ROUQUIER BLOCKS IN CHEVALLEY GROUPS OF TYPE E

HYOHE MIYACHI

Dedicated to Michel Broué on the occasion of his 60-th birthday

1. Preliminary

Definition 1. Let G be a finite group of Lie type. Let A be a unipotent block ideal of G with Φ_{e}-defect torus T and canonical character λ in M = Z_{G}(T). Let W (M, λ) be the inertial group of A. We say that A is a Rouquier block if there exists a Levi subgroup L of G such that

(i) there exists parabolic subgroup P of G with Levi decomposition P = LU_{P}, (ii) L contains M, (iii) H = L · W (M, λ) is a proper subgroup of G, (iv) There exists a block B of H with canonical character λ such that A is Morita equivalent to B.

Remark 2. By L. Puig [Pui90], if ℓ dose not divide the order of Weyl group W of G and dose q − 1, then the principal block of G is Morita (Puig) equivalent to the principal block of T.W. In the case of type A, there is a generalization of this theorem which was conjectured by R.Rouquier [Rou98] in the context of symmetric groups. This generalization is intensively studied by J.Chuang, R. Kessar, K.Tan, W. Turner, A. Hida, James, Lyle, Mathas and the author [CK02], [CT02a], [Tur02], [Tur05], [Tur07],[JLM06], [HM00],[Miy01], [CM07]. In this case there is also an interesting interpretation on these Rouquier blocks in terms of Fock space over a quantum algebra of affine type A [CT02b],[LM02].

One aim of this note is to report that there are Rouquier blocks in type E_{6} and E_{8} (see Theorem 31 and Remark 34 below ) which are not included in [Pui90].

In through this note, we assume that a prime number ℓ and a prime power q satisfy the following condition:

(1)  (i) ℓ ≠ 2, 3, (ii) ℓ divides q^{2} + 1, and (iii) q is odd.

Remark 3. (i) and (ii) are essential in this note. (iii) is needed to use Kawanaka’s generalized Gelfand-Graev character [GP92]. So, this might be removed once one gets the lower unitriangularity of decomposition matrix in Lemma 9 below.

The main strategy (which I learned from T. Okuyama) is analogous to that in [KM00]. Namely, we construct a stable equivalence between two blocks by Broué’s theorem [Bro92, 6.3. Theorem] and checking the assumption, and then we chase the images of simple modules. The most powerful tool in this approach is Linckelmann’s theorem [Lin96].

This note is structured as follows. In Section 2 in order to check the Morita equivalences in local blocks appearing in some centralizers of semisimple ℓ-elements which we need to glue for the global stable equivalence, we recall the structures of principal block ideals in some small unitary groups. In Section 3 we recall some known results on ℓ-modular representations of Chevalley group D_{4}(q) of type D_{4} which is mainly due to Geck-Pfeiffer [GP92]. In Section 4 using the facts in Section 3 we look at representation of D_{4}(q).𝔖_{3}. In Section 5 we recall some numerical information on ℓ-modular representations of E_{6}(q), and then we shall use the main strategy. We shall show that the Green correspondence gives us a stable equivalence between the principal blocks of D_{4}(q).𝔖_{3} and E_{6}(q). After establishing the stable equivalence, we shall chase the images of simple modules by using frequently Linckelmann’s theorem [Lin96]. Then, we shall get the main results in this note. Using Theorem 31, the precise correspondence by the equivalence in this Theorem ( which will be given in Theorem 35) and a non-trivial self Morita equivalence, we will remove some unknown parameters in ℓ-modular decomposition matrices of D_{4}(q) and E_{6}(q) (Theorem 37) which are clearly one of the most important invariants in representation theory. In Section 6 we look at the situation on Iwahori-Hecke algebras. In Section 7 we look at the extension groups between two simple modules of D_{4}(q) and the Loewy layers of some projective indecomposable D_{4}(q)-modules. Before closing this preliminary, we introduce some general notation. For class functions χ and ψ of a finite group, we denote by ⟨ χ, ψ ⟩ the Hermitian inner product. The notation χ_{?} and ϕ_{?} will be used for irreducible ordinary characters. The notation φ_{?} will be used for irreducible Brauer characters. The unipotent characters of a finite Chevalley group G(𝔽_{q}) lying the principal series are labelled by the irreducible characters of the Weyl of G(𝔽_{q}). By this reason, for the unipotent principal series characters we use the same notation with the irreducible characters of the corresponding Weyl group. Namely, if ϕ_{b,d} is a irreducible character of a Weyl group, we write ϕ_{b,d} for the corresponding unipotent principal series character. For a group algebra or a Hecke algebra H, we denote by B_{0}(H) the principal block ideal of H.

Date: June 8, 2007.

1

2. Type A_{1} and ^{2}A_{3}

Keep the assumption (1). Let ^{2}A_{n}(q^{2}) be a unitary group of rank n + 1 (n ≤ 3). Namely, one of U_{n+1}(q^{2}), SU_{n+1}(q^{2}), P SU_{n+1}(q^{2}) or P U_{n+1}(q^{2}). Note that by the condition (1) any two principal blocks of these groups are isomorphic to each other for a fixed n [Alp76],[Dad77]. Since our concern is the structure of unipotent block ideals, we use this ambiguous notation. By Fong-Srinivasan[FS90], we know the following:

Lemma 4. Let D′ be a Sylow ℓ-subgroup of A_{1}(q^{2}).

(i) The principal block of A_{1}(q^{2}) is the Brauer tree algebra given by

χ_{(2)} −χ_{(1^{2})} −χ_{x′}.

(ii) The principal block of ^{2}A_{3}(q^{2}) is the Brauer tree algebra given by

χ_{(4)} −χ_{(2,1^{2})} −χ_{x′} −χ_{(1^{4})} −χ_{(3,1)}.

Here, the exceptional vertex is in the center and x′ runs through a set of representatives of 2A_{3}(q^{2})-conjugacy classes of D′. The corresponding labeling in the Weyl group of type B_{2} is as follows:

ϕ_{(2.)} −ϕ_{(11.)} −ϕ_{x′} −ϕ_{(.11)} −ϕ_{(.2)}.

(iii) Let σ be the field automorphism of 𝔽_{q^{2}} such that σ^{2} induces the Frobenius map of A_{1}(q^{2}). The principal block of A_{1}(q^{2}).⟨ σ ⟩ is the Brauer tree algebra given by

χ_{(2)}^{+} −χ_{(1^{2})}^{+} −χ_{x′} −χ_{(1^{2})}^{−} −χ_{(2)}^{−}.
3. Type D_{4}

In this section we will deal with ℓ-modular representations of CSpin_{8}(𝔽_{q}) in non-defining characteristic. Let q be a prime power p^{f} coprime to 2 such that ℓ divides q^{2} +1 exactly r times for some positive integer r. Let L be a reductive group of type D_{4} with connected centre. Let F be a standard Frobenius map of L defining 𝔽_{q}-structure on L. In this note we would like to consider ℓ-modular representations of the Chevalley groups L^{F} = D_{4}(q). Let the triple (K, O, 𝕜) be a splitting ℓ-modular system for L^{F} where K and 𝕜 are fields such that char(K) = 0 and char(𝕜) = ℓ > 0, and O is a complete discrete valuation ring. Let e_{L} be the principal block idempotent of 𝕜[L^{F}]. We choose the simple roots {r_{1}, r_{2}, r_{3}, r_{4}} of type D_{4} as follows:

1 The main body of this note is already written in 2005. The major difference is that [CM07, Proposition 22] and citations for [Tur05], [Tur07] and [JLM06] were missing in the older version.

ROUQUIER BLOCKS IN CHEVALLEY GROUPS OF TYPE E 3

r_{1}  r_{2}  r_{3} r_{4}

3.1. The Φ_{4}-decomposition matrix for H(D_{4}). By [Gec98] the decomposition matrix of the principal block of the Iwahori-Hecke algebra at a primitive 4-th root of unity is given as follows:

a name
0 .4  1 ….
2 .31  1 1 …
2 2.+ 1 . 1 . .
2 2.− 1 . . 1 .
3 1.21 1 1 1 1 1
6 .211 . 1 . . 1
6 11.+ . . 1 . 1
6 11.− … 1 1
12 .1111 …. 1

By Dipper’s theorem [Dip90], the matrix above is canonically embedded into the decomposition matrix of B_{0}(𝕜L^{F}).

3.2. Conjugacy classes of ℓ-elements. The ℓ-elements of L^{F} are semisimple and of type s_{51} = 1, s_{37}, s_{32}, s_{27}, s_{12} in the Geck-Pfeiffer notation [GP92]. So, our concern in this article is the principal block which has some characters lying in

\begin{matrix}E_{ℓ}(L^{F} , 1) \\ := & E(L^{F} , 1) ∪ ⋃_{s_{37}} E(L^{F} , s_{37}) ∪ ⋃_{s_{32}} E(L^{F} , s_{32}) ∪ ⋃_{s_{27}} E(L^{F} , s_{27}) ∪ ⋃_{s_{12}} E(L^{F} , s_{12}). \end{matrix}

Here, s_{i} runs through a complete set of representatives of semisimple ℓ-classes of L^{F} and we write E(L^{F} , s) for the Lusztig series by [BM89].

3.3. Φ_{4}-Torus and its Weyl group. Let T_{w} be an F -stable maximal torus of L obtained from the maximally split torus T by twisting T with w. Let W be the Weyl group for T. Let w_{12} be α_{1}α_{2}α_{1}α_{3}α_{2}α_{4}. Here, α_{i} is the simple reflection corresponding to the root r_{i}. The torus T_{w_{12}}^{F} is a Sylow Φ_{4}-torus of L^{F} (see [GP92, p.287]). Moreover, it is well-known that N_{L}(T_{w})^{F} /T_{w}^{F} is isomorphic to Z_{W,F} (w) [Car85, p.87 Proposition 3.3.6] and N_{L^{F}} (T_{w}^{F}) is equal to N_{L}(T_{w})^{F}. So, we get the following lemma:

Lemma 5.

\begin{matrix}W_{12} := N_{L}(T_{w_{12}})/T_{w_{12}} ≅ G(4, 2, 2) ◁ G(4, 1, 2), \\ N_{L}^{F} (T_{w_{12}}^{F}) = T_{w_{12}}^{F}.W_{12}. \end{matrix}

Here, G(i, j, k) is a complex reflection group in the same notation in [BMM93].

3.4. Parabolic subgroups, Scott modules and regular characters. Let W be the Weyl group of type D_{4} with simple roots I = {r_{1}, r_{2}, r_{3}, r_{4}}. Let P_{J} (resp. W_{J}) be the standard parabolic subgroup of L^{F} (resp. W) corresponding to a subset J of I.

