SHINTANI DESCENT AND PERFECT ISOMETRIES FOR BLOCKS OF FINITE GENERAL LINEAR GROUPS

RADHA KESSAR

1. Introduction

1.1. Let d be a positive integer, q a prime power and l a prime not dividing q. In [12], A. Watanabe showed that Shintani descent provides a perfect isometry between the principal l-blocks of the finite general linear groups GL_{n}(q) and GL_{n}(q^{d}), provided that d is relatively prime to l and to the order of q modulo l. In this note, we show how Watanabe’s methods can be adapted to prove that under the same hypotheses, for any unipotent l-block of GL_{n}(q), there is a corresponding unipotent l block of GL_{n}(q^{d}), which is perfectly isometric to the first through Shintani descent, and vice versa. Further, we show that as a consequence of C. Bonnafe’s results in [1], one may choose a perfect isometry between the blocks which preserves signs, as long as d is odd. Our result can be combined with the perfect isometries provided by Jordan decomposition of blocks [4] to yield perfect isometries between pairs of non-unipotent blocks of GL_{n}(q) and GL_{n}(q^{d}) as well. Before stating the results we set up notation and briefly recall the relevant background material.

1.2. Blocks and perfect isometries. Let (K, O, k) be an l-modular system which we assume is a splitting l-modular system for all finite groups that we consider in this paper. By a block of a finite group G we will mean a primitive central idempotent of the centre of the group algebra OG. If b is central idempotent of OG, Irr(G, b) will denote the set of ordinary irreducible characters χ of G such that χ(1) = χ(b). If b and c are central idempotents of two finite group algebras OG and OG′ respectively, recall (see [4]) that a perfect isometry between b and c is an isometry (with respect to the usual inner product on the space of K-valued class functions of a finite group) R : ℤ(Irr(G, b)) → ℤ(Irr(G′, c)) , satisfying the following: If μ is the class function on the group G × G′ given by

μ := ∑_{χ∈Irr(G,b)} χ × R(χ),

then for every x ∈ G, y ∈ G′, μ(x, y) is divisible in O by C_{G}(x) and by C_{G′}(y). Further, if exactly one of x and y is l-singular, then μ(x, y) = 0. Note that a perfect isometry is completely determined by the images of the irreducible characters.

1.3. Shintani descent. Let 𝔽 denote the algebraic closure of the field of q elements and G = GL_{n}(𝔽) = GL(𝕍), the finite general linear group on a F-vector space 𝕍 of dimension n. Let F : G → G denote the standard Frobenius morphism on G corresponding to an 𝔽_{q} rational structure for G such that that G^{F} ≅ GL_{n}(q); clearly, G^{F^{d}} ≅ GL_{n}(q^{d}). Denote by σ : G^{F^{d}} → G^{F^{d}} the automorphism of G^{F^{d}} obtained through the restriction of F. Two elements g and h of G^{F^{d}} are called σ-conjugate if h = xg ^{σ}x^{−1} for some x ∈ G^{F^{d}}. If x ∈ G, then x^{−1} ^{F} x ∈ G^{F^{d}} if and only if x ^{F^{d}}x^{−1} ∈ G^{F}. Further, by Lang’s theorem every element y ∈ G can be written as x^{−1} ^{F} x and as z^{−1} ^{F^{d}}z for some y and z in G. The map which sends the σ-conjugacy class of an element g = x^{−1} ^{F} x of G^{F^{d}} to the conjugacy class of x ^{F^{d}}x^{−1} ∈ G^{F} is well defined and bijective. This map is denoted by N_{F^{d}/F}. Let Γ := G^{F^{d}}⋊ < σ >. The G^{F^{d}}-coset, G^{F^{d}}σ, of σ in Γ is a union of conjugacy classes of Γ and for g, h ∈ G^{F^{d}}, gσ and hσ are conjugate in Γ if and only if g and h are σ-conjugate in G^{F^{d}}. The map N_{F^{d}/F} induces a bijection

Sh_{F^{d}/F} : C(G^{F^{d}}σ) → C(G^{F}),

where C(G^{F}) denotes the set of K-valued class functions on G^{F} and C(G^{F^{d}}σ) denotes the set of functions on G^{F^{d}}σ obtained by restriction of K-valued class functions on Γ. If γ is a class function on Γ, by Sh_{F^{d}/F} (γ) we will always mean Sh_{F^{d}/F} (Res_{G^{F^{d}} σ}^{Γ}(γ)). By results of T. Shintani and C. Bonnafe the map Sh_{F^{d}/F} satisfies the following:

Theorem 1.1. [10, Theorem 1 and Lemma 2-11],[1, Theorem 4.3.1] Suppose that χ is an irreducible character of G^{F^{d}}, stable under σ. Then there exists a unique extension χ̃ of χ to Γ such that Sh_{F^{d}/F} (χ̃) is an irreducible character of G^{F}. Furthermore, the map χ → Sh_{F^{d}/F} (χ̃) is a bijection between the set of σ-stable characters of G^{F^{d}} and the set of irreducible characters of G^{F}.

1.4. Unipotent characters and blocks. For each partition λ of n, let L_{λ} be a “block diagonal “ Levi subgroup of G, which corresponds to λ in the usual way, and let P_{λ} be a standard parabolic subgroup of G containing L_{λ} as Levi complement. Note that L_{λ} and P_{λ} are F -stable. Recall that an irreducible character χ of G^{F^{d}} is unipotent if χ is a constituent of the character afforded by the G^{F^{d}}-permutation module KG^{F^{d}}/B^{F^{d}} where B is the subgroup of G consisting of upper triangular matrices. The irreducible unipotent characters of G^{F^{d}} are in bijection with the partitions of n. If λ is a partition of n and ϕ_{λ}^{F^{d}} is the corresponding unipotent irreducible character of G^{F^{d}} then ϕ_{λ}^{F^{d}} = ∑_{α} c_{λ,α}χ(KG^{F^{d}}/P_{α}^{F^{d}}), where α runs over the partitions of n, χ(KG^{F^{d}}/P_{α}^{F^{d}}) denotes the character of the permutation module KG^{F^{d}}/P_{α}^{F^{d}}, and where the c_{λ,α} are integers which depend only on λ and α and which are independent of d (see [7], remarks following Theorem 15.8).

The blocks of finite general linear groups were first described by Fong and Srinivasan in [8] ( see also [3]): To each G^{F^{d}} conjugacy class S of semi-simple l′ elements of G^{F^{d}} is canonically associated a sum of blocks b_{S} of G^{F^{d}} and for each block b of G^{F^{d}}, there exists a unique class S such that bb_{S} = b. A block b of G^{F^{d}} is called unipotent if bb_{(1^{G^{F^{d}}})} = b where (1^{G^{F^{d}}}) denotes the class of the identity element; unipotent blocks are also characterised by the property of containing a unipotent irreducible character of G^{F^{d}}. Two unipotent characters ϕ_{λ}^{F^{d}} and ϕ_{τ}^{F^{d}} belong to the same block if and only if λ and τ have the same e-core where e is the order of q^{d} modulo l. Thus the unipotent blocks of G^{F^{d}} are in one to one correspondence with the set of partitions μ which are their own e-cores and for which n − |μ| is divisible by e, (here |μ| stands for the size of the partition μ). If the block b corresponds to the partition μ, then μ is called the core of b.

Note that if d is relatively prime to e, then e is also the order q modulo l; in particular, the unipotent l-blocks of G^{F^{d}} are in natural bijection with those of G^{F}.

The following is the main result of this paper.

Theorem 1.2. Suppose that d is relatively prime to l and to the order of q modulo l. Let c be a unipotent block of G^{F} and b a unipotent block of G^{F^{d}} such that b and c have the same core. Then all elements of Irr(G^{F^{d}}), b) are σ-stable and for each χ ∈ Irr(G^{F^{d}}, b), there exists a δ_{χ} ∈ {1, −1} such that the map χ → δ_{χ}Sh(χ̃) is a perfect isometry between the blocks b and c. Further, if d is odd, then δ can be chosen so that δ_{χ} = 1 for all χ in Irr(G^{F^{d}}, b).

Note that Theorem 1.2 implies that there is a perfect isometry between b_{(1^{G^{F^{d}}})} and b_{(1^{G^{F}} )}. As a consequence of a result of M. Broue [4, Théorèm 2.3], Theorem 1.2 has the following corollary.

