On the Mackey formula for connected centre groups

1.1.

Let 𝐆 be a connected reductive algebraic group over an algebraic closure 𝔽¯p of the finite field 𝔽p of prime cardinality p. Moreover, let F:𝐆→𝐆 be a Frobenius endomorphism endowing 𝐆 with an 𝔽q-rational structure, where 𝔽q⊆𝔽¯p is the finite field of cardinality q. We assume fixed a prime ℓ≠p and an algebraic closure ℚ¯ℓ of the field of ℓ-adic numbers. If Γ is a finite group then we denote by Class⁡(Γ) the functions f:Γ→ℚ¯ℓ invariant under Γ-conjugation.

1.2.

If 𝐏⩽𝐆 is a parabolic subgroup of 𝐆 with F-stable Levi complement 𝐋 then Deligne–Lusztig have defined a pair of linear maps R𝐋⊂𝐏𝐆:Class⁡(𝐋F)→Class⁡(𝐆F) and R𝐋⊂𝐏𝐆*:Class⁡(𝐆F)→Class⁡(𝐋F) known as Deligne–Lusztig induction and restriction. The Mackey formula, which is an analogue of the usual Mackey formula from finite groups, is then defined to be the equality

of linear maps Class⁡(𝐌F)→Class⁡(𝐋F), where 𝐐⩽𝐆 is a parabolic subgroup with F-stable Levi complement 𝐌⩽𝐐. Here

and (ad⁡g)𝐌F is the linear map Class⁡(𝐌F)→Class⁡(𝐌Fg) induced by the isomorphism 𝐌Fg→𝐌F obtained by restricting the inner automorphism (ad⁡g)𝐆F of 𝐆F defined by conjugation with g.

1.3.

The Mackey formula is a fundamental tool in the representation theory of finite reductive groups. Its importance to ordinary representation theory is made abundantly clear in the book of Digne–Michel [8]. However it also plays a prominent role in modular representation theory via e-Harish-Chandra theory. The formula was first proposed by Deligne in the case where 𝐏 and 𝐐 are both F-stable; a proof of this case appears in [10, Lemma 2.5]. Deligne–Lusztig were also able to establish the formula when either 𝐋 or 𝐌 is a maximal torus, see [7, Theorem 7] and [8, Theorem 11.13]. We note that a consequence of the Mackey formula, namely the inner product formula for Deligne–Lusztig characters, had been shown to hold in earlier work of Deligne–Lusztig, see [6, Theorem 6.8].

1.4.

A possible approach to proving the Mackey formula is suggested by the early work of Deligne–Lusztig, see the proof of [6, Theorem 6.8]. Here the idea is to argue by induction on dim⁡𝐆. In a series of articles [1, 2, 3] Bonnafé made extensive progress on the Mackey formula, specifically establishing criteria that a minimal counterexample must satisfy. In fact, Bonnafé was able to establish the Mackey formula assuming either that q is sufficiently large (with an explicit bound on q) or if all the quasi-simple components of 𝐆 are of type 𝖠. In the latter case Lusztig’s theory of cuspidal local systems [9] plays a prominent role in the proofs.

1.5.

Using the inductive approach mentioned above, together with computer calculations performed with CHEVIE [11], Bonnafé–Michel [5] were able to show the Mackey formula holds assuming either that q>2 or that 𝐆 has no quasi-simple components of type 𝖤6, 𝖤7 or 𝖤8. Our contribution to this problem is to observe that the following holds.

Theorem 1.6.

Assume that q=2 and G is such that Z⁢(G) is connected and G has no quasi-simple component of type E8, then the Mackey formula (1.1) holds.

1.7.

Our approach to proving Theorem 1.6 is exactly the same as that of [5]; namely we argue by induction on dim⁡𝐆. As remarked in [5, Remark 3.10] to show the Mackey formula holds for all tuples (𝐆,F,𝐋,𝐏,𝐌,𝐐) it is sufficient to show the Mackey formula holds when (𝐆,F) is of type 𝖤6sc2⁢(2) and 𝐌 is a Levi subgroup of type 𝖠2⁢𝖠2. Our observation is that by considering the adjoint group 𝖤6ad2⁢(2) the problematic Levi subgroup of type 𝖠2⁢𝖠2 is circumvented.