3.4.1. Scott modules.

Lemma 6. Let B be the set of irreducible characters ϕ of W such that ϕ lies in the principal block of H_{ℚ(\sqrt{−1}),\sqrt{−1}}(W). Let e be the central idempotent of ℚW corresponding to B.

1_{W_{{1,2,3}}} ↑^{ℚ[W]e}= ϕ_{(.4)} + ϕ_{(.31)},
1_{W_{{2,3,4}}} ↑^{ℚ[W]e}= ϕ_{(.4)} + ϕ_{(2.+)},
1_{W_{{1,2,4}}} ↑^{ℚ[W]e}= ϕ_{(.4)} + ϕ_{(2.−)}.

Then, P_{2,3,4}, P_{1,2,3} and P_{1,2,4} have the Levi decomposition U_{i,j,k}L_{i,j,k} such that U_{i,j,k} is a p-group of order q^{6} and L_{i,j,k} is isomorphic to GL_{4}(q). Suppose that {i, j, k} is one of {2, 3, 4}, {1, 2, 3} and {1, 2, 4}.

Let T^{i,j,k} be a Coxeter torus of L_{i,j,k} with generator t_{i,j,k}. Then, T^{i,j,k} is cyclic and has order

(q + 1)(q^{2} + 1). Put x_{i,j,k} := ti,j,k^{\frac{(q+1)(q^{2}+1)}{ℓ^{r}}} and D_{i,j,k} := ⟨x_{i,j,k}⟩. Then, D_{i,j,k} is the Sylow ℓ-subgroup of

T^{i,j,k}. Let

φ_{(.4)}, φ_{(.31)}, φ_{(2.+)}, φ_{(2.−)}φ_{(1.21)}

be the irreducible Brauer characters of L^{F} corresponding to the simple H(D_{4})-modules. For a group M and an ℓ-subgroup Q of M, we denote the Scott 𝕜[M]-module with vertex Q by S_{M}(Q). (See [NT90] for the definition of Scott modules.) Then, by Lemma 6, we know the following:

Proposition 7.

S_{L^{F}} (D_{1,2,3}) = ( \begin{matrix}φ_{(.4)} \\ φ_{(.31)} \\ φ_{(.4)} \end{matrix} ) , S_{L^{F}} (D_{2,3,4}) = ( \begin{matrix}φ_{(.4)} \\ φ_{(2.+)} \\ φ_{(.4)} \end{matrix} ) , S_{L^{F}} (D_{1,2,4}) = ( \begin{matrix}φ_{(.4)} \\ φ_{(2.−)} \\ φ_{(.4)} \end{matrix} ) .

(Note that any two of D_{1,2,3}, D_{2,3,4} and D_{1,2,4} are not L^{F} -conjugate. )

3.4.2. Some regular characters. Let P_{i,j,k} be the standard parabolic subgroup of L^{F} corresponding to the subset {r_{i}, r_{j}, r_{k}} of the simple roots {r_{1}, r_{2}, r_{3}, r_{4}} of G.

Let us recall some known facts on ℓ-local subgroups and ℓ-modular representations of A_{3}(q). (see for [FS80] and [FS82].) Note that the principal block of A_{3}(q) has a cyclic defect group by the assumption in this note, hence is a Brauer tree algebra. We would like to look at the multiplicity at the exceptional vertex in the Brauer (line) tree [FS80].

Note that

N_{L_{i,j,k}}(T_{i,j,k}) = N_{L_{i,j,k}}(D_{i,j,k}) ≅ Z(L^{F}).(ℤ_{(q+1)(q^{2}+1)} ⋊ ℤ_{4}).

In particular, the number of conjugacy classes of ℓ-elements in L_{i,j,k} is \frac{(ℓ^{r}−1)}{4}. So, the Brauer tree of L_{i,j,k} is

(2)  χ_{(4)} −χ_{(3,1)} −χ_{(2,1^{2})} −χ_{(1^{4})} −χ_{x_{i,j,k}}

with multiplicity \frac{(ℓ^{r}−1)}{4} at the vertex χ_{x_{i,j,k}}. Here, χ_{λ} is the unipotent character of L_{i,j,k} corresponding to λ ⊢ 4 and χ_{x_{i,j,k}} is a character with semisimple part x_{i,j,k} and unipotent part 1. Let SS_{i,j,k} be a complete set of representatives of L_{i,j,k}-conjugacy classes of semisimple ℓ-elements of L_{i,j,k}. Let Z_{𝕜}^{i,j,k}(s) be the cuspidal ℓ-modular L_{i,j,k}-module labelled by a semisimple element s. Let P (Z_{𝕜}^{i,j,k}(s)) be the projective cover of Z_{𝕜}^{i,j,k}(s). (Note that the character of P (Z_{𝕜}^{i,j,k}(x_{i,j,k}^{m})) for any m ≢ 0 mod ℓ^{r} is equal to

χ_{(1^{4})} + ∑_{x∈SS_{i,j,k}} χ_{x}.

)

Let Γ_{L} be the Gelfand-Graev character of L. Then,

⟨R_{L_{i,j,k}}^{L^{F}}(Γ_{L_{i,j,k}}), Γ_{L^{F}} e⟩ < ⟨Γ_{L^{F}} , Γ_{L^{F}} e⟩

Indeed, there is a regular character χ such that

⟨χ, Γ_{L^{F}} e⟩ > 0 and ⟨χ, R_{L_{i,j,k}}^{L^{F}}(Γ_{L_{i,j,k}})e⟩ = 0

since x_{2,3,4}, x_{1,2,3} and x_{1,2,4} are not L^{F} -conjugate. So, by using [GP92, p.302-303] we know that R_{L_{i,j,k}}^{L^{F}}(P (Z_{𝕜}^{i,j,k}(x_{i,j,k})))e is indecomposable and projective.

3.5. Some irreducible mixed characters. Let s ∈ {s_{37}, s_{32}, s_{27}}.

3.5.1. Relative ranks and signs. We look at the sign of Z_{L^{F}} (s) determined by its relative rank for s ∈ {s_{51}, s_{37}, s_{32}, s_{27}, s_{12}} as follows: (see [Car85, p.199] for the definition of relative ranks and signs.)

\begin{matrix}s & s_{51} & s_{37} & s_{32} & s_{27} & s_{12} \\ ϵ_{Z(s)} & 1 & −1 & −1 & −1 & 1 \end{matrix}

3.5.2. Let M be a connected reductive algebraic group over 𝔽_{q} with Frobenius map F. Take a maximal torus T of M. For a semisimple element s of M, we denote the linear character ŝ of s regarded as an element of the dual group of M. Suppose s ∈ T. In this setting we denote by R_{T,ŝ}^{M} the Deligne-Lusztig generalized character of M corresponding to T and s [DL76],[Car85],[Lus84]. Then,

R_{T,ŝ}^{Z_{L}(s)} = 1 + St
R_{T_{w_{12}} ,ŝ}^{Z_{L}(s)} = 1 − St

So, we know that

χ_{s,1} = −\frac{1}{2}(R_{T,ŝ}^{L} + R_{T_{w_{12}} ,ŝ}^{L})
χ_{s,St} = −\frac{1}{2}(R_{T,ŝ}^{L} − R_{T_{w_{12}} ,ŝ}^{L})

3.6. Cuspidals. Now, we can find all the cuspidal unipotent modular simple modules as follows:

Lemma 8. (i) The modular Steinberg simple Brauer character φ_{.1111} is cuspidal.

(ii) The ℓ-modular reduction of unipotent cuspidal ordinary character ϕ is cuspidal. (iii) The other unipotent simple modules lying in B_{0}(𝕜L^{F}) are not cuspidal.

Proof. (ii) is obvious by definition. (i) follows from [GHM94, Theorem 4.2]. Clearly, the Hecke algebra part of decomposition matrix corresponds to principal series. The other parts also easily follows from [His93] and the observation in Section 3.4.2. □

3.7. The decomposition matrix for B_{0}(𝕜[L^{F}]). By combining observations in Section 3.1 and Section 3.4, we know the following:

Lemma 9. The decomposition matrix of B_{0}(𝕜[L^{F}]) must have the following shape:

a name ps ps ps ps ps c A_{3} A′_{3} A′′_{3} c
0 .4  1 ….  .  ….
2 .31  1 1 …  .  ….
2 2.+  1 . 1 . .  .  ….
2 2.−  1 . . 1 .  .  ….
3 1.21  1 1 1 1 1  .  ….
3 cusp.ϕ . . . . .  1  ….
6 .211  . 1 . . 1 c_{1}  1 …
6 11.+  . . 1 . 1 c_{2}  . 1 . .
6 11.−  … 1 1 c_{3}  . . 1 .
12 .1111 …. 1 c_{4}  1 1 1 1
χ_{s_{37},1} . . . . . a 1 …
χ_{s_{37},St} . . . . . a + b 1 . . 1
χ_{s_{32},1} . . . . . a . 1 . .
χ_{s_{32},St} . . . . . a + b . 1 . 1
χ_{s_{27},1} . . . . . a . . 1 .
χ_{s_{27},St} . . . . . a + b . . 1 1
χ_{s_{12}}  . . . . .  b  … 1

Here, 0 ≤ c_{i} ≤ q for i = 1, 2, 3 and 0 ≤ c_{4} ≤ q^{3}. Moreover, “ps” means that the corresponding simple module lies in the principal series, “c” means that the corresponding simple module is cuspidal, and “A_{3}^{?}” means that the corresponding simple module lies in Harish-Chandra series (A_{3}^{?}, φ_{St}^{A_{3}^{?}}) for ? = ∅, ′, ′′. Moreover, we find the following character degrees:

\begin{aligned}χ_{t,1}(1) & = (q^{2} − 1)^{3}(q^{4} + q^{2} + 1) \\ χ_{t,St}(1) & = q^{2}(q^{2} − 1)^{3}(q^{4} + q^{2} + 1) \\ χ_{s_{12}}(1) & = (q^{2} − 1)^{4}(q^{4} + q^{2} + 1) \end{aligned}

Here, t ∈ {s_{37}, s_{32}, s_{27}}.

4. Clifford theory :L^{F} .𝔖_{3}

On the triality automorphism and its effects on fusions, we can find some literatures [Car72], [Kle87],[Kle88]. Geck-Pfeiffer determined the character values of unipotent characters explicitly in [GP92]. So, the effects of 𝔖_{3} on the unipotent characters are clear. In particular, combining with the decomposition matrix in Lemma 9, we know the following:

Lemma 10. (i) 𝔖_{3} fixes χ_{(.4)}, χ_{(1.21)}, ϕ, χ_{(.1111)}.

