarly, the one new Green function from each of the two parabolic subgroups with Levi complement of type A_{4} + A_{1} and 2A_{2} + A_{1} follows. We have thus obtained twenty of the twentyfive functions.

By Lusztig’s decomposition of the R_{T_{w},1}, the almost characters R_{6,25} and R_{6,1} coincide with actual irreducible characters. In the formula for the difference R_{6,25} − R_{6,1}, only Green functions occur which are already known. In particular one gets

(R_{6,25} − R_{6,1})(u_{1}) = q^{25} − qΦ_{5}(q^{3} − q + 1).

Using that χ(u)|[u]|/χ(1) is integral for all irreducible characters χ we conclude that q^{25}|R_{6,25}(u_{1}). Moreover certainly |R_{6,1}(u_{1})| ≤ R_{6,1}(1), and these two conditions, coupled with the above formula for the difference, uniquely determine the values of both R_{6,25} and R_{6,1} on u_{1}. Now by Lemma 1(c) the norm of each Q_{w} and hence also of each almost character R_{χ} is known. With this information one checks that R_{6,25}(u_{0}) and R_{6,25}(u_{1}) already give the full contribution for the norm, so that R_{6,25} has to vanish on all classes different from these two. The orthogonality relations and the known decomposition of the Q_{w} then enable us to compute Q_{w}(u_{1}) for all (missing) Q_{w}.

Next we determine Q_{w} for |T_{w}| = Φ_{9}. Define a_{i} := Q_{w}(u_{i}). By Lemma 1 the values of a_{0}, a_{26}, a_{27} are known. Also, a_{1} was calculated above. The orthogonality relations with the known twenty functions yield twenty linear equations in a_{2}, …, a_{25}. We solve these for a_{2}, …, a_{16}, a_{18}, a_{19}, a_{20}, a_{23}, a_{25} and substitute into the quadratic norm equation for Q_{w}. Thus we obtain a quadratic equation in the remaining four unknowns a_{17}, a_{19}, a_{21} and a_{24}.

The expression for a_{25} has the form

a_{25} = \frac{2 − a_{21} − a_{22} + 2q^{4} + 2q^{3} + 4q}{2q(q^{3} + q^{2} + q + 2)} .

As a_{25} has to be integral by Lemma 1(a), this forces

a_{21} = 2 − a_{22} − 2q^{2} + q(q^{3} + q^{2} + q + 2)a, a ∈ ℤ.

We can now decompose the norm equation into a sum of squares with positive coefficients (for positive q) plus a quadratic polynomial in a (with coefficients in ℚ(q)). As the sum of squares is certainly nonnegative, this last summand can be at most zero. But the only integral value for a satisfying this turns out to be a = 0.

As the torus T_{w} of order Φ_{9} contains regular semisimple elements for all q, there exist irreducible Deligne-Lusztig characters R_{T_{w},θ} for all q. So in particular ±Q_{w}(u_{i}) are values of irreducible characters of G. Using again that χ(u)|[u]|/χ(1) is integral, this time applied to χ = R_{T_{w},θ}, we conclude

\frac{(q^{2} − 1)(q^{3} − 1)}{3} | a_{17}.

This forces a_{22} = q + 2 + (q^{2} + q + 1)b/3, for some b ∈ ℤ, via the earlier equations. We can now again write the norm equation as a sum of two squares plus a quadratic term in b negative only for b = −3. In fact, b = −3 annulates this last summand, and so the two previous terms also have to vanish, finally showing a_{17} = (q^{3}−1)(q^{2}−1), a_{24} = −q^{2}+1. The other a_{i} are now determined by the system of linear equations. This completes the calculation of Q_{w} for the torus of order Φ_{9}.

Next we apply the same methods to the Green function for the Coxeter torus of order Φ_{3}Φ_{12}, keeping in mind that we have one additional condition (orthogonality to the new Q_{w}). The arguments are similar but easier. After having found this Green function, the last three follow again by orthogonality and the norm equation. Using the character table of the Weyl group W (E_{6}) the almost characters may then be calculated.

3 The twisted connected groups of type E_{6}

We now turn to the twisted groups G = ^{2}E_{6}(2^{n})_{ad} of type E_{6} in characteristic 2. In good characteristic Beynon and Spaltenstein showed that the Green functions may be obtained from the ones for the untwisted group by simply replacing q by −q. As we will see, this also holds in characteristic 2. But unfortunately we can not just copy the proof of Theorem 1, since for ^{2}E_{6} only sixteen (instead of twenty in case E_{6}) maximal tori lie in F -stable parabolic subgroups, and hence nine have to be found by other arguments.

For Harish-Chandra induction we first have to find the Green functions and unipotent classes of the twisted groups ^{2}D_{4}(2^{n}). The almost characters here have to be formed with respect to σ-class functions of W (D_{4}), where σ denotes the graph automorphism of W (D_{4}) of order two. These may be identified with extensions of σ-stable irreducible characters of W (D_{4}).

Proposition 3: Representatives of the unipotent classes of ^{2}D_{4}(q), q = 2^{n}, and their centralizer orders are as given in Table 7. The values of the almost characters R_{χ}, χ a σ-class function of W (D_{4}) as described above, on these classes are given by Table 8.

Proof : The unipotent classes are calculated as in the previous cases. The proper Levi subgroups of G have type A, so their Green functions are known. All but two of the nine Green functions are then obtained by Harish-Chandra induction. The Green function for the Coxeter torus was determined by Lusztig [6]. The remaining function immediately follows from the expression for the trivial character in terms of the R_{T,1}.

Theorem 2: Representatives of the unipotent classes of G = ^{2}E_{6}(q)_{ad}, q = 2^{n}, and their centralizer orders are as given in Table 9. The Green functions of G are obtained by formally replacing q by −q in the corresponding Green polynomials for E_{6}(2^{n}) as calculated in Section 2.

Proof : The unipotent conjugacy classes in G and in parabolic subgroups may be found using the Steinberg generators and relations. So Table 9 is obtained. The Green functions for the Levi subgroup of type ^{2}A_{5} are known by the result of Lusztig and Srinivasan [9] on Ennola duality for the general linear groups. When the unipotent classes of the corresponding parabolic subgroup and the fusion into G are determined, we may apply Lemma 1 to Harish-Chandra induce these eleven Green functions to G. The Green functions and unipotent classes of the Levi subgroup of type ^{2}D_{4} were calculated in Proposition 3. As in the case of E_{6} we need not determine the fusion of all classes from the corresponding parabolic subgroup into G if we use interpolation of fusion. Thus another three Green functions are obtained by induction. Still two more Green functions belong to isotropic tori. The fusion from the relevant parabolic subgroups may be interpolated, when the fusion of one class is known. This yields two more Green functions. The details are omitted.

Still nine functions are missing. From the known decomposition of the R_{T,1} it follows that the difference R_{2,16′′} − R_{2,4′} of unipotent characters involves only Green functions which have already been determined. In particular we get

R_{2,16′′}(u_{1}) − R_{2,4′}(u_{1}) = q^{25} − qΦ_{10}(q^{3} − q − 1).

As in the case of E_{6} this allows us to determine both values from divisibility properties and again it turns out that the generalized character R_{2,16′′} − R_{2,4′} has to vanish on all unipotent classes different from u_{0} and u_{1}. From the scalar product formula we may then calculate the values of all Q_{w} on u_{1}.

We now turn to Q_{w} for |T_{w}| = Φ_{18}. Denote Q_{w}(u_{i}) =: a_{i}. Then the orthogonality of Q_{w} to the previously determined Green functions yields sixteen linear equations on the a_{i}. Also, a_{0} and a_{26}, a_{27} are known by Lemma 1(d). Integrality considerations on the linear equations and divisibility conditions on the a_{i} (which are character values) show

a_{2} = Φ_{1}^{3}Φ_{2}^{4}Φ_{3}Φ_{4}Φ_{6}(−1 + q − q^{2} − q^{4} − q^{7} + q^{9}a/3)

with a ∈ ℤ. If this is substituted into the norm equation for Q_{w}, then the only integral solution for a turns out to be a = 0. Also one finds

a_{3} = Φ_{1}^{2}Φ_{2}^{3}Φ_{6}(1 + q^{2} + q^{4} + q^{5} + q^{7} + q^{11}b), b ∈ ℤ,

and again the norm equation forces b = 0. This kind of argument also works for a_{4}, a_{6} and a_{7}. With this many unknowns determined, we may now write the norm equation as a sum of squares, all of which must hence vanish. Thus the values of the remaining a_{i} follow.

The next torus to consider is the one of order Φ_{6}Φ_{12}. Write Q_{w}(u_{i}) =: b_{i}. As above one finds

b_{2} = Φ_{1}^{3}Φ_{2}^{4}Φ_{3}Φ_{4}(−1+q−2q^{2}+2q^{3}−3q^{4}+2q^{5}−2q^{6}+q^{7}−q^{8}+q^{9}c), c ∈ ℤ.

The norm condition then forces c = 0. This method may also be applied to b_{3}, b_{20} and b_{10}. From those values and the linear equations on the b_{i} one may then again deduce all missing unknowns.

This method may be successfully applied to all missing Q_{w}, keeping in mind that each additionally determined Green function provides one more condition on the others via orthogonality. We do not go into the details.

Finally one checks that the functions indeed agree with those for the untwisted group E_{6}(q) when q is replaced by −q.

