Counting conjectures and $e$-local structures in finite reductive groups

We prove new results in generalized Harish-Chandra theory providing a description of the so-called Brauer--Lusztig blocks in terms of the information encoded in the $\ell$-adic cohomology of Deligne--Lusztig varieties. Then, we propose new conjectures for finite reductive groups by considering geometric analogues of the $\ell$-local structures that lie at the heart of the local-global counting conjectures. For large primes, our conjectures coincide with the counting conjectures thanks to a connection established by Brou\'e, Fong and Srinivasan between $\ell$-structures and their geometric counterpart. Finally, using the description of Brauer--Lusztig blocks mentioned above, we reduce our conjectures to the verification of Clifford theoretic properties expected from certain parametrisation of generalised Harish-Chandra series.


Introduction
Over the past few decades the research in representation theory of finite groups has been driven by the pursuit of an explanation of the so-called local-global principle. This states that for each prime number ℓ dividing the order of a finite group G, the ℓ-modular representation theory of G is largely determined by the ℓ-local structure of the group G. The local-global principle is supported by numerous conjectural evidences including the McKay Conjecture, the Alperin-McKay Conjecture, Alperin's Weight Conjecture and Brauer's Height Zero Conjecture among others.
In the 1990s, extending a connection made by Knörr and Robinson between local-global conjectures and the Brown complex associated to chains of ℓ-subgroups, Dade introduced a new conjecture known as Dade's Projective Conjecture. This provides a unifying statement which implies all of the local-global conjectures mentioned above [Dad92], [Dad94]. More recently, Dade's Projective Conjecture has been reduced to the verification of the so-called inductive condition for Dade's Conjecture for finite quasi-simple groups [Spä17]. This inductive condition can be stated in terms of Späth's Character Triple Conjecture (see [Spä17,Conjecture 6.3]).
When considering large primes in non-defining characteristic, work of Broué, Fong and Srinivasan shows that the ℓ-local structure of a finite reductive group and the associated Brown complex can be seen as a shadow of geometric objects arising from the underlining linear algebraic group (see Section 7.2). Building on this idea, in this paper we propose new conjectures for finite reductive groups that can be seen as geometric realisations of the local-global counting conjectures (see Sections 5.1 and Section 5.2). These new conjectures imply the counting conjectures under the assumptions considered above (see Section 7.3). Remarkably, our conjectures can be explained within the framework of generalised Harish-Chandra theory and, in fact, they reduce to the verification of Clifford theoretic properties expected from certain parametrisation of generalised Harish-Chandra series. In order to prove these results, we first prove new modular representation theoretic results for finite reductive groups by studying the decomposition of certain virtual representations constructed from the ℓ-adic cohomology of Deligne-Lusztig varieties.

e-Harish-Chandra theory
Let G be a connected reductive group defined over an algebraic closure F of a finite field of characteristic p, F ∶ G → G a Frobenius endomorphism endowing G with an F q -structure for some power q of p and G F the finite reductive group consisting of the F q -rational points. Fix a prime number ℓ not dividing q and denote by e the multiplicative order of q modulo ℓ (modulo 4 if ℓ = 2). All modular representation theoretic notions are considered with respect to the prime ℓ. Let (G * , F * ) be in duality with (G, F ). Blocks of finite reductive groups have been parametrised by work of Fong-Srinivasan [FS82], [FS86], Broué-Malle-Michel [BMM93], Cabanes-Enguehard [CE94], [CE99], Enguehard [Eng00] and Kessar-Malle [KM13], [KM15]. Given this parametrisation, we then need to understand the distribution of characters into such blocks. For this purpose, recall that the set Irr(G F ) of irreducible characters of G F admits a partition where B runs over the set of Brauer ℓ-blocks of G F , s runs over the set of semisimple elements in G * F * up to (rational) conjugation and E(G F , [s]) is the rational Lusztig series associated to s. Using a terminology introduced by Broué, Fong and Srinivasan, we call each non-empty intersection a Brauer-Lusztig block. In particular, each ℓ-block B is a union of Brauer-Lusztig blocks and therefore, in order to understand the distribution of characters into ℓ-blocks, we need to describe Brauer-Lusztig blocks. Our first main result provides such a description in terms of e-Harish-Chandra series defined in terms of the ℓ-adic cohomology of Deligne-Lusztig varieties. where the union runs over the G F -conjugacy classes of (e, s)-cuspidal pairs (L, λ) (see Definition 3.1) such that bl(λ) G F = B via Brauer induction of ℓ-blocks.
We point out that Hypothesis 4.1 is satisfied in most of the cases of interest and, in particular, whenever [G, G] is simply connected with no irreducible rational components of type 2 E 6 (2), E 7 (2), E 8 (2) while considering ℓ ∈ Γ(G, F ) with ℓ ≥ 5 (see Remark 4.2). Moreover, in this case Brauer's induction of blocks is defined (see the discussion preceding Lemma 4.6).
From the perspective of e-Harish-Chandra theory, Theorem A can be seen as an extension of results of Cabanes-Enguehard (see [CE99,Theorem 4.1]) to e-cuspidal pairs associated to ℓ-singular semisimple elements. In addition, Theorem A provides a generalisation of [BMM93, Theorem 3.2 (1)] to non-unipotent characters. This provides a uniform formulation for e-Harish-Chandra theory by considering arbitrary e-cuspidal pairs. In fact in Corollary 4.12, we show that for every e ≥ 1 the set of irreducible characters of G F is partitioned into e-Harish-Chandra series provided that there exists a good prime ℓ for which e is the order of q modulo ℓ. It is worth pointing out that this latter statement does not depend on the choice of the prime ℓ while, relying on block theoretic techniques, it's proof does. However, such a partition should existence for every e ≥ 1 independently on the restrictions considered here.
On the way to prove Theorem A, we obtain two results that are of independent interest. First in Corollary 4.9, we prove an extension of [CE99,Theorem 2.5] that shows how Deligne-Lusztig induction preserves the decomposition into ℓ-blocks. More precisely, under Hypothesis 4.1 we show that for every ℓ-block b L of an e-split Levi subgroup L of G the irreducible constituents of the virtual representations obtained via Deligne-Lusztig induction from any λ ∈ Irr(b L ) belong to a unique ℓblock b G of G F . Secondly in Corollary 4.11, we give a partial solution to a conjecture (see [CE99,Notation 1.11] and Conjecture 3.2) introduced by Cabanes and Enguehard on the transitivity of a certain relation ≤ e defined on the set of e-pairs (see also Proposition 4.5).
As an immediate consequence of Theorem A, we obtain the above-mentioned description of all the characters in any given ℓ-block by considering the union over all conjugacy classes of semisimple elements of G * F * . Namely, for every ℓ-block B of G F there is a partition where the union runs over the G F -conjugacy classes of e-cuspidal pairs (L, λ) such that bl(λ) G F = B via Brauer induction of ℓ-blocks (see Theorem 4.13). Given this partition, the next natural step to understand the distribution of characters into ℓ-blocks of finite reductive groups is to find a parametrisation of the characters in each e-Harish-Chandra series E(G F , (L, λ)). Inspired by classical Harish-Chandra theory and by results of Broué, Malle and Michel for unipotent characters (see [BMM93,Theorem 3.2]), we propose a parametrisation for arbitrary e-Harish-Chandra series which is additionally compatible with Clifford theory and with the action of automorphisms. This Clifford theoretic compatibility is expressed via G F -block isomorphisms of character triples as defined in [Spä17,Definition 3.6].
Parametrisation B. Let G, F , ℓ, q and e be as above and consider an e-cuspidal pair (L, λ) of G. There exists a defect preserving Aut F (G F ) (L,λ) -equivariant bijection in the sense of [Spä17, Definition 3.6] for every ϑ ∈ E G F , (L, λ) and where X ∶= G F ⋊ Aut F (G F ).
It is the author's belief that existence of the above parametrisation should provide an explaination for the validity of the inductive conditions for the local-global conjectures for finite reductive groups in non-defining characteristic. As a matter of fact, similar bijections have been used in [MS16] to verify the McKay Conjecture for the prime ℓ = 2 and then in [Ruh22] to prove the Alperin-McKay Conjecture and Brauer's Height Zero Conjecture for ℓ = 2. Regarding the validity of the above parametrisation, in [Ros22b] we show that in order to obtain Parametrisation B it is enough to verify certain requirements on the extendibility of characters of e-split Levi subgroups. These also appear in the proofs of the inductive conditions for the McKay, the Alperin-McKay and the Alperin Weight conjectures and the checking of these requirements is part of an ongoing project in representation theory of finite reductive groups (see [MS16], [CS17a], [CS17b], [CS19], [BS20], [Spä21], [Bro22]).