(ii) ⟨ (1, 2, 3) ⟩ permutes χ_{(.31)}, χ_{(2.+)}, χ_{(2.−)} (resp. χ_{(.211)}, χ_{(11.+)}, χ_{(11.−)}).

(iii) 𝔖_{3} fixes φ_{(.4)}, φ_{(1.21)}, ϕ, φ_{(.1111)}.

(iv) ⟨ (1, 2, 3) ⟩ permutes φ_{(.31)}, φ_{(2.+)}, φ_{(2.−)} (resp. φ_{(.211)}, φ_{(11.+)}, φ_{(11.−)}).

By Clifford theory, we have

Lemma-Definition 11. (i) For λ ∈ {(.4), (1.21), ϕ, (.1111)} and μ ⊢ 3, we denote by χ_{λ,μ} the irreducible character appearing in Ind_{L^{F}}^{L^{F} .𝔖_{3}}(ϕ_{λ}) corresponding to an irreducible 𝔖_{3}-character χ_{μ}.

(ii) For λ, μ ∈ {(.31), (2.+), (2.−)} (resp. ∈ {(.211), (11.+), (11.−)})
Ind_{L^{F}}^{L^{F} .⟨ (1,2,3) ⟩}(ϕ_{λ}) = Ind_{L^{F}}^{L^{F} .⟨ (1,2,3) ⟩}(ϕ_{μ})

and this is irreducible. We denote this character by α(λ) for λ ∈ {(.31), (.211)}. (iii) For λ ∈ {(.31), (.211)} and μ ⊢ 2, we denote by χ_{λ,μ} the irreducible character appearing in

Ind_{L^{F} .⟨ (1,2,3) ⟩}^{L^{F} .𝔖_{3}}(α(λ)) corresponding to an irreducible 𝔖_{3}-character χ_{μ}.

(iv) For λ ∈ {(.4), (1.21), ϕ, (.1111)} and μ ⊢ 3, we denote by φ_{λ,μ} the irreducible Brauer character

appearing in Ind_{L^{F}}^{L^{F} .𝔖_{3}}(φ_{λ}) corresponding to an irreducible 𝔖_{3}-character χ_{μ}.

(v) For λ, μ ∈ {(.31), (2.+), (2.−)} (resp. ∈ {(.211), (11.+), (11.−)})

Ind_{L^{F}}^{L^{F} .⟨ (1,2,3) ⟩}(φ_{λ}) ≅ Ind_{L^{F}}^{L^{F} .⟨ (1,2,3) ⟩}(φ_{μ})

and this is irreducible. We denote this Brauer character by \overline{α}(φ_{λ}) for λ ∈ \{(.31), (.211)\}. (vi) For λ ∈ {(.31), (.211)} and μ ⊢ 2, we denote by φ_{λ,μ} the irreducible Brauer character appearing

in Ind_{L^{F} .⟨ (1,2,3) ⟩}^{L^{F} .𝔖_{3}}(\overline{α}(φ_{λ})) labelled by μ (according to the eigen space of μ). (vii) We denote by d_{(λ,μ)(ν,σ)} the decomposition number [\overline{χ_{λ,μ}} : φ_{ν,σ}].

By using the decomposition matrix in Lemma 9 and the construction in Lemma-Definition 11, immediately we get the following:

Lemma 12. A ( the unipotent) part of decomposition matrix of B_{0}(𝕜L^{F} .𝔖_{3}) is estimated as follows:

χ_{λ,μ}
Q_{0} ((.4), (3)) 1 . . . . . . . . . . . . . . .
((.4), (21)) . 1 . . . . . . . . . . . . . .
((.4), (1^{3})) . . 1 . . . . . . . . . . . . .
Q_{1} ((.31), (2)) 1 1 . 1 . . . . . . . . . . . .
((.31), (1^{2})) . 1 1 . 1 . . . . . . . . . . .
Q_{2} ((1.21), (3)) 1 . . 1 . 1 . . . . . . . . . .
((1.21), (21)) . 1 . 1 1 . 1 . . . . . . . . .
((1.21), (1^{3})) . . 1 . 1 . . 1 . . . . . . . .
Q_{3} (ϕ, (3)) . . . . . . . . 1 . . . . . . .
(ϕ, (21))  . . . . . . . . . 1 . . . . . .
(ϕ, (1^{3}))  . . . . . . . . . . 1 . . . . .
Q_{4} ((.211), (2)) … 1 . 1 1 . ∗ ∗ ∗ 1 ….
((.211), (1^{2})) …. 1 . 1 1 ∗ ∗ ∗ . 1 …
Q_{5} ((.1111), (3)) . . . . . 1 . . ∗ . . 1 . 1 . .
((.1111), (21)) . . . . . . 1 . . ∗ . 1 1 . 1 .
((.1111), (1^{3})) . . . . . . . 1 . . ∗ . 1 . . 1

∗ means an unknown parameter, but there is a relation

d_{(λ,μ),(σ,ν)} = d_{(λ,sgn⊗μ),(σ,sgn⊗ν)}.
5. Type E_{6}

We keep the assumption (1). Let G be the finite Chevalley group of type E_{6} over 𝔽_{q} with standard Frobenius map F. We will deal with ℓ-modular representations of G^{F}. Note that G contains canonically L in Section 3 as the Levi subgroup corresponding to the roots {r_{2}, r_{3}, r_{4}, r_{5}} in Table E_{6}.

r_{1}  r_{3}  r_{4}  r_{5}  r_{6} r_{2}

Table E_{6}

We consider the principal block of G^{F}. The partition of unipotent characters of G^{F} into the principal block is known by [Gec92]( see also [BMM93]). By [Gec93], we can know the decomposition matrix of Hecke algebra of type E_{6}. By [GH97] we know that the decomposition matrix of B_{0}(𝕜G^{F}) has a lower unitriangular shape and ℓ-reductions of unipotent characters of G^{F} lying in B_{0}(𝕜G^{F}) form a basic set. So, we can know the following:

Lemma 13. The decomposition matrix of B_{0}(𝕜G^{F}) is estimated as follows:

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
a name ps ps ps ps D_{4} ps ps ps D_{4} ps A_{3}  D_{4} D_{4} D_{4}
E_{6}  0 ϕ_{1,0} 1 . . . . . . . . . . . . . . .
E_{6}(a_{1}) 1 ϕ_{6,1} . 1 . . . . . . . . . . . . . .
E_{6}(a_{3}) 3 ϕ_{15,5} . . 1 . . . . . . . . . . . . .
A_{5}  3 ϕ_{15,4} 1 1 . 1 . . . . . . . . . . . .
A_{5}  3 D_{4}, 1 …. 1 . . . . . . . . . . .
A_{4}  6 ϕ_{81,6} . 1 1 . ∗ 1 . . . . . . . . . .
D_{4}(a_{1}) 7 ϕ_{80,7} . 1 . 1 ∗ 1 1 . . . . . . . . .
D_{4}(a_{1}) 7 ϕ_{90,8} . . 1 . ∗ 1 . 1 . . . . . . . .
D_{4}(a_{1}) 7 D_{4}, r …. ∗ … 1 . . . . . . .
2A_{2} + A_{1} 7 ϕ_{10,9} 1 . . 1 ∗ … ∗ 1 . . . . . .
A_{3}  10 ϕ_{81,10} …. ∗ 1 1 1 ∗ . 1 . . . . .
A_{2}  15 ϕ_{15,17} …. ∗ . . 1 ∗ . ∗ 1 ….
3A_{1}  15 ϕ_{15,16} … 1 ∗ . 1 . ∗ 1 ∗ ∗ 1 …
3A_{1}  15 D_{4}, ϵ …. ∗ … ∗ . ∗ ∗ ∗ 1 . .
A_{1}  25 ϕ_{6,25} …. ∗ . 1 . ∗ . ∗ ∗ ∗ ∗ 1 .
1  36 ϕ_{1,36} …. ∗ … ∗ 1 ∗ ∗ ∗ ∗ ∗ 1

Here, ∗ means a non-negative integer.

Let T^{F} be a Sylow Φ_{4}-torus of G^{F}. Let D be the Sylow ℓ-subgroup of T^{F}. Then,

T^{F} .W_{12} = N_{L}^{F} (T^{F}) ⊂ N_{G}^{F}(T^{F}) = T^{F} .G_{8}.

Here, G_{8} is a complex reflection group in the same notation in [BMM93]. Note that W_{12} is a normal subgroup of G_{8} and its quotient is isomorphic to 𝔖_{3}, which causes triality automorphism of D_{4}. In particular, 𝔖_{3} permutes semisimple elements {s_{27}, s_{32}, s_{37}}. So, L^{F} .𝔖_{3} is a subgroup of G^{F}. Let A (resp. B) be the principal block ideal of 𝕜G^{F} (resp. 𝕜L^{F} .𝔖_{3}). Moreover, L^{F} .𝔖_{3} contains N_{G}^{F}(D). Hence,

(3)  there is a Green correspondence \begin{aligned}& f \\ G^{F} & ⇄ & L^{F} .𝔖_{3} \\ & g \end{aligned}

(see [Alp86].) Let Δ(D) be {(x, x) | x ∈ D}. Let X be the Green correspondent of an indecomposable (A, A)-bimodule A in G × H with vertex Δ(D). (In other words, X is the Scott (A, B)-bimodule S_{G^{F} ×(L^{F} .𝔖_{3})}(Δ(D)) with vertex Δ(D). )

5.1. Φ_{4}-Local structures. On semisimple classes of L we refer [GP92]. On semisimple classes of G we refer [Miz77],[DF91],[FJ93]. The fusion of semisimple Φ_{4}-elements of L^{F} into G^{F} is easily deduced by these literatures. We would like to look at structures of centralizers of elements in D up to L^{F} .𝔖_{3}- conjugacy. If x in D is regular semisimple, then Z_{G}^{F}(x) = T^{F}. On the other hand, the other non-trivial semisimple elements in D belong to the semisimple conjugacy class of G^{F} in which a semisimple element of type s_{27} lies. Since 𝔖_{3} permutes {s_{27}, s_{32}, s_{37}}, even in L^{F} .𝔖_{3}, these conjugacy classes are of type s_{27}.

