Generalized Deligne-Lusztig characters

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

Let G^{o} be a connected reductive algebraic group defined over the algebraic closure \overline{𝔽_{q}} of a finite field, and σ an automorphism of G^{o} inducing a graph automorphism of the associated Dynkin diagram. Then the semidirect product G := G^{o}.⟨σ⟩ again is a reductive algebraic group, but no longer connected. So the usual character theory of Deligne and Lusztig gives no information about characters of the finite groups G^{F} on elements x ∈ G^{F} \G^{oF}, where F is a Frobenius endomorphism. But it is possible to generalize the construction of Deligne-Lusztig to disconnected groups of the above type, and it is expected that the results for the connected case hold in similar form for the generalization to disconnected groups. The first steps in this direction could already be proved by Digne and Michel [4]. But the problem of explicit decomposition of the generalized Deligne-Lusztig characters into irreducibles is not yet solved, nor have the generalized Green functions been calculated.

In this paper several aspects of the representation theory of disconnected groups are studied. For this, we need the above mentioned results of Digne and Michel, as well as a generalization of a theorem of Howlett and Lehrer to automorphism groups of groups with a BN-pair, which was proved in [8]. This result and some notation are recalled in the first section. In the second section we prove an explicit formula for an extension of the Steinberg character to disconnected groups. The third section introduces the setting for generalized Deligne-Lusztig characters and collects the results from [4] needed later on. In Section 4–6, the decompositions of the R_{T,1} are completely determined for several small rank groups, including the exceptional cases of D_{4} with a graph automorphism of order three and E_{6} with one of order two. Although strong similarities with the connected case may be noticed, still some differences occur. So for D_{l}(q): 2, l ≤ 8, all extensions of unipotent characters are uniform, while for A_{l}(q): 2, l ≥ 2, not all extensions are uniform, both in contrast to the connected case.

If the disconnected group G contains outer unipotent elements, then new Green functions arise. These are studied in the last three sections, and determined explicitly in several small rank cases. These results and the decompositions in Section 5 make it possible in principal, for example, to write down the character tables of the disconnected groups GO_{8}^{+}(2^{n}).

Also, the results of the last seven sections give informations on the values of unipotent characters of disconnected groups on outer elements, which are of immediate interest in constructive Galois theory (see [8] for example).

I would like to thank François Digne and Jean Michel for numerous conversations about the subject of this paper.

* The author gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft

1 Hecke algebras for automorphism groups of BN-pairs

In this first part we recall some results of [8] where a rather more general situation than that explained in the introduction was studied. Let G^{o} be a finite group with a BN-pair. Further let G be a finite cyclic extension of G^{o}, so G = G^{o}.⟨σ⟩. We moreover assume that σ fixes the BN-pair, i.e., both B and N are normalized by σ. In the case of groups of Lie type this last assumption is always satisfied, because there all Borel subgroups B and inside B all maximal tori T = B ∩ N are conjugate. Hence after multiplication with a suitable element of G^{o}, σ will fix the natural BN-pair. The σ-invariance of B and N immediately implies that σ also acts on the Weyl group W^{o} = N/(B ∩ N). Denote by I = {s_{1}, …, s_{l}} the distinguished generators of W^{o} according to the definition of a BN-pair. Now the set I is uniquely determined in W^{o}. Consequently σ also acts on I, hence also as a (possibly trivial) graph automorphism on the Dynkin diagram of (W^{o}, I). Contrary to all nontrivial elements of W^{o}, σ fixes the subgroup B, so the extension W = W^{o} × ⟨σ⟩ of W^{o} with σ splits.

In [3], II, Theorem 1.6, and in [8], Satz 1.5, it was proved that in analogy to the classical situation for BN-pairs, the Hecke algebra H(G, B) of the permutation character 1_{B}^{G} on the subgroup B is isomorphic to the group algebra ℂW of the extended Weyl group. In particular this yields a bijection between the irreducible complex characters of W and the irreducible constituents of 1_{B}^{G}. In accordance with the classical case we call the irreducible constituents of 1_{B}^{G} the unipotent principal series characters. The bijection with the irreducible characters of W may be chosen to be compatible with the corresponding bijection on the level of the group G^{o}, i.e., compatible with the restriction to W^{o} and H(G^{o}, B).

If G^{o} is a group of Lie type with the natural BN-pair, then the assumptions of the theorem are valid if σ is a field automorphism of G^{o}, in which case all unipotent principal series characters of G^{o} can be extended to G because σ acts trivially on the Weyl group. They also hold if σ acts like a graph automorphism. In that case, it may happen that not all unipotent principal series characters are extendable to G; see [8] for a detailed discussion.

In the application to the decomposition of generalized Deligne-Lusztig characters we will need more precise information. Namely, we want to know the decomposition into principal series characters of an arbitrary principal series unipotent character of any σ-stable Levi subgroup induced to G. For this, first the concept of parabolic subgroup and of Levi subgroup has to be suitably generalized. A parabolic subgroup of G^{o} may be defined by specifying a subset J of I. Now if J is fixed by σ, then σ normalizes the corresponding parabolic subgroup P_{J}^{o}, and we may define P_{J} := P_{J}^{o}.⟨σ⟩ as a parabolic subgroup of G. We now make the additional assumption that G^{o} has a split BN-pair of characteristic p. Then each parabolic subgroup P_{J}^{o} of G^{o} has a Levi decomposition P_{J}^{o} = U_{J} × L_{J}^{o}. If J is σ-stable, the standard Levi complement L_{J}^{o} also is, and we define L_{J} := L_{J}^{o}.⟨σ⟩ as Levi complement for the parabolic P_{J}. As L_{J}^{o} has again a BN-pair, the above results may be applied to L_{J}, and thus yield a bijection between the principal series unipotent characters κ of L_{J} and the irreducible characters of W_{J} := W_{J}^{o}.⟨σ⟩. Moreover we have the canonical bijection from H(L_{J}, B_{J}) to H(P_{J}, B), where κ uniquely corresponds to the inflated irreducible character κ̃ of P_{J} having U_{J} in its kernel. In [8], Satz 1.7, the following was proved about the decomposition of κ̃_{P_{J}}^{G}, which may be thought of as a generalization of the result of Howlett and Lehrer [5] to the disconnected case:

Theorem 1 : Let G^{o} be a finite group with a split BN-pair of characteristic p with Weyl group W^{o}, G = G^{o}.⟨σ⟩ a cyclic extension with W = W^{o}.⟨σ⟩, ζ an irreducible character of G with (1_{B}^{G}, ζ) ≠ 0, φ the corresponding character of W. Let moreover J ⊂ I be a σ-stable subset of the index set attached to W^{o}, κ an irreducible character of the Levi subgroup L_{J} corresponding to ψ of W_{J}, κ̃ the inflation to P_{J}. Then we have

⟨ζ_{φ}, κ̃_{ψ}^{G}⟩_{G} = ⟨φ, ψ^{W} ⟩_{W} .

An alternative formulation for this theorem is: The isometry φ ↦ ζ_{φ} from Char(W) to Char(G) is compatible with induction from parabolic subgroups. This formula will be employed in the sequel to decompose those generalized Deligne-Lusztig characters which may be obtained by Harish-Chandra induction from Levi subgroups.

2 An extension of the Steinberg character

An obvious application of Theorem 1 to the character theory of disconnected groups would consist in trying to express principal series unipotent characters of G as linear combinations of Harish-Chandra induced characters from Levi subgroups. But it is well known that in general for BN-pairs G^{o} of type other than A_{l}, the only characters expressible in this way are just the trivial character and the Steinberg character. In this section we show that in the situation discussed in the first section an extension of the Steinberg character may also be obtained as linear combination of such induced characters.

So let G^{o} be a finite group with a split BN-pair of characteristic p, and G, σ as in the first section. We will assume moreover that the action of ⟨σ⟩ on the Dynkin diagram for G^{o} is faithful. The most interesting case occurs when σ induces a nontrivial graph automorphism on a connected component of the diagram. Then by the classification of Dynkin diagrams, σ must have order two, except for G^{o} of type D_{4}, where an automorphism of order three may occur. We will restrict ourselves to the case σ^{2} = ι of an extension of degree two. For the single case of D_{4} with σ^{3} = ι, a similar formula for an extension of the Steinberg character as below may easily be derived from the explicit decomposition of the induced principal series characters using Theorem 1.

Let V := ∑_{i=1}^{l} ℝα_{i} be the ℝ-vector space spanned by the roots α_{i} with the natural W^{o}- action. This may be extended to a (not necessarily faithful) W -action, by letting σ(α_{i}) := αī, i = 1, …, l, where ī is defined by s_{i}^{σ} = sī. The sign character ϵ of W^{o} is given by ϵ(w) := det_{V} (w) ([1], Proposition 6.2.1), and hence possesses via ϵ̂(w) := det_{V} (w) an extension to W. In the bijection between irreducible characters of W^{o} and principal series characters of G^{o}, ϵ corresponds uniquely to the Steinberg character St of G^{o}. As ϵ may be extended to W, the remarks in Section 1 imply that St also possesses an extension to G. It turns out that this extension may again be obtained as a linear combination of Harish-Chandra induced characters.

In G^{o} the Steinberg character is defined by the alternating sum St = ∑_{J⊆I}(−1)^{|J|}1_{P_{J}^{o}}^{G^{o}}. The idea for the right formula for an extension of St to G will be that elements on the coset G^{o}.σ ‘belong’ to the fixed group G^{σo}. Therefore on outer elements St should be defined like the Steinberg character for G^{σo}. In particular, this requires an adjustment of the sign, which sort of replaces G^{o} by G^{σo}, and moreover the summation should extend only over the σ-stable subsets of I.

\underline{Definition 1}: Let

ϵ̃:= ∑_{{J⊆I \atop Jσ=J}} (−1)^{|J|}(−1_{σ})_{W_{J}}^{W} + ∑_{{[J]⊂I \atop Jσ≠J}} (−1)^{|J|} 1_{W_{J}^{o}}^{W},

where the summation ranges over the σ-orbits [J] of subsets J ⊆ I. Here for σ-stable J we let W_{J} := ⟨W_{J}^{o}, σ⟩ = W_{J}^{o}.⟨σ⟩, and with (−1_{σ}) we denote that extension of the trivial character of W_{J}^{o} to W_{J} which takes value (−1)^{(l−l′+|J|−|J_{σ}|)} on outer elements, where J_{σ} := J/⟨σ⟩ is the orbit set in J under σ. Finally l = |I| denotes the rank of the BN-pair and l′ = |I_{σ}| that of G^{σo}.

Obviously ϵ̃ is a virtual character of W.

Proposition 1: We have ϵ̃ = ϵ̂, i.e., ϵ̃ is an irreducible character of W extending the sign character of W^{o}.

Proof : On inner elements w ∈ W^{o} we obviously have (−1_{σ})_{W_{J}}^{W} = 1_{W_{J}}^{W} = 1_{W_{J}^{o}}^{W^{o}} for σ-stable J, and 1_{W_{J}^{o}}^{W} = 1_{W_{J}^{o}}^{W^{o}} + (1_{W_{J}^{o}}^{W^{o}})^{σ} = 1_{W_{J}^{o}}^{W^{o}} + 1_{W_{Jσ}^{o}}^{W^{o}}. So on inner elements

ϵ̃(w) = _{J⊆I}^{∑}(−1)^{|J|} 1_{W_{J}^{o}}^{W^{o}},

and this is just equal to ϵ(w) (see for example [1], 6.2).