Counting conjectures via e-local structures
Our next aim is to provide a geometric realisation of the local-global principle for finite reductive groups by replacing ℓ-local structures with e-local structures. For this purpose we first need some notation. We keep G, F , q, ℓ and e as previously defined. Denote by L e (G, F ) >0 the set of nontrivial descending chains of e-split Levi subgroups σ = {G = L 0 > L 1 ⋅ ⋅ ⋅ > L n } and define the length of σ as σ ∶= n > 0. We denote by G F σ the stabiliser of a chain σ under the action of G F . Then, as explained in Section 5.1, we can associate to every ℓ-block b of G F σ an ℓ-block R G Gσ (b) of G F . For every ℓ-block B of G F and d ≥ 0, we denote by k d (B σ ) the number of irreducible characters ϑ of G F σ with ℓ-defect d and such that the ℓ-block of ϑ in G F σ correspond to B via R G Gσ . Moreover, let k d (B) and k d c (B) be the number of irreducible and e-cuspidal characters respectively, with ℓ-defect d and belonging to the ℓ-block B. Our first conjecture proposes a formula to count the number of characters of any given defect in any ℓ-block in terms of e-local data.
Conjecture C. Let B be an ℓ-block of G F and d ≥ 0. Then where σ runs over a set of representatives for the action of G F on L e (G, F ) >0 .
Conjecture 5.1 provides a geometric form of Dade's Conjecture for finite reductive groups. In fact, in Section 7 we show that the two statements coincide when the prime ℓ is large for G (see Proposition 7.10). We expect this connection to hold for good primes as well and we suspect that this could be connected to the existence of a homotopy equivalence between the Brown complex and the simplicial complex associated to L e (G, F ) >0 . In the spirit of Conjecture C, in Section 5 we introduce a statement (see Conjecture 5.2) which relates to Alperin's Weight Conjecture. As for the Alperin-McKay Conjecture and its inductive condition, at least for large primes, we identify a geometric counterpart in Parametrisation B (see Proposition 7.11 and Proposition 7.15).
The numerical phenomena proposed in Conjecture C as well as in the local-global counting conjectures are believed to be consequences of a deeper underlying theory. For blocks with abelian defect groups, Broué has suggested a structural explanation which predicts the existence of certain derived equivalences [Bro90]. In a different direction, work of Isaacs, Malle and Navarro [IMN07], followed by Navarro and Späth [NS14,Theorem 7.1] (see also [Ros22c]) and Späth [Spä17, Conjecture 1.2], suggests a description of Clifford theoretic properties hidden behind the counting conjectures by studying certain relations on sets of character triples. Exploiting this second approach, we now introduce a more conceptual background for Conjecture C by introducing G F -block isomorphisms of character triples in this context.
In Section 5.2, for every ℓ-block B, d ≥ 0 and ǫ = ±, we introduce a set consisting of a chain σ ∈ L e (G, F ) and a certain irreducible character ϑ of G F σ . With this notation, we can now present our second conjecture.
As a first application of Theorem A, we establish a connection between Conjecture C and Conjecture D. More precisely, we show that for a fixed σ ∈ L d (G, F ) >0 the number k d (B σ ) coincides with the number of quadruples ω for which there is a character ϑ ∈ Irr(G F σ ) such that (σ, ϑ) ∈ ω • . On the other hand k d (B) − k d c (B) coincide with the number of quadruples ω for which there exists a character χ of G F such that (σ 0 , χ) ∈ ω • and where σ 0 = {G} is the trivial chain. In particular we see that Conjecture D implies Conjecture C.
Theorem E. Assume Hypothesis 4.1 and consider an ℓ-block B and d ≥ 0. If Conjecture D holds for B and d ≥ 0, then Conjecture C holds for B and d ≥ 0.
In analogy with the inductive conditions for the local-global counting conjectures, Conjecture D provides a more conceptual explanation for the numerical phenomenon proposed by Conjecture C and, in particular, yields a description of the Clifford theoretic properties naturally arising in this context. Furthermore, in Section 7 we show that Conjecture D implies Späth's Character Triple Conjecture and the inductive condition for Dade's Conjecture for large primes under suitable assumptions (see Proposition 7.13 and Corollary 7.14).
Remarkably, Conjecture D can be explained within the framework of e-Harish-Chandra theory. Our final theorem shows that Conjecture D (and hence Conjecture C) is a consequence of Parametrisation B via an application of Theorem A. In what follows, we say that Parametrisation B holds for (G, F ) at the prime ℓ if it holds for every e-cuspidal pair (L, λ) of G where q is the prime power associated to F and e is the order of q modulo ℓ. Moreover, we refer the reader to Section 6 for the definition of irreducible rational components (see Definition 6.4).
Theorem F. Assume Hypothesis 4.1 and suppose that G is simply connected with Frobenius endomorphism F . If Parametrisation B holds at the prime ℓ for every irreducible rational component (H, F ) of every e-split Levi subgroup of G, then Conjecture D holds for the prime ℓ.
As a consequence of Theorem E, Theorem F and the results obtained in [Ros22b], our conjectures are now reduced to the verification of technical requirements on character extendibility from e-split Levi subgroups in finite reductive groups of irreducible rational type.

Reader's guide
The paper is organised as follows. Section 2 contains the main notation and preliminary results on finite reductive groups. In Section 3, and more precisely in Proposition 3.6, we study the transitivity of a certain relation defined on the set of e-pairs and provide a solution to a fundamental case of a conjecture proposed by Cabanes-Enguehard in [CE99,Notation 1.11]. This is then extended to the ℓ-singular case in Corollary 4.11. Furthermore, in Proposition 4.5 we prove the conjecture inside e-Harish-Chandra series associated with certain good semisimple elements. Most importantly, in Section 4 we obtain a description of the distribution of characters into ℓ-blocks of finite reductive groups (see Theorem 4.13) and prove Theorem A (see Theorem 4.17). As a by-product, in Corollary 4.9 we prove an extension of [CE99, Theorem 2.5] to arbitrary e-cuspidal pairs while Corollary 4.12 show the existence of an e-Harish-Chandra partition of characters under the existence of a prime ℓ satisfying certain properties. In section 5, we introduce Conjecture C (see Conjecture 5.1), a similar statement relating to Alperin's Weight Conjecture (see Conjecture 5.2), Conjecture D (see Conjecture 5.4) and prove Theorem E (see Theorem 5.6). Here, we also consider Parametrisation B (see Parametrisation 5.8) in more detail and consider an interesting consequence in Remark 5.9. Section 6 is dedicated to the proof of Theorem F. This relies on Theorem A and requires a careful study of the interaction between Parametrisation B and Conjecture D. Finally, in Section 7 we make use of ideas of Broué, Fong and Srinivasan in order to show that our conjectures agree with the local-global counting conjectures for finite reductive groups and large primes.

Preliminaries
Throughout this paper, G is a connected reductive linear algebraic group defined over an algebraic closure F of a finite field of characteristic p and F ∶ G → G is a Frobenius endomorphism endowing G with an F q -structure for a power q of p. We denote by (G * , F * ) a group in duality with (G, F ) with respect to a choice of an F -stable maximal torus T of G and an F * -stable maximal torus T * of G * . In this case, there exists a bijection L ↦ L * between the set of Levi subgroups of G containing T and the set of Levi subgroups of G * containing T * (see [CE04,p.123]). This bijection induces a correspondence between the set of F -stable Levi subgroups of G and the set of F * -stable Levi subgroups of G * . Moreover, it is compatible with the action of G F and G * F * .

Automorphisms
Let G and F be as above. If σ ∶ G → G is a bijective morphism of algebraic groups satisfying σ ○ F = F ○ σ, then the restriction of σ to G F , which by abuse of notation we denote again by σ, is an automorphism of the finite group G F . We denote by Aut F (G F ) the set of those automorphisms of G F obtained in this way. As mentioned in [CS13, Section 2.4], a morphism σ ∈ Aut F (G F ) is determined by its restriction to G F up to a power of F . It follows that Aut F (G F ) acts on the set of F -stable closed connected subgroups H of G. In particular, for any F -stable closed connected subgroup H of G, there is a well defined set Aut F (G F ) H whose elements are the restrictions to G F of those morphisms σ as above that stabilize H. When G is a simple algebraic group of simply connected type such that G F Z(G F ) is a non-abelian simple group, then we have Aut F (G F ) = Aut(G F ) (see [GLS98,Section 1.15] and the comments in [CS13, Section 2.4]).
Assume now that G is simple of simply connected type. Fix a maximally split torus T 0 contained in an F -stable Borel subgroup B 0 of G. This choice corresponds to a set of simple roots ∆ ⊆ Φ ∶= Φ(G, T 0 ). For every α ∈ Φ consider a one-parameter subgroup x α ∶ G a → G. Then G is generated by the elements x α (t), where t ∈ G a and α ∈ ±∆. Consider the field endomorphism F 0 ∶ G → G given by F 0 (x α (t)) ∶= x α (t p ) for every t ∈ G a and α ∈ Φ. Moreover, for every symmetry γ of the Dynkin diagram of ∆, we have a graph automorphism γ ∶ G → G given by γ(x α (t)) ∶= x γ(α) (t) for every t ∈ G a and α ∈ ±∆. Then, up to inner automorphisms of G, any Frobenius endomorphism F defining an F q -structure on G can be written as F = F m 0 γ, for some symmetry γ and m ∈ Z with q = p m (see [MT11,Theorem 22.5]). One can construct a regular embedding G ≤G in such a way that the Frobenius endomorphism F 0 extends to an algebraic group endomorphism F 0 ∶G →G defining an F p -structure onG. Moreover, every graph automorphism γ can be extended to an algebraic group automorphism ofG commuting with F 0 (see [MS16,Section 2B]). If we denote by A the group generated by γ and F 0 , then we can construct the semidirect productG F ⋊ A. Finally, we define the set of diagonal automorphisms of G F to be the set of those automorphisms induced by the action ofG F on G F . If G F Z(G F ) is a non-abelian simple group, then the groupG F ⋊ A acts on G F and induces all the automorphisms of G F (see, for instance, the proof of [Spä12, Proposition 3.4] and of [CS19, Theorem 2.4]).
We conclude this section by recalling an important property that is needed in Section 6.
Lemma 2.1. Let G,G, F and A as in the above paragraph and suppose that G F is the universal Proof. By the above paragraph, we know thatG F A CG F A (G F ) ≃ Aut(G F ) and therefore, using the fact that On the other hand, since the third isomorphism theorem yields the desired isomorphism.