Being aware of the above remarks and using some methods in [SS70], we have the following:

Lemma 14. The table on the structure of Z_{L^{F} .𝔖_{3}}(E) and Z_{G}^{F}(E) for 1 ≠ E < D is given as follows:

E = ⟨ x ⟩  Z_{G}^{F}(E)  Z_{L^{F} .𝔖_{3}}(E)
x : non-regular  ℤ_{(q−1)(q^{2}+1)}.^{2}A_{3}(q^{2})  ℤ_{(q−1)(q^{2}+1)}.(ℤ_{(q−1)}.A_{1}(q^{2})).𝔖_{2}
x : regular ℤ_{(q−1)(q^{2}+1)} × ℤ_{(q−1)(q^{2}+1)}  ℤ_{(q−1)(q^{2}+1)} × ℤ_{(q−1)(q^{2}+1)}

5.2. A stable equivalence.

Lemma 15. X induces a stable equivalence of Morita type between A and B.

Proof. For a subgroup E of D let X(E) be the Green correspondent of the principal block ideal of Z_{G}^{F}(E) as bimodules in Z_{G}^{F}(E)×Z_{L^{F} .𝔖_{3}}(E) with respect to (Z_{G}^{F}(E)×Z_{G^{F}} (E), Δ(D), Z_{G}^{F}(E)×Z_{L^{F} .𝔖_{3}}(E)). By Lemma 4 and Lemma 14 we know that for any proper non-trivial subgroup E of D the principal blocks of Z_{G}^{F}(E) and Z_{L^{F} .𝔖_{3}}(E) are Morita equivalent by X(E) ⊗ −. Then, by Broué’s theorem [Bro92, 6.3. Theorem](see also [Ric96, Theorem 4.1]) we know that X induces a stable equivalence of Morita type between A and B. □

Remark 16. In our situation, for any indecomposable B-module V with vertex D, X ⊗_{B} V ≅ g(V) up to projective modules.

5.3. Trivial source simple modules. Let M be a trivial source module. Then, M is uniquely lifted to a module over O [Sco73]. We write this unique lift by M_{O}. In this situation, we also write M_{K} for M_{O} ⊗_{O} K. Moreover, X_{K} enjoys the following properties (on these properties we refer [Bro90]) : For any χ and ϕ ∈ Irr(B),

(4)  X_{K} ⊗_{B_{K}} (χ + ϕ) = X_{K} ⊗_{B_{K}} χ + X_{K} ⊗_{B_{K}} ϕ.

(5)  X_{K} ⊗_{B_{K}} χ is ℕ_{0}-linear a combination of irreducible characters lying in A

(6)  X_{K} ⊗_{B_{K}} χ is non-zero

(7)  X^{∨} enjoys the same properties as above.

we know that following:

Lemma 17. (i) R_{L}^{G}(φ_{(.4)}) is semisimple.

(ii) e_{A}R_{L}^{G}(φ_{(.4)}) ≅ φ_{1,0} ⊕ 2 · φ_{6,1} ⊕ φ_{15,5} as A-modules.

(iii) g(φ_{(.4),(3)}) ≅ φ_{1,0}, g(φ_{(.4),(21)}) ≅ φ_{6,1}, g(φ_{(.4),(1^{3})}) ≅ φ_{15,5}.

(iv) X ⊗_{B} φ_{(.4),(3)} ≅ φ_{1,0}, X ⊗_{B} φ_{(.4),(21)} ≅ φ_{6,1}, X ⊗_{B} φ_{(.4),(1^{3})} ≅ φ_{15,5}.

(v) X_{K} ⊗_{B_{K}} ϕ_{(.4),(3)} = ϕ_{1,0}, X_{K} ⊗_{B_{K}} ϕ_{(.4),(21)} = ϕ_{6,1}, X_{K} ⊗_{B_{K}} ϕ_{(.4),(1^{3})} = ϕ_{15,5}.

(vi) The duals of (iv) and (v) are true for X^{∨}.

Proof. (i)(ii) Let b be the principal block idempotent of a Levi subgroup M of G corresponding to the roots {r_{1}, r_{2}, ⋯, r_{5}}. By the branching rules in 8.1.1, 8.1.2 and 8.1.3 in Appendix and the transitivity of Harish-Chandra induction, we know that

e_{A}R_{L}^{G}(ϕ_{(.4)}) ≅ e_{A}R_{M}^{G}bR_{L}^{M}(ϕ_{(.4)})

and

e_{A}R_{L}^{G}(φ_{.4}) ≅ e_{A}R_{M}^{G}bR_{L}^{M}(φ_{.4})

More precisely by the decomposition matrix of Hecke algebra of type D_{5} in 8.2 and the selfduality of bR_{L}^{M}(φ_{(.4)}) , we know that

bR_{L}^{M}(φ_{(.4)}) ≅ φ_{(.5)} ⊕ φ_{(1.4)}.

Again by the same reason as above and the decomposition matrix in Lemma 13, we have

e_{A}R_{M}^{G}(φ_{(.5)}) ≅ φ_{1,0} ⊕ φ_{6,1}

and

e_{A}R_{M}^{G}(φ_{(1.4)}) ≅ φ_{6,1} ⊕ φ_{15,5}.

(iii) (A usual technique.) g(φ_{(.4),(3)}) ≅ φ_{1,0} is clear. e_{A}R_{L}^{G}(φ_{(.4)}) is a direct summand of e_{A}Ind_{L}^{G}(φ_{(.4)}) since we can define a Green correspondence with respect to (L^{F} , D, P^{F}) where P^{F} is a parabolic subgroup of G^{F} with Levi decomposition LU_{L} = P. On the other hand,

Ind_{L}^{G}(φ_{(.4)})
≅ Ind_{L.𝔖_{3}}^{G}Ind_{L}^{L.𝔖_{3}}(φ_{(.4)})
≅ Ind_{L.𝔖_{3}}^{G}(⨁_{λ⊢3} dim(λ)φ_{(.4),λ})
≅ ⨁_{λ⊢3} dim(λ)Ind_{L.𝔖_{3}}^{G}(φ_{(.4),λ})

and

Ind_{L.𝔖_{3}}^{G}(φ_{(.4),λ}) ≅ g(φ_{(.4),λ}) ⊕ P_{λ}

where P_{λ} is a module whose indecomposable direct summand has vertex strictly smaller than D ( or possibly P_{λ} is zero). By (i),(ii) and counting the multiplicity we conclude (iii).

(iv) By Linckelmann’s theorem [Lin96] and (iii), the result follows.

□

5.4. Harish-Chandra induced cuspidals from D_{4}.

Lemma 18. (i) For X ∈ {φ_{(0123|)}, φ_{.1111}} R_{L}^{G}(X) is semisimple (ii) X ⊗_{B} φ_{ϕ,(21)} ≅ φ_{(D_{4},r)}, (X ⊗_{B} φ_{ϕ,(3)}, X ⊗_{B} φ_{ϕ,(1^{3})}) = (φ_{(D_{4},1)}, φ_{(D_{4},ϵ)}) or (φ_{(D_{4},ϵ)}, φ_{(D_{4},1)}), (iii) X⊗_{B} φ_{(.1111),(21)} ≅ φ_{6,25}, (X⊗_{B} φ_{(.1111),(3)}, X⊗_{B} φ_{(.1111),(1^{3})}) = (φ_{15,17}, φ_{1,36}) or (φ_{1,36}, φ_{15,17}). (iv) X_{K} ⊗_{B_{K}} ϕ_{(.1111),(21)} = ϕ_{6,25}, (X_{K} ⊗_{B_{K}} ϕ_{(.1111),(3)}, X_{K} ⊗_{B_{K}} ϕ_{(.1111),(1^{3})}) = (ϕ_{15,17}, ϕ_{1,36}) or

(ϕ_{1,36}, ϕ_{15,17}).

(v) The duals of (ii), (iii) and (iv) are true for X^{∨}. (vi) [X ⊗_{B} φ_{(1.21),λ} : φ_{81,6}] = 0 for λ ⊢ 3.

Proof. By [GHM94, Proposition 2.9], its proof and Lemma 8, we have (i) and (ii). By arguments similar to the proof of Lemma 17 we can prove (iii) and (iv).

(v): Note that

(I)

e_{A}R_{L}^{G}(ϕ_{(.1111)}) = ϕ_{15,17} + 2 · ϕ_{6,25} + ϕ_{1,36}

by the branching rule 8.1.1 in Appendix. (II) (X_{K} ⊗_{B_{K}} −) ∘ Ind_{L^{F}}^{L^{F} .𝔖_{3}} is a direct summand of e_{A}R_{L}^{G}.

Noticing (I) and (II), and using the properties (4) and (5), we know that

(8)  ∑_{λ⊢3} dim(λ)⟨ X_{K} ⊗_{B_{K}} ϕ_{(.1111),λ}, χ ⟩ ≤ 1 for χ ∈ {ϕ_{15,17}, ϕ_{1,36}}

(9)  ∑_{λ⊢3} dim(λ)⟨ X_{K} ⊗_{B_{K}} ϕ_{(.1111),λ}, ϕ_{6,25} ⟩ ≤ 2

and

∑_{λ⊢3} dim(λ)⟨ X_{K} ⊗_{B_{K}} ϕ_{(.1111),λ}, χ ⟩ = 0 for χ ≠ ϕ_{15,17}, ϕ_{6,25}, ϕ_{1,36}

However, we have equalities in (8) and (9) because we can replace “at least” by “precisely” by (6). Hence,

X_{K} ⊗_{B_{K}} ϕ_{(.1111),λ} ∈ {ϕ_{15,17}, ϕ_{6,25}, ϕ_{1,36}} for any λ ⊢ 3.

More precisely, by counting the multiplicities we know that X_{K} ⊗_{B_{K}} ϕ_{(.1111),(21)} = ϕ_{6,25}. So, we are done.

(vi) :

By taking the modular reduction of equations in (iv), and by the known parts of decomposition matrices in Lemma 12 and Lemma 13, we get the desired result. □

5.5. Scott modules and relative projective covers. In this subsection we consider a Scott module S_{L^{F} .𝔖_{3}}(E) for a Sylow ℓ-subgroup E of L_{1,2,3}^{F}. In Proposition 7, we have discussed some Scott module in L^{F}.

Proposition 19. (i) [S_{L^{F} .𝔖_{3}}(E)_{K}] = χ_{(.4),(3)} + χ_{(.4),(21)} + χ_{(.31),(2)}, (ii)

S_{L^{F} .𝔖_{3}}(E) = ( \begin{matrix}φ_{(.4),(3))} φ_{(.4),(21)} \\ φ_{(.31),(2)} \\ φ_{(.4),(3)} φ_{(.4),(21)} \end{matrix} ) .