For the values of ϵ̃ on outer elements the second sum in the definition visibly yields no contribution. So let wσ ∈ W with w ∈ W^{o} (remember that σ has order two). Then

(−1_{σ})_{W_{J}}^{W} (wσ) = \frac{(−1)^{(l−l′+|J|−|J_{σ}|)}}{|W_{J}|} ∑_{{x∈W \atop (wσ)x∈WJ}} 1.

In order to evaluate this further, we introduce the Coxeter complex of W^{o} in V ([1], 2.3). The simplizes C_{J} for J ⊆ I are defined by

C_{J} := {v ∈ V | ⟨α_{i}, v⟩ = 0 ∀i ∈ J, ⟨α_{i}, v⟩ > 0 ∀i ∉ J}.

Hence for each σ-stable J we see that C_{J} remains stable under σ. By [1], Proposition 2.3.4, W_{J}^{o} is the full stabilizer of C_{J} in W^{o}, and so for σ-stable J, W_{J} is the full stabilizer of C_{J} in W. The condition (wσ)^{x} ∈ W_{J} may hence be reformulated as: wσ stabilizes x(C_{J}). We now denote by n_{J}(w) the number of different simplizes x(C_{J}) fixed by wσ. Then

(−1_{σ})_{W_{J}}^{W} (wσ) = (−1)^{(l−l′+|J|−|J_{σ}|)}n_{J}(w).

As in the classical case of the Steinberg character of G^{o} ([1], 6.2), n_{J}(w) may now be interpreted as the number of simplizes of a fixed dimension in a certain simplicial complex, thus making possible the evaluation of the formula for ϵ̃(wσ). For that, let U := {v ∈ V | wσv = v} denote the eigenspace of wσ for the eigenvalue 1. The intersections U ∩ x(C_{J}), J ⊆ I, form a simplicial complex in U. We now prove that its simplizes correspond bijectively to the wσ-stable simplizes of the form x(C_{J}) for σ-stable J in V.

Let J, K be σ-stable with wσx(C_{J}) = x(C_{J}), wσy(C_{K}) = y(C_{K}) and x(C_{J})_{wσ} = y(C_{K})_{wσ}, where the lower index wσ denotes the fixed points under wσ. Then (wσ)^{x} is contained in W_{J}, hence (wσ)^{x} = w′σ with some w′ ∈ W_{J}^{o}. By [1], Proposition 2.3.4, W_{J}^{o} is even the pointwise stabilizer of C_{J}, and so the fixed point set (C_{J})_{(wσ)^{x}} of C_{J} under (wσ)^{x} is the same as that under σ. The same argument applies to C_{K}, consequently we obtain (C_{J})_{σ} = x^{−1}y(C_{K})_{σ^{y−1x}}. The orbit map, which to every J associates the set of σ-orbits contained in J, is obviously bijective on σ-stable subsets J ⊆ I. As the fixed point sets (C_{J})_{σ} and (C_{K})_{σ} are conjugate via x^{−1}y, we must already have that C_{J} and C_{K} are conjugate by x^{−1}y. This shows x(C_{J}) = y(C_{K}), and the map x(C_{J}) ↦ x(C_{J}) ∩ U is injective on wσ-stable simplizes with σ-stable J. But the surjectivity is clear.

By the above considerations, the dimension of the simplex x(C_{J}) ∩ U is equal to the one of (C_{J})_{σ}. With this we have proved dim(C_{J})_{σ} = l′ − |J_{σ}|. Moreover the number n_{i} of simplizes in U of dimension i is calculated to be n_{i} = ∑_{|J_{σ}|=l′−i} n_{J}(w), where the summation ranges over all σ-stable J with |J_{σ}| = l′ − i. The formula ∑_{i}(−1)^{i}n_{i} = (−1)^{dim U} for the simplicial complex defined by a finite set of hyperplanes in a vector space U thus yields

(−1)^{dim U} = _{i}^{∑}(−1)^{i} ∑_{{J \atop |Jσ|=l′−i}} n_{J}(w) = _{J}^{∑}(−1)^{l′−|J_{σ}|}n_{J}(w) = (−1)^{l}ϵ̃(wσ).

Now the elements wσ are orthogonal transformations of the vector space V, and hence their determinant may easily be calculated if one knows the dimension of the eigenspace U for the eigenvalue 1:

det(wσ) = (−1)^{l−dim U} = ϵ̃(wσ).

This completes the proof of the proposition.

Theorem 2: Let G^{o}, G be as above. Then

Ŝt := ∑_{{J⊆I \atop Jσ=J}} (−1)^{|J|}(−1_{σ})_{P_{J}}^{G} + ∑_{{[J]⊂I \atop Jσ≠J}} (−1)^{|J|} 1_{P_{J}^{o}}^{G}

gives an extension of the Steinberg character of G^{o} to G. Here we let P_{J} := P_{J}^{o}.⟨σ⟩, and (−1_{σ}) denotes that extension of the trivial character from P_{J}^{o} to P_{J} which takes value (−1)^{(l−l′+|J|−|J_{σ}|)} on outer elements, where l = |I| and l′ = |I_{σ}|.

Proof : Obviously the formula for Ŝt defines a virtual character of G which coincides with the Steinberg character on inner elements g ∈ G^{o} (see the argument in the proof of the preceding proposition). Finally, by Theorem 1 and Proposition 1 we have

⟨Ŝt, Ŝt⟩_{G} = ⟨ϵ̃, ϵ̃⟩_{W} = 1,

and hence Ŝt is an irreducible character of G, extending St.

3 Generalized Deligne-Lusztig characters

The finite groups of Lie type all have a natural split BN-pair, so may be taken as the groups G^{o} in the first two sections. The extension groups G = G^{o}.⟨σ⟩ studied there then become disconnected groups. This type of groups was always excluded in previous work about Lie type groups, and only the case of connected groups was studied. Now the close resemblance of the previous two theorems for groups G = G^{o}.⟨σ⟩ to those for G^{o} make it reasonable to expect that similar generalizations of the Deligne-Lusztig character theory to disconnected groups should be possible. Indeed this is true, as is shown in the paper [4] of Digne and Michel. Here we will only be interested in the following special case: Let G^{o} be a connected reductive group defined over the algebraic closure 𝔽̄_{q} of a finite field, and σ an outer automorphism of G^{o} which comes from an automorphism of the Dynkin diagram of G^{o}. Denote by G the semidirect product of G^{o} by σ. Assume moreover given a Frobenius map F with respect to an 𝔽_{q}-structure, which commutes with σ. We denote the finite fixed group G^{F} by G, and similarly G^{o} := G^{o F}. The idea is now to define Deligne-Lusztig characters for G which do not vanish on outer elements, hence which are not induced from G^{o}. For this let T^{o} be an F - and σ-stable maximal torus of G^{o}, lying in a σ-stable (but not necessarily F -stable) Borel subgroup B^{o} of G^{o}. Denote by B the normalizer of B^{o} in G, and by T the normalizer of T^{o} in B. Let U be the unipotent radical of B^{o}. In generalization of the construction of Deligne-Lusztig [2] we define the variety

X_{T} := {g ∈ G | g^{−1}F (g) ∈ U}.

Then the l-adic cohomology groups with compact support H_{c}^{i}(X_{T}, ℚ̄_{l}) become G, T -bi-modules, where T := T^{F}. For every linear character θ of T we then obtain a virtual character R_{T,θ} of G by the alternating sum

R_{T,θ}(g) := _{i}^{∑}(−1)^{i} Tr(g, H_{c}^{i}(X_{T}, ℚ̄_{l})_{θ})

which we will call generalized Deligne-Lusztig character. These characters are studied by Digne and Michel in [4]. One of the first observations is the following:

Proposition 2 [4]: The restriction of R_{T,θ} to G^{o} is equal to an ordinary Deligne-Lusztig character for G^{o}:

R_{T,θ}|_{G^{o}} = R_{T^{o},θ|_{T o}}.

By this property, the characters defined above are extensions of the characters of Deligne and Lusztig, whence their name. This means that by the work of Lusztig the decomposition into irreducibles of the restrictions of R_{T,θ} to the connected component G^{o} is completely known. In other words, on the coset G^{o} of G, the R_{T,θ} yield nothing new. Hence it remains to study them on the cosets G^{o}.σ^{i}, 1 ≤ i ≤ o(σ) − 1. If σ is of prime order (as is the case for example if the Dynkin diagram of G^{o} is connected and σ acts faithfully on it), then it suffices to consider just G^{o}.σ because the character values on the other cosets may be obtained by multiplication with linear characters. So we interpret the R_{T,θ} on the coset G^{o}.σ as a σ-class function on G^{o} and denote the corresponding scalar product of σ-class functions by ⟨·, ·⟩_{σ}.

It is shown in [4] that for a suitable choice of σ ∈ G\G^{o} (namely, σ quasi-central rational) the classes of σ, F -stable maximal tori T of G are in bijection with the conjugacy classes of the fixed group W^{σ} of the Weyl group under σ, so we may write T = T_{w} with w ∈ W^{σ}. From now on we assume that σ is chosen in that special way (otherwise, multiply σ by a suitable inner element to obtain a quasi-central outer element). With these notations, Digne and Michel prove the following properties of the R_{T,θ}, completely analogous to the ones in the connected case:

Theorem 3 [4]: (1) R_{T_{w},1} only depends on the class of w in W^{σ}. (2)If T is already contained in a proper parabolic subgroup P = P^{o}.⟨σ⟩ of G, then R_{T,θ}^{G} (in G) may be obtained by Harish-Chandra induction from a Levi subgroup L = L^{o}.⟨σ⟩ < P:

R_{T,θ}^{G} = (R_{T,θ}^{L})_{P}^{G}.

(3) The R_{T,θ} satisfy a scalar product formula similar to the one for the connected case; if θ = 1, and T_{v}, T_{w} are parametrized by the classes of v, w in W^{σ} we have

⟨R_{T_{v},1}, R_{T_{w},1}⟩_{σ} = \{ \begin{matrix}|C_{W^{σ}}(w)| & if v ∼ w in W^{σ} \\ 0 & otherwise \end{matrix}

(4) For the trivial characters 1 of T and 1_{G} of G we have

⟨R_{T,1}, 1_{G}⟩_{σ} = 1.

Despite these results, the main problem of the decomposition of the virtual characters R_{T,1} into irreducible characters of G is not yet solved in general. But this also resembles the situation in the connected case, where the explicit decomposition of the Deligne-Lusztig characters was a very hard problem.

In [4], an analogue of the character formula is proved, which expresses the values of the R_{T,θ} in terms of sums over Green functions. In the case that the characteristic p of G divides the order of σ, the outer cosets contain unipotent classes. The Green functions on these classes have also not yet been determined in general; some particular cases will be studied in later sections.