1.8.

In the very first step of the proof of [5, Theorem 3.9] one encounters the following problem. If Z⁢(𝐆) is connected then it is not necessarily the case that Z⁢(C𝐆∘⁢(s)) is connected for all semisimple elements s∈𝐆. This means one cannot apply directly, to C𝐆∘⁢(s), any induction hypothesis which relies on the centre being connected. However, in the cases under consideration we have enough control over the structure of C𝐆∘⁢(s) to make use of the induction hypothesis, see Lemma 2.4. Let us note now that our proof of Theorem 1.6 relies on all the previously established cases of the Mackey formula.

1.9.

Unfortunately we cannot push our argument through to the case where 𝐆F is 𝖤8⁢(2). Here there exists a semisimple element s∈𝐆F such that C𝐆∘⁢(s)F is a product 𝖤6sc2⁢(2)⋅𝖠2sc2⁢(2). Thus we arrive back to the problem of dealing with the case of 𝖤6sc2⁢(2). However, we can establish one general statement about (1.1) assuming Z⁢(𝐆) is connected. For this we need the following notation. Let 𝐆uni⊆𝐆 be the variety of all unipotent elements in 𝐆. By Classuni⁡(𝐆F)⊆Class⁡(𝐆F) we mean the space of unipotently supported class functions of 𝐆F, i.e., those functions f∈Class⁡(𝐆F) for which f⁢(g)≠0 implies g∈𝐆uniF.

Theorem 1.10.

Assume Z⁢(G) is connected, then the Mackey formula (1.1) holds on Classuni⁡(MF).

2.1.

Throughout we assume that 𝐆 and F:𝐆→𝐆 are as in §1.1. In what follows we will write 𝐆 as a product 𝐆1⁢⋯⁢𝐆n⁢Z⁢(𝐆) where 𝐆1,…,𝐆n are the quasi-simple components of 𝐆. With this notation in place we have the following.

Lemma 2.2.

Let H=H1⁢⋯⁢Hn⁢Z⁢(G)⩽G be an F-stable subgroup of G where Hi⩽Gi is a closed connected reductive subgroup of Gi. If π:H→H/Z∘⁢(H) denotes the natural quotient map and Z⁢(G)⩽Z∘⁢(H) then we have a bijective morphism of varieties

which is defined over Fq. Moreover, if Z⁢(Hi)⩽Z∘⁢(H) then π⁢(Hi) has a trivial centre.

Proof.

Recall that for any 1⩽i⩽n we have 𝐆i∩∏j≠i𝐆j⩽Z⁢(𝐆)⩽Z∘⁢(𝐇). As 𝐇i∩∏j≠i𝐇j⩽𝐆i∩∏j≠i𝐆j, we have π⁢(𝐇i)∩∏j≠iπ⁢(𝐇j)={1} which establishes the bijective morphism. Now, let us consider the case where Z⁢(𝐇i)⩽Z∘⁢(𝐇). We know that 𝐇i/Z⁢(𝐇i)⩽𝐇/Z⁢(𝐇i) has a trivial centre and we have a surjective homomorphism

which restricts to a bijective homomorphism 𝐇i/Z⁢(𝐇i)→π⁢(𝐇i). Thus π⁢(𝐇i) also has a trivial centre. ∎

2.3.

Our application of Lemma 2.2 will be to the case where 𝐇 is the connected centraliser of a semisimple element of 𝐆. Specifically we will need the following.

Lemma 2.4.

Assume that p=2 and G is such that Z⁢(G) is connected and all the quasi-simple components of G are of type A, E6, or E7. If s∈GF is a semisimple element then there exist F-stable closed connected reductive subgroups H1,H2⩽CG∘⁢(s) with the following properties:

1. 𝐇1 has a trivial centre and has no quasi-simple component of type 𝖤8,

2. all the quasi-simple components of 𝐇2 are of type 𝖠 or 𝖣,

3. there exists a bijective homomorphism of algebraic groups

   𝐇1×𝐇2→C𝐆∘⁢(s)/Z∘⁢(C𝐆∘⁢(s))

   which is defined over 𝔽q.

Proof.