Corollary 1.3. Suppose that d is relatively prime to l and to the order of q modulo l. Let s be a semi-simple l′-element of G^{F}. If all eigen values of s, when s is considered as an element of G = GL(𝕍), lie in ∪_{t∈ℕ,(t,d)=1}𝔽_{q^{t}}, then there is a perfect isometry between b_{(s^{G^{F^{d}}})} and b_{(s^{G^{F}} )}.

2. Proof of 1.2 and 1.3

2.1. Our proof of Theorem 1.3 will follow that of Watanabe for principal blocksthe extra subtlety in our case lies in showing that the block b in the statement of the theorem and all the characters in it are σ-stable and in detecting the images of unipotent characters under Shintani descent. For this, recall that for a l-subgroup P of a finite group H, the Brauer homomorphism Br_{P}^{OH} is the algebra homomorphism from the subalgebra (OH)^{P} of OH consisting of elements which are fixed by the conjugation action of P on OH to the group algebra kC_{H}(P) which sends an element ∑_{g∈H} α_{g}g of (OH)^{P} to ∑_{g∈C_{H}(P)} ᾱ_{g}g (here ᾱ_{g} is the image of α_{g} under the canonical surjection of O onto k). If b is a block of H and P is a defect group of b, then Br_{P}^{OH}(b) is a sum of blocks of kC_{H}(P), which are transitively permuted by N_{G}(P). Further, if f is a block of kC_{H}(P) such that Br_{P}^{OH}(b)f = f, then Z(P) is a defect group of f, and in particular kC_{H}(P)f has a unique simple module. For details, we refer the reader to [11]. We will need the following proposition.

Proposition 2.1. Let Γ = H ⋊ S be a finite group and b be a block of H. Suppose that S is cyclic of order prime to l, S stabilises b and centralises a defect group P of b. Suppose further that Br_{P}^{OH}(b) is a single block of C_{G}(P) and let M denote the unique (up to isomorphism) simple module of kC_{H}(P)Br_{P}^{OH}(b). Finally, suppose that there is a direct product decomposition N_{H}(P) = A × B such that A ⊂ C_{H}(P), C_{B}(D) acts as the identity on M and σ normalises both A and B. Then every irreducible character of H in b is Γ-stable. Furthermore, there are |S| blocks of Γ covering b and if b̃ is a block of Γ covering b, the block algebras OHb and OΓb̃ are isomorphic.

Proof. Let d := |S|. Since M is the unique (up to isomorphism ) simple module of kC_{H}(P)Br_{P}^{OH}(b), the hypothesis implies that M is S-stable, and since S is cyclic, there are d extensions of M to C_{Γ}(P) = C_{H}(P)S. Since C_{G}(P) has l′ index in C_{Γ}(P), it follows that Br_{P}^{OΓ}(b) = Br_{P}^{OH}(b) is a sum of d blocks of C_{Γ}(P) each with defect group Z(P), and each with an extension of M as simple module. Let M̃ be an extension of M to C_{Γ}(P) and let g ∈ B, x ∈ C_{H}(P), y ∈ S. Since x and y centralise P and g normalises P, [g, x] ∈ C_{H}(P) ∩ B and [g, y] ∈ B ∩ C_{Γ}(P). Thus by hypothesis for m ∈ M̃, ^{g}(xy)m = [g, x]x[g, y]ym = xym. Hence, M̃ is N_{Γ}(P) = N_{H}(P)S-stable and it follows that each of the d-blocks of C_{Γ}(P) in Br_{P}^{OΓ}(b) is N_{Γ}(P)-stable. On the other hand, since each block of Γ covering b has P as a defect group, by Brauer’s first main theorem the number of blocks of Γ covering b is the number of N_{Γ}(P) classes of blocks f of C_{Γ}(P) with defect group Z(P) such that Br_{P}^{OΓ}(b)f = f and such that P C_{Γ}(P) is of l′ index in N_{Γ}(P, f). Thus there are d blocks of Γ covering b. In particular, if χ is a character of G in b, there are d characters of Γ covering χ and this is only possible if Γ stabilises χ and Ind_{H}^{Γ}(χ) consists of d characters , each lying in a different covering block of b. Finally, let b̃ be a block of Γ covering b. The above analysis shows that restriction provides a bijection between Irr(Γ, b̃) and Irr(H, b). Hence OHb and OΓb̃ are isomorphic by a result of Broué [5, Théorèm 0.1]. □