But first we will calculate the decomposition of the generalized Deligne-Lusztig characters R_{T,1} for several low rank cases, using Theorem 3. For an elegant formulation of the results it is convenient to introduce the so called almost characters, again in analogy to the connected case but now on a coset G^{o}.σ. It was already mentioned above that the classes of σ, F -stable maximal tori of G are parametrized by the conjugacy classes of W^{σ}.

\underline{Definition 2}: Let χ be an irreducible character of W^{σ}. Then we define the almost character R_{χ} to be the σ-class function

R_{χ} := \frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑} χ(w)R_{T_{w},1},

where T_{w} denotes a σ, F -stable maximal torus parametrized by w ∈ W^{σ} and T_{w} := T_{w}^{F}.

Now it makes no sense to consider the class function defined by this recipe on all of G, because by the work of Lusztig it is known that if W^{σ} ≠ W the restriction to G^{o} will be far from an irreducible character. But we may interpret it as a σ-class function on G^{o}, i.e., as restriction of a character to the coset G^{o}.σ. It will turn out in all examples that the R_{χ} are close to being (extensions to G^{o}.σ of) irreducibles, as is the case with the almost characters for connected groups. The approach resembles very much the one in the case of the extension of the Steinberg character in Section 2, where we also had two ‘recipes’, according to whether we look at the inner or an outer coset.

In accordance with the connected case we may subdivide the extensions to G^{o}.σ of unipotent characters into families. Namely the families are defined by the transitive hull of the relation generated by: ρ_{1} ∼ ρ_{2} if there exists an almost character R_{χ} with respect to W^{σ} such that ⟨R_{χ}, ρ_{i}⟩_{σ} ≠ 0 for i = 1, 2. This induces an obvious subdivision into families of the corresponding almost characters R_{χ}. If {R_{χ_{i}}}, {ρ_{j}} are related families of almost characters (with respect to W^{σ}) and (extensions to G^{o}.σ of) unipotent characters, we define the Fourier transform matrix associated to this family to be the matrix

F = (⟨R_{χ_{i}}, ρ_{j}⟩_{σ})_{i,j} .

The outer automorphism σ of G^{o} acts on the Dynkin diagram of G^{o}. If σ permutes different connected components of this diagram, then the finite group G = G^{F} will be a wreath product, and the characters of such wreath products are well understood. Hence the more interesting cases occur when σ induces a nontrivial automorphism on a connected component of the Dynkin diagram. We may then also assume that the diagram for G^{o} has just one connected component. By the classification of reductive algebraic groups over 𝔽̄_{q}, this just leaves the following possibilities for the pair (G^{o}, o(σ)): (A_{l}, 2) for l ≥ 2, (D_{l}, 2) for l ≥ 4, (D_{4}, 3) and (E_{6}, 2). In the next three sections we consider a number of low rank cases and for each determine the decomposition of the R_{T,1}. From the results obtained we formulate a conjecture how the situation should look in the general case.

4 Almost characters for A_{l}(q): 2 and ^{2}A_{l}(q): 2, l ≤ 5

Let G^{o} be a connected reductive algebraic group of type A_{l} over the algebraic closure of a finite field. For l ≥ 2, G^{o} has an outer automorphism σ of order two coming from the graph automorphism of the Dynkin diagram. Let F be a Frobenius map on G^{o} commuting with σ. Then G^{F} is either A_{l}(q).⟨σ⟩ or the twisted version ^{2}A_{l}(q).⟨σ⟩, where in the second case σ acts like the graph-field automorphism on ^{2}A_{l}(q). In the previous section, generalized Deligne-Lusztig characters R_{T_{w},1} were defined for G = G^{F}, where the T_{w} are indexed by elements of the fixed group W^{σ} of the Weyl group. To describe their decompositions, we first have to study their possible constituents.

Let first G^{o} = A_{l}(q). Then the extension of the Weyl group W^{o} ≅ S_{l+1} with σ is isomorphic to a direct product. So as all unipotent characters of A_{l}(q) lie in the principal series, they all extend to G by Theorem 1. The usual almost characters for the connected group A_{l}(q) are exactly the unipotent characters of G^{o}. This gives a bijection ψ ↦ ρ_{ψ} between the irreducible characters of S_{l+1} and the unipotent characters of A_{l}(q). The irreducible characters of S_{l+1} are parametrized by partitions λ of l + 1. So ρ_{ψ} may also be denoted by ρ_{λ} with λ ⊢ l + 1. To each partition λ is associated its 2-core and 2-quotient. If l+1 is even, then this gives a bijection between partitions of l+1 with trivial 2-core and their 2-quotients, which are pairs of partitions (λ_{1}, λ_{2}) ⊢ (l_{1}, l_{2}) with l_{1} + l_{2} = (l + 1)/2. Similarly for odd l + 1, we obtain a bijection between partitions with 2-core consisting of one point and again pairs of partitions of l/2. Now the latter parametrize the irreducible characters of the Weyl group of type B_{[\frac{l+1}{2}]}. Moreover the characters of S_{l+1} indexed by partitions with trivial 2-core are precisely those not vanishing on the longest element w_{0} of W (A_{l}). Summarizing the previous discussion, we have a bijection from the unipotent characters ρ_{ψ} of A_{l}(q) with ψ not vanishing on w_{0} and the irreducible characters of W (B_{[\frac{l+1}{2}]}) = W (A_{l})^{σ}. The decomposition of the generalized Deligne-Lusztig characters in terms of unipotent characters may then be described as follows:

Theorem 4: For G = A_{l}(q).⟨σ⟩, l ≤ 5, the set of almost characters {R_{χ} | χ ∈ Irr(W^{σ})} form a subset of the set of restrictions to the coset G^{o}.σ of the extensions of the unipotent characters. More precisely, if χ is indexed by (λ_{1}, λ_{2}) then R_{χ} is equal to an extension of ρ_{λ} on G^{o}.σ, where λ has trivial 2-core and 2-quotient (λ_{1}, λ_{2}).

Proof : We have to determine the decomposition of the R_{T_{w},1} into extensions of unipotent characters on G^{o}.σ. The theorem then follows by calculating the R_{χ} according to Definition 2. The classes of σ, F -stable maximal tori of G are parametrized by conjugacy classes of W^{σ} = W (B_{[\frac{l+1}{2}]}). If T_{w} is already contained in a parabolic subgroup P, then by Theorem 3(2), R_{T_{w},1} can be obtained by ordinary induction from P. Inductively we may assume that the decomposition of R_{T,1}^{L} is already known for all proper Levi subgroups L of G. Then with Theorem 1, the decomposition of the induced virtual character can be computed from the corresponding information inside the Weyl group.

With this, all but 1 (resp. 2, 2, 3) R_{T_{w},1} for l = 2 (3, 4, 5 respectively) are known. In case l = 2, the remaining decomposition follows directly from the scalar product formula in Theorem 3(3). The same is true for l = 3, 4, if we also use Theorem 3(4). Finally for l = 5, we apply Proposition 2, which says that the decomposition numbers are known at least mod 2. These restrictions together with the scalar product formula uniquely determine the missing decompositions.

For groups G^{o} = ^{2}A_{l}(q), the unipotent principal series characters are indexed by irreducible characters of the Weyl group W = W (B_{[\frac{l+1}{2}]}) (where W is the Weyl group in the sense of BN-pairs, not the one of G^{o}). Here σ acts trivially on the Weyl group, so all these characters extend to G. We obtain the same picture as in the untwisted case:

Theorem 5: For G = ^{2}A_{l}(q).⟨σ⟩, l ≤ 5, the set of almost characters {R_{χ} | χ ∈ Irr(W^{σ})} coincides with the set of (restrictions to G^{o}.σ of the) extensions of unipotent principal series characters. More precisely, R_{χ} equals ρ_{χ} on G^{o}.σ. (Remember that here the R_{χ} and the ρ_{χ} are both indexed by the irreducible characters of W (B_{[\frac{l+1}{2}]})).

The proof is exactly analogous to the previous one.

It is clear what kind of conjecture might be made for l ≥ 6 in both cases. But note the big difference to the connected case. There all unipotent characters are obtained as almost characters, while here only the subset of those which correspond to characters of W (B_{[\frac{l+1}{2}]}) occur. In particular, not all σ-class functions are uniform. It is not at all clear how to obtain the remaining extensions of unipotent characters, which do not occur among the R_{χ}.

5 Almost characters for D_{4}(q): 2, ^{2}D_{l}(q): 2, l ≤ 5, and D_{4}(q): 3, ^{3}D_{4}(q): 3

We first consider the case that G^{o} is of type D_{4} and σ acts as the graph automorphism of order three on G^{o}. From the action of σ on the Weyl group W = W (D_{4}) one sees that precisely eight unipotent characters of G^{o} extend to G in case D_{4}(q) and that all eight unipotent characters of ^{3}D_{4}(q) extend to ^{3}D_{4}(q): 3. The notation for the unipotent characters is taken from [1], 13.8 and 13.9.

Theorem 6: For G = D_{4}(q): 3 and G = ^{3}D_{4}(q): 3 the decomposition of the R_{T_{w},1} is given in Table 1. (For the untwisted case, the columns are indexed by |T^{σo}| and |T^{o}| in the first two rows and by σ-stable unipotent characters in the first column, for the twisted case, the columns are indexed by |T^{o}| in the last row and the unipotent characters in the last column). In particular, four almost characters with respect to W^{σ} = W (G_{2}) coincide with extensions of unipotent characters to G. The remaining four unipotent characters form a family, to which correspond the remaining two almost characters via the Fourier transform matrix

\frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & −1 & 1 & −1 \end{matrix} ) .

Table 1: Decomposition of R_{T,1} for D_{4}(q): 3 and ^{3}D_{4}(q): 3.

|T^{σo}|  Φ_{1}^{2}  Φ_{1}Φ_{2}  Φ_{1}Φ_{2}  Φ_{3}  Φ_{6}  Φ_{2}^{2}
D_{4}(q): 3  Φ_{1}^{4}  Φ_{1}^{3}Φ_{2}  Φ_{1}Φ_{2}^{3} Φ_{1}^{2}Φ_{3} Φ_{2}^{2}Φ_{6} Φ_{2}^{4}
ϕ_{4,−}  1  1  1  1  1  1  ϕ_{1,0}
ϕ_{3,1}  1  −1  1  1  −1  −1  ϕ′_{1,3}
ϕ_{22,−}  2  .  .  −1  .  .  ϕ_{2,1}
ϕ_{2,11}  .  .  .  .  1  −2  ^{3}D_{4}[−1]
ϕ_{21,1}  2  .  .  −1  .  .  ϕ_{2,2}
D_{4}  .  .  .  .  1  −2  ^{3}D_{4}[1]
ϕ_{111,1}  1  1  −1  1  −1  −1  ϕ′′_{1,3}
ϕ_{1111,−}  1  −1  −1  1  1  1  ϕ_{1,6}
Φ_{1}^{2}Φ_{3} Φ_{1}Φ_{2}Φ_{3} Φ_{1}Φ_{2}Φ_{6} Φ_{3}^{2}  Φ_{6}^{2} Φ_{2}^{2}Φ_{6}  ^{3}D_{4}(q): 3

(We have used the notation Φ_{i} := Φ_{i}(q), the i-th cyclotomic polynomial evaluated at q. Also, in this and in the following tables, we choose the extensions of the unipotent characters so that their multiplicities in R_{T,1}, for T^{σo} maximally split in G^{σo} are nonnegative.)