4 The disconnected groups of type E_{6}

We now consider the disconnected groups of type E_{6} in characteristic 2. Let G := E_{6}(2^{n})_{ad}.2 be the extension of G^{o} := E_{6}(2^{n})_{ad} by the graph automorphism of order 2. Spaltenstein determined the unipotent conjugacy classes in the outer coset for the algebraic group [15, p.160], but unfortunately his table for the representative elements contains several misprints; also, the splitting of the classes in the finite group was not previously known. Therefore we first have to calculate the unipotent outer classes.

Proposition 4: The unipotent classes in the coset G^{o}.σ of G = E_{6}(q)_{ad}: 2, q = 2^{n}, with representative uσ, and their centralizer orders are as given in Table 10. The last column indicates the corresponding representative in the algebraic group according to [15, p.160].

Proof : The outer conjugacy classes of G may be determined in the same way as in the previous cases, using the Steinberg relations and the known action of the graph automorphism on the root subgroups. The group G^{o} has a σ-stable maximal parabolic subgroup P with σ-stable Levi decomposition P = UL, where L has type A_{5}(2^{n}). First we determine the unipotent classes in P. The outer unipotent classes of the extension A_{5}(2^{n}).2 by the graph automorphism of A_{5} were determined in [11, Table 5]. All unipotent conjugacy classes in P.σ have a representative of type u_{1}u_{2}σ, where u_{1} lies in U and u_{2} in A_{5}. Moreover u_{2}σ may be chosen to lie in a fixed set of representatives of outer unipotent classes of A_{5}(2^{n}).σ. It therefore suffices to determine the P -conjugacy classes of unipotent elements in the U-coset for a representative of each unipotent conjugacy class of A_{5}(2^{n}).σ. Then the unipotent classes in P.σ are known. In the next step the fusion of these classes in G is calculated, and this yields the required outer unipotent classes. The details of these calculations are rather straightforward but tiresome, so we will not give them. Similar computations may be found in [12], for example.

The proof of the preceding proposition already yielded the fusion of unipotent classes from P.σ ≅ [q^{21}].A_{5}(q)(q − 1).σ into G. This allows us to determine certain Harish-Chandra induced Green functions. The matrix relating the generalized Deligne-Lusztig characters and the unipotent characters, i.e., the decomposition of the R_{T,1}, was calculated in [11]. It turned out that almost characters formed with respect to W^{σ} are close to being irreducible unipotent characters on the coset G^{o}.σ. Instead of writing down the values of the Green functions, we give the equivalent information on the values of the almost characters, because these have a more compact form and moreover are nearer to the values of unipotent characters.

Theorem 3: For G = E_{6}(2^{n}): 2, the values of the almost characters R_{χ}, χ ∈ Irr(W^{σ}), on the unipotent classes in G^{o}.σ are given by Table 11.

Proof : The outer Green functions of the disconnected group A_{5}(2^{n}).2 are given in [11, Table 7]. Having determined the classes of P in the proof of Proposition 3, we may thus apply Lemma 2(b) to obtain ten induced Green functions for G^{o}.σ. Next we want to use the maximal σ-stable parabolic subgroup P_{2} = V M with Levi complement M of type D_{4}. The outer unipotent classes and the Green functions for the disconnected group of type D_{4} were found in [11, Tables 10,11]. It now suffices to determine the classes in V - cosets of four outer D_{4}.2-classes in P_{2}. To avoid more cumbersome calculations with generators and relations we may then apply the interpolation of fusion described in the proof of Proposition 2. So four more Green functions follow by Harish-Chandra induction. The remaining two maximal σ-stable parabolic subgroups of G contain only one new σ-stable maximal torus each. Again by interpolation of fusion the corresponding two Green functions are obtained.

The situation now is much worse than for the connected group, since nine functions remain to be determined after using induction. But still similar ideas as in that case lead to success. In [11, Theorem 8] we decomposed the generalized Deligne-Lusztig characters on G^{o}.σ into extensions of unipotent characters. The decomposition was described in terms of almost characters with respect to W^{σ} = W (F_{4}). In particular, R_{2,16′′} and R_{2,4′} were proved to coincide with extensions of unipotent characters (where the characters of W (F_{4}) are denoted as in [2]). Moreover, the difference R_{2,16′′} −R_{2,4′} of almost characters is independent of the missing nine Green functions. It turns out that

(R_{2,16′′} − R_{2,4′})(u_{1}) = q^{16} − q^{4}.

Considerations as in the proof of Theorem 1 on divisibility and absolute value of the character values now force R_{2,16′′}(u_{1}) = q^{16} and R_{2,4′}(u_{1}) = q^{4}. Calculation of the norm of R_{2,16′′} then reveals that this extension has to vanish on all outer unipotent classes different from u_{0} and u_{1} (the value on the quasicentral element u_{0} is known by Lemma 2(d)). By taking scalar products with the Green functions and using Lemma 2(c) the values of all functions on u_{1} follow as in the proof for the connected group.

We now consider Q_{w} for T_{w}^{σ} of order Φ_{12}. Write Q_{w}(u_{i}) =: a_{i}. The orthogonality relations of Lemma 2(c) to the first sixteen Green functions yield sixteen linear equations in a_{2}, …, a_{25} (a_{26} = a_{27} = 1 follows from Lemma 2(d)). The norm condition for Q_{w} leads to a further quadratic equation which may be decomposed into a sum of squares and a term quadratic in a_{24} whose only nonpositive integral solution is a_{24} = −q + 1. The same procedure yields a_{25} ∈ {q + 1, q + 3}, where the second case might only occur for q = 4. (Here we used that a_{25} ≡ 1 (mod 2) by well known congruences for character values.) Trying to continue the calculations with a_{25} = q + 3 leads to no integer solution for a_{14}, so a_{25} = q + 1. Then we get a_{23} ∈ {1, q + 2}, and the congruence mod 2 excludes the second possibility. After having determined these unknowns, the remaining values follow rather easily by suitable decomposition of the norm equation.

We next turn to the torus with σ-fixed point group of order Φ_{8}, and write Q_{w}(u_{i}) =: b_{i}. Again a suitable decomposition of the norm equation shows b_{25} = 1, then b_{24} = 1, b_{23} = −q^{2} + 1 = b_{22}, and then leads to unique integral solutions for all other missing b_{i}.

The third anisotropic torus to consider is the one with |T_{w}^{σ}| = Φ_{3}^{2}. Here a small case by case analysis for q = 2, 4 and 8 has to be carried through to exclude the possibility of certain degenerate solutions for these small values of q. It turns out that again just one integral solution exists. Having determined these new three Green functions, we have three more equations by orthogonality for the other Green functions at our disposal. This facilitates their computation, which is rather straightforward with the above method. We omit the details. The almost characters may now be computed from the character table of W (F_{4}) in [2].

5 Twisted disconnected groups

We will use a Shintani descent argument to handle the Green functions for the twisted disconnected groups. Let G be a disconnected algebraic group with connected component G^{o} of prime index. Let σ ∈ G be such that G = G^{o}.⟨σ⟩. Assume further given a Frobenius map F of G commuting with σ. If σ acts nontrivially on G^{o}, we obtain two different Frobenius maps and two distinct finite groups of fixed points, namely G := G^{F} and G̃ := G^{σF}. Then there exists a nice bijection between the outer conjugacy classes of these two finite groups, as was first shown in [3].

Namely let g ∈ G^{o}. Then by the theorem of Lang-Steinberg applied to the Frobenius map σF of the connected group G^{o} there exists x ∈ G^{o} with g = x^{σF} x^{−1}. Define ψ_{x}(g) := x^{−1}x^{F}.

Proposition 5: Let C be a G^{o}-conjugacy class in G^{o}.σ with representative gσ, where g ∈ G^{o}. As above write g = x^{σF} x^{−1}.

(a) Then ψ_{x}(g) ∈ G̃^{o}, and the G̃^{o} conjugacy class of ψ_{x}(g)σ is uniquely determined by C. We write ψ(C) := [ψ_{x}(g)σ] for this class, where gσ ∈ C.

(b) The map ψ defines a bijection between the G^{o}-conjugacy classes in G^{o}.σ and the G̃^{o}-conjugacy classes in G̃^{o}.σ.

(c) We have |C_{G}(g_{1}σ)| = |CG̃(g_{2}σ)| if ψ([g_{1}σ]) = [g_{2}σ], so the map ψ preserves centralizer orders.

Proof : (a) As gσ ∈ G we have x^{σF} x^{−1}σ = (x^{σF} x^{−1}σ)^{F} = x^{σF^{2}}x^{−F} σ, so x^{σF} x^{−1} = x^{σF^{2}}x^{−F}. Then

ψ_{x}(g)^{σF} = (x^{−1}x^{F})^{σF} = x^{−σF} x^{σF^{2}} = x^{−1}x^{F} = ψ_{x}(g),

so ψ_{x}(g) ∈ G̃^{o}. Now let g = y^{σF} y^{−1} be another decomposition of g via the Lang-Steinberg theorem. Then g = x^{σF} x^{−1} = y^{σF} y^{−1} and hence

ψ_{y}(g)σ = y^{−1}y^{F} σ = y^{−σF} x^{σF} x^{−1}y^{F} σ = (x^{−1}y)^{−σF} ψ_{x}(g)σ(x^{−1}y)^{σF} ,

where (x^{−1}y)^{σF} = x^{−1}y, so the G̃^{o}-conjugacy class of the image is independent of the choice of x. Finally choose another representative hσ ∈ C. Then there exists k ∈ G^{o} with gσ = (hσ)^{k}, i.e., h = kgk^{−σ^{−1}} = kx^{σF} x^{−1}k^{−σ^{−1}}, so the element y := k^{σ^{−1}}x is a preimage of h under Langs map. We obtain

ψ_{y}(h) = y^{−1}y^{F} = x^{−1}k^{−σ^{−1}}k^{σ^{−1}}x^{F} = ψ_{x}(g).