Good primes and e-split Levi subgroups
For the rest of this section we consider the following setting.
Notation 2.2. Let G be a connected reductive linear algebraic group defined over an algebraic closure F of a finite field of characteristic p and F ∶ G → G a Frobenius endomorphism defining an F q -structure on G, for a power q of p. Consider a prime ℓ different from p and denote by e the multiplicative order of q modulo ℓ (modulo 4 if ℓ = 2). All blocks are considered with respect to the prime ℓ.
In what follows we make some restrictions on the prime ℓ. First, recall that ℓ is a good prime for G if it is good for each simple factor of G, while the conditions for the simple factors are as follows: A n ∶ every prime is good We say that ℓ is a bad prime for G if it is not a good prime. Next, we introduce the set of primes Γ(G, F ) from [CE94, Notation 1.1].
As a consequence, if a connected reductive group G has no simple components of type A, then ℓ ∈ Γ(G, F ) if and only if ℓ is good for G and ℓ ≠ p.
In this paper we make use of the terminology of Sylow Φ e -theory introduced in [BM92] (see also [BMM93]). For a set of positive integers E, we say that an F -stable Levi subgroup T of G is a Φ Etorus if its order polynomial is of the form P (T,F ) = ∏ n∈E Φ an n for some integers a n and where Φ n denotes the n-th cyclotomic polynomial (see [CE04,Definition 13.3]). The centralisers of Φ E -tori are called E-split Levi subgroups. If E = {e}, we call Φ {e} -tori and {e}-split Levi subgroups simply Φ e -tori and e-split Levi subgroups respectively. When ℓ ∈ Γ(G, F ), some significant consequences on the structure of e-split Levi subgroups can be drawn.
Proof. The first statement follows directly from the definition. In fact, since Z ○ (L) is a torus, we deduce that Z ○ (L) Φ E is a Φ E -torus and therefore C G (Z ○ (L) Φ E ) is E-split. Conversely, assume that L is E-split. Then there exists a Φ E -torus T such that L = C G (T). Since T is abelian, we deduce that T ≤ Z(L). Then, as T is connected, we have T ≤ Z ○ (L) and therefore T ≤ For the second statement see [CE04,Proposition 13.19].
Lemma 2.6. Let ℓ ∈ Γ(G, F ) and consider an ℓ-subgroup Y of G F . Then: We say that a prime ℓ is large for (G, F ) if there exists a unique integer e 0 such that Φ e 0 divides the order polinomial P (G,F ) and ℓ divides Φ e 0 (q) (see [BMM93,  For any finite ℓ-group H and positive integer n we define the subgroup Ω n (H) ∶= ⟨h ∈ H h ℓ n = 1⟩. In particular, when H is abelian, Ω 1 (H) is the largest ℓ-elementary abelian subgroup of H.
Proposition 2.7. Suppose that ℓ is large for G and consider an e-split Levi subgroup L of G and an ℓ-subgroup Y of G F . Then: Proof. By [CE04, Proposition 13.16 (ii)] and Lemma 2.6 (ii) we know that is an e 0 -split Levi subgroup. This proves (i).
Next, set S ∶= Z ○ (L) Φe . It is enough to show that S ≰ Z(G) implies Ω 1 (S F ℓ ) ≰ Z(G). Since S is a Φ e -torus, we have S ≰ Z(G) if and only if S ≰ Z ○ (G) Φe while, using the fact that ℓ is large, we deduce that Z(G) F ℓ = (Z ○ (G) Φe ) F ℓ and therefore Ω 1 (S F ℓ ) ≰ Z(G) if and only if Ω 1 (S F ℓ ) ≰ Z ○ (G) Φe . Hence, in order to obtain (ii) we need to show that S ≰ Z ○ (G) Φe implies Ω 1 (S F ℓ ) ≰ Z ○ (G) Φe . Assume that S ≰ Z ○ (G) Φe =∶ T e and consider the canonical morphism π e ∶ G → G T e . Observe, that T e ≤ S and that S and π e (S) ≠ 1 are Φ e -tori. If ℓ a is the largest power of ℓ dividing Φ e (q), then T F ℓ is the direct product of copies of C ℓ a for every Φ e -torus T (see [BM92,Proposition 3.3]). Let y ∈ π e (S) F ℓ be an element of order ℓ a . Proceeding as in the proof of [CE04, Lemma 13.17 (i)] and noticing that (S T e ) F = S F T F e , we deduce that π e (S) F ℓ = π e (S F ℓ ) and hence there exists x ∈ S F ℓ such that π e (x) = y. Now, the order of y divides the order of x while the order of x must be less or equal than ℓ a by the description of S F ℓ given above. We conclude that x has order ℓ a . Then s ∶= x ℓ a−1 ∈ Ω 1 (S F ℓ ) and π e (s) = y ℓ a−1 ≠ 1. This shows that Ω 1 (S F ℓ ) ≰ T e = Z ○ (G) Φe . To prove (iii), we proceed by induction on the dimension of G. Set S ∶= Z ○ (L) Φe and notice that Observe that K is a Levi subgroup of G by [CE04,Proposition 13.16 (ii)]. If S ≤ Z(G), then K = G = L. Therefore, we can assume S ≰ Z(G). By the above paragraph, we obtain Ω 1 (S F ℓ ) ≰ Z(G) and therefore dim(K) < dim(G). Noticing that ℓ is large for K and that L is an e-split Levi subgroup of K, the inductive hypothesis yields L = C ○ K (Ω 1 (S F ℓ )). The result follows by noticing that C ○ K (Ω 1 (S F ℓ )) = K.

Deligne-Lusztig induction and blocks
Let G, F , q, ℓ and e be as in Notation 2.2 and consider an F -stable Levi subgroup of a (not necessarily F -stable) parabolic subgroup P of G. By tensoring with a (G F , L F )-bimodules arising from the ℓ-adic cohomology of Deligne-Lusztig varieties, Deligne-Lusztig [DL76] (in the case where L is a maximal torus) and Lusztig [Lus76] (in the general case) have defined a map with adjoint * R G L≤P ∶ ZIrr G F → ZIrr L F that we call Deligne-Lusztig induction and restriction respectively. Notice that these maps are often referred to simply as Lusztig induction and restriction, and the terms Deligne-Lusztig induction and restriction are only used when considering the case of a maximal torus. Nonetheless we believe that the contribution of Deligne should be acknowledged. It is expected that the map R G L≤P does not depend on the choice of the parabolic subgroup P and this would, for instance, follow from the Mackey formula which has been proved whenever G F does not have components of type 2 E 6 (2), E 7 (2) or E 8 (2) [BM11]. For simplicity, we just write R G L when the results are known not to depend on the choice of P. Similar remarks apply for Deligne-Lusztig restriction.
Recall that (L, λ) is an e-cuspidal pair of (G, F ) (or simply of G when no confusion arises) if L is an e-split Levi subgroup of G and λ ∈ Irr(L F ) satisfies * R L M≤Q (λ) = 0 for every e-split Levi subgroup M < L and every parabolic subgroup Q of L containing M as Levi complement.
To fix our notation, we now review the parametrisation of blocks given in [CE99]. Let's assume ℓ ≥ 5 with ℓ ≥ 7 if G has a component of type E 8 . Then for every B ∈ Bl(G F ) there exists a unique e-cuspidal pair (L, λ) up to G F -conjugation such that λ lies in a rational Lusztig series associated with an ℓ-regular semisimple element and all the irreducible constituents of R G L≤P (λ) belongs to the block B for every parabolic subgroup P of G having L as Levi complement. In this case we write B = b G F (L, λ). Moreover, [CE99, Theorem 2.5] implies that bl(λ) G F = B whenever ℓ ∈ Γ(G, F ) (see Lemma 2.5). See [KM15] for a generalisation of these results to all primes.