Proof. By recalling the construction of some simple modules over K[L^{F} .𝔖_{3}] and 𝕜[L^{F} .𝔖_{3}] in Lemma-Definition 11 and the definition of Scott module we obtain this result. □

Definition 20. We denote S_{L^{F} .𝔖_{3}}(E) ⊗ sgn_{L^{F} .𝔖_{3}} by S_{L^{F} .𝔖_{3}}^{sgn}(E). Here, sgn_{L^{F} .𝔖_{3}} is the 1-dimensional representation of L^{F} .𝔖_{3} induced by the sgn representation of 𝔖_{3}. Namely,

Ind_{L^{F}}^{L^{F} .𝔖_{3}}(S_{L^{F}} (E)) ≅ S_{L^{F} .𝔖_{3}}(E) ⊕ S_{L^{F} .𝔖_{3}}^{sgn}(E).

Proposition 21. Let M be a Levi subgroup of G corresponding to roots {r_{1}, r_{3}, r_{4}, r_{5}, r_{6}}, which is of type A_{5}. Then,

(i) e_{A}R_{M}^{G}(χ_{(.6)}) = ϕ_{1,0} + ϕ_{6,1} + ϕ_{15,4}, (ii)

S_{G^{F}} (E) = ( \begin{matrix}φ_{(1,0)} φ_{(6,1)} \\ φ_{(15,4)} \\ φ_{(1,0)} φ_{(6,1)} \end{matrix} )

For a subgroup Q of D and a module M we denote the Q-projective cover of M by

0 → Ω_{Q}(M) → P_{Q}(M) → M → 0

(see [Kun00, Definition 2.1] for the definition and her references.) By Proposition 19, Proposition 21 and [Kun00, Proposition 2.2], we have

Lemma 22. (i) P_{E}(φ_{1,0}) ≅ S_{G^{F}} (E), P_{E}(φ_{(.4),(3)}) ≅ S_{L^{F} .𝔖_{3}}(E).

(ii) Ω_{E}(φ_{1,0})/φ_{1,0} ≅ g (Ω_{E}(φ_{(.4),(3)})/φ_{(.4),(3)}).

Lemma 23. X ⊗_{B} φ_{(.31),(2)} ≅ φ_{15,4} . X_{K} ⊗_{B_{K}} ϕ_{(.31),(2)} = ϕ_{15,4}.

Proof. By Proposition 19, Proposition 21 and Lemma 22, we know that Ω_{E}(φ_{(.4),(3)})/φ_{(.4),(3)} and Ω_{E}(φ_{1,0})/φ_{1,0} have the following Loewy layers respectively:

Ω_{E}(φ_{(.4),(3)})/φ_{(.4),(3)} = ( \begin{matrix}φ_{(.4),(21)} \\ φ_{(.31),(2)} \\ φ_{(.4),(21)} \end{matrix} ) , Ω_{E}(φ_{1,0})/φ_{1,0} = ( \begin{matrix}φ_{(6,1)} \\ φ_{(15,4)} \\ φ_{(6,1)} \end{matrix} ) .

Note that the canonical map of Ω_{E}(φ_{(.4),(3)})/φ_{(.4),(3)} onto φ_{((.4),(21))} is not a projective homomorphism. By the exactness of X, the stable equivalence between A and B by X and Lemma 17(iv), we can remove the tops from both indecomposable modules above. Then, we know that

X ⊗_{B} ( \begin{matrix}φ_{(.31),(2)} \\ φ_{(.4),(21)} \end{matrix} ) = ( \begin{matrix}φ_{(15,4)} \\ φ_{(6,1)} \end{matrix} ) ⊕ Q.

Here, Q is a projective (possibly zero) A-module. By the same argument, we remove the socle in the both sides. Then, we have

X ⊗_{B} φ_{(.31),(2)} ≅ φ_{15,4} ⊕ Q.

Finally by Linckelmann’s theorem [Lin96] we deduce that Q = 0. □

As in the proof of Lemma 23 we frequently use the key, due to M. Linckelmann [Lin96], that X⊗_{B} S ≅ T ⊕ Q where S, T are simple means that Q = 0. So, it is useful to introduce the following notation:

Definition 24. For modules M and N, we write M ∼ N if there exists a projective module (possibly zero) Q such that

M ≅ N ⊕ Q or N ≅ M ⊕ Q.

Lemma 25. Let M_{1} be the Levi subgroup corresponding to roots {r_{3}, r_{4}, r_{5}}, which is a Levi subgroup of L. Let M be the Levi subgroup corresponding to roots {r_{1}, r_{3}, r_{4}, r_{5}, r_{6}}, which is of type A_{5}.

(i) R_{M_{1}}^{M}(ϕ_{(4)}) = ϕ_{(6)} + (2ϕ_{(5,1)} + ϕ_{(4,2)}) + ϕ_{(4,1^{2})}.

(ii) R_{M_{1}}^{M}(φ_{(4)}) = φ_{(6)} ⊕ (φ_{(5,1)} ⊕ P (φ_{(5,1)})) ⊕ P (φ_{(4,1^{2})}).

(iii) e_{A}R_{M}^{G}(ϕ_{(5,1)}) = ϕ_{6,1} + ϕ_{15,5} + ϕ_{81,6}. (iv) φ_{(5,1)} is a trivial source module. (v)

e_{A}R_{M}^{G}(φ_{(5,1)}) = ( \begin{matrix}φ_{(6,1)}φ_{(15,5)} \\ φ_{(81,6)} \\ φ_{(6,1)}φ_{(15,5)} \end{matrix} ) .

(vi) R_{M_{1}}^{L}(ϕ_{(4)}) = (ϕ_{(.4)} + ϕ_{(2.−)}) + ϕ_{(1.3)}. Here, ϕ_{(1.3)} is a Φ_{4}-defect zero unipotent character.

(vii) Let E be a Sylow ℓ-subgroup of M_{1}.

e_{A}R_{L}^{G}(S_{L^{F}} (E)) ∼ S_{G^{F}} (E) ⊕ e_{A}R_{M}^{G}(φ_{(5,1)})

We denote e_{A}R_{M}^{G}(φ_{5,1}) by S_{G}^{sgn}(E).

Lemma 26. (i)

(X ⊗_{B} −) ∘ Ind_{L^{F}}^{L^{F} .𝔖_{3}}(S_{L^{F}} (E)) ∼ S_{G^{F}} (E) ⊕ S_{G^{F}}^{sgn} (E)

(ii)

X ⊗_{B} φ_{(.31),(1^{2})} ≅ φ_{81,6}.

(iii)

X_{K} ⊗_{B_{K}} ϕ_{(.31),(1^{2})} = ϕ_{81,6}.

Proof. (i) Note that (X ⊗_{B} −) ∘ Ind_{L^{F}}^{L^{F} .𝔖_{3}}e_{B_{0}(𝕜L^{F})} is a direct summand of e_{A}R_{L}^{G}e_{B_{0}(𝕜L^{F})} as bimodules. So, by Lemma 25 we have

(X ⊗_{B} −) ∘ Ind_{L^{F}}^{L^{F} .𝔖_{3}}(S_{L^{F}} (E)) ∼ S_{G^{F}} (E) ⊕ e_{A}R_{M}^{G}(φ_{5.1})

(ii) By the construction of some simple L^{F} .𝔖_{3}-modules in Lemma-Definition 11,

Ind_{L^{F}}^{L^{F} .𝔖_{3}}(S_{L^{F}} (E)) ≅ S_{L^{F} .𝔖_{3}}(E) ⊕ S_{L^{F} .𝔖_{3}}^{sgn}(E)

Since X⊗_{B} − is exact and we have already observed that X⊗_{B} S_{L^{F} .𝔖_{3}}(E) ≅ S_{G^{F}} (E), we deduce that

(10)  X ⊗_{A} S_{L^{F} .𝔖_{3}}^{sgn}(E) ∼ S_{G^{F}}^{sgn} (E).

On the other hand, we have a canonical exact sequence

0 → ker(π) → S_{L^{F} .𝔖_{3}}^{sgn}(E) →^{π} φ_{(.4),(21)} ⊕ φ_{(.4),(1^{3})} → 0

Here, the epimorphism π is defined by the canonical quotient

S_{L^{F} .𝔖_{3}}^{sgn}(E)/ Rad(S_{L^{F} .𝔖_{3}}^{sgn}(E)) ≅ φ_{(.4),(21)} ⊕ φ_{(.4),(1^{3})}.

Moreover, π is not projective by definition. Since X⊗_{B} is exact and induces a stable equivalence, we know that

ker((X ⊗_{B} −)(π)) ∼ X ⊗_{B} (ker(π)).

Moreover, by Schur’s lemma, (X ⊗_{B} −)(π) and the canonical map

ξ : S_{G^{F}}^{sgn} (E) → S_{G^{F}}^{sgn} (E)/ Rad(S_{G^{F}}^{sgn} (E))

are identified since X ⊗_{B} φ_{(.4),λ} is known by Lemma 17(iv) for λ ⊢ 3. So,

ker(ξ) ∼ X ⊗_{B} ker(π).

By the argument similar to this and the fact that both ker(π) and ker(ξ) have the same Loewy structure in the desired correspondence, we also know that

(11)  ker(ξ)/ Soc(ker(ξ)) ∼ X ⊗_{B} (ker(π)/ Soc(ker(π)).

So, this equation (11) means that

X ⊗_{B} φ_{(.31),(1^{2})} ∼ φ_{81,6}.

So, by Linckelmann’s theorem [Lin96], we are done. (iii) By the proof of (ii), we have

X ⊗_{B} S_{L^{F} .𝔖_{3}}^{sgn}(E) ≅ S_{G^{F}}^{sgn} (E).

By this equation, properties (4), (5) and (6), and Lemma 17(v) the result follows.

□

Lemma 27.

(12)  X ⊗_{B} Ind_{L^{F}}^{L^{F} .𝔖_{3}}(P (.4)) ≅ P (φ_{1,0}) ⊕ 2 · P (φ_{6,1}) ⊕ P (φ_{15,5}).

Proof. Let P be the RHS in (12). Recall that Ind_{L^{F}}^{L^{F} .𝔖_{3}}(P (.4)) ≅ ⨁_{λ⊢3} P ((.4), λ). Since X ∘ Ind_{L^{F}}^{L^{F} .𝔖_{3}} is a direct summand of e_{A}R_{L}^{G}, by Lemma 17 P is certainly a direct summand of the LHS in (12). Moreover, X is exact. So, the LHS in (12) is projective.

On the other hand, the branching rule 8.1.1 and the decomposition matrix in Lemma 13 tell us that

(13)  e_{A}R_{L}^{G}(P (.4)) ≅ P ⊕ 6 · P (φ_{81,6}).