Thus the G̃^{o}-class of ψ_{x}(g)σ is uniquely determined by C.

(b) and (c) are similar in spirit and we omit them (see [3, I.7.2], and note that σF -classes of G^{o} are the same as outer classes in G^{o}.σ).

Now assume in addition that the order of σ coincides with the defining characteristic of G. Then ψ preserves unipotent classes. In fact, let U be the unipotent radical of a σ-stable Borel subgroup of G. If gσ ∈ G^{o}.σ is unipotent, then after taking a suitable conjugate we already have gσ ∈ U.σ, where U = U^{F}. Now apply Proposition 5 to the algebraic group U to find that ψ([gσ]) intersects U^{σF} .σ, so also contains unipotent elements.

This property already gives a strong hint for a connection between the outer Green functions for the twisted and the untwisted case. Namely we may also apply the above ideas to a σ, F -stable maximal parabolic subgroup P of G^{o}, thus obtaining a bijection between the unipotent conjugacy classes in P^{F} .σ and P^{σF} .σ. Moreover that map preserves the fusion pattern from the parabolic subgroup into the reductive group by the same arguments as in the proof of Proposition 5. Or, to put it differently, ψ is compatible with the restriction to connected subgroups.

So if a function on the level of P is known to agree in the twisted and the untwisted case, the induced function also agrees on the level of G. A σ, F -stable maximal torus of G clearly also is σ, σF -stable and hence the families of generalized Deligne-Lusztig characters and the Green functions are parametrized in the same way. These observations enable us to determine the outer Green functions in the twisted cases rather easily.

Theorem 4: Let G be one of A_{l}.2, 2 ≤ l ≤ 5, or D_{4}.2 in characteristic 2 or D_{4}.3 in characteristic 3, and denote by F, σF the untwisted and the twisted Frobenius map. Then there exists a bijection between the unipotent conjugacy classes in (G^{o})^{F} .σ and those in (G^{o})^{σF} .σ as defined in Proposition 5, such that the outer Green functions for G^{σF} are the same as the ones for G^{F}, given in [11].

Proof : The proof uses the above remark on induced Green functions. Namely, we may inductively assume to have shown the assertion for all groups of smaller rank. Then the Green functions induced according to Lemma 2 also agree in the twisted and the untwisted case. Now one checks that the arguments used in [11] to find the remaining functions are purely arithmetical, or use properties of character values, so they carry over immediately to our case. So the results have to be the same, and the Theorem is proved.

Theorem 5: Let G = G^{o}.⟨σ⟩ = ^{2}E_{6}(2^{n})_{ad}: 2. Then there exists a bijection as in Proposition 5 between the unipotent conjugacy classes in G^{o}.σ and those in E_{6}(2^{n}).2 as given in Table 10 such that the outer Green functions for G^{o}.σ are the same as the ones for E_{6}(2^{n}).2 in Table 11.

Proof : Again we may use induction and Proposition 5 to see that the statement holds at least for the Harish-Chandra induced Green functions. Then the calculations in the proof of Theorem 3 can be copied, as they only rely on the orthogonality relations and integrality. Note that the decomposition of the generalized Deligne-Lusztig characters in the twisted case were also obtained in [11, Theorem 8], and that the results are the same for those almost characters which we used in the proof of Theorem 3. Thus we end up with the same Green functions as in the untwisted case.

6 The groups of type F_{4}

For groups of type F_{4} the Green functions in good characteristic were determined by Shoji [14] under the additional assumption that q is large enough. Lusztig later showed that this last assumption could be dropped. Also, in bad characteristic he could prove that the Deligne-Lusztig Green functions coincide with those defined via character sheaves if q is large enough [8]. Here we compute the Green functions for these groups in characteristic 2. The unipotent conjugacy classes of G := F_{4}(2^{n}) and of B_{3}(2^{n}), which occurs in a Levi subgroup of G were described by Shinoda [13]. We use his notation and numbering of the classes. In a first step, the Green functions in even characteristic for B_{3}(2^{n}) have to be found. The notation for the characters χ ∈ Irr(W (B_{3})) is taken from [2].

Proposition 6: The values of the almost characters R_{χ}, χ ∈ Irr(W (B_{3})), on the unipotent classes of B_{3}(2^{n}) are given by Table 12.

Proof : All but three Green functions are obtained by Harish-Chandra induction. The Green function for the Coxeter torus follows from [6]. The remaining two functions are easily computed using the orthogonality relations, as described in the previous proofs.

Theorem 6: For G = F_{4}(2^{n}), the values of the almost characters R_{χ}, χ ∈ Irr(W (F_{4})), on the unipotent classes of G are given by Table 13.

Proof : We proceed in analogy to the previously treated cases. The group G has a maximal parabolic subgroup P with Levi complement of type B_{3}. The Green functions for B_{3}(2^{n}) were computed in Proposition 6. The fusion of unipotent classes from P to G can be determined from the Steinberg relations as usually. A list of unipotent conjugacy classes of G is given in [13]. To achieve a uniform notation, we will write u_{i} for Shinoda’s elements x_{i}. Having determined the fusion, we may induce the ten Green functions given in Table 12. Next we make use of the exceptional outer automorphism σ of groups of type F_{4} in characteristic 2. It acts on the classes of maximal tori and on the unipotent conjugacy classes, and by applying this action to the first ten Green functions, we obtain another four new functions. Still two more Green functions belong to isotropic tori, so may be obtained by Harish-Chandra induction. They are seen to be conjugate under the graph automorphism, so take equal value on all σ-stable unipotent classes. The difference St − 1 of the Steinberg character and the trivial character may be expressed as a linear combination of Green functions for isotropic tori, as follows from Lusztig’s decomposition of the R_{T,1}. As the values of this difference are known on all unipotent classes, we may solve for the sum of the two still unknown Green functions, and determine the values on all σ-stable classes. The remaining values then follow by orthogonality and integrality properties. Thus just the nine Q_{w} for anisotropic tori T_{w} remain to be found.

Using Lusztig’s decomposition of the R_{T,1} we may express the fake characters belonging to the four element families of unipotent characters in terms of the Q_{w}. Certain sums of them do not involve the unknown functions, and by arguments on divisibility and absolute value, as in the previous proofs, we may thus determine the values on u_{1}, u_{2} and u_{3}. We omit the details.

Write a_{i} := Q_{w}(u_{i}) for the values of the Green function belonging to the torus of order |T_{w}| = Φ_{8}. By orthogonality relations and using the integrality of χ(u)|[u]|/χ(i) the norm equation for Q_{w} may now be used to obtain −Φ_{1}^{2}Φ_{2}^{2}(q^{6} − q^{2} − 1) as the only integral solution for a_{4}. Similarly one finds a_{20} + a_{21} = −2q^{2} + 2. At this point, the investigation of the norm equation yields no more information, so a new method has to be applied.

The root subgroups for the short roots of G are seen to generate a subgroup H of type D_{4}(2^{n}). Although this is not a Levi subgroup, we may obtain some new information by restricting our Green functions to H. The fusion of unipotent classes from H into G is easily determined from Table 1 and [13]. For convenience we reproduce it in Table 14.

Thus we may restrict the Green functions of G to the unipotent classes of H. By Lusztig’s results, the irreducible characters of H are linear combinations over 1/|W (D_{4})| · ℤ of the Deligne-Lusztig characters plus some nonuniform functions orthogonal to all R_{T,θ}. From Table 2 and the verification of Lusztig’s conjecture for D_{4}(2^{n}) in Section 7 it follows that all nonregular unipotent classes of H are uniform, i.e., their characteristic function is a linear combination of Deligne-Lusztig characters. Moreover the two classes of regular unipotent elements are the only nonuniform conjugacy classes of H. There can hence exist up to multiples only one nonuniform function orthogonal to all R_{T,θ}, and by orthogonality to the characteristic function for the uniform classes, and to the trivial character, it has to take opposite values on the two regular unipotent classes. By inspection of Table 2 its restriction to the unipotent classes hence remains orthogonal to the Green functions. We may now use the restricted Green functions in the following way: Their scalar products with all Q_{w}^{H} have to lie in 1/|W (D_{4})|ℤ. This provides further congruences and divisibility properties for the missing values in F_{4}. For example we conclude

a_{5} = (q^{3} − 1)(q^{2} − 1)(1 + q^{3} − q^{4} + aq^{5}), a ∈ ℤ.

Inserting this strong condition into the norm equation now forces a = 0. Similarly we get

a_{11} = (q^{2} − 1)(−1 + q^{4} + bq^{6}/8), b ∈ ℤ,

yielding b = 0. We are left with the norm equation in a_{21}, a_{26}, a_{28}, a_{30}, a_{10}, and this may be written as a sum of squares, every term of which has hence to vanish, giving the other missing values.

We next turn to Q_{w} for |T_{w}| = Φ_{12}. Again the restriction of this Green function to D_{4} yields enough conditions on the unknown values to determine them uniquely from the norm equation for Q_{w}. We have now found two of the remaining nine missing functions. Orthogonality to them imposes still more conditions on the last seven functions, and calculations as above lead to a complete determination of all values.