(e, ℓ ′ )-pairs and transitivity
In this section we provide new evidence for a conjecture proposed by Cabanes and Enguehard in [CE99, Notation 1.11]. Consider G, F , q, ℓ and e as in Notation 2.2. We start by defining the notion of e-pair and (e, s)-pair.
Definition 3.1. An e-pair of (G, F ) (or simply of G when no confusion arises) is a pair (L, λ) where L is an e-split Levi subgroup of G and λ ∈ Irr(L F ). For any semisimple element s ∈ G * F * , we say that an e-pair (L, λ) is an (e, s)-pair of (G, F ) if λ ∈ E(L F , [s ′ ]) for some s ′ ∈ L * F * that is G * F * -conjugate to s. Finally, we say that (L, λ) is an (e, ℓ ′ )-pair if it is an (e, s)-pair for some ℓ-regular semisimple element s ∈ G * F * .
If P e (G, F ) is the set of e-pairs of (G, F ), then there exists a binary relation on P e (G, F ) denoted by ≤ e (see [CE99,Notation 1.11]). Namely, write (L, λ) ≤ e (K, κ) provided that L ≤ K are e-split Levi subgroups of G and there exists a parabolic subgroup P of K containing L as a Levi complement such that κ is an irreducible constituent of the generalised character R K L≤P (λ). Noticing that Deligne-Lusztig induction sends characters to generalised characters, we observe that the relation ≤ e might not be transitive at first glance. We denote by ≪ e the transitive closure of ≤ e . Since P e (G, F ) is finite, we deduce that two e-pairs (L, λ) and With this notation, a pair (L, λ) is e-cuspidal if and only if it is a minimal element in the poset (P e (G, F ), ≪ e ). We denote by CP e (G, F ) the subset of P e (G, F ) consisting of e-cuspidal pairs.
Observe that, by [CE04,Proposition 15.7] the relations ≤ e and ≪ e restrict to the set of (e, s)-pairs for every s ∈ G * F * ss . A minimal element in the induced poset of (e, s)-pairs is called (e, s)-cuspidal. The following conjecture has been proposed in [CE99, Notation 1.11] and is inspired by [BMM93, Theorem 3.11].
In this section, we show that this conjecture holds when considering (e, ℓ ′ )-cuspidal pairs in groups of simply connected type under certain assumptions on ℓ. Before proceeding with the proof of this result, we point out an important consequence of Conjecture 3.2. Let (L, λ) be an e-pair of G. If Conjecture 3.2 holds, then is the e-Harish-Chandra series determined by (L, λ), that is the set of irreducible constituents of R G L≤P (λ) for every parabolic subgroup P of G having L as a Levi complement. In addition, if Deligne-Lusztig induction does not depend on the choice of a parabolic subgroup, then , where we recall that, for any finite group X and χ ∈ ZIrr(X), we denote by Irr(χ) the set of irreducible constituent of χ. Because this remark is used multiple times in Section 4, we introduce the following condition. Condition 3.3. Consider G, F , q, ℓ and e as in Notation 2.2 and assume that Deligne-Lusztig induction does not depend on the choice of parabolic subgroups and for every F -stable Levi subgroup K of G and every (e, ℓ ′ )-cuspidal pair (L, λ) of K.
Observe that Conjecture 3.2 is known for (e, 1)-pairs by [ In the next section we extend this result to e-pairs associated with ℓ-singular semisimple elements (see Corollary 4.11). Moreover, in Proposition 4.5 we prove Conjecture 3.2 inside e-Harish-Chandra series associated to certain semisimple elements. Notice that our proof does not depend on [Eng13] in any way.
Lemma 3.4. Let L be an e-split Levi subgroup of a connected reductive group G and consider G 0 ∶= Proof. First observe that L 0 is an e-split Levi subgroup of G 0 . By [GM20, Proposition 3.3.24] (see also the proof of [GM20, Corollary 3.3.25]) and since G = Z ○ (G)G 0 , it follows that Suppose now that (L, λ) ≤ e (G, χ) and let λ 0 be an irreducible constituent of λ L F 0 . Since Deligne-Lusztig induction and restriction are adjoint with respect to the usual scalar product, we deduce that The following result shows that Condition 3.3 holds when G has only components of classical types and ℓ ≥ 5 or when G is simple, K = G and λ lies in a rational Lusztig series associated with a quasiisolated element. Recall that a semisimple element s of a reductive group G is called quasi-isolated if C G (s) is not contained in any proper Levi subgroup of G.
Lemma 3.5. Let G be connected reductive, χ ∈ Irr(G F ) and consider an e-cuspidal pair (L, λ) ≪ e (G, χ), where λ ∈ E(L F , [s]) for some ℓ-regular semisimple element s ∈ L * F * . Suppose that ℓ ≥ 5 is good for G and that the Mackey formula holds for (G, F ). If either G has only components of classical types and F does not induce the triality automorphism on components of type D 4 or G is simple and s is quasi-isolated in G * , then (L, λ) ≤ e (G, χ).
Proof. Consider a regular embedding i ∶ G →G. By applying [BMM93, 3.11] together with [GM20, Theorem 4.7.2 and Corollary 4.7.8] toG, it follows that Conjecture 3.2 holds inG unless s is quasiisolated in G and G is simple of simply connected type E 6 or E 7 or G F = 3 D 4 (q). However, in these excluded cases the result holds by [Hol22, Theorem 1.1] and we can therefore assume that Conjecture 3.2 holds inG. Now, the result follows by applying [CE99, Proposition 5.2].
We can now prove the main result of this section. Recall that for a connected reductive group G, we say that G is simply connected if the semisimple group [G, G] is simply connected.
Without loss of generality we can hence assume Now, G is a direct product of simple algebraic groups H 1 , . . . , H n (see [Mar91,Proposition 1.4.10]). The action of F induces a permutation on the set of simple components H i . For every orbit of F we denote by G j , j = 1, . . . , t, the direct product of the simple components in such an orbit. Then G j is F -stable and Define L j ∶= L ∩ G j and observe that L j is an e-split Levi subgroup of G j and that Then we can write Without loss of generality, we may thus assume that F is transitive on the set of simple components H i or equivalently that t = 1.

Now, consider a simple component H of G and observe that there are isomorphisms
and where n is the number of simple components H i of G. Let M ∶= L ∩ H and notice that M is an e-split Levi subgroup of (H, F n ) and that the isomorphism from (3.3) restricts to an isomorphism correspond to λ ∈ Irr(L F ) via the isomorphism (3.5). Since (L, λ) << e (G, χ), we deduce that (M, µ) << e (H, ψ). Moreover, as s is quasi-isolated in G * , it follows that the semisimple element t ∈ H * F * n obtained via the isomorphism (3.4) is quasi-isolated in H * . Finally, Lemma 3.5 implies that (M, µ) ≤ e (H, ψ) and we hence conclude that (L, λ) ≤ e (G, χ).
Since the hypotheses of the above proposition are inherited by Levi subgroups, it follows that Condition 3.3 holds whenever G is a simply connected reductive group.
Corollary 3.7. Let G be a simply connected reductive group. Then Condition 3.3 holds for every F -stable Levi subgroup K of G and every (e, ℓ ′ )-cuspidal pair (L, λ) of K.

Brauer-Lusztig blocks and e-Harish-Chandra series
In this section we prove Theorem A and hence obtain a description of the distribution of characters into blocks for finite reductive groups in non-defining characteristic under suitable assumptions on ℓ.
As already mentioned in Section 2.3, under certain assumptions on ℓ, the results of [CE99, Theorem 4.1] show that for every block B ∈ Bl(G F ) there exists a unique G F -conjugacy class of e-cuspidal pairs (L, λ) such that λ ∈ E(L F , [s]) for some ℓ-regular semisimple element s ∈ L * F * and every irreducible constituent of R G L≤P (λ) is contained in Irr(B) for every parabolic subgroup P of G containing L as Levi complement. [CE99,Theorem 4.1] also provides a characterisation of the set of so-called ℓ ′ -characters in the block B as (4.1) On the other hand, [BM89, Theorem 2.2] shows that where t runs over the elements of C G * (s) F * ℓ up to conjugation and E(G F , B, [st]) is a Brauer-Lusztig block (see Definition 4.15). In particular, in order to obtain all the characters in Irr(B), we have to describe the Brauer-Lusztig blocks E(G F , B, [st]). Now, by using Corollary 3.7, the equality (4.1) can be restated as showing that those Brauer-Lusztig blocks associated with ℓ-regular semisimple elements coincide with e-Harish-Chandra series. Our aim is to provide a similar description for arbitrary Brauer-Lusztig blocks and remove the restriction on s being ℓ-regular.