2.2. Proof of Theorem 1.2. Let us keep all the notation introduced for the statement of the theorem. In addition, by abuse of notation, from now on, if χ is a σ-stable irreducible character of G^{F^{d}}, by Sh(χ) we will mean the irreducible character of G^{F} corresponding to χ under bijection given in 1.1.

let L be a ”block diagonal” Levi subgroup of G and let P be a standard parabolic containing L as a Levi complement. Let τ be the character of the G^{F^{d}}- module K(G^{F^{d}}/P^{F^{d}}) and τ̄ the character of the G^{F} -module K(G^{F} /P^{F}). Since K(G^{F^{d}}/P^{F^{d}}) has a basis permuted by the elements of G^{F^{d}}, and since P^{F^{d}} is σstable, K(G^{F^{d}}/P^{F^{d}}) can naturally be considered as a module for G^{F^{d}}⋊ < σ > . Denote by τ̃ the character of K(G^{F^{d}}/P^{F^{d}}) considered as G^{F^{d}}⋊ < σ >-module. It follows by the proof of Theorem 1 of [10] that τ̃(gσ) = τ̄(i ∘ N_{F^{d}/F} (g)) for every g ∈ G^{F^{d}}. But since K(G^{F} /P^{F}) is a permutation module for G^{F}, this means that τ̃(gσ) = τ̄(N_{F^{d}/F} (g)) for every g ∈ G^{F^{d}}.

Let ϕ_{λ}^{F^{d}} = ∑_{α} c_{λ,α}χ(KG^{F^{d}}/P_{α}^{F^{d}}), be the unipotent irreducible character of G^{F^{d}} corresponding to the partition λ. By the above remarks we have that ϕ_{λ}^{F^{d}} is σstable. Also, denoting by ϕ̃_{λ}^{F^{d}} the natural extension of ϕ_{λ}^{F^{d}} to G^{F^{d}}⋊ < σ > (by extending the action of G^{F^{d}} on each KG^{F^{d}}/P_{α}^{F^{d}} to G^{F^{d}}⋊ < σ > ) we have that

ϕ̃_{λ}^{F^{d}}(gσ) = ϕ_{λ}^{F} (N_{F^{d}/F} (g)) ∀g ∈ G^{F^{d}}.  (1)

In other words, Sh(ϕ_{λ}^{F^{d}}) = ϕ_{λ}^{F}. In particular, b is a σ-stable block of G^{F^{d}} and for any unipotent character ϕ of G^{F^{d}} in b, Sh(ϕ) lies in the block c of G^{F}.

Note that the conditions on d imply that for any natural number t, the multiplicity of l as a divisor of q^{l} − 1 is the same as the multiplicity of l as a divisor of q^{d}^{t} − 1 and consequently the index of G^{F} in G^{F^{d}} is relatively prime to l. Thus, we may choose a defect group P of the block b of G^{F^{d}} which lies in G^{F}. By the description of blocks of finite general linear groups and their defect groups as given in [8] (see also [4]), P may be described as follows. Let V be the 𝔽_{q} form for 𝕍 so that G^{F} = GL(V). Let V^{P} be the subspace of V on which P acts as the identity, and let [V, V^{P}] denote the P -stable complement of V^{P} in V. Denote by V̂, V̂^{P} and { \atop [V, V^{P}]} the extension of V, V^{P} and [V, V^{P}] respectively to 𝔽_{q^{d}}. Then

C_{G^{F^{d}}} (P) = GL(V̂^{P}) × C_{GL({ \atop [V,V^{P}]})}(P) (here GL(V̂^{P}) × GL({ \atop [V, V^{P}]}) is considered

a subgroup of G^{F^{d}} in the natural way). P is a Sylow l-subgroup of GL({ \atop [V, V^{P}]}) and if μ is the core of b, then V̂^{P} has 𝔽_{q^{d}} dimension |μ|. The unipotent character

of GL(V̂^{P}) labelled by μ thus lies in a defect 0 block of GL({ \atop [V, V^{P}]}). Let M be the tensor product of the unique simple module of kGL(V̂^{P}) lying in this defect

0 block with the trivial module of kC_{GL({ \atop [V,V^{P}]})}(P). Br_{P}^{OG^{F^{d}}} (b) is a single block

of C_{G^{F^{d}}} (P) and M is the unique simple module of kC_{G^{F^{d}}} (P)Br_{P}^{OG^{F^{d}}} (b). Also, N_{G^{F^{d}}} (P) = GL(V̂^{P})×N_{GL({ \atop [V,V^{P}]})}(P). By applying the previous proposition, with

H replaced by G^{F^{d}}, S by < σ >, A by GL(V̂^{P}) and B by N_{GL({ \atop [V,V^{P}]})}(P) it follows

all characters of G^{F^{d}} in b are σ-stable.