7 A conjecture of Lusztig

The Green functions calculated in the previous sections enjoy properties which are expected to hold for all Green functions, but for which in bad characteristic no a priori proof is known. Namely the following are seen to hold:

Remarks:

(1) The Green functions are polynomials in q.

(2) The constant terms of these polynomials are always equal to 1.

(3) The Green polynomials satisfy an Ennola type duality: If q is replaced by −q in the Green polynomials for one type of torus, then one obtains the Green polynomials for the Ennola dual of the original torus (after performing a permutation on the unipotent classes, which only affects classes fusing in the algebraic group).

All of this is true in those cases where the Green functions are explicitly known. Clearly the a priori knowledge of, say, properties (1) and (3) would simplify the determination of Green functions considerably.

The Green polynomials also satisfy a conjecture originally made by Lusztig (for connected groups only) [7, 2.16]. For this let C be a unipotent conjugacy class in G, i.e., in the algebraic group, let C := C^{F}, and denote by

Q_{w}(C) := \frac{1}{|C|} _{u∈C}^{∑} Q_{w}(u)

the average value of Q_{w} on C. Tables 6,10 and 13 and Theorems 2 and 5 then immediately imply:

Proposition 7: The Green functions in characteristic 2 for connected and disconnected, untwisted and twisted groups of type E_{6} and for groups of type F_{4} satisfy the second orthogonality relations

\frac{1}{|W |} _{w∈W}^{∑} |T_{w}| Q_{w}(C) Q_{w}(C′) = \{ \begin{matrix}\frac{|G|}{|C|}, & if C = C′ \\ 0, & else \end{matrix} ,

where C := C^{F} as defined above. In particular, the characteristic function of each unipotent class C in G is uniform.

8 The tables

In this final section we have collected the tables of the Green functions for groups of types E_{6} and F_{4} in characteristic 2, and of some other results needed in the proofs. We denote by Φ_{i} the i-th cyclotomic polynomial evaluated at q (the relevant q should always be clear from the context).

Table 1: Unipotent classes of D_{4}(2^{n}).

|C_{G}(u)| [15]  |C_{G}(u)| [15]
u_{0} q^{12}Φ_{1}^{4}Φ_{2}^{4}Φ_{3}Φ_{4}^{2}Φ_{6} 1^{8}  u_{7}  2q^{8}Φ_{2}^{2} 3^{2}1^{2}
u_{1}  q^{12}Φ_{1}^{3}Φ_{2}^{3} 2_{0}^{2}1^{4}  u_{8}  q^{6}Φ_{1}Φ_{2} 4_{0}^{2}
u_{2}  q^{10}Φ_{1}^{2}Φ_{2}^{2}Φ_{4} 2_{0}^{4}  u_{9}  q^{6}Φ_{1}Φ_{2} 4_{0}^{2}
u_{3}  q^{10}Φ_{1}^{2}Φ_{2}^{2}Φ_{4} 2_{0}^{4}  u_{10} q^{6}Φ_{1}Φ_{2} 421^{2}
u_{4}  q^{10}Φ_{1}^{2}Φ_{2}^{2}Φ_{4} 2^{2}1^{4}  u_{11}  q^{6} 4^{2}
u_{5}  q^{10}Φ_{1}Φ_{2} 2^{4}  u_{12}  2q^{4} 62
u_{6}  2q^{8}Φ_{1}^{2} 3^{2}1^{2}  u_{13}  2q^{4} 62

The third and sixth column of Table 1 indicate the G-class of u_{i} in the notation of [15, p.239].

Table 2: Almost characters for D_{4}(2^{n}).

χ  u_{0}  u_{1}  u_{2} u_{3} u_{4} u_{5} u_{6} u_{7} u_{8} u_{9} u_{10} u_{11}u_{12,13}
4, −  1  1  1 1 1 1 1 1 1 1 1 1 1
3, 1  qΦ_{4}^{2} q(2q^{2}+1) qΦ_{4} qΦ_{4} qΦ_{4} q qΦ_{2} –qΦ_{1} q q q q .
2, 2′  q^{2}Φ_{3}Φ_{6} q^{2}Φ_{4}  q^{2} q^{2}Φ_{4} q^{2} q^{2} q^{2} q^{2} . . q^{2} . .
2, 2  q^{2}Φ_{3}Φ_{6} q^{2}Φ_{4}  q^{2} q^{2} q^{2}Φ_{4} q^{2} q^{2} q^{2} . q^{2} …
31, − q^{2}Φ_{3}Φ_{6} q^{2}Φ_{4} q^{2}Φ_{4} q^{2} q^{2} q^{2} q^{2} q^{2} q^{2} ….
21, 1  q^{3}Φ_{4}^{3} q^{3}(3q^{2}+1) q^{3}Φ_{4} q^{3}Φ_{4} q^{3}Φ_{4} q^{3} q^{3} q^{3} . . . . .
2, 11 2q^{4}Φ_{3}Φ_{6} q^{4}(q^{2}+2) q^{4} q^{4} q^{4} . q^{3} –q^{3} . . . . .
22, −  q^{4}Φ_{8}  q^{4}  q^{4} q^{4} q^{4} q^{4} . . . . . . .
11, 11 q^{6}Φ_{3}Φ_{6}  q^{6}  .  . q^{6} . . . . . . . .
11, 11′ q^{6}Φ_{3}Φ_{6}  q^{6}  . q^{6} . . . . . . . . .
211, − q^{6}Φ_{3}Φ_{6}  q^{6}  q^{6} .  . . . . . . . . .
111, 1  q^{7}Φ_{4}^{2}  q^{7}  .  .  . . . . . . . . .
1111, − q^{12}  .  .  .  . . . . . . . . .

Table 3: Unipotent classes of D_{5}(2^{n}).

|C_{G}(u)| [15]  |C_{G}(u)| [15]
u_{0} q^{20}Φ_{1}^{5}Φ_{2}^{4}Φ_{3}Φ_{4}^{2}Φ_{5}Φ_{6}Φ_{8} 1^{10}  u_{10}  q^{11}Φ_{1}Φ_{2} 42^{3}
u_{1}  q^{20}Φ_{1}^{4}Φ_{2}^{3}Φ_{3}Φ_{4} 2_{0}^{2}1^{6}  u_{11}  q^{10}Φ_{1}^{2}Φ_{2} 4_{0}^{2}1^{2}
u_{2}  q^{18}Φ_{1}^{3}Φ_{2}^{2}Φ_{4} 2_{0}^{4}1^{2}  u_{12}  2q^{10}Φ_{1} 4^{2}1^{2}
u_{3}  q^{17}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6} 2^{2}1^{6}  u_{13}  2q^{10}Φ_{2} 4^{2}1^{2}
u_{4}  q^{17}Φ_{1}^{2}Φ_{2}^{2} 2^{4}1^{2}  u_{14}  q^{8}Φ_{1} 5^{2}
u_{5}  2q^{14}Φ_{1}^{3}Φ_{2}^{2} 3^{2}1^{4}  u_{15} 2q^{7}Φ_{1}Φ_{2} 621^{2}
u_{6}  2q^{14}Φ_{1}Φ_{2}^{2}Φ_{4} 3^{2}1^{4}  u_{16} 2q^{7}Φ_{1}Φ_{2} 621^{2}
u_{7}  q^{14}Φ_{1}^{2}Φ_{2} 3^{2}2_{0}^{2}  u_{17}  q^{7} 64
u_{8}  q^{13}Φ_{1}Φ_{2} 3^{2}2^{2}  u_{18}  2q^{5} 82
u_{9}  q^{11}Φ_{1}^{2}Φ_{2}^{2}Φ_{4} 421^{4}  u_{19}  2q^{5} 82

The third and sixth column give the corresponding class in the algebraic group in the notation of [15, p.246].

The notation for Table 4 follows the one introduced in [1]: All Green functions and almost characters are polynomials in q with integer coefficients. For each character χ ∈ Irr(W (D_{5})) we print a table, where the columns are indexed by (the indices of) the unipotent classes, and the rows by powers of q. Missing entries are zero. The value of R_{χ} on u_{i} is obtained by adding up the entries in the corresponding column, each multiplied by the appropriate power of q. If a column is indexed by two numbers, the values on these two unipotent classes agree for all χ. (In particular, neither of these classes can be uniform.)