Proof : We may obtain both results simultaneously. As in the proofs of the preceding theorems, the first three columns are obtained by Harish-Chandra induction and application of Theorem 1, because the corresponding tori already lie in proper σ-stable parabolic subgroups. Namely all their Levi complements are of type A_{1}, and there we know the decomposition of the R_{T_{w},1} by trivial arguments. Further from the orthogonality relations we have ⟨R_{T_{w},1}, R_{T_{w},1}⟩_{σ} = 6 for the fourth and fifth torus T_{w} in the table. Moreover the restriction to G^{o} by Proposition 2 gives congruences modulo 3 for the entries, and this together already uniquely determines the two columns. Orthogonality now leaves just one possibility for the last column, proving the theorem.

In analogy to the terminology for the connected case we might say that G^{o}.σ (in both cases) has two cuspidal characters, ϕ_{2,11} and D_{4} (respectively the two usual cuspidals 3D_{4}[−1] and ^{3}D_{4}[1]), namely those which do not occur in R_{T_{1},1}.

We now turn to the case of G^{o} of type D_{4} and σ of order two. From the action of σ on the Weyl group we deduce that for G^{o} = D_{4}(q) exactly ten unipotent characters extend to G, and for G^{o} = ^{2}D_{4}(q) all ten unipotent characters extend to G. The noncuspidal unipotent characters of D_{4}(q) are indexed by irreducible characters of W (D_{4}), the unipotent characters of ^{2}D_{4}(q) all lie in the principal series and are indexed by irreducible characters of W (B_{3}). Contrary to the cases so far, the results may not be expressed in a uniform fashion for the twisted and for the untwisted groups:

Theorem 7: (1) For D_{4}(q): 2 the decomposition of the R_{T_{w},1} is given by Table 2. In particular, six almost characters with respect to W^{σ} = W (B_{3}) coincide with extensions of unipotent characters to G. The other four almost characters form a family, and are related by the Fourier transform matrix

\frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix} )

to the remaining four unipotent characters. So all extensions of unipotent characters are uniform.

(2) For ^{2}D_{4}(q): 2 the decomposition of the R_{T_{w},1} is obtained by replacing the middle four rows of Table 2 by those appearing in Table 3. This time the corresponding Fourier transform matrix has the form

\frac{1}{2} ( \begin{matrix}1 & 1 & 1 & −1 \\ 1 & 1 & −1 & 1 \\ 1 & −1 & 1 & 1 \\ −1 & 1 & 1 & 1 \end{matrix} ) .

So again all extensions of unipotent characters are uniform.

(3) For ^{2}D_{5}(q): 2 the extensions of the twelve unipotent characters of G^{o} lying in one element families are obtained as almost characters on G^{o}.σ. The other eight fall in two families, each with corresponding Fourier transform matrix as under (2).

Table 2: Decomposition of the R_{T,1} for D_{4}(q): 2.

\begin{matrix}|T | & 1^{4} & 1^{3}2 & 1^{2}2^{2} & 1^{2}2^{2} & 124 & 12^{3} & 1^{2}3 & 12^{3} & 124 & 2^{2}6 \\ |T^{σ}| & 1^{3} & 1^{2}2 & 1^{2}2 & 12^{2} & 14 & 12^{2} & 13 & 2^{3} & 24 & 26 \\ ϕ_{4,−} & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ ϕ_{3,1} & 2 & . & 2 & 2 & . & . & −1 & 2 & . & −1 \\ ϕ_{31,−} & 3 & 1 & 1 & −1 & 1 & −1 & . & −3 & −1 & . \\ ϕ_{21,1} & 4 & . & . & . & −2 & . & 1 & . & . & . \\ ϕ_{22,−} & 2 & . & . & −2 & . & 2 & −1 & . & . & . \\ ϕ_{2,11} & . & 2 & −2 & . & . & . & . & 2 & . & −1 \\ D_{4} & . & . & . & . & . & . & . & 4 & −2 & 1 \\ ϕ_{211,−} & 3 & −1 & −1 & −1 & 1 & −1 & . & 3 & 1 & . \\ ϕ_{111,1} & 2 & . & −2 & 2 & . & . & −1 & −2 & . & 1 \\ ϕ_{1111,−} & 1 & −1 & −1 & 1 & 1 & 1 & 1 & −1 & −1 & −1 \end{matrix}

Table 3: Part of the decomposition of the R_{T,1} for ^{2}D_{4}(q): 2.

\begin{matrix}1^{3}2 & 1^{3}2 & 1^{2}2^{2} & 1^{2}2^{2} & 1^{2}4 & 12^{3} & 123 & 12^{3} & 2^{2}4 & 126 & |T | \\ 1^{3} & 1^{2}2 & 1^{2}2 & 12^{2} & 14 & 12^{2} & 13 & 2^{3} & 24 & 26 & |T^{σ}| \\ 3 & 1 & −1 & −1 & −1 & 1 & . & −1 & 1 & −1 & ϕ_{1,2} \\ 3 & −1 & 1 & −1 & −1 & 1 & . & 1 & −1 & 1 & ϕ_{11,1} \\ 1 & 1 & −1 & 1 & −1 & −1 & 1 & 3 & −1 & . & ϕ_{−,3} \\ 1 & −1 & 1 & 1 & −1 & −1 & 1 & −3 & 1 & . & ϕ_{111,−} \end{matrix}

Proof : (1) The first seven columns are obtained by Harish-Chandra induction from Levi subgroups, by Theorem 3(2) using Theorem 1. From comparison with the restriction to G^{o} it follows that the last column has to contain six odd entries, and because the norm of the last R_{T,1} is equal to six, they all have to be ±1. Orthogonality then determines the signs. The other two missing columns are found in the same way.

The proof of the second part is entirely similar to the previous one. As for (3), again all decompositions are obtained by the above arguments.

Again note the difference to the connected case. Here both for G^{F} and for G^{σF}, all unipotent characters are uniform, which is not the case for the connected groups of types D_{4}, ^{2}D_{4} and ^{2}D_{5}. Indeed it is possible to show that this holds for all l ≤ 8. To prove this, we have to look more closely at the parametrization of the unipotent characters of D_{l}(q) and 2D_{l}(q), which was established by Lusztig (see [1]).

A symbol is a pair of sequences of natural numbers

S = ( \begin{matrix}λ_{1} & … & λ_{a} \\ μ_{1} & … & μ_{b} \end{matrix} )

with 0 ≤ λ_{1} < λ_{2} … < λ_{a} and 0 ≤ μ_{1} < μ_{2} … < μ_{b}. The symbols S and

S′ = ( \begin{matrix}0 & λ_{1} + 1 & … & λ_{a} + 1 \\ 0 & μ_{1} + 1 & … & μ_{b} + 1 \end{matrix} )

are defined to be equivalent. Also exchanging the two rows of S by definition gives an equivalent symbol. The rank of S is defined by rank(S) := ∑ λ_{i} + ∑ μ_{i} − [(\frac{a+b−1}{2})^{2}]. Note that this only depends on the equivalence class of S. Denote by S^{+}(l) the set of classes of symbols of rank l with a − b ≡ 0 (mod 4) and for which the two rows are different, and by S^{−}(l) the classes of symbols of rank l with a − b ≡ 2 (mod 4). Then the unipotent characters of D_{l}(q) invariant under the graph automorphism of order two are parametrized by S^{+}(l), and those of ^{2}D_{l}(q) by S^{−}(l) ([1], 13.8).

We now define a map ψ from S^{−}(l) to S^{+}(l). For this let S ∈ S^{−}(l). Denote by j the smallest index for which λ_{i} ≠ μ_{i} (where we agree that λ_{i} for i > a should be understood to be ∞, and the same for μ_{i}). If necessary, interchange the two rows of S so that λ_{j} < μ_{j}. Then delete λ_{j} from the top row and insert it after μ_{j−1} in the bottom row. For the resulting symbol ψ(S) we now obviously have a − b ≡ 0 (mod 4), and the rank has stayed the same. It is easily seen that ψ only depends on classes, and so ψ is a well defined map from S^{−}(l) to S^{+}(l).

Proposition 3: The map ψ defines a bijection between S^{−}(l) and S^{+}(l).

Proof : As S^{+}(l) does not contain a symbol with two equal rows, the definition of ψ also makes sense for S ∈ S^{+}(l). One easily checks that ψ^{2} is the identity, and the proposition is proved.

The irreducible characters of W (B_{l−1}) may be parametrized by pairs of partitions (α, β) ⊢ (l_{1}, l_{2}) with l_{1} + l_{2} = l − 1. Let α = (α_{1}, …, α_{r+1}) with α_{1} ≤ … ≤ α_{r+1}, and correspondingly β = (β_{1}, …, β_{r}) with β_{1} ≤ … ≤ β_{r}, where we have introduced a suitable number of zeroes in α or β such that α contains precisely one part more than β. Then to (α, β) is associated the symbol

S(α, β) := \binom{α_{1} α_{2} + 1 … α_{r} + r − 1 α_{r+1} + r}{β_{1} β_{2} + 1 … β_{r} + r − 1} ,

and it follows easily that equivalent symbols lead to the same pair of partitions of l − 1 (when we agree that the row with more entries always is the top one). One calculates rank(S) := ∑ λ_{i} + ∑ μ_{i} − r^{2} = l − 1. Let S^{0}(l − 1) be the set of classes of symbols of rank l − 1 with a − b = 1. Hence the irreducible characters of W (B_{l−1}) are parametrized by S^{0}(l − 1) ([1], 11.4). We now define a map φ : S^{0}(l − 1) → S^{−}(l) by the following recipe: Let

S = \binom{λ_{1} … λ_{r} λ_{r+1}}{μ_{1} … μ_{r}}

be a symbol in S^{0}(l − 1). We put

φ(S) := \binom{0 λ_{1} + 1 … λ_{r} + 1 λ_{r+1} + 1}{μ_{1} … μ_{r}} .

Now the upper row has precisely two more entries than the lower one. Moreover the entries in each row are strictly increasing. The rank of φ(S) is equal to ∑ λ_{i}+∑ μ_{i}+r+1−(r^{2}+r) = l − 1 + 1 = l. So the new symbol lies in S^{−}(l), and one checks that φ is well defined. But unfortunately this map is not in general bijective. If we denote with S_{2}^{−}(l) the subset of S^{−}(l) of (classes of) symbols of defect d := |a − b| = 2, then at least we have:

Proposition 4: The map φ defines a bijection between S^{0}(l − 1) and S_{2}^{−}(l).

Proof : First, from the definition we have φ(S^{0}(l − 1)) ⊆ S_{2}^{−}(l). Now let

S = ( \begin{matrix}λ_{1} & & … & λ_{r+2} \\ μ_{1} & … & μ_{r} \end{matrix} )

be a symbol in S_{2}^{−}(l), with the first entry in the top row a zero (which may be assumed by passing to an equivalent symbol). Denote by κ(S) the symbol we obtain by deleting the zero and decreasing all other λ_{i} by one. Then obviously κ(S) ∈ S^{0}(l − 1). Moreover the two maps φ and κ are inverse to each other, as is immediately seen from the definition. So the proposition follows.