e-Harish-Chandra series and ℓ-blocks
Throughout this section we assume the following conditions. Hypothesis 4.1. Let G, F ∶ G → G, q, ℓ and e be as in Notation 2.2. Assume that: Observe that the Mackey formula and Condition 3.3 are expected to hold for any connected reductive group. Moreover, it follows from Remark 2.4 that Hypothesis 4.1 is inherited by F -stable Levi subgroups.
We now start working towards a proof of Theorem A. First, we show how to associate to every (e, s)-pair an (e, s ℓ ′ )-pair via Jordan decomposition. This can be used to extend some of the results of [CE99] from (e, ℓ ′ )-pairs to arbitrary e-pairs.
Proof. Under our assumptions, Lemma 2.6 (iii) implies that Next, we show that the relation ≪ e is preserved under the construction of Proof. First, using [Hol22, Proposition 2.12] notice that every e-split Levi subgroup of G(s ℓ ) is of the form M(s ℓ ) for some e-split Levi subgroup M of G. Then, since ≪ e is a pre-order on a finite poset, proceeding by induction it is enough to show that Suppose first that (L, λ) ≤ e (K, κ) and assume without loss of generality that s ∈ L * . We know that κ is an irreducible constituent of R K L (λ). By the transitivity of Deligne-Lusztig induction (see [GM20, Theorem 3.3.6]), we have Moreover, by [CE04,Proposition 15.7], every irreducible constituent of R [Bon06,10.2]). The reverse implication follows from a similar argument.  Proof. Let (L, λ) ≪ e (K, κ) be (e, s)-pairs. Replacing s with a G F -conjugate, we can assume that s ∈ L * . Since no bad prime divide o(s), a repeated application of [CE04,Proposition 13.16 . Now proceeding as in Lemma 4.3 we construct unipotent e-pairs (L(s), λ(s)) and (K(s), κ(s)) of G(s). Arguing as in Lemma 4.4 we deduce from (L, λ) ≪ e (K, κ) that (L(s), λ(s)) ≪ e (K(s), κ(s)). Applying [BMM93, Theorem 3.11] we get (L(s), λ(s)) ≤ e (K(s), κ(s)) and again proceeding as in the proof of Lemma 4.4 we conclude that (L, λ) ≤ e (K, κ).
The following lemma is a fundamental ingredient to understand the distribution of characters into blocks. The proof is based on an idea used first in [CE94] in order to deal with unipotent blocks. Notice that, if ℓ ∈ Γ(G, F ) and L is an e-split Levi subgroup of G, then L F = C G F (Q) for some abelian ℓ-subgroup Q ≤ G F by Lemma 2.5 together with Lemma 2.6 (i). Therefore, block induction from L F to G F is defined by [Nav98, Theorem 4.14].
As a corollary we deduce that the construction given in Lemma 4.3 preserves the decomposition of characters into blocks. Proof. Let c be the block of L(s ℓ ) containing λ 1 (s ℓ ) and λ 2 (s ℓ ) and consider an e-cuspidal pair (M, µ) such that c = b L(s ℓ ) F (M, µ). Then, Lemma 4.6 implies that bl(λ 1 ) = bl(µ) L F = bl(λ 2 ).
The next corollary is basically a restatement of Proposition 4.8.
Finally, we show that for every e-pair (K, κ) there exists a unique e-cuspidal pair (L, λ) up to K Fconjugation satisfying (L, λ) ≤ e (K, κ). Observe that our next results also extends Proposition 3.6 to e-pairs associated with ℓ-singular semisimple elements.
Proposition 4.10 shows in particular that Conjecture 3.2 holds when working above e-cuspidal pairs. Corollary 4.11. Assume Hypothesis 4.1. Then the relation ≤ e and ≪ e from Conjecture 3.2 coincide on the subset CP e (G, F ) × P e (G, F ) ⊆ P e (G, F ) × P e (G, F ).
As an immediate consequence of Proposition 4.10 we deduce that the set Irr(K F ) is a disjoint union of e-Harish-Chandra series. This should be compared with the classical Harish-Chandra theory (see [GM20, Corollary 3.1.17]) and with the analogous result for unipotent characters [GM20, Theorem 4.6.20]. These two results, can be recovered by considering (1, s)-pairs and (e, 1)-pairs respectively.
Corollary 4.12. Let G be a connected reductive group with a Frobenius endomorphism F defining an F q -structure on G and consider an integer e ≥ 1. For every e-split Levi subgroup K of G there is a partition where the union runs over a K F -transversal in the set of e-cuspidal pairs of K, provided that there exists a prime ℓ such that Hypothesis 4.1 is satisfied with respect to (G, F, q, ℓ, e).
Observe that although the proof of the above corollary depends on the choice of a certain prime ℓ, its statement does not. This is due to the fact that our result is obtained as a consequence of ℓ-modular representation theoretic techniques. Nonetheless, a partition of characters into e-Harish-Chandra series is expected to hold without the restrictions considered here.
Next, combining Corollary 4.12 and Proposition 4.8 we can describe all the characters in the blocks of K F in terms of e-Harish-Chandra series.

Brauer-Lusztig blocks
We now extend Theorem 4.13 in order to obtain Theorem A. To start, following Broué, Fong and Srinivasan, we define the Brauer-Lusztig blocks of G F . where B is a block of G F and s is a semisimple element of G * F * . In this case, we say that (G, B, [s]) is the associated Brauer-Lusztig triple of G F . Moreover, we denote by BL(G, F ) the set of all Brauer-Lusztig triples of G F . We also define the set where L runs over all e-split Levi subgroups of G.
Next, assume ℓ ∈ Γ(G, F ). Recall from the discussion preceding Lemma 4.6 that, for every e-split Levi subgroup L of G and b ∈ Bl(L F ), the Brauer induced block b G F is defined. Then, we can introduce a partial order relation on BL ∨ (G, F ) by defining In the next lemma we compare the relation ≤ on Brauer-Lusztig triples with the relations ≪ e and ≤ e on e-pairs.
Lemma 4.16. Assume Hypothesis 4.1. Let L and K be e-split Levi subgroups of G and consider semisimple elements s ∈ L * F * and t ∈ K * F * .    As we have mentioned before, the results obtained by Cabanes and Enguehard have been extended to all primes by Kessar and Malle in [KM15] and the reader might wonder why we are not considering this more general situation. Unfortunately, many of the techniques used in this section fail for bad primes and a different proof needs to be found in this case.

Defect zero characters and e-cuspidal pairs
Recall that for an irreducible character χ of a finite group G, the ℓ-defect of χ is the non-negative integer d(χ) defined by ℓ d(χ) χ(1) ℓ = G ℓ . Our next result shows a necessary condition for an e-cuspidal character of G to be of ℓ-defect zero.

This implies
Combining (4.5) and (4.6) we see that it is enough to show that Z ∶= Z ○ (H) F * ℓ = 1. To do so, observe that Z ○ (G * ) Φe = Z ○ (H) Φe by [CE99, Proposition 1.10]. In particular, for every e-split Levi subgroup K * of G * containing H, we have K * = G * . Notice that H ≤ C G * (Z) and that C G * (Z) is an e-split Levi subgroup of G * by Proposition 2.7 (i). Therefore C G * (Z) = G * and Z ≤ Z(G * ) F * ℓ = 1. This shows that χ has defect zero. To conclude, we notice that χ is the only character in its block B while using [Hiß90] we obtain Irr(B) ∩ E(G F , ℓ ′ ) ≠ ∅. It follows that χ ∈ E(G F , ℓ ′ ).

New conjectures for finite reductive groups
For any prime number ℓ, it is expected that the ℓ-modular representation theory of a finite group is strongly determined by its ℓ-local structure. This belief is supported by numerous results and conjectural evidences. If G F is a finite reductive group defined over F q with ℓ not dividing q, then its ℓ-local structure is closely related to its e-local structure where e is the order of q modulo ℓ (modulo 4 if ℓ = 2). For instance, under suitable restrictions on the prime ℓ, every e-split Levi subgroup L of G gives rise to an ℓ-local subgroup of G F . Namely, L F = C G F (Z(L) F ℓ ) is the centraliser of an ℓ-subgroup in G F (see Lemma 2.5 and Lemma 2.6 (i)). Using this idea, we can then try to determine a link between the ℓ-modular representation theory of G F and the e-local structure of G. In this section we propose new conjectures that can be seen as analogues for finite reductive groups of the so-called counting conjectures. In Section 7 we compare our statements with the counting conjectures and show that they coincide whenever the prime ℓ is large enough.

Counting characters e-locally
Let G, F , q, ℓ and e be as in Notation 2.2. Denote by L(G, F ), or simply by L(G) when F is clear from the context, the set of descending chains of e-split Levi subgroups σ = {G = L 0 > L 1 ⋅ ⋅ ⋅ > L n } and define the length of σ as σ ∶= n. We denote by L(σ) the smallest term of the chain σ. Consider the subset L(G) >0 of all chains of positive length. Notice that G F acts on L(G) and on L(G) >0 and denote by G F σ the stabiliser in G F of any chain σ. Next, using [CE99, Theorem 2.5] and [KM15, Theorem 3.4], we associate to every block b of G F σ a uniquely determined block of G F . In order to apply [KM15, Theorem 3.4], we assume that G is an F -stable Levi subgroup of a simple simply connected group whenever ℓ is bad for G. Let L ∶= L(σ) be the smallest term of σ and consider a block b L of L F covered by b. By [CE99, Theorem 2.5] and [KM15, Theorem 3.4], we deduce that there exists a unique block of G F , denoted by R G L (b L ) (see [CE99, Notation 2.6]), containing all irreducible constituents of R G L≤P (λ) for every parabolic subgroup P of G with Levi complement L and every λ ∈ Irr(b L ) ∩ E(L F , ℓ ′ ). Since b L is uniquely determined by b up to G F σ -conjugation, the block of G F is well defined. Now, for every ℓ-block B of G F and d ≥ 0, we denote by k d (B σ ) the cardinality of the set Moreover, we denote by k d (B) the cardinality of the set Irr d (B) consisting of the irreducible characters of B with defect d and by k d c (B) be the number of e-cuspidal characters in Irr d (B). With the above notation, we can present our conjecture which proposes a formula to count the number of characters of a given defect in a block in terms of e-local data. Notice that, since ecuspidal characters are minimal with respect to the e-structure of G, they should be interpreted as e-local objects. Here, we use the term e-local structure to indicate the collection of (proper) e-split Levi subgroups of G together with their normalisers and their intersections.
Conjecture 5.1. Let B be an ℓ-block of G F and d ≥ 0. Then where σ runs over a set of representatives for the action of G F on L(G) >0 .
Our statement can be considered as an adaptation of Dade's Conjecture to finite reductive groups. In fact, in Section 7 we show the two statements coincide when the prime ℓ is large for G (see Proposition 7.10). We also notice that by considering the contribution of characters of any defect, we can state a weak version of Conjecture 5.1 which could be interpreted as an analogue of Alperin's Weight Conjecture in the formulation given by Knörr where σ runs over a set of representatives for the action of G F on L(G) >0 .
In Section 7 we show that Conjecture 5.2 is equivalent to Alperin's Weight Conjecture for finite reductive groups and large primes. Indeed, in this case these two statment coincide with both Conjecture 5.1 and Dade's Conjecture (see Proposition 7.10). As it was the case the Knörr-Robinson reformulation of Alperin's Weight Conjecture, it is natural to ask whether the alternating sums presented in Conjecture 5.1 and Conjecture 5.2 can be expressed as the Euler characteristic of a chain complex. Such a complex has been constructed by using Bredon cohomology for Alperin's Weight Conjecture (see [Sym05] and [Lin05]). We claim that in this case we even have k(B σ ) = 0 for every σ ∈ L(G) >0 . In fact, if σ has smallest term L ∶= L(σ) < G and k d (B σ ) ≠ 0, then there exists a block b L of L F such that Irr(B) contains all constituent of R G L (λ) for any λ ∈ Irr(b L ) ∩ E(L F , ℓ ′ ). Since χ is the only character in B, it follows that (L, λ) ≤ e (G, χ). Since χ is e-cuspidal, this is a contradiction and hence k d (B σ ) = 0 for every σ ∈ L(G) >0 . Notice that, if ℓ is (G, F, e 0 )-adapted for some e 0 ≠ e, then G has no proper e-split Levi subgroups and Conjecture 5.1 holds trivially.