Table 4: Almost characters for D_{5}(2^{n}).

character ϕ_{5,0}

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17 18
16 19

character ϕ_{4,1}

7 1

6

5 1 1 1
4 1 1 1
3 1 1 1 1 1 2 1 1
2  1 1 −1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 17

16

character ϕ_{15,01}

8 1

7

6 1 1 1

5

4 1 1 1 1 1 1 1  1

3

2 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

16

character ϕ_{3,2}

10 1
9 1
8 1
7 1 1
6 2 2 1 1
5 1 1 1
4 1 1 1 1 1 2 1
3 1 1 1 1 1 1 1 1 1  1
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

character ϕ_{14,02}

13 1

12

11 2
10 1
9 3 1 1
8 1 1
7 3 3 1 2
6 1 1 1
5 2 2 2 2 2 3 1 1 1

4

3 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13

character ϕ_{04,12}

12 1
11 1
10 1
9 1 1
8 2 1 1
7 1 1 1
6 1 1 1 1 −1
5 1 1 1
4 1 1 1 1 1 2  1 1
3  1 1 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13

character ϕ_{24,01}

12 1

11

10 1

9

8 1 1 1

7

6 1 1 1 1 1

5

4 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10
14 1
13 1
12 2
11 2
10 3 1
9 2 1
8 3 3 1 1
7 2 2 1 1
6 2 2 2 1 1 1 1
5 1 1 1 1 1 2 1
4 1 1 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9 10 11

character ϕ_{23,02}

15 1

14

13 1
12 1
11 2

10

9 2 2 1
8 1 1 1
7 1 1 1 1 1

6

5 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8

character ϕ_{125,012}

14 1

13

12 1

11

10 2 1 1

9

8 1 1 1

7

6 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9

character ϕ_{13,12}

16 1
15 1
14 2
13 2
12 3
11 2 1
10 3 2 1
9 2 2 1
8 2 2 1 1
7 1 1 1
6 1 1 1 1 1 2 1
0 1 2 3 4 5 6 7

character ϕ_{124,013}

17 1

16

15 2
14 1
13 3
12 1
11 3 2 1
10 1 1
9 2 2 1 1

8

7 1 1 1 1 1 1 1
0 1 2 3 4 5 6

character ϕ_{123,014}

16 1
15 1
14 1
13 1
12 2 1
11 1
10 1 1
9 1 1 1
8 1 1 1
7  1 −1
0 1 2 3 4 5 6

character ϕ_{134,012}

16 1

15

14 1

13

12 1

11

10 1 1 1

9

8 1 1 1 1 1
0 1 2 3 4

character ϕ_{123,023}
18 1
17 1
16 1
15 1
14 2
13 1
12 1 1
11 1 1
(10 1 1 1) / (0 1 2)

character ϕ_{1235,0123}

18 1

17

16 1

15

14 1

13

12 1 1 1
0 1 2 3

character ϕ_{0124,1234}

19 1

18

17 1
16 1
15 1

14

(13 1 1) / (0 1)

character ϕ_{12345,01234}
(20 1) / 0

Table 5: Unipotent classes of E_{6}(2^{n}).

[12]  [12]  [12]  [12]
u_{0}  1 u_{7}  x_{9}  u_{14}  x_{42}  u_{21}  x_{18}
u_{1}  x_{1}  u_{8}  x_{10}  u_{15}  x_{7}  u_{22}  x_{44}
u_{2}  x_{3}  u_{9}  x_{4}  u_{16}  x_{14}  u_{23}  x_{16}
u_{3}  x_{6}  u_{10}  x_{12}  u_{17}  x_{41}  u_{24}  x_{43}
u_{4}  x_{2}  u_{11}  x_{8}  u_{18}  x_{11}  u_{25}  x_{19}
u_{5}  x_{40}  u_{12}  x_{15}  u_{19}  x_{17}  u_{26}  x_{20}
u_{6}  x_{5}  u_{13} x_{24}, x_{29}  u_{20}  x_{13}  u_{27}  x_{45}

In Table 5 we give the correspondence between Mizuno’s notation [12] for the unipotent classes of E_{6}(2^{n}) and ours. For our class [u_{13}] he gives two different representatives depending on whether q ≡ 1 or −1 (mod 3).

Table 6: Almost characters for E_{6}(2^{n}).

character ϕ_{1,0}

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23 25 26
17  24 27

character ϕ_{6,1}

11 1

10 9

8 1 1
7 1 1 1

6

5 1 1 1 1 2 1 1
4 1 1 1 1 1 1 1 1 1
3  1  1 2 −1  1
2  1  1  1 −1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23 25
17  24

character ϕ_{20,2}

16 1
15 1
14 1
13 1
12 2 1
11 1 1
10 2 2 1
9 2 2 1
8 2 2 2 1 1 1
7 1 1 1 1 2 1
6 2 2 2 2 3 1 2 1 1 1
5 1 1 1 1 1 1 1 1 1
4 1 1 1 1 1 1 1 1 1 2 1 2 3  1 1 1
3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23
17  24

character ϕ_{15,5}

19 1
18 1

17

16 1
15 2 1

14

13 1 1
12 3 2 1
11 1 1 1
10  1 −1
9 2 2 1 1 2  1
8 1 1 1
7  1
6 1 1 1 1 2 1 1 1 1 −1
5 1 1 1 1 1 1 1 1 1
4  1  1 2 −1  1
3  1  1  1 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22

17

character ϕ_{30,3}

21 1

20

19 1
18 1
17 2
16 1
15 3 1
14 2 1
13 3 2 1
12 2 2
11 3 3 2
10 2 2 1
9 3 3 3 2 3 1 1 1
8 1 1 1 1 2 1
7 2 2 2 2 3 1 2 1 1 1
6 1 1 1 1 1 1 1 1 1
5 1 1 1 1 1 1 1 1 1 2 1 2 3  1 1

4

3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22

17 character ϕ_{15,4}

20 1

19

18 1
17 1
16 1

15

14 2 1
13 1 1
12 1 1 1
11 1 1
10 2 2 2 1

9

8 1 1 1 1 2 1 1
7 1 1 1 1 1 1 1 1
6 1 1 1 1 1 1 1 1 1 1 1 1

5

4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20

17

character ϕ_{64,4}

23 1
22 1
21 1
20 3
19 3
18 3
17 5 1
16 5 2
15 4 2
14 6 4 1
13 6 5 2
12 4 4 2
11 5 5 3 1 2
10 5 5 4 2 3 1 1
9 3 3 3 2 3 1 1
8 3 3 3 3 5 1 3 1 1 1
7 3 3 3 3 4 2 3 1 2 2
6 1 1 1 1 1 1 1 1 2  1 2 −1
5 1 1 1 1 1 1 1 1 1 2 1 2 3  1 1 1
4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19

17 character ϕ_{60,5}

25 1
24 1
23 1
22 2
21 3
20 2
19 4
18 5 1
17 4 1
16 4 2
15 6 4 1
14 4 4 1
13 4 4 2
12 5 5 3 1
11 4 4 4 2 1 1
10 2 2 2 2 3 1 1
9 3 3 3 3 5 1 3 1 1
8 2 2 2 2 2 2 2 2 1
7 1 1 1 1 1 1 1 1 1 1 1 1
6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18

17

character ϕ_{24,6}

24 1

23

22 1
21 1
20 1
19 1
18 3 1
17 1
16 2 1
15 2 1
14 2 2 1
13 1 1
12 3 3 2 1 1 1
11 1 1 1
10 1 1 1 1 1 1
9 1 1 1 1 2 1
8 1 1 1 1 1 1 1 1 1

7

6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

17 character ϕ_{81,6}

26 1
25 1
24 2
23 2
22 4
21 4
20 5
19 5 1
18 7 2
17 6 3
16 7 4 1
15 6 5 1
14 7 6 3
13 5 5 2
12 5 5 4 1 2
11 4 4 3 2 3 1 1
10 4 4 4 3 5 1 2 1
9 2 2 2 2 3 1 2 1 1
8 2 2 2 2 3 1 2 1 1 1
7 1 1 1 1 1 1 1 1 1
6 1 1 1 1 1 1 1 1 1 2 1 2 3  1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

character ϕ_{20,10}

26 1

25

24 1
23 2
22 1

21

20 3 1
19 2 1

18

17 2 1
16 3 2 1

15

14 1 1  1 −1
13 2 2 1
12 1 1 1
11  1 −1
10 1 1 1 1 2  1

9

8  1
7  1 1 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

character ϕ_{80,7}

29 1

28

27 1
26 2
25 3
24 2
23 6
22 4
21 5
20 6 1
19 8 3
18 4 2
17 8 5 1
16 6 5 1
15 5 5 3
14 4 4 1
13 6 6 5 2 1 1
12 2 2 2 1 1 1
11 3 3 3 3 5 1 2 1
10 2 2 2 2 3 1 2 1
9 1 1 1 1 1 1 1 1 1

8

7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

character ϕ_{90,8}

28 1
27 2
26 1
25 3
24 4
23 4
22 5
21 8 1
20 6 1
19 7 2
18 8 4
17 7 5 1
16 6 5 1
15 8 7 3
14 5 5 3
13 4 4 3 1 2
12 4 4 3 2 4 1
11 3 3 3 2 3 1 1
10 1 1 1 1 2 1
9 2 2 2 2 3 1 2 1 1 1
8 1 1 1 1 1 1 1 1 1
7 1 1 2 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13

28 1
27 1
26 2
25 1
24 3
23 3
22 4
21 3
20 6 1
19 4 1
18 4 2
17 4 3
16 6 5 2
15 3 3 1
14 4 4 3 1
13 3 3 2 1
12 3 3 3 2 1 1
11 1 1 1 1 2 1
10 2 2 2 2 3 1 2 1 1
9 1 1 1 1 1 1 1 1
8 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11

character ϕ_{10,9}

27 1

26 25 24

23 1

22

21 1
20 1
19 1

18

17 1 1
16 1 1
15 1 1 1

14

13 1 1 1 1

12 11 10

9 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10

character ϕ_{81,10}

30 1
29 1
28 2
27 2
26 4
25 4
24 5
23 5
22 7 1
21 6 1
20 7 3
19 6 3
18 7 5 1
17 5 4 1
16 5 5 2
15 4 4 2
14 4 4 3 1 1 1
13 2 2 2 1 2
12 2 2 2 2 3 1 1
11 1 1 1 1 2 1
10 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9

character ϕ_{60,11}
31 1
30 1
29 1
28 2
27 3
26 2
25 4
24 5
23 4
22 4
21 6 2
20 4 2
19 4 3
18 5 4 1
17 4 4 2
16 2 2 1
15 3 3 2 1
14 2 2 2 1
13 1 1 1 1 1 1
12 1 1 1 1 2 1
11 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8

character ϕ_{24,12}

30 1

29

28 1
27 1
26 1
25 1
24 3
23 1
22 2
21 2 1
20 2 1
19 1 1
18 3 2 1
17 1 1
16 1 1 1
15 1 1
14 1 1 1