The symbol of smallest rank for ^{2}D of defect more than 2 is

\binom{0 1 2 3 4 5}{−}

of rank 9. Hence for l ≤ 8 (and only for those) the map ψ ∘ φ is a bijection between S^{0}(l − 1) and S^{+}(l), and this gives the desired conclusion:

Proposition 5: For l ≤ 8 the extensions of the unipotent characters of D_{l}(q) to D_{l}(q): 2 all lie in the space of uniform functions on the coset D_{l}(q).σ. The same holds for ^{2}D_{l}(q): 2. On the other hand, this is false for all l ≥ 9.

Remark: In the cases D_{3} (=A_{3}), D_{4} and D_{5}, for which the decomposition of the R_{T,1} was explicitly calculated above, the map ψ ∘ φ is precisely the right one, in the following sense: If the almost character R_{S} with respect to the Weyl group W (B_{l−1}) coincides on G^{o}.σ with an extension of a unipotent character ζ, then ζ = ζ_{ψ∘φ(S)} in Lusztig’s parametrization. Moreover ψ also maps the four element families of unipotent characters of ^{2}D_{l} to families in D_{l}.σ.

6 Almost characters for E_{6}(q): 2 and ^{2}E_{6}(q): 2

By the remark at the end of Section 3 the only remaining case for G with connected Dynkin diagram and σ inducing a nontrivial diagram automorphism apart from the infinite families (A_{l}, 2) and (D_{l}, 2), l ≥ 6, is (E_{6}, 2). So let G^{o} be of type E_{6} and σ act as the graph automorphism. Then it is again possible to determine the decomposition of all R_{T,1} into extensions of unipotent characters. As E_{6}(q) possesses 30 unipotent characters, which all extend to G, and W^{σ} = W (F_{4}) has 25 conjugacy classes, the complete table of decomposition would become rather large. Instead we give the equivalent information on the decomposition of the almost characters R_{χ}.

For this we introduce the following notation. Let ρ be a unipotent character of E_{6}(q). Then its degree ρ(1) may uniformly be written as f_{ρ}(q) where f_{ρ}(X) is a polynomial in X. If we replace X by −X in this polynomial, then ±f_{ρ}(−X) is the order polynomial of a well defined unipotent character of the twisted group ^{2}E_{6}(q) (except for an ambiguity at the cuspidal characters, which does not matter). This defines a natural bijection from unipotent characters of E_{6}(q) to those of ^{2}E_{6}(q) (see [1], 13.9). The principal series unipotent characters of ^{2}E_{6} are indexed by irreducible characters of W (F_{4}). Via the above bijection, this also indexes a subset of the unipotent character of E_{6}(q). The notation for the characters of W (F_{4}) is taken from [1], p.413.

Theorem 8: (1) For G = E_{6}(q): 2 the almost characters R_{χ} with χ ∈ {ϕ_{1,0}, ϕ′_{2,4}, ϕ_{4,1}, ϕ′_{4,7}, ϕ′′_{8,3}, ϕ′_{9,6}, ϕ′′_{9,6}, ϕ′_{8,9}, ϕ′′_{4,7}, ϕ_{4,13}, ϕ′′_{2,16}, ϕ_{1,24}} coincide with restrictions to G^{o}.σ of the unipotent characters indexed by these χ as explained above. The remaining almost characters divide up into three families, the first and second both containing four almost characters and four unipotent characters, {ϕ′′_{2,4}, ϕ_{9,2}, ϕ′_{1,12}, ϕ′_{8,3}}, resp. {ϕ′_{2,16}, ϕ_{9,10}, ϕ′′_{1,12}, ϕ′′_{8,9}}, related by the Fourier transform matrix

\frac{1}{2} ( \begin{matrix}1 & 1 & 1 & 1 \\ 1 & 1 & −1 & −1 \\ 1 & −1 & 1 & −1 \\ 1 & −1 & −1 & 1 \end{matrix} ) ,

and the third family containing five almost characters and the unipotent characters {ϕ_{80,7}, ϕ_{20,10}, ϕ_{60,8}, ϕ_{10,9}, ϕ_{90,8}, D_{4,r}, E_{6}[θ], E_{6}[θ^{2}]}, related by the Fourier transform matrix

\frac{1}{6} ( \begin{matrix}1 & −1 & 3 & 2 & 2 & −3 & 2 & 2 \\ 3 & 3 & −3 & 0 & 0 & −3 & 0 & 0 \\ 2 & −2 & 0 & 4 & −2 & 0 & −2 & −2 \\ 2 & −2 & 0 & −2 & 4 & 0 & −2 & −2 \\ 3 & 3 & 3 & 0 & 0 & 3 & 0 & 0 \end{matrix} ) .

(2) For G = ^{2}E_{6}(q): 2, the almost characters R_{χ} with χ ∈ {ϕ_{1,0}, ϕ′_{2,4}, ϕ_{4,1}, ϕ′_{4,7}, ϕ′′_{8,3}, ϕ′_{9,6}, ϕ′′_{9,6}, ϕ′_{8,9}, ϕ′′_{4,7}, ϕ_{4,13}, ϕ′′_{2,16}, ϕ_{1,24}} coincide with restrictions to G^{o}.σ of the unipotent characters indexed by these χ. The remaining almost characters divide up into three families, the first and second both containing four almost characters and four unipotent characters, {ϕ′′_{2,4}, ϕ_{9,2}, ϕ′_{1,12}, ϕ′_{8,3}}, resp. {ϕ′_{2,16}, ϕ_{9,10}, ϕ′′_{1,12}, ϕ′′_{8,9}}, related by the Fourier transform matrix

\frac{1}{2} ( \begin{matrix}1 & 1 & 1 & −1 \\ 1 & 1 & −1 & 1 \\ 1 & −1 & 1 & 1 \\ −1 & 1 & 1 & 1 \end{matrix} ) ,

and the third family containing five almost characters and the unipotent characters {^{2}E_{6}[1], ϕ_{12,4}, ϕ_{4,8}, ϕ′_{6,6}, ϕ′′_{6,6}, ϕ_{16,5}, ^{2}E_{6}[θ], ^{2}E_{6}[θ^{2}]}, related by the Fourier transform matrix

\frac{1}{6} ( \begin{matrix}1 & 1 & −3 & 2 & 2 & 3 & 2 & 2 \\ 3 & −3 & 3 & 0 & 0 & 3 & 0 & 0 \\ 2 & 2 & 0 & 4 & −2 & 0 & −2 & −2 \\ 2 & 2 & 0 & −2 & 4 & 0 & −2 & −2 \\ −3 & 3 & 3 & 0 & 0 & 3 & 0 & 0 \end{matrix} ) .

Proof : (1) The maximal σ-stable parabolic subgroups of G have Levi subgroups of types A_{l}, l ≤ 5, or D_{4}, extended by the graph automorphism of order two. Hence the decompositions of the R_{T,1} for all proper Levi subgroups are known by Theorems 4–7. With Harish-Chandra induction using Theorem 1 we may thus compute the decomposition of sixteen of the R_{T_{w},1} for G. Now this is the first case for which there exists more than one solution in integers for the remaining columns, taking into account orthogonality and the congruence modulo 2 obtained by restriction. So some new arguments have to be used to deduce the complete decomposition.

First for the torus T^{σo} of order Φ_{8} the orthogonality relations give ⟨R_{T,1}, R_{T,1}⟩_{σ} = 8, and the restriction to G^{o} imposes congruences mod 2 on the unknown coefficients. This is enough to determine the decomposition of this Deligne-Lusztig character, up to one ambiguity. Namely it is not possible to distinguish between the extensions of the unipotent characters D_{4,1} and D_{4,ϵ}. The conditions only imply that either both occur with multiplicity 1, or both with multiplicity −1. So for the moment we write ϵ for their common multiplicity in this R_{T,1}, with ϵ ∈ {±1}. This ambiguity will only be resolved at the end of the proof.

In a similar manner the orthogonality relations yield the decomposition of R_{T,1} for T^{σo} of order Φ_{12}, up to a dependence on ϵ. This leaves seven columns to be determined. Here for the first time the methods applied so far do not suffice to complete the decomposition. Some more information on the generalized Deligne-Lusztig characters is needed. By [4], the value of R_{T_{w},1} on the quasi-central element σ is given by

R_{T_{w},1}^{G}(σ) = R_{T_{w}^{σ o},1}^{G^{σ o}}(1) = ϵ_{G^{σo}} ϵ_{T^{σo}} \frac{|G^{σ o}|_{p′}}{|T^{σ o}|} ,

and the characteristic function on the conjugacy class [σ] is uniform. So for ρ an extension of a unipotent character we have

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

An inspection of the |T^{σo}| shows that only one of them, say for T = T_{1}, is divisible by Φ_{4}^{2}. So when working modulo Φ_{4} only those summands with T_{w}^{σ} ∼ T_{1}^{σ} contribute, hence

ρ(σ) ≡ \frac{1}{|C_{W^{σ}}(w)|}⟨ρ, R_{T_{1},1}⟩_{σ} R_{T_{1},1}(σ) (mod Φ_{4}).

Now |C_{W^{σ}}(w)| = 96 and one easily calculates R_{T_{1},1}(σ) = ϵ_{G^{σo}} ϵ_{T^{σo}}|G^{σ o}|_{p′}/|T_{1}^{σ o}| ≡ 96 (mod Φ_{4}), so that finally

ρ(σ) ≡ ⟨ρ, R_{T_{1},1}⟩_{σ} (mod Φ_{4}).

It is well known that the quotient

\frac{|[σ]|ρ(σ)}{ρ(ι)}

is an algebraic integer for any irreducible character ρ and any conjugacy class [σ] of elements. Using the character degrees in [1], 13.9, this implies in our special case that ρ(σ) ≡ 0 (mod Φ_{4}) for ρ ∈ {ϕ_{20,2}, ϕ_{64,4}, ϕ_{60,5}, ϕ_{24,6}, ϕ_{24,12}, ϕ_{60,11}, ϕ_{64,13}, ϕ_{20,20}, ϕ_{30,3}, ϕ_{30,15}, ϕ_{20,10}, ϕ_{60,8}} and for the cuspidal characters E_{6}[θ] and E_{6}[θ^{2}]. Now let l be a primitive prime divisor of Φ_{4}(q) in the sense of Zsigmondy. It is easily seen that l ≥ 5. So the congruence on ⟨ρ, R_{T_{1},1}⟩_{σ} implies that l divides this multiplicity for all characters mentioned above. But we also know the restriction of R_{T_{1},1} to G^{o}, and this shows that those multiplicities are congruent to zero mod 2, hence divisible by an integer ≥ 10. As the σ-norm of R_{T_{1},1} is only equal to 96, all these multiplicities are actually equal to zero. Using this additional information, the decomposition for T_{1} follows by orthogonality and restriction.