As above we write 𝐆 as a product 𝐆1⁢⋯⁢𝐆n⁢Z⁢(𝐆) where the 𝐆i are the quasi-simple components of 𝐆. Similarly we may write s as a product s1⁢⋯⁢sn⁢z where si∈𝐆i and z∈Z⁢(𝐆). We then have C𝐆∘⁢(s)=C𝐆1∘⁢(s1)⁢⋯⁢C𝐆n∘⁢(sn)⁢Z⁢(𝐆), see [4, Equation (2.2)] for instance. By assumption each 𝐆i is of type 𝖠, 𝖤6, or 𝖤7 which implies one of the following holds:

1. all the quasi-simple components of C𝐆i∘⁢(si) are of type 𝖠 or 𝖣,

2. 𝐆i is of type 𝖤7 and C𝐆i∘⁢(si) is a Levi subgroup of type 𝖤6,

3. 𝐆i=C𝐆i∘⁢(si) is of type 𝖤6 or 𝖤7.

As p=2, we have in the second case that Z⁢(𝐆i)={1} which implies Z⁢(C𝐆i∘⁢(si)) is connected because C𝐆i∘⁢(si) is a Levi subgroup of 𝐆i. In particular, we have Z⁢(C𝐆i∘⁢(si))⩽Z∘⁢(C𝐆∘⁢(s)). In the third case we have

because, by assumption, Z⁢(𝐆) is connected. The statement now follows from Lemma 2.2. ∎

3.1.

Assume we are given a tuple (𝐆,F,𝐋,𝐏,𝐌,𝐐) as in §1.1. Then we set

The Mackey formula (1.1) is therefore equivalent to the statement Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆=0. Note that Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆 is a linear map Class⁡(𝐌F)→Class⁡(𝐋F). In what follows we will say that the Mackey formula holds for (𝐆,F), or for short that it holds for 𝐆, if Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆=0 for all possible quadruples (𝐋,𝐏,𝐌,𝐐).

3.2.

Recall that a homomorphism ι:𝐆→𝐆~ is said to be isotypic if the following hold: 𝐆 and 𝐆~ are connected reductive algebraic groups, the kernel Ker⁡(ι) is central in 𝐆 and the image Im⁡(ι) contains the derived subgroup of 𝐆~. If ι is defined over 𝔽q then this restricts to a homomorphism ι:𝐆F→𝐆~F and we have a corresponding restriction map

If 𝐊⩽𝐆 is a closed subgroup of 𝐆 then we denote the subgroup ι⁢(𝐊)⁢Z⁢(𝐆~)⩽𝐆~ by 𝐊~. With this notation we have by [5, Equation (3.7)] that

The following is an easy consequence of (3.1).

Lemma 3.3.

If ι:G→G~ is a bijective morphism of algebraic groups defined over Fq then the Mackey formula holds for (G,F) if and only if it holds for (G~,F).

3.4.

Now assume s∈𝐆F is a semisimple element. Then for any class function f∈Class⁡(𝐆F) we define a function ds𝐆⁢(f):C𝐆∘⁢(s)F→ℚ¯ℓ by setting

Note that ds𝐆⁢(f)∈Classuni⁡(C𝐆∘⁢(s)F) is a unipotently supported class function so we have defined a ℚ¯ℓ-linear map ds𝐆:Class⁡(𝐆F)→Classuni⁡(C𝐆∘⁢(s)F). In particular, if z∈Z⁢(𝐆)F then we obtain a ℚ¯ℓ-linear map

Now, if s∈𝐋F is a semisimple element then by [5, Equation (3.5)] we have

Moreover, if s∈Z⁢(𝐆)F⩽𝐋F∩𝐌F then it follows that

see [5, Equation (3.6)].

Lemma 3.5.

Assume ι:G→G~ is a surjective isotypic morphism such that Ker⁡(ι)⩽Z∘⁢(G), then the map

restricts to an isomorphism Classuni⁡(G~F)→Classuni⁡(GF).

Proof.