Now let χ ∈ Irr(G^{F^{d}}, b) and let χ̃ denote the canonical extension of χ to Γ as defined in 1.1. Let wχ̃ denote the central character of G^{F^{d}}⋊ < σ > corresponding to χ̃ and denote by (σ) the sum of the Γ-conjugates of σ. Then, by the description of Sh(χ), we get

wχ̃((σ)) = \frac{|G^{F^{d}}|}{|G^{F} |} \frac{Sh(χ)(1)}{χ(1)}.  (2)

In particular, wχ̃((σ)) is a rational integer, and wχ̃((σ)) = wχ̃((σ^{−1})).

Pick a unipotent character ϕ := ϕ_{λ}^{G^{F^{d}}} in b, and let b̃ be the block of G^{F^{d}}⋊ < σ > containing the canonical extension ϕ̃ of ϕ satisfying 1.1. Since b̃ is a covering block of b, by the proof of the above proposition, for each χ ∈ Irr(G^{F^{d}}, b) there exists an irreducible character τ_{χ} of < σ > such that regarding τ_{χ} as a character of Γ via inflation, τ_{χ}χ̃ is in the block b̃. Then τ_{χ}(σ)wχ̃((σ)) is congruent to w_{ϕ}((σ)) modulo the valuation ideal J(O) of O. On the other hand since ϕ and Sh(ϕ) have the same unipotent label, it follows from the description of the degrees of the unipotent characters of finite general linear groups (see for example equation 1.15 of [8]) and the nature of d that ϕ and Sh(ϕ) have the same height. Equation (2) thus yields that wϕ̃((σ)) = wϕ̃((σ^{−1})) is a rational integer not congruent to zero modulo l. Hence, τ_{χ}(σ)wχ̃((σ)) is congruent to τ_{χ}(σ^{−1})wχ̃((σ^{−1})) modulo J(O) and each of these is non-zero modulo J(O). Now wχ̃((σ)) = wχ̃((σ^{−1})) forces τ_{χ}(σ) = ±1.

For χ ∈ Irr(G^{F^{d}}, b), put δ_{χ} = τ_{χ}(σ). We claim that the map χ → δ_{χ}Sh(χ) is a perfect isometry from b to c. The proof of this is taken almost verbatim from Watanabe’s article-we reproduce the argument for the convenience of the reader. Then for any g, h ∈ G^{F^{d}}, we have

∑_{χ∈Irr(G^{F^{d}} ,b)} τ_{χ}χ(gσ)τ_{χ}χ(hσ) = ∑_{χ∈Irr(G^{F^{d}} ,b)} Sh(χ)(N_{F^{d}/F} (g))Sh(χ)(N_{F^{d}/F} (h)).

It is easy to check that N_{F^{d}/F} (g) = (gσ)^{d} for any element g of G^{F^{d}}. Thus, since l does not divide d, N_{F^{d}/F} (g) is l-regular if and only if gσ is l-regular. Since every element of G^{F} is conjugate in G^{F} to N_{F^{d}/F} (g) for some element g of G^{F^{d}}, it follows from the above equation that

∑_{χ∈Irr(G^{F^{d}} ,b)} Sh(χ)(x)Sh(χ)(y) = 0

for all elements x and y of G^{F} such that exactly one of x and y is l-regular. A theorem of Osima [9, Theorem 3] now allows us to deduce that {Sh(χ) : χ ∈ b} is a union of l-blocks of G^{F}. This union contains c as Sh(χ) and χ have the same label for all unipotent characters χ ∈ Irr(G^{F^{d}}, b). Let μ be the generalised character of G^{F^{d}} × G^{F} given by

μ = ∑_{χ∈Irr(G^{F^{d}} ,b)} δ_{χ}(χ × Sh(χ)).

Let g ∈ G^{F^{d}}, and x ∈ G^{F}, and suppose that x is conjugate in G^{F} to N_{F^{d}/F} (h), h ∈ G^{F^{d}}. We have

μ(g, x) = ∑_{χ∈Irr(G^{F^{d}} ,b)} χ(g) × τ_{χ}χ(hσ) = ∑_{χ∈Irr(G^{F^{d}} ,b)} τ_{χ}χ(g) × τ_{χ}χ(hσ).