Introducing G F -block isomorphisms of character triples
Counting conjectures for finite groups provide numerical evidence that is believed to be consequence of an underlying structural theory. In this regard, Broué's Abelian Defect Group Conjecture [Bro90] proposes a structural explanation when considering the case of blocks with abelian defect groups. In a similar fashion, although perhaps on a more superficial level, the introduction by Isaacs, Malle and Navarro [IMN07] of the so-called inductive conditions for the counting conjectures has initiated a study of stronger conjectures which suggest a way to control Clifford theory via the use of relations on the set of character triples. This idea has been exploited further in [NS14,Theorem 7.1] for the Alperin-McKay Conjecture (see also [Ros22c] for the simpler McKay Conjecture) and in [Spä17, Conjecture 1.2] for Dade's Conjecture. In this section, we introduce G F -block isomorphisms of character triples in the context of Conjecture 5.1 and Conjecture 5.2. The equivalence relation on character triples that we consider here is denoted by ∼ G F and has been introduced in [Spä17, Definition 3.6]. We refer the reader to that paper for further details. Before proceeding further, we mentioned that in order to obtain the conditions on defect groups necessary to define G F -block isomorphisms of character triples we must assume ℓ ∈ Γ(G, F ).
Recall from Section 3 that, for a finite reductive group G F , we denote by CP e (G, F ) the set of all e-cuspidal pairs (L, λ) of (G, F ) and by CP e (G, F ) < the subset of e-cuspidal pairs (L, λ) with L < G. When ℓ ∈ Γ(G, F ) and B is a block of G F , we define the subsets CP e (B) and CP e (B) < consisting of those pairs (L, λ) in CP e (G, F ) and CP e (G, F ) < respectively such that bl(λ) G F = B. Recall that block induction is defined in this case as explained in the discussion preceding Lemma 4.6. As in [BMM93, Definition 2.18], let AbIrr(L F ) be the set of (linear) characters of L F containing [L, L] F in their kernel and, for a fixed character λ ∈ Irr(L F ), define , where L(G) ǫ is the subset of L(G) consisting of those chains σ satisfying (−1) σ = ǫ1 and the set Irr d (B σ E(L(σ) F , (M, Ab(µ)))) consists of those characters ϑ ∈ Irr(G F σ ) lying over some character in E(L(σ) F , (M, Ab(µ))) and such that d(ϑ) = d and bl(ϑ) G F = B. Notice that the group G F acts by conjugation on L d (B) ǫ and denote by L d (B) ǫ G F the corresponding set of G F -orbits. As usual, for (σ, M, Ab(µ), ϑ) ∈ L d (B) ǫ we denote the corresponding G F -orbit by (σ, M, Ab(µ), ϑ). Moreover, for every ω ∈ L d (B) ǫ G F we denote by ω • the G F -orbit of pairs (σ, ϑ) such that (σ, M, Ab(µ), ϑ) ∈ ω for some e-cuspidal pair (M, µ).
With the notation introduced above, we can now present a more conceptual framework for the conjectures presented in the previous section by considering bijections inducing G F -block isomorphisms of character triples.
Conjecture 5.4. Let ℓ ∈ Γ(G, F ) and consider a block B of G F and d ≥ 0. There exists an In analogy with the inductive conditions for the counting conjectures, the above statement should be understood as a version of Conjecture 5.1 compatible with Clifford theory and with the action of automorphisms. Although not completely satisfactory from a structural point of view, Conjecture 5.4 suggest a deeper explanation for the numerical phenomena proposed in Conjecture 5.1. By considering the contribution given by characters of any defect, we could introduce G F -block isomorphisms in the context of Conjecture 5.2. In Section 7, we show that Conjecture 5.4 implies Späth's Character Triple Conjecture for finite reductive groups and large primes (see Proposition 7.13).
We now explain in more details the connection between Conjecture 5.1 and Conjecture 5.4. First, we provide a more explicit description of the blocks and irreducible characters of stabilisers of chains.
Lemma 5.5. Consider a chain of e-split Levi subgroups σ ∈ L(G) with final term L ∶= L(σ). If ℓ ∈ Γ(G, F ), then: (i) every block of G F σ is L F -regular (see [Nav98,p.210]). In particular, for b ∈ Bl(L F ), the induced block b G F σ is defined and is the unique block of G F σ that covers b; (ii) if ϑ ∈ Irr(G F σ ), then bl(ϑ) G F is defined and R G Gσ (bl(ϑ)) = bl(ϑ) G F ; (iii) assume Hypothesis 4.1. There is a partition of the irreducible characters of G F σ given by where the union runs over the e-cuspidal pairs (M, µ) of L up to G F σ -conjugation.
We can now prove Theorem E as a consequence of the above considerations.
Theorem 5.6. Assume Hypothesis 4.1 and consider B ∈ Bl(G F ) and d ≥ 0. If Conjecture 5.4 holds for B and d, then Conjecture 5.1 holds for B and d.
Proof. If there exists a bijection between the sets L d (B) where the sum runs over G F -conjugacy classes of triples (σ, M, Ab(µ)) with σ ∈ L(G) and (M, µ) ∈ CP e (B) < with M ≤ L(σ). By Lemma 5.5 (ii)-(iii) we deduce that whenever σ ∈ L(G) >0 . On the other hand, since we are only considering (M, µ) ∈ CP e (B) < , the contribution given by the trivial chain {G} is Using (5.2) and (5.3), we deduce that (5.1) may be rewritten as and therefore Conjecture 5.1 holds for B and d.
In the following concluding remark we discuss the definition of the sets of quadruples L d (B) ǫ and show that some of the conditions imposed above are redundant.

A parametrisation of e-Harish-Chandra series
In section 4 we have shown how to describe the characters in a block of a finite reductive group in terms of e-Harish-Chandra theory. More precisely, Theorem 4.13 shows that the set of characters of a block B of G F is the disjoint union of e-Harish-Chandra series E(G F , (L, λ)) associated to certain e-cuspidal pairs (L, λ). The next natural step to understand the distribution of characters in the block B is to find a parametrisation of the characters in each series. Inspired by the results of [BMM93] and by classical Harish-Chandra theory, we propose a parametrisation of the series E(G F , (L, λ)) in terms of data analogue to the one encoded in the relative Weyl group W G (L, λ) F . At the same time, this parametrisation suggests an explanation for the Clifford theoretic and cohomological requirements imposed by the inductive conditions for the counting conjectures.
Parametrisation 5.8. Let ℓ ∈ Γ(G, F ) and consider an e-cuspidal pair (L, λ) of G. There exists a defect preserving Aut F (G F ) (L,λ) -equivariant bijection We say that Parametrisation 5.8 holds for (G, F ) at the prime ℓ if it holds for every e-cuspidal pair (L, λ) of G where q is the prime power associated to F and e is the order of q modulo ℓ.
As we have said before, the above parametrisation should provide an explanation for the inductive conditions for the counting conjectures for finite reductive groups. Analogously, in Section 6 we show that Conjecture 5.4, and hence Conjecture 5.1 and Conjecture 5.2, holds once we assume the existence of Parametrisation 5.8 (see Theorem 6.13). We are then left to prove Parametrisation 5.8. The results of [Ros22b] shows that the bijections Ω G (L,λ) can be constructed once we assume certain technical conditions on the extendibility of characters of e-split Levi subgroups. These conditions also appear in the proofs of the inductive conditions for the McKay, the Alperin-McKay and the Alperin Weight conjectures (see [MS16], [CS17a], [CS17b], [CS19], [BS20], [Spä21], [Bro22]).
Remark 5.9. To conclude this section we derive an interesting consequence of Parametrisation 5.8. For every e-cuspidal pair (L, λ) of G and any d ≥ 0 we denote by k d (G F , (L, λ)) the number of characters χ ∈ E(G F , (L, λ)) with d(χ) = d and by k d (N G (L) F , λ) the number of characters ψ ∈ Irr(N G (L) F λ) with d(ψ) = d. Since the bijection Ω G (L,λ) preserves the defect of characters, assuming Parametrisation 5.8 we obtain On the other hand, under Hypothesis 4.1, the partition given by Theorem 4.13 implies that for any block B of G F and where (L, λ) runs over a set of representatives for the action of G F on CP e (B) < . Then, combining (5.4) and (5.5) we obtain where as before (L, λ) runs over a set of representatives for the action of G F on CP e (B) < . The formula given in (5.6) suggests another way of counting the number k d (B) in terms of e-local data.
In particular, if we believe Conjecture 5.1, then under the above hypothesis we must have where (L, λ) and σ run over a set of representatives for the action of G F on CP e (B) < and on L(G) >0 respectively.