Now R_{T,1} for T^{σo} of order Φ_{2}^{2}Φ_{4}, which has σ-norm 32, may be determined. Finally, arguments similar to the above, but now with Φ_{3} and the torus of order Φ_{3}^{2} determine the corresponding R_{T,1}, and then the missing four columns of the decomposition matrix are uniquely determined by the orthogonality relations and the restrictions to G^{o}.

All of the last nine decompositions still depend on ϵ. Now we may calculate the almost characters, also depending on ϵ. As the class of σ is uniform [4], we can also compute ρ(σ) for each unipotent character. For the choice ϵ = −1, the value of the unipotent character ρ = D_{4,1} on σ becomes larger than ρ(ι), which is not possible. So this finally determines ϵ, and all decompositions are known.

(2) In case of ^{2}E_{6} we again may compute the decomposition of sixteen R_{T,1} by Harish-Chandra induction. Next, the columns for |T^{σo}| = Φ_{8} and |T^{σo}| = Φ_{12} follow from orthogonality and restriction. The case of |T^{σo}| = Φ_{4}^{2} may be handled as in the untwisted group. Then the decomposition for |T^{σo}| = Φ_{2}^{2}Φ_{4} follows again. Also, the column for Φ_{3}^{2} is computed by using congruences mod Φ_{3}, using that a primitive prime divisor of Φ_{3} is at least seven. Now just four columns remain. One finds that orthogonality and restriction allow only three sets of solutions, and in particular, the column for Φ_{2}^{4} is uniquely determined. To find the right solution among the remaining three possibilities, we first observe that ϕ′′_{2,16} − ϕ′_{2,4} is a uniform function because the multiplicities of both characters in the last three R_{T,1} agree (this is a consequence of orthogonality to the first sixteen columns). Hence we may calculate

ϕ′′_{2,16}(σ) − ϕ′_{2,4}(σ) = qΦ_{4}Φ_{8}Φ_{12}(q^{12} − 1).

By elementary properties of character values (see Lemma 2.3 in [8]) this implies ϕ′′_{2,16}(σ) = q^{13}Φ_{4}Φ_{8}Φ_{12}. Only one of the three possibilities for the decomposition of the last R_{T,1} yields this value for ϕ′′_{2,16}(σ), and this completes the proof of the Theorem.

It should be noted that again, as for A_{l}(q): 2 and^{2}A_{l}(q): 2, not all extensions of unipotent characters occur in one of the families. Indeed, the subcuspidal characters induced from the Levi factor ^{2}A_{5} have scalar product zero with each almost character, hence are orthogonal to the space of uniform functions.

7 A conjecture

Let G = G^{o}.⟨σ⟩ be a reductive group, where the Dynkin diagram of the connected component G^{o} is connected and σ acts like a graph automorphism of this diagram. Let F be a split Frobenius map of G commuting with σ. In the previous sections we considered the decomposition of the generalized Deligne-Lusztig characters for the untwisted group G^{F} and the twisted group G^{σF}. Here we will point out some connection between the two cases, which can be observed in the above results, and formulate a general conjecture.

Denote by W the Weyl group of G. The principal series characters of G^{o} := (G^{o})^{F} are parametrized by the irreducible characters of W, as are the usual almost characters. The almost characters of the twisted group (G^{o})^{σF} are parametrized by σ-class functions of W, and the principal series characters by the irreducible characters of W^{σ}. The following result can be found in [7], 4.19.

Proposition 6 (Lusztig): There exists a bijection Ψ between the σ-stable unipotent characters of (G^{o})^{F} and the unipotent characters of (G^{o})^{σF}.

We now write R_{χ} (resp. R_{χ}^{σ}) for the almost characters on the coset (G^{o})^{F} .σ (resp. (G^{o})^{σF} .σ), with χ ∈ Irr(W^{σ}). The results of the previous sections now lead to:

Conjecture: (1) A set {R_{χ}} is a family of almost characters for (G^{o})^{F} .σ iff the set {R_{χ}^{σ}} is a family for (G^{o})^{σF} .σ.

(2) The set {R_{χ}^{σ}} is a family for (G^{o})^{σF} .σ iff the set of restrictions to (G^{o})^{σF} of the corresponding unipotent characters is a family for (G^{o})^{σF}.

(3) The bijection Ψ may be chosen such that in particular we have: If R_{χ} = ρ is an irreducible unipotent character (on (G^{o})^{F} .σ), then R_{χ}^{σ} = Ψ(ρ) is irreducible (on (G^{o})^{σF} .σ), and moreover Ψ(ρ) is an extension to (G^{o})^{σF} .σ of the principal series character of (G^{o})^{σF} indexed by χ.

Note that the conjecture involves three different parametrizations: the almost characters for the untwisted and the twisted disconnected group and the principal series characters for the connected twisted group. It was pointed out by Jean Michel that at least the first part of the conjecture has a convenient formulation in terms of Shintani descent from (G^{o}.⟨σF ⟩)^{F} to (G^{o}.⟨F ⟩)^{σF}. The conjecture does not make any statement about the behavior inside families. Indeed the example of D_{4} and ^{2}D_{4} already shows that the decomposition inside families is not always the same in the untwisted and the twisted groups. Finally the problem of non uniform extensions, as encountered for A_{l} and E_{6}, is still completely open.

This conjecture is true for the groups treated in the previous sections, so it remains open for the families (A_{l}, 2) and (D_{l}, 2), l ≥ 6. For l = 3, 4 in type D_{l}, the maps ψ and φ constructed in Propositions 3 and 4 are compatible with the conjecture in the following sense: The first part of (3) in the Conjecture holds with Ψ := ψ^{−1}, and φ gives precisely the parametrization of the principal series characters of (G^{o})^{σF} by irreducible characters of W^{σ}.

8 Green functions

In the next three sections we will study the Green functions on outer cosets of disconnected groups. The character formula in [4], 2.4, expresses the value of a generalized Deligne-Lusztig character R_{T,θ} on an outer element g with Jordan decomposition g = us as

R_{T,θ}(g) = \frac{1}{|T |} _{\{h∈G|s∈^{h}T \}}^{∑} \frac{1}{|C_{G}(s)^{o}|} _{v∈C_{hT} (s)_{u}}^{∑} Q_{(^{h}T)^{o}}^{C_{G}(s)^{o}}(u, v^{−1})θ(sv).

If g = u is unipotent, this simplifies to

R_{T,θ}(u) = \frac{1}{|T |} _{h∈G}^{∑} \frac{1}{|G^{o}|} _{v∈^{h}T_{u}}^{∑} Q_{(^{h}T)^{o}}^{G^{o}}(u, v^{−1})θ(v).

By [4], Q_{T^{o}}^{G^{o}}(u, v^{−1}) vanishes if uv^{−1} is not contained in G^{o}. So the last sum is over unipotent elements in^{h}T \^{h}T^{o}. By Schur-Zassenhaus every coset of^{h}T^{o} in^{h}T contains just one unipotent class, and by 1.35 in [4] all these elements are conjugate to σ. This gives

R_{T,θ}(u) = \frac{1}{|T^{σ}^{o}|}Q_{T^{o}}^{G^{o}}(u, σ^{−1})θ(σ).

If we only look at such extensions θ of characters of T^{o} which take value 1 on σ, then this expression does not depend on θ, and we may define the Green functions on unipotent elements

Q_{T} (u) := R_{T,1}(u) = \frac{1}{|T^{σ}^{o}|}Q_{T^{o}}^{G^{o}}(u, σ^{−1}), u ∈ G^{o}.σ unipotent,

to be the common value of the R_{T,θ}. This may also be interpreted as a collection of class functions Q(u) on the Weyl group W^{σ}, parametrized by the unipotent classes in G^{o}.σ, via

Q(u) : w ↦ Q_{T_{w}}(u).

As in the connected case the Green functions take values in the integers, being rational fractions of Lefschetz numbers and algebraic integers. One of the main problems in the representation theory of reductive groups is the determination of all Green functions. Even for connected groups, this is not completely solved at present.

In the connected case, the Green functions satisfy some nice orthogonality properties. Analogues of them hold for our new functions as well:

Proposition 7: Let G = G^{o}.⟨σ⟩ be a disconnected group where the quasi-central element σ induces a graph automorphism on the connected component G^{o}. Then for all σ-stable maximal tori T contained in a σ-stable Borel subgroup of G^{o} we have

∑_{u∈Go.σ_{unipotent}} Q_{T} (u) = \frac{|G^{o}|}{|T^{σ}^{o}|}.

Proof : It was kindly pointed out by the referee that this follows from the next proposition and Theorem 3(4). Namely the latter implies that 1/|W^{σ}| ∑_{w∈W^{σ}} Q_{T_{w}} = 1. Taking the appropriate sum in Proposition 8 then proves the result.

Proposition 8: Let G be as in Proposition 7. Then for all pairs T, T′ of σ-stable maximal tori of G contained in σ-stable Borel subgroups we have

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

Proof : The proof follows closely the one in the connected case, see [1], 7.6.2. To obtain a uniform expression for the scalar product, write N_{G}(T, T′) := {g ∈ G | ^{g}T = T′}, so the right hand side of our formula becomes |N_{G^{σ}}^{o}(T, T′)|/|T^{σ}^{o}|^{2}. Now by [4], 4.7, (see Theorem 3(3)) we have

⟨R_{T,1}, R_{T′,1}⟩_{σ} = |N_{G^{σ}}^{o}(T, T′)|/|T^{σ}^{o}|.

Writing out the scalar product and substituting the character formula of [4] given above for the value of R_{T,1} on g we obtain

|G^{o}| · |N_{G^{σ}}^{o}(T, T′)|/|T^{σ}^{o}| =

_{su∈G^{o}.σ}^{∑} \frac{1}{|T^{σ}||T^{′σ}||C_{G}(s)^{o}|^{2}} _{\{h∈G|s∈^{h}T \}}^{∑} _{\{h′∈G|s∈^{h′}T′\}}^{∑} Q_{(^{h}T)^{o}}^{C_{G}(s)^{o}}(u,^{h}σ^{−1})Q_{(^{h′}T′)^{o}}^{C_{G}(s)^{o}}(u,^{h′}σ^{−1}),

where we have used that the semisimple part of an outer element g in a disconnected group with G/G^{o} unipotent lies in G^{o}. From the first sum we split off the contribution by s ∈ Z(G), which is equal to

|Z(G)| _{u∈(G^{o}.σ)_{u}}^{∑} Q_{T} (u)Q_{T′}(u).