Note that ι restricts to a bijection ι:𝐆uniF→𝐆~uniF. We will denote the inverse of this map by ι-1:𝐆~uniF→𝐆uniF. Now, if f∈Classuni⁡(𝐆F) then we define f~:𝐆~F→ℚ¯ℓ by setting

The proof of [5, Lemma 3.8] shows that u,v∈𝐆uniF are 𝐆F-conjugate if and only if ι⁢(u),ι⁢(v)∈𝐆~uniF are 𝐆~F-conjugate because Ker⁡(ι)⩽Z∘⁢(𝐆) and ι is surjective. This implies f~∈Class⁡(𝐆~F) so we are done. ∎

Proof of Theorem 1.6.

We will denote by ⪯ the lexicographic order on ℕ×ℕ. With this we assume that (𝐆,F,𝐋,𝐏,𝐌,𝐐) is a tuple such that the following hold:

1. Z⁢(𝐆) is connected and 𝐆 has no quasi-simple component of type 𝖤8,

2. Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆≠0,

3. (dim⁡𝐆,dim⁡𝐋+dim⁡𝐌) is minimal, with respect to ⪯, amongst all the tuples satisfying (H1) and (H2).

Arguing on the minimality of (dim⁡𝐆,dim⁡𝐋+dim⁡𝐌), we aim to show that such a tuple cannot exist. We follow precisely the argument used in the proof of [5, Theorem 3.9].

As p=2 and (H1) holds, there exist F-stable closed connected reductive subgroups 𝐆1,𝐆2⩽𝐆 such that the following hold:

1. all the quasi-simple components of 𝐆1 are of type 𝖠, 𝖤6, or 𝖤7,

2. all the quasi-simple components of 𝐆2 are of type 𝖡, 𝖢, 𝖣, 𝖥4, or 𝖦2,

3. the product map 𝐆1×𝐆2→𝐆 is a bijective morphism of algebraic groups defined over 𝔽q.

As (H2) holds for 𝐆, we have by Lemma 3.3 that the same must be true of the direct product 𝐆1×𝐆2. Now, by [5, Theorem 3.9], the Mackey formula holds for 𝐆2 so as Deligne–Lusztig induction is compatible with respect to direct products we can assume that the Mackey formula fails for 𝐆1. Applying (H3) and Lemma 3.3, we may thus assume that all the quasi-simple components of 𝐆 are of type 𝖠, 𝖤6, or 𝖤7.

Let us denote by μ∈Class⁡(𝐌F) a class function such that

By [5, Equation (3.2)] there must exist a semisimple element s∈𝐋F such that ds𝐋⁢(Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ))≠0. Applying (3.2) there thus exists an element g∈𝐆F such that

We set 𝐌′=𝐌g, 𝐐′=𝐐g and λ=ds𝐌′⁢(μg)∈Classuni⁡(C𝐌′∘⁢(s)F).

If 𝐊⩽𝐆 is a closed subgroup of 𝐆, we denote by 𝐊s the subgroup C𝐊∘⁢(s)⩽𝐆 and by 𝐊¯s the image of 𝐊s⁢Z∘⁢(C𝐆∘⁢(s)) under the natural quotient map C𝐆∘⁢(s)→C𝐆∘⁢(s)/Z∘⁢(C𝐆∘⁢(s)). Note this quotient map is a surjective isotypic morphism with connected kernel. Therefore, by Lemma 3.5, there exists a unique unipotently supported class function λ¯∈Classuni⁡(𝐆¯sF) such that

Applying (3.1) and (3.3), we see that

so

Let us now assume that 𝐇1,𝐇2⩽𝐆¯s are closed subgroups as in Lemma 2.4. By [5, Theorem 3.9] the Mackey formula holds for 𝐇2 so, arguing as above, we may assume the Mackey formula fails for 𝐇1. Now, we have dim⁡𝐇1⩽dim⁡𝐆s⩽dim⁡𝐆 and 𝐇1 satisfies (H1). Thus by (H3) we can assume these inequalities are equalities. In particular, this implies that 𝐆s=𝐆 and the quotient map

is bijective.