Thus μ(g, x) = 0 if exactly one g and x is l-regular. Furthermore, μ(g, x) is divisible in O by C_{G^{F^{d}} ⋊<σ>}(g) and by C_{G^{F^{d}} ⋊<σ>}(hσ). But since |C_{G^{F^{d}} ⋊<σ>}(hσ)| = |C_{G^{F}} (N_{F^{d}/F} (h)|, we have that μ(g, x) is divisible in O by C_{G^{F^{d}}} (g) and by C_{G^{F}} (x). Thus μ defines a perfect isometry between b and the sum of the blocks defined by {Sh(χ) : χ ∈ b}. Since perfect isometries between two algebras induce isomorphisms between the centers of the algebras, it follows that {Sh(χ) : χ ∈ b} defines a single block, namely c.

Finally, since δ_{χ} is the value at σ of an irreducible character of < σ >, δ_{χ} must equal 1 when d is odd. □

2.3. Proof of Corollary 1.3. Let V and V̂ be as in the proof of Theorem 1.2. and let s ∈ G^{F} be as in the statement of the corollary. Let V = ∑_{i} V_{i} be a decomposition of V into simple (non-zero) 𝔽_{q} < s >-modules.

If X is a simple 𝔽_{q} < s >-module, let r_{X} be the 𝔽_{q} < s >-dimension of X and let n_{X} be the number of V_{i} in the above decomposition which are isomorphic to X as 𝔽_{q} < s >-module. Then C_{G^{F}} (s) ≃ ∏_{X} GL_{n_{X}} (q^{r_{X}}) where X runs over a system of representatives of isomorphism classes of simple 𝔽_{q} < s >-modules.

For each i appearing in the above decomposition of V, let V̂_{i} denote the extension of V_{i} to an 𝔽_{q^{d}} vector space. The hypothesis on s implies that V̂_{i} is a simple 𝔽_{q^{d}} < s > module, whence C_{G^{F^{d}}} (s) ≃ ∏_{X} GL_{n_{X}} (q^{d}^{r_{X}}).

For each X, let u_{X}^{F} and u_{X}^{F^{d}} denote the sum of the unipotent blocks of GL_{n_{X}} (q^{r_{X}}) and GL_{n_{X}} (q^{d}^{r_{X}}) respectively appearing in the two products above. The order of q^{r_{X}} -modulo l is a divisor of the order of q modulo l, so is relatively prime to d. Thus by Theorem 1.2 there is a perfect isometry between u_{X}^{F} and u_{X}^{F^{d}} (which preserves signs when d is odd). Let u^{F} denote the central idempotent of OC_{G^{F}} (s) which corresponds to the product of the u_{X}^{F} under the isomorphism between OC_{G^{F}} (s) and the tensor product of the algebras OGL_{n_{X}} (q^{r_{X}}) and let u^{F^{d}} have an analogous meaning for OC_{G^{F^{d}}} (s). The above shows that there is a perfect isometry between OC_{G^{F}} (s)u^{F} and OC_{G^{F^{d}}} (s)u^{F^{d}}. On the other hand , by Théorèm 2.3 of [4], there is a perfect isometry between u^{F} and b_{(s^{G^{F}} )} and between u^{F^{d}} and b_{(s^{G^{F^{d}}})}. The result now follows from the fact that the composition of two perfect isometries is a perfect isometry. □

2.4. Remark. Keep the notation of the above proof. By Théorèm 11.8 of [2], there is a Morita equivalence between the algebras OC_{G^{F}} (s)u^{F} and OG^{F} b_{(s^{G^{F}} )}

and between the algebras OC_{G^{F^{d}}} (s)u^{F^{d}} and OG^{F^{d}}b_{(s^{G^{F^{d}}})}. Thus, when d is odd, there is in fact a sign preserving perfect isometry between b_{(s^{G^{F}} )} and b_{(s^{G^{F^{d}}})}.

Acknowledgements. I should like to thank Cedric Bonnafé, Francois Digne and Jean Michel for useful discussions.

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R.K.: University College, Oxford, OX1 4BH, England

 E-mail address: R.K.:  kessar@maths.ox.ac.uk