Note that it is irrelevant whether we sum over s ∈ Z(G) or s ∈ Z(G^{o}), because for s ∈ Z(G^{o})\Z(G), the sum over outer unipotent elements is empty. For the remaining elements the induction hypotheses for C_{G}(s)^{o}.⟨τ⟩ may be substituted, using that we have |T^{σ}^{o}||N_{G^{o}}(T, T′)| = |T^{o}||N_{G^{σ}}^{o}(T, T′)| by [4], 1.36, which yields

∑_{{s∈G{o \atop ss}\Z(G) \atop CG(s)∩Go.σ≠∅}} \frac{1}{|T^{σ}||T^{′σ}||C_{G}(s)^{o}|} _{\{h∈G|s∈^{h}T \}}^{∑} _{\{h′∈G|s∈^{h′}T′\}}^{∑} _{\{x∈C_{G}(s)^{o}|^{xh}T=^{h′}T′\}}^{∑} \frac{|T^{σ}^{o}|}{|T^{o}|}

= ∑_{{s∈G{o \atop ss}\Z(G) \atop CG(s)∩Go.σ≠∅}} \frac{1}{|T^{σ}||T^{′σ}||C_{G}(s)^{o}|} _{\{h∈G|s∈^{h}T \}}^{∑} _{x∈C_{G}(s)^{o}}^{∑} _{\{h′∈G|^{xh}T=^{h′}T′\}}^{∑} \frac{|T^{σ}^{o}|}{|T^{o}|}

= ∑_{{s∈G{o \atop ss}\Z(G) \atop CG(s)∩Go.σ≠∅}} \frac{1}{|T^{σ}|^{2}} _{\{h∈G|s∈^{h}T \}}^{∑} |N_{G^{σ}}(T, T′)|.

Adding and subtracting the corresponding contribution from s ∈ Z(G) yields

_{h∈G}^{∑} ∑_{{s∈hT o \atop CG(s)∩Go.σ≠∅}} \frac{1}{|T^{σ}|^{2}} |N_{G^{σ}}(T, T′)| − |Z(G)|\frac{|G|}{|T^{σ}|^{2}} |N_{G^{σ}}(T, T′)|.

Now the required identity follows when writing down the new expressions for the terms in the first equation.

Proposition 8 has also been obtained by Digne and Michel.

We now mention a criterion for a unipotent class to be uniform, i.e., for the characteristic function of that class to lie in the space spanned by the R_{T,θ}. This is again an analogue of a similar formula for connected groups, see [6], remark after 2.16.

Proposition 9: Let G be as above. The unipotent class [u] ⊂ G^{o}.σ is uniform if and only if

\frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑} |T_{w}^{σ}| Q_{T_{w}}(u)^{2} = \frac{|G|}{|[u]|}.

Proof : This follows easily from the formula 4.10 in [4] for the projection p(χ) of a σ-class function χ to the space of uniform functions, namely

p(χ) = \frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑} R_{T_{w}}^{G} ∘^{∗}R_{T_{w}}^{G}(χ)

(where ^{∗}R_{T}^{G} denotes the adjoint of R_{T}^{G}). With this the σ-norm of p(χ) becomes

⟨p(χ), p(χ)⟩_{σ} = \frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑}⟨p(χ), R_{T_{w}}^{G} ∘^{∗}R_{T_{w}}^{G}(χ)⟩_{σ} = \frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑}⟨χ, R_{T_{w}}^{G} ∘^{∗}R_{T_{w}}^{G}(χ)⟩_{σ},

the last equality because p(χ) is the projection of χ to the space of uniform functions. If χ is the characteristic function of the unipotent class [u], the character formula 2.4 in [4] for R_{T} and ^{∗}R_{T} gives

⟨p(χ), p(χ)⟩_{σ} = \frac{1}{|W^{σ}|} _{w∈W^{σ}}^{∑} \frac{|[u]|}{|G^{o}|} · \frac{|[u]|}{|G|}|T_{w}^{σ}|Q_{T} (u)^{2}.

The σ-class function χ is uniform iff ⟨p(χ), p(χ)⟩_{σ} = ⟨χ, χ⟩_{σ} = |[u]|/|G^{o}|. The Proposition follows.

9 Green polynomials for A_{l}(2^{n}): 2, l ≤ 5

We first consider the case of A_{l} with the graph automorphism of order two. If G is defined over a field of characteristic two, the outer coset contains unipotent elements, so new Green functions arise. Some examples are studied in this section. For this we first have to determine the unipotent classes in the coset G^{o}.σ, where as usually we write G := G^{F}. The unipotent classes of the algebraic groups G were described by Spaltenstein [9], I.2. Here we will also need the classes in the finite groups and the classes in various parabolic subgroups.

Let B^{o} be a Borel subgroup of G^{o}. We identify the simple roots α_{i} of A_{l} with their indices (where the Dynkin diagram is numbered in the obvious way), and denote a generator of the corresponding root subgroups in B^{o} by x_{i}(t), 1 ≤ i ≤ l. If α = α_{i_{1}} + … + α_{i_{r}} is a positive root, a generator of the root subgroup will be denoted by x_{i_{1}+...+i_{r}}(t). Finally choose a fixed element λ ∈ 𝔽_{2^{n}} such that x^{2} + x + λ has no root in 𝔽_{2^{n}}.

Proposition 10: The unipotent classes in the coset G^{o}.σ of G = A_{l}(q): 2, q = 2^{n}, 3 ≤ l ≤ 5, with representative uσ, and their centralizers are as given in Tables 4 and 5. For l = 2, there are three unipotent classes in G^{o}.σ, with representatives u_{1}σ = σ, u_{2}σ = x_{2}(1)σ and u_{3}σ = x_{2}(λ)σ, and centralizer orders qΦ_{1}Φ_{2}, 2q, 2q respectively.

Table 4: Outer unipotent classes for A_{3}(2^{n}): 2 and A_{4}(2^{n}): 2.
u  |C_{G^{o}}(u)|_{u_{1} = 1  q^{4}Φ_{1}^{2}Φ_{2}^{2}Φ_{4}}_{u_{2} = x_{1+2+3}(1)  q^{4}Φ_{1}Φ_{2}}_{u_{3} = x_{2+3}(1)  2q^{3}Φ_{1}}_{u_{4} = x_{2}(1)x_{2+3}(1)x_{1+2+3}(λ)  2q^{3}Φ_{2}}_{u_{5} = x_{2}(1)x_{3}(1)  q^{2}} u  |C_{G^{o}}(u)|
u_{1} = 1  q^{4}Φ_{1}^{2}Φ_{2}^{2}Φ_{4}
u_{2} = x_{2+3+4}(1)  q^{4}Φ_{1}Φ_{2}
u_{3} = x_{3+4}(1)  2q^{4}Φ_{1}Φ_{2}
u_{4} = x_{3+4}(1)x_{1+2+3+4}(λ)  2q^{4}Φ_{1}Φ_{2}
u_{5} = x_{3+4}(1)x_{2+3+4}(1)  q^{4}
u_{6} = x_{3}(1)x_{4}(1)  2q^{2}
u_{7} = x_{3}(1)x_{4}(1)x_{2+3}(λ)  2q^{2}

Table 5: Outer unipotent classes for A_{5}(2^{n}): 2.

u  |C_{G^{o}}(u)|
u_{1} = 1  q^{9}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}
u_{2} = x_{1+2+3+4+5}(1)  q^{9}Φ_{1}^{2}Φ_{2}^{2}Φ_{4}
u_{3} = x_{2+3+4+5}(1)  2q^{8}Φ_{1}^{2}Φ_{2}
u_{4} = x_{2+3+4}(1)x_{2+3+4+5}(1)x_{1+2+3+4+5}(λ)  2q^{8}Φ_{1}Φ_{2}^{2}
u_{5} = x_{2+3+4}(1)x_{3+4+5}(1)  q^{7}Φ_{1}Φ_{2}
u_{6} = x_{3+4}(1)x_{4+5}(1)  q^{5}Φ_{1}Φ_{2}
u_{7} = x_{5}(1)x_{2+3+4}(1)  q^{5}Φ_{1}Φ_{2}
u_{8} = x_{5}(1)x_{3+4}(1)x_{2+3+4}(1)  2q^{5}
u_{9} = x_{3}(1)x_{3+4}(1)x_{4+5}(1)x_{2+3+4}(λ)  2q^{5}
u_{10} = x_{3}(1)x_{4}(1)x_{5}(1)  2q^{3}
u_{11} = x_{3}(1)x_{4}(1)x_{5}(1)x_{3+4+5}(λ)  2q^{3}

Proof : As a first step, one determines the unipotent classes in the coset B^{o}.σ of the Borel subgroup of G. The commutator relations for B^{o} are well known, as is the action of σ on G^{o}. It is then straightforward, though not very pleasant, to calculate representatives for the unipotent classes in the coset of the Borel subgroup. Computations of that type were done by various authors, for example for all connected exceptional groups, so we omit the details in our case. We just state that the number of unipotent classes in the coset B^{o}.σ is equal to 3, 8, 13, q + 41 for l = 2, 3, 4, 5 respectively. From the first step, the conjugacy classes in G^{o}.σ follow easily using the Steinberg relations. This yields the results in the tables.

It is possible to calculate the Green functions for all the groups in Proposition 10. Instead of writing down their values, we will give the values of the almost characters with respect to W^{σ} on the outer unipotent classes. This is equivalent to giving the Green functions, but shows more clearly the decomposition of the Q(u) into class functions on W^{σ}. Also, the values of the almost characters tend to vanish rather frequently, while all values of Green functions are nonzero. In the case of A_{l}, l ≤ 5, the almost characters moreover coincide with extensions of unipotent characters by Theorem 4.

Theorem 9: For G = A_{l}(2^{n}): 2, 3 ≤ l ≤ 5, the values of the almost characters R_{χ}, χ ∈ Irr(W^{σ}), on the unipotent classes in G^{o}.σ are given by Tables 6 and 7. For l = 2 the decomposition of the Green functions into characters of W^{σ} = Z_{2} agrees with that in A_{1}(q).

Table 6: Almost characters for A_{3}(2^{n}): 2 and A_{4}(2^{n}): 2.

\begin{aligned}& u_{1} & u_{2} & u_{3} & u_{4} & u_{5} \\ ϕ_{2,−} & 1 & 1 & 1 & 1 & 1 \\ ϕ_{−,2} & q^{2} & . & q & −q & . \\ ϕ_{1,1} & qΦ_{4} & q & q & q & . \\ ϕ_{11,−} & q^{2} & q^{2} & . & . & . \\ ϕ_{−,11} & q^{4} & . & . & . & . \end{aligned}  \begin{aligned}& u_{1} & u_{2} & u_{3}, u_{4} & u_{5} & u_{6}, u_{7} \\ ϕ_{2,−} & 1 & 1 & 1 & 1 & 1 \\ ϕ_{11,−} & q^{2} & . & q^{2} & . & . \\ ϕ_{1,1} & qΦ_{4} & q & q & q & . \\ ϕ_{−,2} & q^{2} & q^{2} & . & . & . \\ ϕ_{−,11} & q^{4} & . & . & . & . \end{aligned}

Table 7: Almost characters for A_{5}(2^{n}): 2.

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

Proof : The almost characters are obtained from the Green functions by Definition 2. Hence it suffices to calculate the Green functions. According to Theorem 3(2), Q_{T} (u) may be obtained by usual Harish-Chandra induction if T is contained in a proper σ-stable parabolic subgroup. The unipotent classes in the Borel subgroup, and hence in the parabolic subgroups, are known from Proposition 10, and the Green functions for the corresponding Levi subgroups of type A_{m}, m < l, may inductively also be assumed to be known. So the Harish-Chandra induction for these tori can be calculated explicitly.