13

12 1 1 1 1 2 1 1
0 1 2 3 4 5 6 7

32 1
31 1
30 1
29 3
28 3
27 3
26 5
25 5
24 4
23 6 1
22 6 2
21 4 2
20 5 3
19 5 4 1
18 3 3 1
17 3 3 1
16 3 3 2
15 1 1 1
14 1 1 1 1 2
13 1 1 1 1 2 1
0 1 2 3 4 5 6

character ϕ_{15,17}

31 1
30 1

29

28 1
27 2

26

25 1
24 3 1
23 1

22

21 2 1
20 1 1

19

18 1 1
17 1 1 1

16

15  1 −1
0 1 2 3 4 5

character ϕ_{30,15}

33 1

32

31 1
30 1
29 2
28 1
27 3
26 2
25 3
24 2
23 3 1
22 2 1
21 3 2
20 1 1
19 2 2 1
18 1 1
17 1 1 1

16

15 1 1 1 1 1 1
0 1 2 3 4 5

character ϕ_{15,16}

32 1

31

30 1
29 1
28 1

27

26 2
25 1
24 1
23 1
22 2 1

21

20 1 1
19 1 1
18 1 1 1

17

16 1 1 1 1
0 1 2 3

character ϕ_{20,20}
34 1
33 1
32 1
31 1
30 2
29 1
28 2
27 2
26 2
25 1
24 2 1
23 1 1
22 1 1
21 1 1
(20 1 1 1) / (0 1 2)

character ϕ_{6,25}

35 1

34 33

32 1
31 1

30

29 1
28 1

27 26

(25 1 1) / (0 1)

character ϕ_{1,36}
(36 1) / 0

Table 7: Unipotent classes of ^{2}D_{4}(2^{n}).

|C_{G}(u)|  |C_{G}(u)|
u_{0} = 1  q^{12}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}Φ_{8} u_{5} = x_{110}(1)x_{012}(1)  2q^{8}Φ_{1}Φ_{2}
u_{1} = x_{122}(1)  q^{12}Φ_{1}^{2}Φ_{2}^{2}Φ_{4} u_{6} = x_{100}(1)x_{011}(1)  q^{6}Φ_{1}Φ_{2}
u_{2} = x_{111}(1)  q^{10}Φ_{1}^{2}Φ_{2}^{2}Φ_{4} u_{7} = x_{001}(1)x_{110}(1)x_{011}(ξ)  q^{6}
u_{3} = x_{111}(1)x_{012}(1)  q^{10}Φ_{1}Φ_{2} u_{8} = x_{100}(1)x_{010}(1)x_{001}(1)  2q^{4}
u_{4} = x_{011}(1)x_{111}(ξ) 2q^{8}Φ_{1}Φ_{2} u_{9} = x_{100}(1)x_{010}(1)x_{001}(1)x_{012}(λ) 2q^{4}

If seen as a BN-pair, the Weyl group of ^{2}D_{4}(q) has type B_{3}. Hence its roots and root subgroups may be indexed by the roots of B_{3}. Note however that the root subgroups corresponding to short roots have order q^{2}. We label the diagram of B_{3} so that the long roots have numbers 1 and 2. A positive root α = ∑_{i} n_{i}α_{i} is abbreviated by the sequence of coefficients n_{1} … n_{3}. Elements of the corresponding root subgroup X_{α} are then denoted by x_{n_{1}...n_{3}}. This explains the notation for the representative elements in Table 7. Also, ξ denotes a fixed element of 𝔽_{q^{2}}\𝔽_{q} and λ is a fixed element of 𝔽_{q} such that the polynomial x^{2} + x + λ is irreducible over 𝔽_{q}.

Table 8: Almost characters for ^{2}D_{4}(2^{n}).

χ  u_{0}  u_{1}  u_{2} u_{3} u_{4}  u_{5} u_{6} u_{7} u_{8} u_{9}
4, −  1  1  1  1 1  1  1 1 1 1
3, 1  qΦ_{8}  q  qΦ_{4} q qΦ_{2} −qΦ_{1} q q . .
31, −  q^{2}Φ_{3}Φ_{6} q^{2}Φ_{4} q^{2}Φ_{4} q^{2} q^{2}  q^{2} q^{2} …
21, 1  q^{3}Φ_{4}Φ_{8} q^{3}Φ_{4} q^{3}Φ_{4} q^{3} q^{3}  q^{3}  ….
2, 11  .  q^{6} −q^{4} . −q^{3} q^{3}  ….
22, −  q^{4}Φ_{8}  q^{4}  q^{4} q^{4} .  .  ….
211, − q^{6}Φ_{3}Φ_{6} q^{6}  q^{6} .  .  .  ….
111, 1  q^{7}Φ_{8}  q^{7}  .  .  .  .  ….
1111, −  q^{12}  .  .  .  .  .  ….

The notation for the σ-stable characters χ of W (D_{4}) is as in [2].

Table 9: Unipotent classes of ^{2}E_{6}(2^{n})_{ad}.

u |C_{G^{o}}(u)|

u_{0} = 1 q^{36}Φ_{1}^{4}Φ_{2}^{6}Φ_{3}^{2}Φ_{4}^{2}Φ_{6}^{3}Φ_{8}Φ_{10}Φ_{12}Φ_{18}

u_{1} = x_{2342}(1) q^{36}Φ_{1}^{3}Φ_{2}^{5}Φ_{3}Φ_{4}Φ_{6}^{2}Φ_{10}

u_{2} = x_{1232}(1) q^{33}Φ_{1}^{3}Φ_{2}^{4}Φ_{3}Φ_{4}Φ_{6}

u_{3} = x_{1222}(1)x_{1231}(1) q^{31}Φ_{1}^{2}Φ_{2}^{3}Φ_{6}

u_{4} = x_{0121}(1)x_{1221}(ξ) 2q^{26}Φ_{1}^{2}Φ_{2}^{4}Φ_{6}^{2}

u_{5} = x_{0122}(1)x_{1220}(1) 2q^{26}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{4}Φ_{6}

u_{6} = x_{1121}(1)x_{0122}(1)x_{1221}(ξ) q^{26}Φ_{1}Φ_{2}^{3}Φ_{6}

u_{7} = x_{1111}(1)x_{0121}(1) q^{22}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{6}

u_{8} = x_{1121}(1)x_{0122}(1)x_{1220}(1) q^{25}Φ_{1}Φ_{2}^{2}

u_{9} = x_{1110}(1)x_{0122}(1) q^{19}Φ_{1}^{2}Φ_{2}^{3}Φ_{4}

u_{10} = x_{1111}(1)x_{0121}(1)x_{1220}(1) q^{22}Φ_{1}Φ_{2}

u_{11} = x_{1110}(1)x_{1111}(ξ)x_{0121}(1) q^{19}Φ_{1}Φ_{2}^{2}

u_{12} = x_{0011}(1)x_{0110}(1)x_{1220}(1)x_{1221}(ξ) 6q^{18}Φ_{2}^{2}

u_{13} = x_{0011}(1)x_{0110}(1)x_{1120}(1)x_{1221}(ξ)x_{1222}(ζ) 3q^{18}Φ_{6}

u_{14} = x_{1100}(1)x_{0121}(1)x_{1121}(ξ) 2q^{18}Φ_{1}Φ_{2}

u_{15} = x_{0011}(1)x_{0111}(ξ)x_{1110}(1) q^{15}Φ_{1}Φ_{2}^{2}

u_{16} = x_{1000}(1)x_{0111}(1)x_{0120}(1) 2q^{13}Φ_{1}Φ_{2}^{2}Φ_{6}

u_{17} = x_{1000}(1)x_{0111}(1)x_{0120}(1)x_{1222}(λ) 2q^{13}Φ_{1}Φ_{2}^{2}Φ_{6}

u_{18} = x_{0011}(1)x_{0111}(ξ)x_{1110}(1)x_{0120}(1) q^{15}Φ_{2}

u_{19} = x_{0011}(1)x_{1100}(1)x_{0111}(ξ)x_{0120}(1) q^{13}Φ_{2}

u_{20} = x_{0011}(1)x_{1100}(1)x_{0110}(1) q^{12}Φ_{1}Φ_{2}

u_{21} = x_{0100}(1)x_{0011}(1)x_{1110}(1)x_{1120}(1) 2q^{12}

u_{22} = x_{0001}(1)x_{0010}(1)x_{0110}(ξ)x_{1220}(1) 2q^{12}

u_{23} = x_{1000}(1)x_{0100}(1)x_{0011}(1)x_{0120}(1) 2q^{9}Φ_{2}

u_{24} = x_{1000}(1)x_{0100}(1)x_{0011}(1)x_{0120}(1)x_{1111}(λ) 2q^{9}Φ_{2}

u_{25} = x_{0001}(1)x_{0010}(1)x_{1100}(1)x_{0110}(ξ) q^{8}

u_{26} = x_{0001}(1)x_{1000}(1)x_{0010}(1)x_{0100}(1) 2q^{6}

u_{27} = x_{0001}(1)x_{1000}(1)x_{0010}(1)x_{0100}(1)x_{0120}(λ) 2q^{6}

The notation for the representatives is as explained for ^{2}D_{4}. In addition to that, let ζ be a fixed element in 𝔽_{q} such that the polynomial x^{3} + x + ζ is irreducible over 𝔽_{q}. We identify the Weyl group of ^{2}E_{6}(q) with the Weyl group of F_{4} and thus obtain an indexing of the root subgroups by roots of F_{4}. Again the subgroups for short roots are isomorphic to 𝔽_{q^{2}}^{+}. We label the roots so that the long roots are the two first ones.