By the decomposition matrix in Lemma 12, Lemma 17(iv),Lemma 18(vi),Lemma 23 and Lemma 26(ii), and by comparing the multiplicity of φ_{81,6} in P with that in

[X ⊗_{B} Ind_{L^{F}}^{L^{F} .𝔖_{3}}(P (.4))] = ∑_{λ⊢3} ∑_{μ} c_{((.4),λ),μ}[X ⊗_{B} φ_{μ}]

where in the second sum μ runs through the indices of simple B-modules, we deduce that claimed equation by (13). □

Corollary 28. X_{K} ⊗_{B_{K}} ϕ_{(1.21),(3)} = ϕ_{10,9}, X_{K} ⊗_{B_{K}} ϕ_{(1.21),(21)} = ϕ_{80,7} and X_{K} ⊗_{B_{K}} ϕ_{(1.21),(1^{3})} = ϕ_{90,8}.

Proof. Clear by the decomposition matrix in Lemma 12, Lemma 17(iv)(v), Lemma 23, Lemma 26(iii), Lemma 27. □

Lemma 29.

(14)  X ⊗_{B} Ind_{L^{F}}^{L^{F} .𝔖_{3}}(P (.31)) ≅ P (φ_{15,4}) ⊕ P (φ_{81,6}).

Proof. Let P be the RHS in (14). Recall that Ind_{L^{F}}^{L^{F} .𝔖_{3}}(P (.31)) ≅ ⨁_{λ⊢2} P ((.31), λ). Since (X ⊗_{B} −) ∘ Ind_{L^{F}}^{L^{F} .𝔖_{3}} is a direct summand of e_{A}R_{L}^{G}, by Lemma 17 P is certainly a direct summand of the LHS in (14). Moreover, X is exact. So, the LHS in (14) is projective.

On the other hand, the branching rule 8.1.1 and the decomposition matrix in Lemma 13 tell us that

(15)  e_{A}R_{L}^{G}(P (.31)) ≅ P ⊕ 5 · P (φ_{81,6}).

By the decomposition matrix in Lemma 13, the multiplicity of ϕ_{81,6} in P is 1. On the other hand by the decomposition matrix in Lemma 12, Lemma 18,Lemma 23(iii),Corollary 28, and the branching rule 8.1.1, the multiplicity of ϕ_{81,6} in

[X_{K} ⊗_{B_{K}} Ind_{L^{F}}^{L^{F} .𝔖_{3}}(P (.31))] = ∑_{λ⊢2} ∑_{μ} d_{μ,((.31),λ)}X_{K} ⊗_{B_{K}} ϕ_{μ}

is at most 1. Here, in the second sum μ runs through the indices of simple B_{K}-modules. So, we are done. □

Corollary 30.

X_{K} ⊗_{B_{K}} ϕ_{(.211),(2)} = ϕ_{15,16}, X_{K} ⊗_{B_{K}} ϕ_{(.211),(1^{2})} = ϕ_{81,10}.

Theorem 31. The functor X ⊗_{B} − induces a Morita equivalence between A and B.

Proof. By Lemma 17, Lemma 18, Lemma 23, Lemma 26, Lemma 27, Lemma 28 we know that for any unipotent character χ lying in B X_{K}⊗_{B_{K}} χ is a unipotent character lying in A. These unipotent characters induce their basic sets for A and B, The decomposition matrices for A and B are lower unitriangular as in Lemma 12 and Lemma 13. It means that by the exactness of X ⊗_{B} −, the decomposition matrices for A and B are coincident with each other via the correspondence induced by X ⊗_{B} −, and X ⊗_{B} S is simple for any simple B-module S. Hence by Linckelmann’s theorem [Lin96] we are done. □

Remark 32. As in the proof of Theorem 31, (in particular, Lemma 14) one can not deduce Theorem 31 by the method developed in [Pui90] and [CK02] since the projective module P (φ_{81,6}) appears if one uses their methods. But the module X completely looks like the one in [Pui90] and [CK02]. In our proof, the key from Linckelmann’s theorem [Lin96] is particularly important not only to deduce the final proof but also to remove the projective summand P (φ_{81,6}) in the previous steps.

By [Mar96], we have

Corollary 33. If Broué’s abelian defect conjecture is true for the principal Φ_{4}-block of L^{F}, then so is true for the principal Φ_{4}-block of G^{F}.

Remark 34. By [Miy03] we know that the principal block B_{0}(𝕜G^{F}) is Morita equivalent to the unipotent block ideal of E_{8}(q) with canonical character ϕ_{23,01} in the notation [BMM93]. Moreover, thanks to Broué’s abelian defect conjecture and our main chart [BMM93], our main result is expected to be useful to settle Broué’s abelian defect conjecture for the following unipotent Φ_{4}-blocks:

(i) Group E_{7}(q) : canonical characters ϕ_{2}^{3}, ϕ_{11}^{3},
(ii) Group E_{8}(q) : canonical characters ϕ_{3.1}, ϕ_{123,013}, ϕ_{12,03}.

These will be discussed in elsewhere.

We summarize the correspondences induced by X.

Theorem 35. (i) The character correspondence via X_{K} ⊗_{B_{K}} − is given as follows:

ϕ_{((.4),(3))} → ϕ_{1,0}, ϕ_{((.4),(21))} → ϕ_{6,1}, ϕ_{((.4),(1^{3}))} → ϕ_{15,5},
ϕ_{(.31),(2)} → ϕ_{15,4}, ϕ_{(.31),(1^{2})} → ϕ_{81,6},
ϕ_{(1.21),(3)} → ϕ_{10,9}, ϕ_{(1.21),(21)} → ϕ_{80,7}, ϕ_{(1.21),(1^{3})} → ϕ_{90,8},
ϕ_{ϕ,(3)} → ϕ_{D_{4},(1^{3})}, ϕ_{ϕ,(21)} → ϕ_{D_{4},(21)}, ϕ_{ϕ,(1^{3})} → ϕ_{D_{4},(3)},
ϕ_{(.211),(2)} → ϕ_{15,16}, ϕ_{(.211),(1^{2})} → ϕ_{81,10},
ϕ_{(.1111),(3)} → ϕ_{1,36}, ϕ_{(.1111),(21)} → ϕ_{16,25}, ϕ_{(.1111),(1^{3})} → ϕ_{15,17}

(ii) The module correspondence via X ⊗_{B} − is given as follows:

φ_{((.4),(3))} → φ_{1,0}, φ_{((.4),(21))} → φ_{6,1}, φ_{((.4),(1^{3}))} → φ_{15,5},
φ_{(.31),(2)} → φ_{15,4}, φ_{(.31),(1^{2})} → φ_{81,6},
φ_{(1.21),(3)} → φ_{10,9}, φ_{(1.21),(21)} → φ_{80,7}, φ_{(1.21),(1^{3})} → φ_{90,8},
φ_{φ,(3)} → φ_{D_{4},(1^{3})}, φ_{φ,(21)} → φ_{D_{4},(21)}, φ_{φ,(1^{3})} → φ_{D_{4},(3)},
φ_{(.211),(2)} → φ_{15,16}, φ_{(.211),(1^{2})} → φ_{81,10},
φ_{(.1111),(3)} → φ_{1,36}, φ_{(.1111),(21)} → φ_{16,25}, φ_{(.1111),(1^{3})} → φ_{15,17}

(iii) We denote by x(χ) for the image of χ in simple modules over B_{K} or B. We have

d_{λ,μ} = d_{x(λ),x(μ)}.

Proof. The proof has not been done for X_{K} ⊗_{B_{K}} ϕ_{ϕ,λ} = ϕ_{D_{4},λ} , λ ∈ {(3), (1^{3})} . By Lemma 13, we deduce that the unipotent character part of [P (D_{4}, ϵ)] must be ϕ_{D_{4},ϵ} + xϕ_{6,25} + yϕ_{1,36} where at least one of x and y is non-zero. Indeed, if (x, y) = (0, 0), then Res_{L^{F}}^{L^{F} .𝔖_{3}}(X^{∨} ⊗_{A} P (φ_{D_{4},ϵ})) has no unipotent character except the unipotent cuspidal character. However, by Lemma 12 and Theorem 31, we know that x = 0 and y > 0. Since X_{K} ⊗_{B_{K}} (P (φ_{φ,λ})) is P (φ_{D_{4},λ}) or P (φ_{D_{4},λ′}) by Lemma 18, we deduce the desired result by Lemma 12 and Theorem 31. □

Remark 36. The branching rule by (X_{K} ⊗_{B_{K}} −)∘Ind_{L^{F}}^{L^{F} .𝔖_{3}} is identical with the branching rule bewtween G(4, 2, 2) and G_{8}.

5.6. Decomposition matrices for the principal blocks of L^{F}, L^{F} .𝔖_{3} and G^{F}. In this subsection, we shall determine the decomposition matrices of principal blocks of L^{F}, L^{F} .𝔖_{3} and G^{F} as possible. By Theorem 35(iii) and Remark 5.5 we have

(16)  d_{((.211),λ),(ϕ,(3))} = 0 for λ ⊢ 2.

By Clifford theory, we have

(17)  c_{1} = c_{2} = c_{3} = 0 and c_{4} = d_{(1,36),(D_{4},ϵ)}

in the decomposition matrix in Lemma 9. This means that

(18)  d_{((.211),λ),(ϕ,μ)} = 0 for any λ ⊢ 2, μ ⊢ 3.

Moreover, by (17) we have

(19)  a = 1, c_{4} = 2 + b

since for s ∈ {s_{37}, s_{32}, s_{27}}

\begin{aligned}χ_{s,1}(1) − dim_{𝕜} φ_{(.211)} & = & χ_{s,1}(1) − (ϕ_{.211}(1) − ϕ_{1.21}(1) + ϕ_{2.+}(1) + ϕ_{2.−}(1) − ϕ_{.4}(1)) \\ & = & ϕ_{∅}(1) \end{aligned}

and

ϕ_{.1111}(1) − dim_{𝕜}(φ_{.211} + φ_{11.+} + φ_{11.−} + φ_{1.21} + 2φ_{∅}) = ϕ_{s_{12}}(1).

So, by Theorem 35, we have

(20)  d_{(15,6),(D_{4},λ)} = 0, d_{(81,10),(D_{4},λ)} = 0 for any λ ⊢ 3.

So, more precisely, by Theorem 35,

(21)  d_{μ,(D_{4},λ)} = 0 for any λ ⊢ 3, μ ∉ {(15, 17), (16, 25), (1, 36)}.