Counting characters via e-Harish-Chandra theory
The results obtained in Section 4 together with the parametrisation proposed in Section 5.3 constitute crucial properties of e-Harish-Chandra theory. As an application of these powerful tools we show that Conjecture 5.4, and hence Conjecture 5.1 and Conjecture 5.2, holds if we assume Parametrisation 5.8 (see Theorem 6.13). First, we prove some preliminary showing how to lift isomorphisms of character triples.

Bijections and N-block isomorphic character triples
The following proposition is an adaptation of [Ros22a, Proposition 2.10] to finite reductive groups. Recall that, for Y ⊴ X and S ⊆ Irr(Y ), we denote by Irr(X S) the set of irreducible characters of X whose restriction to Y has an irreducible constituent contained in S. Moreover, we define X S ∶= {x ∈ X S x = S}.
Proposition 6.1. Let K ≤ G ≤ A be finite groups with G ⊴ A, consider A 0 ≤ A. and set H 0 ∶= H ∩ A 0 for every H ≤ A. Consider S ⊆ Irr(K) and S 0 ⊆ Irr(K 0 ) and suppose there exists K ≤ V ≤ X ≤ N A (K) and U ≤ X 0 such that: Assume there exists a U -equivariant bijection for every ϑ ∈ S. If K ≤ J ≤ X ∩ G and C X (Q) ≤ X 0 for every radical ℓ-subgroup Q of J 0 , then there exists an N U (J)-equivariant bijection for every χ ∈ Irr(J S).
Proof. Consider an N U (J)-transversal S in S and define S 0 ∶= {Ψ(ϑ) ϑ ∈ S}. Since Ψ is Uequivariant, it follows that S 0 is an N U (J)-transversal in S 0 . For every ϑ ∈ S, with ϑ 0 ∶= Ψ(ϑ) ∈ S 0 , we fix a pair of projective representations (P (ϑ) , P Now, let T be an N U (J)-transversal in Irr(J S) such that every character χ ∈ T lies above a character ϑ ∈ S (this can be done by the choice of S). Moreover, using Clifford's theorem together with hypotheses (i) and (iii), it follows that every χ ∈ T lies over a unique ϑ ∈ S.
Remark 6.2. Consider the setup of Proposition 6.1. Then, the bijection Φ J is defect preserving if and only if Ψ is defect preserving.
In [Spä17, Lemma 3.8 (c)] it is shown that N -block isomorphisms of character triples are compatible with the action of inner automorphisms. It is straightforward to extend this compatibility to arbitrary automorphisms.
Proof. The claim follows directly from the definition of ∼ N (see [Spä17, Definition 3.6]).

Proof of Theorem F
We now start working towards a proof of Theorem F. In order to apply the results on e-Harish-Chandra theory obtain in Section 4, we assume throughout this section that Hypothesis 4.1 holds for our choice of G, F , q, ℓ and e as in Notation 2.2.
Definition 6.4. Let G be a connected reductive group with Frobenius endomorphism F ∶ G → G.
Recall that [G, G] is the product of simple algebraic groups G 1 , . . . , G n and that F acts on the set {G 1 , . . . , G n }. For any orbit O of F , we denote by G O the product of those simple algebraic groups in the orbit O. Notice that G O is F -stable and, by abuse of notation, denote by F the restriction of F to G O . Then, we say that (G O , F ) is an irreducible rational component of (G, F ).
Recall that a connected reductive group G is called simply connected if the semisimple algebraic group [G, G] is simply connected.
Proposition 6.5. Assume Hypothesis 4.1 and suppose that G is simply connected. Consider an e-split Levi subgroup K of G and suppose that Parametrisation 5.8 holds at the prime ℓ for every irreducible rational component of (K, F ). Let K 0 ∶= [K, K] and consider an e-cuspidal pair (L 0 , λ 0 ) of K 0 . Then there exists a defect preserving Aut F (K F 0 ) (L 0 ,λ 0 ) -equivariant bijection Proof. Since G is simply connected, we deduce that K 0 is a simply connected semisimple group (see [MT11,Proposition 12.14]). By [Mar91, Proposition 1.4.10], K 0 is the direct product of simple algebraic groups K 1 , . . . , K n and the action of F induces a permutation on the set of simple components K i . For every orbit of F we denote by H j , j = 1, . . . , t, the direct product of the simple components in such an orbit. Then H j is F -stable and where by abuse of notation we denote the restriction of F to H j again by F . Observe that the (H j , F )'s are the irreducible rational components of (K, F ). Define M j ∶= L 0 ∩ H j and observe that M j is an e-split Levi subgroup of H j and that Then, we can write [DM91,Proposition 10.9 (ii)]), it follows that (M j , µ j ) is an e-cuspidal pair of H j for every j = 1, . . . , t and, using our assumption, there exist bijections as in Parametrisation 5.8. Since ) coincides with the set of characters of the form ϑ 1 × ⋅ ⋅ ⋅ × ϑ t with ψ j ∈ E(H F j , (M j , µ j )), while it is not hard to see that Irr(N K 0 (L 0 ) F λ 0 ) coincides with the set of characters of the form ξ 1 × ⋅ ⋅ ⋅ × ξ t with ξ j ∈ Irr(N H j (M j ) F µ j ). Hence, we obtain a bijection We now show that Ω K 0 (L 0 ,λ 0 ) satisfies the required properties.
First, consider the partition {1, . . . , t} = ∐ l A l given by j, k ∈ A l if there exists a bijective morphism ϕ ∶ H j → H k commuting with F such that ϕ(M j , µ j ) = (M k , µ k ). Fix j l ∈ A l . By Lemma 6.3, we may assume without loss of generality that and and Ω To prove (6.3), observe that ϑ A l is Aut F (H F A l ) (M A l ,µ A l ) -conjugate to a character of the form × u ϑ u such that for every u, v we have either ϑ u = ϑ v or ϑ u and ϑ v are not Aut F (H F A l )-conjugate. By Lemma 6.3, we may assume without loss of generality that ϑ A l = × u ν mu u , where for every u ≠ v the characters ν u and ν v are distinct and not Aut F (H A l )-conjugate while m u are non-negative integers such that A l = ∑ u m u . Then and hence (6.3) follows from the properties of the bijections (6.2) by applying [Spä17,Theorem 5.2]. A similar argument shows that the bijection Moreover Ω K 0 (L 0 ,λ 0 ) preserves the defect of characters by the analogous property of the bijections (6.2).
We now prove an easy lemma which we use to combine bijections Ω K 0 (L 0 ,λ 0 ) given by Proposition 6.5 for various e-cuspidal pairs (L 0 , λ 0 ). Lemma 6.6. Let X ≤ Y ≤ Z be finite groups with X, Y ⊴ Z and Y X abelian. Consider η ∈ Irr(Y ) and define the set , we conclude that Y z ⊆ Y and the result follows.
Next, using Proposition 6.5, for every λ 0 ξ ∈ T, χ ∈ T λ 0 ξ and x ∈ Aut F (G F ) K,L,Y 0 we define The remaining properties follow directly from the corresponding properties of the bijections Ω K 0 (K 0 ,λ 0 ξ) given by Proposition 6.5.