We show how to find the remaining functions for A_{5}(2^{n}): 2, the other cases being similar and easier. Firstly, the values on the class of u_{1}σ = σ are known by 4.12 of [4]. They agree with the values of the Green functions on the identity for the group (G^{σ})^{o}. This leaves ten unknown values for each of the three T_{w} which can not be obtained by Harish-Chandra induction. We first consider the case with |T_{w}^{σo}| = Φ_{2}Φ_{6}. The condition in Proposition 8 on the norm of the corresponding Green function shows that R_{T} (u_{10})^{2} + R_{T} (u_{11})^{2} ≤ 12. Moreover, both these values have to be odd, as the values of the (inner) Green functions on the squares of u_{10} and u_{11} are odd. The orthogonality of this Green function to the previous seven yields a system of linear equations, in addition to the norm equation. Of the twelve possible choices for the pair (R_{T} (u_{10}), R_{T} (u_{11})), only (1, 1) gives real solutions for R_{T} (u_{9}). Then q+1 is the only integer solution for R_{T} (u_{9}). From this, the remaining entries of R_{T} are uniquely determined by the system of linear equations. For the other two missing functions, we now have one more condition (orthogonality to the R_{T} determined above), and the same arguments may be applied. This completes the computation of the Green functions.

10 Green polynomials for D_{4}(3^{n}): 3 and D_{4}(2^{n}): 2

We first consider the case of D_{4} with a graph automorphism of order three. For the outer cosets to contain unipotent elements, G has to be defined over a field of characteristic three. The unipotent classes in the coset G^{o}.σ may then be determined as in the last section. For the algebraic group G^{o}.σ they were described by Spaltenstein [9], I.3. (There are, however, several misprints in his definitions of representative elements; compare with the following Proposition.) We will also need the unipotent classes in the finite groups G and in the Borel subgroup. For the definition of the classes we fix a nonsquare μ ∈ 𝔽_{3^{n}}.

Table 8: Outer unipotent classes for D_{4}(3^{n}): 3.

u  |C_{G^{o}}(u)|
u_{1} = 1  q^{6}Φ_{1}^{2}Φ_{2}^{2}Φ_{3}Φ_{6}
u_{2} = x_{α+β+γ+2δ}(1)  q^{6}Φ_{1}Φ_{2}
u_{3} = x_{β+γ+δ}(1)  q^{4}Φ_{1}Φ_{2}
u_{4} = x_{γ+δ}(1)x_{α+β+γ+δ}(1)  6q^{4}
u_{5} = x_{γ+δ}(1)x_{α+β+γ+δ}(μ)  2q^{4}
u_{6} = x_{δ}(1)x_{β+γ+δ}(1)x_{α+β+γ+δ}(1)  3q^{4}
u_{7} = x_{γ}(1)x_{δ}(1)  q^{2}

Proposition 11: There are seven unipotent classes in the coset G^{o}.σ of G = D_{4}(3^{n}): 3, with representatives uσ and centralizers as given in Table 8. The notation for the root subgroups is taken from [9].

Proof : Generators and relations for the Borel subgroup B = B^{o}.⟨σ⟩ of G may be deduced from the Steinberg relations for B^{o} and the known action of the automorphism σ on G^{o}. It is then a standard procedure to enumerate all unipotent classes of the Borel subgroup and determine their centralizers, see the remarks in the proof of Proposition 10. One finds that B^{o}.σ contains q + 15 unipotent classes. With that information one easily computes the unipotent classes of G^{o}.σ, and this gives the result in Table 8.

Again we give the values of the almost characters with respect to W^{σ} = W (G_{2}) rather than the Green functions.

Theorem 10: For G = D_{4}(3^{n}): 3 the values of the almost characters R_{χ}, χ ∈ Irr(W^{σ}), on the unipotent classes in G^{o}.σ are given by Table 9.

Table 9: Almost characters for D_{4}(3^{n}): 3.

\begin{matrix}χ & u_{1} & u_{2} & u_{3} & u_{4} & u_{5} & u_{6} & u_{7} \\ ϕ_{1,0} & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ ϕ′_{1,3} & q^{3} & . & q & 2q & . & −q & . \\ ϕ_{2,1} & qΦ_{8} & q & q & q & q & q & . \\ ϕ_{2,2} & q^{2}Φ_{4} & q^{2} & q^{2} & . & . & . & . \\ ϕ′′_{1,3} & q^{3} & q^{3} & . & . & . & . & . \\ ϕ_{1,6} & q^{6} & . & . & . & . & . & . \end{matrix}

Proof : Three of the six Green functions may be calculated by usual Harish-Chandra induction from proper σ-stable parabolic subgroups, the unipotent classes in the Borel subgroup being known from Proposition 11. The values on the class of u_{1}σ = σ are again known by 4.12 of [4] to agree with the values of the Green functions on the identity for the group (G^{σ})^{o} = G_{2}(3^{n}). From Proposition 8 and the centralizer order for u_{7}σ in Proposition 11 it follows that |Q_{T} (u_{7}σ)|^{2} ≤ 8 for the tori T with |T^{σ}^{o}| ∈ {Φ_{3}, Φ_{6}}. Moreover the values of the Q_{T} lie in ℤ, and by properties of character values we know Q_{T} (u_{7}σ) ≡ Q_{T} ((u_{7}σ)^{3}) ≡ 1 (mod 3), using the fact that the Green functions for the connected group are known to be congruent 1 modulo q. The possibility Q_{T} (u_{7}σ) = −2 does not lead to an integer solution for the system of linear equations derived from the scalar products with the first three Green functions. So it follows that Q_{T} (u_{7}σ) = 1 for these two tori. The remaining entries in these rows are then uniquely determined by the orthogonality relations of Proposition 8 and the fact that the Q_{T} (u) lie in ℤ. The last Green function may finally be calculated via orthogonality to the previous rows.

We now turn to groups of type D_{4} extended by the graph automorphism of order two.

We assume that G is defined over a field of characteristic two. The following result is proved in the same way as the previous two propositions (λ ∈ 𝔽_{2^{n}} was defined in Section 9).

Proposition 12: There are ten unipotent classes in the coset G^{o}.σ of G = D_{4}(2^{n}): 2, with representatives uσ and centralizers as given in Table 10. The notation for the root subgroups is taken from [9].

Table 10: Outer unipotent classes for D_{4}(2^{n}): 2.

u  |C_{G^{o}}(u)|
u_{1} = 1  q^{9}Φ_{1}^{3}Φ_{2}^{3}Φ_{3}Φ_{4}Φ_{6}
u_{2} = x_{α+β+γ+2δ}(1)  q^{9}Φ_{1}^{2}Φ_{2}^{2}
u_{3} = x_{α+γ+δ}(1)  2q^{7}Φ_{1}^{2}Φ_{2}^{2}
u_{4} = x_{α+δ}(1)x_{α+γ+δ}(1)x_{α+β+γ+δ}(λ)  2q^{7}Φ_{1}Φ_{2}Φ_{4}
u_{5} = x_{α+γ+δ}(1)x_{β+γ+δ}(1)  q^{7}Φ_{1}Φ_{2}
u_{6} = x_{α+δ}(1)x_{β+γ+δ}(1)  q^{7}Φ_{1}Φ_{2}
u_{7} = x_{α+δ}(1)x_{α+γ+δ}(1)x_{β+γ+δ}(1)  q^{7}
u_{8} = x_{γ}(1)x_{α+δ}(1)  2q^{4}Φ_{1}
u_{9} = x_{α}(1)x_{δ}(1)x_{γ+δ}(1)x_{β+γ+δ}(λ)  2q^{4}Φ_{2}
u_{10} = x_{α}(1)x_{γ}(1)x_{δ}(1)  q^{3}

Here the coset B^{o}.σ of the Borel subgroup contains 2q + 41 unipotent classes.

Theorem 11: For G = D_{4}(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.

Table 11: Almost characters for D_{4}(2^{n}): 2.

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

Proof : Seven of the ten Green functions are obtained by Harish-Chandra induction from σstable parabolic subgroups, using the calculations for Proposition 12. The values on the class of u_{1}σ are the same as in B_{3}(q), hence known. This leaves nine columns to be determined. We start by determining the Green functions R_{T} for |T^{σo}| = Φ_{2}Φ_{6}. The condition of Proposition 8 on the norm of this Green function shows immediately that |R_{T} (u_{10})| ≤ 2. Moreover this value has to be congruent to 1 modulo 2, by the knowledge of the Green functions on the square of u_{10}, which is an inner element. So two possibilities arise for R_{T} (u_{10}). If R_{T} (u_{10}) = −1, then the system of linear equations for the remaining values obtained from orthogonality to the first seven rows together with the norm condition yields only nonreal solutions for R_{T} (u_{9}). Hence R_{T} (u_{10}) = 1, and of the two solutions for R_{T} (u_{9}) which now arise, only one is an integer. The remaining entries for this Green function follow by orthogonality. The other two missing rows are then again determined by the orthogonality relations in Proposition 8.

From Theorems 7 and 11 it is possible in principle to write down all character values of the disconnected groups GO_{8}^{+}(2^{n}).

The Green functions calculated in the previous two sections enjoy properties similar to those in the connected case. Namely the following hold: Remark: (1) The Green functions are polynomials in q.

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

(3) The Green polynomials take value 1 on the regular unipotent elements (in the sense of [9], p.41).

(4) 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.

All of this is true for connected groups as well, at least in those cases where the Green functions are explicitly known. Only property (3) has an a priori proof in the connected case. It has recently been generalized to disconnected groups by Digne and Michel. Clearly the a priori knowledge of, say, properties (1) and (4) would simplify the determination of Green functions considerably.

Our Green functions also satisfy a conjecture originally made by Lusztig for connected groups [6], 2.16. For this let C be a unipotent conjugacy class in G^{o}.σ, i.e., in the algebraic group, let C := C^{F}, and denote by

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

the average value of Q_{T} on C.

Proposition 13: For disconnected groups of types (A_{l}, 2), l ≤ 5, (D_{4}, 2) and (D_{4}, 3) the Green functions on the outer cosets satisfy the second orthogonality relations

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

In particular, the characteristic function of each unipotent class C in G.σ is uniform.

Proof : This follows immediately from Tables 6, 7, 9 and 11.

For the moment we have to leave open the case of E_{6} in characteristic 2 with the graph automorphism of order two. The determination of the unipotent classes in the Borel subgroup seems to be quite a hard computational problem. Moreover nine of the Green functions can not be obtained by Harish-Chandra induction, so their determination will probably require more than the orthogonality relations. We just remark that even for the connected groups of type E_{6} in characteristic 2 the Green functions are not yet known.

References

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

[2] P. Deligne, G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), p.103–161.

[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] R. B. Howlett, G. I. Lehrer, Representations of generic algebras and finite groups of Lie type, Trans. Am. Math. Soc. 280 (1983), p.753–779.

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

[7] G. Lusztig, Characters of reductive groups over a finite field, Annals of Math. Studies 107 (1984), Princeton University Press.

[8] G. Malle, Darstellungstheoretische Methoden bei der Realisierung einfacher Gruppen vom Lie Typ als Galoisgruppen, in: G. Michler, C. Ringel, eds., Representation theory of finite groups and finite dimensional algebras, Progress in Mathematics 95, Birkhäuser Verlag 1991.

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

Heidelberg June 1991