In the next table, which describes the unipotent classes in the outer coset of E_{6}(2^{n})_{ad}: 2, λ denotes a fixed element of 𝔽_{q} such that the polynomial x^{2}+x+λ is irreducible over 𝔽_{q}. Also ζ is chosen in 𝔽_{q} such that x^{3} + x + ζ is irreducible over 𝔽_{q}. We number the simple roots α_{i}, i = 1, …, 6, of the Dynkin diagram of E_{6} as shown:

1  3  4  5  6
•  •  •  •  •
• 2

A positive root α = ∑_{i} n_{i}α_{i} is abbreviated by the sequence of coefficients n_{1} … n_{6}. Elements of the corresponding root subgroup X_{α} are then denoted by x_{n_{1}...n_{6}}. This explains the notation for the representative elements in Table 10.

Table 10: Outer unipotent classes of E_{6}(2^{n})_{ad}: 2.

u  |C_{G^{o}}(u)| [15]
u_{0} = 1  q^{24}Φ_{1}^{4}Φ_{2}^{4}Φ_{3}^{2}Φ_{4}^{2}Φ_{6}^{2}Φ_{8}Φ_{12} u_{17}
u_{1} = x_{122321}(1)  q^{24}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6} u_{16}
u_{2} = x_{111221}(1)  2q^{21}Φ_{1}^{3}Φ_{2}^{2}Φ_{3}Φ_{4} u_{15}
u_{3} = x_{111211}(1)x_{111221}(1)x_{112221}(λ)  2q^{21}Φ_{1}^{2}Φ_{2}^{3}Φ_{4}Φ_{6} u_{15}
u_{4} = x_{111211}(1)x_{011221}(1)  q^{20}Φ_{1}^{2}Φ_{2}^{2} u_{13}
u_{5} = x_{011210}(1)x_{111111}(1)  q^{20}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{6} u_{14}
u_{6} = x_{011210}(1)x_{111111}(1)x_{111221}(1)  q^{20}Φ_{1}Φ_{2} u_{12}
u_{7} = x_{101111}(1)x_{011111}(1)x_{011210}(1)  q^{16}Φ_{1}Φ_{2} u_{10}
u_{8} = x_{010111}(1)x_{001111}(1)  q^{14}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{6} u_{11}
u_{9} = x_{010111}(1)x_{001111}(1)x_{011210}(1)  q^{14}Φ_{1}Φ_{2} u_{9}
u_{10} = x_{010110}(1)x_{101111}(1)  2q^{12}Φ_{1}^{2}Φ_{2}^{2} u_{8}
u_{11} = x_{010100}(1)x_{010110}(1)x_{011110}(λ)x_{101111}(1)  2q^{12}Φ_{1}Φ_{2}Φ_{4} u_{8}
u_{12} = x_{010110}(1)x_{010111}(1)x_{001111}(1)  2q^{12}Φ_{1}Φ_{2} u_{7}
u_{13} = x_{001110}(1)x_{010111}(1)x_{001111}(1)x_{101111}(λ)  2q^{12}Φ_{1}Φ_{2} u_{7}
u_{14} = x_{010100}(1)x_{010110}(1)x_{010111}(1)x_{001111}(1)  24q^{12} u_{6}
u_{15} = x_{010100}(1)x_{010110}(1)x_{010111}(1)x_{001111}(1)x_{011110}(λ)x_{111111}(λ) 8q^{12} u_{6}
u_{16} = x_{010100}(1)x_{010110}(1)x_{010111}(1)x_{001111}(1)x_{111111}(λ)  4q^{12} u_{6}
u_{17} = x_{010100}(1)x_{001110}(1)x_{010111}(1)x_{001111}(1)x_{101111}(λ)x_{011111}(λ) 4q^{12} u_{6}
u_{18} = x_{000110}(1)x_{000011}(1)x_{011110}(1)x_{011211}(1)x_{111211}(ζ)  3q^{12} u_{6}
u_{19} = x_{000001}(1)x_{001110}(1)x_{010110}(1)  2q^{8}Φ_{1}Φ_{2} u_{5}
u_{20} = x_{000001}(1)x_{001110}(1)x_{010110}(1)x_{011221}(λ)  2q^{8}Φ_{1}Φ_{2} u_{5}
u_{21} = x_{000001}(1)x_{010100}(1)x_{001110}(1)  2q^{9}Φ_{1} u_{4}
u_{22} = x_{000001}(1)x_{010100}(1)x_{000110}(1)x_{001110}(1)x_{011221}(λ)  2q^{9}Φ_{2} u_{4}
u_{23} = x_{000001}(1)x_{010100}(1)x_{010110}(1)x_{001110}(1)  q^{8} u_{3}
u_{24} = x_{010000}(1)x_{000001}(1)x_{000110}(1)x_{001110}(1)  2q^{6} u_{2}
u_{25} = x_{010000}(1)x_{000100}(1)x_{000110}(1)x_{000011}(1)x_{001110}(λ)  2q^{6} u_{2}
u_{26} = x_{010000}(1)x_{000100}(1)x_{000010}(1)x_{000001}(1)  2q^{4} u_{1}
u_{27} = x_{010000}(1)x_{000100}(1)x_{000010}(1)x_{000001}(1)x_{000111}(λ)  2q^{4} u_{1}

Table 11: Almost characters for E_{6}(2^{n}): 2.

character ϕ_{1,0}

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25 26
20  27

character ϕ_{4,1}

11 1

10 9 8

7 1 1 1 1

6

5 1 1 1 1 1 1 1 1

4

3 1 1 1 1 2 2 −1

2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25

20

character ϕ′_{2,4}

8 1

7 6

5  1 –1
4 1 1 1 1 1  1

3

2 1 1 1 –1 2 3 –1 1 –1 1
1  1 –1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23 24 25

20

character ϕ′′_{2,4}

8 1 1  1

7 6 5

4 1 1 1 1 1 1 1 1  1 1
3  1 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22

20

character ϕ_{9,2}

14 1

13

12 1

11

10 2 1

9

8 1 1 1 1
7  1 −1
6 2 2 2 2 2 1 1 1

5

4 1 1 1 1 1 1 1 1 1 1 2  1 1

3

2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22 23

20

character ϕ′′_{8,3}

15 1

14

13 1

12

11 1 1

10

9 2 2 1 1 1

8

7 1 1 1 1 1 1 1

6

5 1 1 1 1 1 1 1 1  1 1

4

3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 21 22

20

character ϕ′_{8,3}

15 1

14

13 1

12

11 1

10

9 2 1 1 1
8  1 −1
7 1 1 1 1 1  1
6  1 −1
5 1 1 1 1 1  1

4

3 1 1 1 1 1 1 1 1 2 2 2 3 1 4 2 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

20

character ϕ′′_{6,6}

18 1

17

16 1

15 14 13

12 2 1 1 1

11 10 9

8 1 1 1 1

7

6 1 1 1 1 1 1 1 1

5

4 1 1 1 1 2 2 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

20 1

19 18 17

16 2

15

14 2

13

12 2 1

11

10 2 2 2 2
9  1 −1
8 2 2 2 2 2 1 1 1

7 6 5

4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

character ϕ′_{1,12}

12 1

11 10

9  1 −1

8 7

6  1

5

4  1 −1 3 −1 −1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

18 1

17

16 1

15

14 2

13

12 1
11  1 −1
10 2 1 1 1
9  1 −1
8 1 1 1 1 1  1
7  1 −1
6 1 1 1 1 1  1

5

4 1 1 1 −1 2 3 −1 1 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

character ϕ′_{4,7}

17 1

16 15 14

13 1

12

11 1
10  1 −1

9

8  1 −1
7 1 1 1 1 1  1

6

5 1 1 1 −1
0 1 2 3 4 5 6 7 8 9 10 11 12 13

19 1

18

17 2

16

15 2

14

13 3 1

12

11 3 2 1 1
10  1 −1
9 2 2 2 2 1
8  1 −1
7 2 2 2 2 2 1 1 1

6

5 1 1 1 1 1 1 1 1 1 1 2  1 1
0 1 2 3 4 5 6 7 8 9 10 11 12 13

character ϕ′′_{9,6}

18 1

17

16 1

15

14 2 1

13

12 1 1

11

10 2 2 1 1 1

9

8 1 1 1 1 1 1 1

7

6 1 1 1 1 1 1 1 1  1 1
0 1 2 3 4 5 6 7 8 9 10 11

character ϕ_{4,8}

16 1

15 14 13

12 2 1

11 10

9  1 −1
8 1 1 1 1 1

7

6  1 −1
0 1 2 3 4 5 6 7 8 9 10 11

character ϕ′_{6,6}

18 1

17

16 1

15 14 13

12 2 1

11 10

9  1 −1
8 1 1 1 1 1

7

6 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9

21 1

20

19 1

18

17 1

16

15 2

14

13 1
12  1 −1
11 1 1 1 1
10  1 −1
9 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8

character ϕ′′_{4,7}

17 1

16 15 14

13 1 1

12

11 1 1  1

10 9 8

7 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7

character ϕ′′_{8,9}

21 1

20

19 1

18

17 1

16

15 2 1

14

13 1 1

12

11 1 1 1 1

10

9 1 1 1 1 1 1 1
0 1 2 3 4 5 6

character ϕ_{9,10}

22 1

21

20 1

19

18 2

17

16 1

15

14 2 1

13

12 1 1 1 1
11  1 −1
10 1 1 1 1 1
0 1 2 3 4

character ϕ′′_{1,12}
12 1 1  1
0 1 2 3 4 5

character ϕ_{4,13}

23 1

22 21 20

19 1

18

17 1

16 15 14

13 1 1 1 1
0 1 2 3

character ϕ′_{2,16}

20 1

19 18 17

16 1

15 14

13  1 −1
0 1 2 3

character ϕ′′_{2,16}

20 1

19 18 17

(16 1 1) / (0 1)

character ϕ_{1,24}
(24 1) / 0

Table 12: Almost characters for B_{3}(2^{n}).