Theorem 37. The unipotent part of decomposition matrix of B_{0}(𝕜G^{F}) is given as follows:

No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
a name ps ps ps ps D_{4} ps ps ps D_{4} ps A_{3} D_{4} A_{3} D_{4} D_{4} D_{4}
E_{6}  0 ϕ_{1,0} 1 . . . . . . . . . . . . . . .
E_{6}(a_{1}) 1 ϕ_{6,1} . 1 . . . . . . . . . . . . . .
E_{6}(a_{3}) 3 ϕ_{15,5} . . 1 . . . . . . . . . . . . .
A_{5}  3 ϕ_{15,4} 1 1 . 1 . . . . . . . . . . . .
A_{5}  3 D_{4}, 1 …. 1 . . . . . . . . . . .
A_{4}  6 ϕ_{81,6} . 1 1 . . 1 . . . . . . . . . .
D_{4}(a_{1}) 7 ϕ_{80,7} . 1 . 1 . 1 1 . . . . . . . . .
D_{4}(a_{1}) 7 ϕ_{90,8} . . 1 . . 1 . 1 . . . . . . . .
D_{4}(a_{1}) 7 D_{4}, r . . . . . . . . 1 . . . . . . .
2A_{2} + A_{1} 7 ϕ_{10,9} 1 . . 1 . . . . . 1 . . . . . .
A_{3}  10 ϕ_{81,10} . . . . . 1 1 1 . . 1 . . . . .
A_{2}  15 ϕ_{15,17} …. ∗ . . 1 . . 1 1 ….
3A_{1}  15 ϕ_{15,16} … 1 . . 1 . . 1 . . 1 …
3A_{1}  15 D_{4}, ϵ . . . . . . . . . . . . . 1 . .
A_{1}  25 ϕ_{6,25} . . . . . . 1 . ∗ . 1 . 1 . 1 .
1  36 ϕ_{1,36} . . . . . . . . . 1 . . 1 ∗ . 1

Here, ∗ is equal to the decomposition number d_{(.1111),φ} in type D_{4}.

6. Remarks on Hecke algebras

By Theorem 35 and [CM07, Proposition 22], we can know that

Theorem 38. Let F be a field with an invertible element q. We assume that (i) The characteristic of F is not 2, 3. (ii) q^{4} = 1, q^{2} ≠ 1. (iii) F contains q^{\frac{1}{2}} . (iv) If the characteristic of F is positive, then, q lie in the prime field of F.

Then, B_{0}(H_{F,q}(E_{6})) and B_{0}(H_{F,q}(D_{4})).𝔖_{3} are Morita equivalent.

We can construct B_{0}(H_{F,q}(D_{4})).𝔖_{3} as a block ideal of a well-known Iwahori-Hecke algebra in the following way. H_{F,q}(D_{4}) is a q-deformation of the group algebra of Weyl group W (D_{4}) of type D_{4}. And, in our situation, W (D_{4}).𝔖_{3} is nothing but the Weyl group W (F_{4}) of type F_{4}. Moreover, W (D_{4}) is realized as a reflection subgroup of W (F_{4}) generated by all the reflections of W (F_{4}) whose roots are long. So, let us recall the definition of Iwahori-Hecke algebra of type F_{4}.

Definition 39. Let R be an integral domain with invertible elements u^{\frac{1}{2}} , v^{\frac{1}{2}} . The Iwahori-Hecke algebra H_{R,u,v}(F_{4}) over R with parameter u, v is an associative algebra with generators T_{1}, T_{2}, T_{3}, T_{4} and relations

(T_{i} − u)(T_{i} + 1) = 0 for i = 1, 2, (T_{j} − v)(T_{j} + 1) = 0 for j = 3, 4,

T_{i}T_{i}T_{i} = T_{i+1}T_{i}T_{i+1} for i = 1, 3,
T_{2}T_{3}T_{2}T_{3} = T_{3}T_{2}T_{3}T_{2},
T_{i}T_{j} = T_{j}T_{i} for 1 ≤ i < j − 1 ≤ 3.

From now on, we consider the Iwahori-Hecke algebra H_{𝕜,q,1}(F_{4}) of type F_{4} with parameter q and 1. Put a_{1}, a_{2}, a_{3}, a_{4} respectively to be

a_{1} = T_{2}, a_{2} = T_{1}, a_{3} = T_{3}T_{2}T_{3}, a_{4} = T_{4}T_{3}T_{2}T_{3}T_{4}.

H_{𝕜,q}(D_{4}) is isomorphic to the subalgebra H′ of H_{𝕜,q,1}(F_{4}) generated by a_{1}, a_{2}, a_{3}, a_{4}. One can easily check that a_{i}’s satisfy the quadratic relations and braid relations. Moreover, clearly, 𝕜⟨ T_{3}, T_{4} ⟩ is isomorphic to the group algebra 𝕜𝔖_{3} since T_{3}^{2} = 1 = T_{4}^{2}. By definition, the action of T_{3} and T_{4} on H′ is also clear. Since the principal block idempotent of H_{𝕜,q}(D_{4}) is normalized by 𝕜⟨ T_{3}, T_{4} ⟩, it is lifted to the idempotent of the whole algebra H_{𝕜,q,1}(F_{4}). The decomposition matrix of H_{𝕜,q,1}(F_{4}) is first calculated by Bremke [Bre94, p.342]. So, it is worth saying the correspondences among characters, simple modules, PIM’s, etc over B_{0}(H_{𝕜,q}(E_{6})) and B_{0}(H_{𝕜,q,1}(F_{4})). The correspondence is given as follows:

\begin{matrix}E_{6} \\ ϕ_{1,0} & 1 & . & . & . & . & . & . & . \\ ϕ_{6,1} & . & 1 & . & . & . & . & . & . \\ ϕ_{15,5} & . & . & 1 & . & . & . & . & . \\ ϕ_{15,4} & 1 & 1 & . & 1 & . & . & . & . \\ ϕ_{81,6} & . & 1 & 1 & . & 1 & . & . & . \\ ϕ_{90,8} & . & . & 1 & . & 1 & 1 & . & . \\ ϕ_{80,7} & . & 1 & . & 1 & 1 & . & 1 & . \\ ϕ_{10,9} & 1 & . & . & 1 & . & . & . & 1 \\ ϕ_{81,10} & . & . & . & . & 1 & 1 & 1 & . \\ ϕ_{15,16} & . & . & . & 1 & . & . & 1 & 1 \\ ϕ_{15,17} & . & . & . & . & . & 1 & . & . \\ ϕ_{6,25} & . & . & . & . & . & . & 1 & . \\ ϕ_{1,36} & . & . & . & . & . & . & . & 1 \end{matrix} ↔ \begin{matrix}F_{4} \\ ϕ_{1,0} & 1 & . & . & . & . & . & . & . \\ ϕ′′_{2,4} & . & 1 & . & . & . & . & . & . \\ ϕ′′_{1,12} & . & . & 1 & . & . & . & . & . \\ ϕ_{9,2} & 1 & 1 & . & 1 & . & . & . & . \\ ϕ′′_{9,6} & . & 1 & 1 & . & 1 & . & . & . \\ ϕ′′_{8,9} & . & . & 1 & . & 1 & 1 & . & . \\ ϕ_{16,5} & . & 1 & . & 1 & 1 & . & 1 & . \\ ϕ′_{8,3} & 1 & . & . & 1 & . & . & . & 1 \\ ϕ_{9,10} & . & . & . & . & 1 & 1 & 1 & . \\ ϕ′_{9,6} & . & . & . & 1 & . & . & 1 & 1 \\ ϕ_{1,24} & . & . & . & . & . & 1 & . & . \\ ϕ′_{2,16} & . & . & . & . & . & . & 1 & . \\ ϕ′_{1,12} & . & . & . & . & . & . & . & 1 \end{matrix}

Here, dot · stands for zero.

7. Ext^{1}

In this section we try to calculate the dimension of extension groups Ext_{𝕜L^{F}}^{1} (φ_{λ}, φ_{μ}) for two simple 𝕜L^{F} -modules φ_{λ} and φ_{μ} . By using Clifford theory, Theorem 31 and Theorem 35 , we can construct all the homological information on B_{0}(𝕜G^{F}) from that on B_{0}(𝕜L^{F}). So, we omit the details for type E_{6}.

7.1. Geck-Pfeiffer. By [GP92], we can know the values of unipotent characters of L^{F}. Let B_{0} be the subset of unipotent characters of L^{F} lying in the principal Φ_{4}-block. Then, by the direct computation, we can prove the following

Lemma 40. Let ϕ be the unipotent cuspidal character of L^{F}. For any unipotent character χ of L^{F}

⟨ ϕ ⊗ ϕ, χ ⟩ = 0

except the symbol of χ is one of

( \begin{matrix}4 \\ 0 \end{matrix} ) , ( \begin{matrix}2, 3 \\ 0, 1 \end{matrix} ) , ( \begin{matrix}0, 1, 2, 3 \\ − \end{matrix} ) , ( \begin{matrix}1, 2, 3, 4 \\ 0, 1, 2, 3 \end{matrix} ) .

More precisely,

⟨ ϕ ⊗ ϕ, ϕ ⟩ = \frac{q − 1}{2}, ⟨ ϕ ⊗ ϕ, ϕ_{(23|01)} ⟩ = \frac{q + 1}{2}, ⟨ ϕ ⊗ ϕ, St ⟩ = \frac{(q − 1)^{2}}{4} .

Lemma 41. For any unipotent character χ of L^{F},

⟨ ϕ_{∅} ⊗ ϕ_{(1.21)}, χ ⟩ = 0

except χ is the Steinberg character of L^{F}.

Lemma 42. The diagonal Cartan entries are strictly greater than 2.

By combining [Web82],[Erd95],[Oku87] and [Kaw97] we have

Corollary 43. All the simple modules of B_{0}(𝕜L^{F}) lie in the end of the Auslander-Reiten quivers. In particular, the heart of any projective indecomposable modules over B_{0}(𝕜L^{F}) is indecomposable.

Corollary 44. Let S and T be simple modules over B_{0}(𝕜L^{F}). If the Cartan entry c_{S,T} ≤ 1, then dim_{𝕜} Ext_{𝕜L^{F}}^{1} (S, T) = 0.

18 HYOHE MIYACHI

Let Q_{(.4)→} be a subset

{S | S : a simple 𝕜L^{F} -module, Ext_{𝕜L^{F}}^{1} (φ_{(.4)}, S) ≠ {0}}

of a complete set of isomorphism classes of simple 𝕜L^{F} -modules. Lemma 45. (i)

Ext_{𝕜L^{F}}^{1} (φ_{(.4)}, φ_{(1.21)}) = 0

(ii)

Ext_{𝕜L^{F}}^{1} (S, T) = 0 for S, T ∈ {φ_{(.31)}, φ_{(2.+)}, φ_{(2.−)}}.