As the kernel of an adjoint quotient of 𝐆 is the same as the kernel of the map 𝐆→𝐆/Z⁢(𝐆), any adjoint quotient of 𝐆 is bijective so by Lemma 3.3 we can assume 𝐆 is semisimple and adjoint. In particular, we have 𝐆 is a direct product of its quasi-simple components and by compatibility with direct products we can assume that F cyclically permutes these quasi-simple components. Finally, we can assume that either all the quasi-simple components are of type 𝖤6 or they are all of type 𝖤7 because the Mackey formula holds if they are of type 𝖠 by [5, Theorem 3.9]. To recapitulate we may assume the following holds:

1. 𝐆 is semisimple and adjoint and F permutes cyclically the quasi-simple components of 𝐆,

2. all the quasi-simple components of 𝐆 are of type 𝖤6 or 𝖤7.

Recall that μ∈Class⁡(𝐌F) is a class function satisfying Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ)≠0 and s∈𝐋F is a semisimple element such that ds𝐋⁢(Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ))≠0. The above argument shows that we must have 𝐆s=𝐆 so s∈Z⁢(𝐆)={1}. We therefore have

by (3.3) so replacing μ by d1𝐌⁢(μ), we can assume that there exists a unipotently supported class function μ∈Classuni⁡(𝐌F) satisfying Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ)≠0.

Now let (𝐆⋆,F⋆) be a pair dual to (𝐆,F) and let 𝐌⋆⩽𝐆⋆ be a Levi subgroup dual to 𝐌⩽𝐆. We note that 𝐆⋆ is simply connected as 𝐆 is adjoint. Arguing exactly as in the proof of [5, Theorem 3.9], we may assume that the following properties hold:

1. 𝐌 is not a maximal torus and 𝐌≠𝐆,

2. there exists an F-stable unipotent class of 𝐌 which supports an F-stable cuspidal local system, in the sense of [9, Definition 2.4],

3. 𝐐 is not contained in an F-stable proper parabolic subgroup of 𝐆,

4. there exists a semisimple element s∈𝐌⋆F⋆ which is quasi-isolated in both 𝐌⋆ and 𝐆⋆ such that sz is 𝐆⋆F⋆-conjugate to s for every z∈Z⁢(𝐌⋆)F⋆.

Indeed, (P3) follows immediately from the fact that the Mackey formula holds if either 𝐋 or 𝐌 is a maximal torus. Moreover, (P5) follows from the formula in [5, Equation (3.4)] together with the fact that the Mackey formula holds if both 𝐏 and 𝐐 are F-stable. The remaining properties (P4) and (P6) are established by using the fact that (H1) holds for all proper Levi subgroups of 𝐆 and that there exists a unipotently supported class function μ∈Classuni⁡(𝐌F) satisfying Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ)≠0. In particular, the Mackey formula holds for all proper Levi subgroups of 𝐆.

It is already established in [5, Lemma (E7)] that if the quasi-simple components of 𝐆 are of type 𝖤7 then there is no pair (𝐌,𝐐) satisfying (P3) to (P6). Hence we can assume that all the quasi-simple components are of type 𝖤6. As 𝐆 is adjoint and p=2, the only possible choice for 𝐌 satisfying (P3) and (P4) is a Levi subgroup of type 𝖣4, see [9, Section 15.1]. However the exact same argument used in the proof of [5, Lemma 2.2 (f)] shows that no such Levi subgroup can satisfy both (P5) and (P6). This completes the proof. ∎

Proof of Theorem 1.10.

Assume for a contradiction that μ∈Classuni⁡(𝐌F) is a unipotently supported class function satisfying Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ)≠0. By [5, Theorem 3.9] we can assume that q=2. By Theorem 1.6 and Lemma 3.3 and compatibility with direct products we can assume that all the quasi-simple components of 𝐆 are of type 𝖤8 and that F cyclically permutes these quasi-simple components. Note that 𝐆 is necessarily semisimple and simply connected.

We note that any proper F-stable Levi subgroup of 𝐆 has connected centre and has no quasi-simple component of type 𝖤8. Thus by Theorem 1.6 the Mackey formula holds for any proper F-stable Levi subgroup. With this we may argue as above, and exactly as in the proof of [5, Theorem 3.9], that the pair (𝐌,𝐐) satisfies the properties (P1) to (P6) of [5, Proposition 2.1]. However, [5, Proposition 2.1] establishes precisely that there is no such pair (𝐌,𝐐) satisfying these properties, so we must have Δ𝐋⊂𝐏,𝐌⊂𝐐𝐆⁢(μ)=0. ∎