χ  u_{0}  u_{1}  u_{2} u_{3} u_{4} u_{5} u_{6} u_{7} u_{8} u_{9} u_{10} u_{11}
3, −  1  1  1  1 1 1 1 1 1 1 1 1
2, 1  qΦ_{3}Φ_{6} qΦ_{4} qΦ_{4} qΦ_{4} q q q q q q . .
21, −  q^{2}Φ_{4} q^{2}Φ_{4} q^{2} q^{2} q^{2} .  . q^{2} q^{2} …
−, 3  q^{3}  .  q^{3}  .  . q^{2} −q^{2} . . . . .
1, 2  q^{2}Φ_{3}Φ_{6} q^{2} q^{2}Φ_{4} q^{2} q^{2} q^{2} q^{2} . . . . .
11, 1  q^{3}Φ_{3}Φ_{6} q^{3}Φ_{4} q^{3} q^{3} q^{3} .  .  . . . . .
1, 11  q^{4}Φ_{3}Φ_{6} q^{4}  q^{4} q^{4} . .  .  . . . . .
−, 21  q^{5}Φ_{4}  .  q^{5}  .  . .  .  . . . . .
111, −  q^{6}  q^{6}  .  .  . .  .  . . . . .
−, 111  q^{9}  .  .  .  . .  .  . . . . .

Table 13: Almost characters for F_{4}(2^{n}).

character ϕ_{1,0}

0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22 24 25 27 28 29 31
10  21 23 26  30 ...

character ϕ_{4,1}

11 1

10 9 8

7 1 1 1 1

6

5 1 1 1 1 1 1 1 1 1

4

3 1 1 1 1 2 –1

2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22 24 25 27 28 29
10  21 23 26  30

8 1 1

7 6

5  1 –1
4 1 1 1 1 1 1 1 1 1 1

3

2 1 1 1 –1 –1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22 24 25 27 28
10  21 23 26

character ϕ′′_{2,4}

8 1 1

7 6

5  1 –1
4 1 1 1 1 1 1 1 1 1 1

3

2  1 1 –1 1 –1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22 24 25 27 28
10  21 23 26

character ϕ_{9,2}

14 1

13

12 1

11

10 2 1 1

9

8 1 1 1 1

7

6 2 2 2 2 2 1 1 1 1 1

5

4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3

2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22 24 25 27 28
10  21 23 26

15 1

14

13 1

12

11 1 1

10

9 2 2 1 1

8

7 1 1 1 1 1  1 1
6  1 –1
5 1 1 1 1 1 1 1 1 1 1

4

3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20 22
10  21 23

character ϕ′_{8,3}

15 1

14

13 1

12

11 1 1

10

9 2 1 2 1

8

7 1 1 1 1 1 1 1
6  1 –1
5 1 1 1 1 1 1 1 1 1 1

4

3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19 20
10  21

18 1

17

16 1

15 14 13

12 2 1 1 1

11 10 9

8 1 1 1 1

7

6 1 1 1 1 1 1 1 1 1

5

4 1 1 1 1 2 –1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19

10

character ϕ_{12,4}

20 1

19 18 17

16 2

15

14 2

13

12 2 1 1

11

10 2 2 2 2

9

8 2 2 2 2 2 1 1 1 1

7 6 5

4 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16 17 18 19

10

19 1

18

17 2

16

15 2

14

13 3 1 1

12

11 3 2 2 1

10

9 2 2 2 2 1

8

7 2 2 2 2 2 1 1 1 1 1

6

5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15 16

10

character ϕ′_{6,6}

18 1

17

16 1

15 14 13

12 2 1 1

11 10 9

8 1 1 1 1 1

7

6 1 1 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14 15

10 character ϕ′′_{9,6}

18 1

17

16 1

15

14 2 1

13

12 1 1

11

10 2 2 1 1

9

8 1 1 1 1 1  1 1
7  1 –1
6 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13 14

10

character ϕ′_{9,6}

18 1

17

16 1

15

14 2 1

13

12 1 1

11

10 2 1 2 1

9

8 1 1 1 1 1 1 1
7  1 –1
6 1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12 13

10

17 1

16 15 14

13 1 1

12

11 1 1

10 9

8  1 –1
7 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9 11 12

10 character ϕ′_{4,7}

17 1

16 15 14

13 1 1

12

11 1 1

10 9

8  1 –1
7 1 1 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9 11

10

character ϕ_{4,8}

16 1

15 14 13

12 2 1 1

11 10 9

8 1 1 1 1 1  1
0 1 2 3 4 5 6 7 8 9

10 character ϕ′′_{8,9}

21 1

20

19 1

18

17 1

16

15 2 1

14

13 1 1

12

11 1 1 1 1

10

9 1 1 1 1 1  1 1
0 1 2 3 4 5 6 7 8

character ϕ′′_{1,12}

12 1 1

11 10

9  1 –1
0 1 2 3 4 5 6 7 8

character ϕ′_{1,12}

12 1 1

11 10

9  1 –1
0 1 2 3 4 5 6

21 1

20

19 1

18

17 1

16

15 2 1

14

13 1 1

12

11 1 1 1 1

10

9 1 1 1 1 1 1 1
0 1 2 3 4 5 6

character ϕ_{9,10}

22 1

21

20 1

19

18 2

17

16 1

15

14 2 1 1

13

12 1 1 1 1

11

10 1 1 1 1 1
0 1 2 3 4

character ϕ_{4,13}

23 1

22 21 20

19 1

18

17 1

16 15 14

13 1 1 1 1
0 1 2 3

character ϕ′_{2,16}

20 1

19 18 17

(16 1 1) / (0 1 2)

character ϕ′′_{2,16}

20 1

19 18 17

(16 1 1) / (0 1)

character ϕ_{1,24}
(24 1) / 0

Table 14: Fusion from D_{4}(2^{n}) into F_{4}(2^{n}).

[13]  [13]  [13]  [13]
u_{0}  x_{0}  u_{4}  x_{3}  u_{8}  x_{13}  u_{12}  x_{20}
u_{1}  x_{2}  u_{5}  x_{4}  u_{9}  x_{13}  u_{13}  x_{21}
u_{2}  x_{3}  u_{6}  x_{5}  u_{10}  x_{13}
u_{3}  x_{3}  u_{7}  x_{6}  u_{11}  x_{17}

References

[1] W. M. Beynon, N. Spaltenstein, Green functions of finite Chevalley groups of type E_{n} (n = 6, 7, 8), J. Algebra 88 (1984), 584–614.

[2] R. W. Carter, Finite groups of Lie type: Conjugacy classes and complex characters, John Wiley and Sons 1985, Chichester.

[3] F. Digne, J. Michel, Fonctions L des variétés de Deligne-Lusztig et descente de Shintani, Mémoires Soc. Math. France 20 (1985).

[4] F. Digne, J. Michel, Groupes réductifs non connexes, preprint.

[5] J. A. Green, Characters of the finite general linear groups, Trans. Am. Math. Soc. 80 (1955), 402–447.

[6] G. Lusztig, On the Green polynomials of classical groups, Proc. London Math. Soc. (3) 33 (1976), 443–475.

[7] G. Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics 39 (1977).

[8] G. Lusztig, Green functions and character sheaves, Annals of Math. 131 (1990), 355–408.

[9] G. Lusztig, B. Srinivasan, The characters of the finite unitary groups, J. Algebra 49 (1977), 167–171.

[10] G. Malle, Die unipotenten Charaktere von ^{2}F_{4}(q^{2}), Comm. Algebra 18 (1990), 2361–2381.

[11] G. Malle, Generalized Deligne-Lusztig characters, to appear in J. Algebra.

[12] K. Mizuno, The conjugate classes of Chevalley groups of type E_{6}, J. Fac. Sci. Univ. Tokyo 24 (1977), 525–563.

[13] K. Shinoda, The conjugacy classes of Chevalley groups of type (F_{4}) over finite fields of characteristic 2, J. Fac. Sci. Univ. Tokyo 21 (1974), 133–159.

[14] T. Shoji, On the Green polynomials of a Chevalley group of type F_{4}, Comm. Algebra 10 (1982), 505–543.

[15] N. Spaltenstein, Classes unipotents et sous-groupes de Borel, Springer Lecture Notes 946, Berlin - Heidelberg - New York 1982.