Proposition 46. The Loewy layer of P (.4) is

P (.4) = ( \begin{matrix}φ_{(.4)} \\ φ_{(.31)}φ_{(2.+)}φ_{(2.−)} \\ φ_{(.4)}φ_{(.4)}φ_{(.4)}φ_{(1.21)} \\ φ_{(.31)}φ_{(2.+)}φ_{(2.−)} \\ φ_{(.4)} \end{matrix} )

In particular, Q_{(.4)→} = {φ_{(.31)}, φ_{2.+}, φ_{2.−}} . Proof. By Lemma 7, we can find some partial information of the Loewy layer of P (.4) as follows:

P (.4) = ⌈ φ_{(.4)} ; φ_{(.31)}φ_{(2.+)}φ_{(2.−)} ; φ_{(.4)}φ_{(.4)}φ_{(.4)} ; ⋮ ; φ_{(.4)} ⌉

By Lemma 45, we have completed the second Loewy layer of P (.4). By Waki’s lemma[Wak93, Lemma 3.5] we complete the proof. □ Proposition 47. For φ_{λ} ∈ Q_{(.4)→} the Loewy layer of P (φ_{λ}) is given as follows:

( \begin{matrix}φ_{λ} \\ φ_{(.4)}φ_{(1.21)} \\ φ_{(.31)}φ_{(2.+)}φ_{(2.−)} \\ φ_{(.4)}φ_{(1.21)} \\ φ_{λ} \end{matrix} ) .

Lemma 48.

dim_{𝕜} Ext_{𝕜L^{F}}^{1} (φ, φ) = 0

Proof.

(22)  Ext_{𝕜L^{F}}^{1} (φ, φ) ≅ Ext_{𝕜L^{F}}^{1} (φ_{(.4)}, φ ⊗ φ)

By the decomposition matrix in Lemma 9, Lemma 40 and Lemma 46 we know that there is no composition factor in φ ⊗ φ lying in Q_{(.4)→}. So, the LHS of (22) is zero. Therefore, we complete the proof. □ Lemma 49. dim_{𝕜} Ext_{𝕜}^{1}(S, φ_{∅}) = 0 for S ∈ {φ_{(.4)}, φ_{(.31)}, φ_{(2.+)}, φ_{(2.−)}, φ_{(1.21)}}. Proof. ⟨ χ, ϕ_{(1.21)} ⊗ ϕ ⟩ = 0 for any unipotent character χ of L^{F} except the Steinberg character. More precisely, ⟨ St, ϕ_{(1.21)} ⊗ ϕ ⟩ = \frac{q^{2}−1}{4} . It means that Ext_{𝕜L^{F}}^{1} (φ_{(.4)}, \overline{ϕ_{(1.21)} ⊗ ϕ}) = 0. □ Lemma 50. Let L_{i(1),i(2),i(3)} be the Levi subgroup of L^{F} corresponding to {r_{i(1)}, r_{i(2)}, r_{i(3)}} for i(k) ∈ {1, 2, 3, 4}. Let e_{0} be the principal block idempotent of 𝕜L^{F}.

(i) e_{0}R_{L_{1,2,3}}^{L^{F}}(D(2, 1^{2})) = ( \begin{matrix}φ_{(1.21)} \\ φ_{(.31)}φ_{(11.+)}φ_{(11.−)} \\ φ_{(1.21)} \end{matrix} )

(ii) e_{0}R_{L_{2,3,4}}^{L^{F}}(D(2, 1^{2})) = ( \begin{matrix}φ_{(1.21)} \\ φ_{(2.−)}φ_{(.211)}φ_{(11.+)} \\ φ_{(1.21)} \end{matrix} )

(iii) e_{0}R_{L_{1,3,4}}^{L^{F}}(D(2, 1^{2})) = ( \begin{matrix}φ_{(1.21)} \\ φ_{(2.+)}φ_{(.211)}φ_{(11.−)} \\ φ_{(1.21)} \end{matrix} )

(iv) dim_{𝕜} Ext_{𝕜L^{F}}^{1} (φ_{(1.21)}, φ_{(1.21)}) = 0.

Theorem 51. The matrix (dim_{𝕜} Ext_{𝕜L^{F}}^{1} (φ_{λ}, φ_{μ}))_{λ,μ} has the following shape:

(.4) (.31) (2.+) (2.−) (1.21) ∅ (.211) (11.+) (11.−) (.1111)
(.4)  .  1  1  1  .  .  .  .  .  .
(.31)  1  .  .  .  1  .  .  .  .  .
(2.+) 1  .  .  .  1  .  .  .  .  .
(2.−) 1  .  .  .  1  .  .  .  .  .
(1.21) .  1  1  1  .  .  1  1  1  .
∅  .  .  .  .  .  .  ∗  ∗  ∗  ?
(.211) .  .  .  .  1  ∗  ?  .  .  ?
(11.+) .  .  .  .  1  ∗  .  ?  .  ?
(11.−) .  .  .  .  1  ∗  .  .  ?  ?
(.1111) .  .  .  .  .  ?  ?  ?  ?  ?

Here, ∗ stands for an unknown integer greater than 0 and ? means an unknown non-negative integer.

Acknowledgements

I would like to thank M. Broué for his works, abelian defect conjecture and encouraging me. My chart in this note comes from [BMM93] which says what should be true in ℓ-modular representation theory of finite Chevalley groups. I am indebted to T. Shoji for his continuous encouragement and several lectures on representations of Chevalley groups. I am grateful to A. Hida for stimulating discussions.

8. Appendix

8.1. Branching rules.

8.1.1. Branching rule of some unipotent characters lying in B_{0}(𝕜D_{4}(q)) and B_{0}(𝕜E_{6}(q)).

.4 .31 2.+ 2.− 1.21 .211 11.+ 11.− .1111
ϕ_{1,0} 1 .  .  .  .  .  .  .  .
ϕ_{6,1} 2 .  .  .  .  .  .  .  .
ϕ_{15,5} 1 .  .  .  .  .  .  .  .
ϕ_{15,4} . 1 1  1  .  .  .  .  .
ϕ_{81,6} . 2 2  2  3  1  1  1  .
ϕ_{80,7} . 1 1  1  5  1  1  1  .
ϕ_{90,8} . 1 1  1  4  1  1  1  .
ϕ_{10,9} . .  .  .  1  .  .  .  .
ϕ_{81,10} . 1 1  1  3  2  2  2  .
ϕ_{15,16} . .  .  .  .  1  1  1  .
ϕ_{15,17} . .  .  .  .  .  .  .  1
ϕ_{6,25} . .  .  .  .  .  .  .  2
ϕ_{1,36} . .  .  .  .  .  .  .  1

20 HYOHE MIYACHI

8.1.2. Branching rule of unipotent characters lying in B_{0}(𝕜D_{4}(q)) and B_{0}(𝕜D_{5}(q)).

.4 .31 2.+ 2.− 1.21 .211 11.+ 11.− .1111
.5  1 .  .  .  .  .  .  .  .
1.4 1 .  .  .  .  .  .  .  .
2.3  . 1  .  1  .  .  .  .  .
1.31 . .  1  .  1  .  .  .  .
.32  . .  1  .  .  .  .  .  .
1.22 . .  .  .  1  .  .  .  .
.311 . .  1  .  .  .  1  .  .
1.211 . .  .  .  1  .  1  .  .
.221 . .  .  .  .  .  1  .  .
11.111 . .  .  .  .  1  .  1  .
1.1111 . .  .  .  .  .  .  .  1
.11111 . .  .  .  .  .  .  .  1

8.1.3. Branching rule of unipotent characters lying in B_{0}(𝕜D_{5}(q)) and B_{0}(𝕜E_{6}(q)).

.5 1.4 2.3 1.31 .32 1.22 .311 1.211 .221 11.111 1.1111 .11111
ϕ_{1,0} 1 .  .  .  .  .  .  .  .  .  .  .
ϕ_{6,1} 1 1 .  .  .  .  .  .  .  .  .  .
ϕ_{15,5} . 1 .  .  .  .  .  .  .  .  .  .
ϕ_{15,4} . . 1  .  1  .  .  .  .  .  .  .
ϕ_{81,6} . . 1  1  .  .  1  .  .  .  .  .
ϕ_{80,7} . .  .  1  .  1  .  1  .  .  .  .
ϕ_{90,8} . .  .  1  .  .  .  1  .  .  .  .
ϕ_{10,9} . .  .  .  .  1  .  .  .  .  .  .
ϕ_{81,10} . .  .  .  .  .  1  1  .  1  .  .
ϕ_{15,17} . .  .  .  .  .  .  .  .  .  1  .
ϕ_{15,16} . .  .  .  .  .  .  .  1  1  .  .
ϕ_{6,25} . .  .  .  .  .  .  .  .  .  1  1
ϕ_{1,36} . .  .  .  .  .  .  .  .  .  .  1
.5 1.4 2.3 .32 .311 1.31 1.22 1.211 .221 11.111 1.1111 .11111
ϕ_{1,0} 1 .  .  .  .  .  .  .  .  .  .  .
ϕ_{6,1} 1 1 .  .  .  .  .  .  .  .  .  .
ϕ_{15,5} . 1 .  .  .  .  .  .  .  .  .  .
ϕ_{15,4} . . 1 1  .  .  .  .  .  .  .  .
ϕ_{81,6} . . 1 .  1  1  .  .  .  .  .  .
ϕ_{90,8} . .  .  .  .  1  .  1  .  .  .  .
ϕ_{80,7} . .  .  .  .  1  1  1  .  .  .  .
ϕ_{10,9} . .  .  .  .  .  1  .  .  .  .  .
ϕ_{81,10} . .  .  .  1  .  .  1  .  1  .  .
ϕ_{15,16} . .  .  .  .  .  .  .  1  1  .  .
ϕ_{15,17} . .  .  .  .  .  .  .  .  .  1  .
ϕ_{6,25} . .  .  .  .  .  .  .  .  .  1  1
ϕ_{1,36} . .  .  .  .  .  .  .  .  .  .  1

8.2. Φ_{4}-decomposition matrix for H(D_{5}).

a name
0 .5  1 . . . . . .
1 1.4  . 1 . . . . .
2 2.3  1 1 1 ….
3 .32  1 . . 1 …
3 1.31  . 1 1 . 1 . .
5 1.22  1 . 1 1 . 1 .
6 .311  …. 1 . .
7 .221  … 1 . 1 .
7 1.211 . . 1 . 1 . 1
10 11.111 . . 1 . . 1 1
13 1.1111 . . . . . . 1
20 .11111 . . . . . 1 .

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