Now [Spä17, Theorem 5.3] implies that
and this concludes the proof.
Corollary 6.10. Assume Hypothesis 4.1 and suppose that G is simply connected, let K be an e-split Levi subgroup of G and suppose that Parametrisation 5.8 holds at the prime ℓ for every irreducible rational component of (K, F ). Let (L, λ) be an e-cuspidal pair of K and consider Ab(λ) as defined in Section 5.2. Then there exists a defect preserving Aut F (G F ) K,L,Ab(λ) -equivariant bijection for every ϑ ∈ E(K F , (L, Ab(λ))) and where X ∶= Observe that assumptions (ii) and (iii) of Proposition 6.1 are satisfied by Proposition 4.10 and Lemma 6.6. Consider the bijection between S and S 0 given by Corollary 6.7 and Corollary 6.8. In order to apply Proposition 6.1 with J ∶= K F we need to show that C X (Q) ≤ X 0 for every radical ℓ-subgroup Q of J 0 = N K (L) F . By Lemma 2.5 (ii), we know that Since Q is a radical ℓ-subgroup of J 0 , it follows that E ≤ Q (see [Dad92,Proposition 1.4]) and therefore C X (Q) ≤ C X (E) ≤ N X (E) = N X (L) = X 0 . We can thus apply Proposition 6.1 together with Corollary 6.7, Corollary 6.8 and Lemma 6.9 to obtain an Aut F (G F ) K,L,Y 0 -equivariant bijection for every ϑ ∈ E(K F , (L, Ab(λ))). Moreover, Ψ K (L,λ) preserves the defect of characters by Remark 6.2. To conclude, notice that by a Frattini argument and using Clifford's theorem and Lemma 6.6 we have Now, applying Proposition 6.1, we show how to lift the bijection given by Corollary 6.10 to a bijection for every χ ∈ Irr(H E(K F , (L, Ab(λ)))) and where X ∶= (G F ⋊ Aut F (G F )) K .
Proof. We apply Proposition 6.1 to the bijection given by Corollary 6.10. We consider , V ∶= X S and J ∶= H. By Proposition 4.10 and Lemma 6.6 we deduce that conditions (ii) and (iii) of Proposition 6.1 hold. Next, let Q be a radical ℓ-subgroup of N H (L). Set E ∶= Z ○ (L) F ℓ and notice that under our assumptions L = C ○ G (E) by Lemma 2.5. Then E ≤ O ℓ (N H (L)) ≤ Q because Q is radical and we conclude that C X (Q) ≤ C X (E) ≤ N X (L) = X 0 . We can therefore apply Proposition 6.1 to obtain an Aut F (G F ) H,K,L,Ab(λ) -equivariant bijection Ω K,H (L,λ) as in the statement. Moreover Ω K,H (L,λ) preserves the defect of characters by Remark 6.2.
Remark 6.12. It is worth pointing out that, in the previous results, the assumption that K is an e-split Levi subgroup of G can be weakened by only requiring K to be an F -stable Levi subgroup of G.
We can finally prove Theorem F.
Theorem 6.13. Assume Hypothesis 4.1 and suppose that G is simply connected. If Parametrisation 5.8 holds at the prime ℓ for every irreducible rational component of any e-split Levi subgroup of (G, F ), then Conjecture 5.4 holds with respect to the prime ℓ.

A connection with the local-global counting conjectures
In this section we show that if ℓ is large, (G, F, e)-adapted and Z(G * ) F * ℓ = 1, then Conjecture 5.1 is equivalent to Dade's Conjecture. Moreover, since in this case we are dealing with blocks of abelian defect, the main result of [KM13] where (L, λ) runs over a set of representatives for the action of G F on CP e (B) and k(N G (L) F , λ) is the number of characters of Irr(N G (L) F λ). Since the Alperin-McKay Conjecture holds in this case as a consequence of [BMM93, Theorem 5.24], the above equality provides evidence for the validity of Parametrisation 5.8 as discussed in Remark 5.9.
We need to make a small remark: in the discussion following [   Inspired by [IMN07] and by the inductive Alperin-McKay condition for quasi-simple groups introduced in [Spä13], a stronger version of the above conjecture has been considered in [NS14]. such that for every χ ∈ Irr 0 (B).
The main result of [NS14] shows that Conjecture 7.2 reduces to quasi-simple groups and, together with [KM13], implies Brauer's Height Zero Conjecture. Now, consider the set P(G) of ℓ-chains of G with initial term O ℓ (G). These are the ℓ-chains where D i is an ℓ-subgroup of G and n is a non-negative integer. If we denote by D the integer n, called the length of D, then we obtain a partition of P(G) into the sets P(G) + and P(G) − consisting of ℓ-chains of even and odd length respectively. Notice that G acts by conjugation on the sets P(G), P(G) + and P(G) − and we denote by Observe that G acts on C d (B) ǫ and denote by (D, ϑ) the G-orbit of (D, ϑ) ∈ C d (B) ǫ and by C d (B) ǫ G a set of representatives for the G-orbits on C d (B) ǫ . The following statement has been

An idea of Broué, Fong and Srinivasan
For every finite group G, recall that the set E(G) of ℓ-elementary abelian chains of G (starting at O ℓ (G)) consists of those chains Consider G, F , q, ℓ and e as in Notation 2.2.
Definition 7.7 (Broué-Fong-Srinivasan). Let E be an ℓ-elementary abelian subgroup of G F . Then E is said to be good if and bad otherwise. An ℓ-elementary abelian chain E ∈ E(G F ) is said to be good if E i is good for every i, while it is bad otherwise. The set of good and bad ℓ-elementary abelian chains of G F is denote by E g (G F ) and E b (G F ) respectively.
When ℓ is large and (G, F, e)-adapted there exists a bijection between chains of e-split Levi subgroups of G and good ℓ-elementary abelian chains of G F . Recall from Section 2.1 that every automorphism α ∈ Aut F (G F ) extends to a bijective endomorphism of G commuting with F . Then Aut F (G F ) acts on the set of F -stable closed connected subgroups of G.
Lemma 7.8. Suppose that ℓ is large, (G, F, e)-adapted and O ℓ (G F ) = 1. Then the maps are mutually inverse Aut F (G F )-equivariant length preserving bijections.
Proof. First, consider a chain of e-split Levi subgroups σ = (G = L 0 > ⋅ ⋅ ⋅ > L n ). Since ℓ is large for G, Proposition 2.7 (iii) implies that E i ∶= Ω 1 (O ℓ (Z ○ (L i ) F )) is a good ℓ-elementary abelian subgroup such that L i = C ○ G (E i ). Since L i > L i+1 , this also shows that E i < E i+1 for every i = 0, . . . , n − 1. Moreover, as O ℓ (G F ) = 1, we deduce that E 0 = O ℓ (G F ). On the other hand, if D = (O ℓ (G F ) = D 0 < ⋅ ⋅ ⋅ < D n ) is a good ℓ-elementary abelian chain, then all terms D i are elementary abelian (since O ℓ (G F ) = 1) and Proposition 2.7 (i) shows that K i ∶= C ○ G (D i ) is an e-split Levi subgroup. Furthermore D i = Ω 1 (O ℓ (Z ○ (K i ) F )), because D i is good in the sense of Definition 7.7, and K 0 = G. As a consequence, since D i < D i+1 , we obtain that K i > K i+1 for every i = 0, . . . , n − 1. It follows that the above maps are inverses of each other and preserve the length of chains. To show that the maps are Aut F (G F )-equivariant, observe that Ω 1 (O ℓ (Z ○ (L) F )) α = Ω 1 (O ℓ (Z ○ (L α ) F )) and C ○ G (E) α = C ○ G (E α ) for every e-split Levi subgroup L of G, every ℓelementary abelian subgroup E of G F and every α ∈ Aut F (G F ).
Next, we show that there exists a self inverse Aut F (G F )-equivariant bijection on the set of bad ℓ-elementary abelian chains.
Next, consider the bijection described in Lemma 7.8 and suppose that σ ∈ L(G) corresponds to D ∈ E g (G F ). Since the bijection is Aut F (G F )-equivariant, it follows that G F D = G F σ and therefore k d (B D ) = k d (B σ ). Then (7.3) is equivalent to where (L, λ) runs over a set of representatives for the action of G F on CP e (B). Moreover, this is equivalent to (5.6) whenever B has non-trivial defect and Z(G * ) F * ℓ = 1.
Proof. Let D be a defect group of B and consider its Brauer correspondent b in N G F (D). Notice that D is abelian since ℓ is large and hence k(B) = k 0 (B) and k(b) = k 0 (b) (see [KM13]). To prove the first statement we need to show that Fix an N G (L) F -transversal T ′ in the set of e-cuspidal characters λ ∈ Irr(L F ) such that bl(λ) G F = B. By the above paragraph we deduce that T ∶= {(L, λ) λ ∈ T ′ } is an N G (L) F -transversal in CP e (L, F ) ∩ CP e (B) and so, applying Lemma 5.5 together with Brauer's first main theorem, we get Irr(b) = ∐ (L,λ)∈T Irr N G (L) F λ . Now, (7.5) follows by noticing that T is also a G F -transversal in CP e (B).
To conclude, assume that B has non-trivial defect and that Z(G * ) F * ℓ = 1. In this case, Proposition 4.18 implies that no e-cuspidal character belong to the block B. In particular CP e (B) = CP e (B) < and k d c (B) = 0 for every d ≥ 0. Furthermore, the main result of [KM13] shows that k(B) = k d (B) and that k d (N G (L) F , λ) = k(N G (L) F , λ) for every (L, λ) ∈ CP e (B). Now, (7.4) is equivalent to (5.6), that is, where (L, λ) runs over a set of representatives for the action of G F on CP e (B) < .
By Remark 5.9 (see (5.6)), the following corollary provides evidence for Parametrisation 5.8. where (L, λ) runs over a set of representatives for the action of G F on CP e (B). Moreover, (5.6) holds whenever B has non-trivial defect and Z(G * ) F * ℓ = 1.
Proof. By [BMM93, Theorem 5.24] and the subsequence discussion, under our assumptions the Alperin-McKay Conjecture holds for any block B of G F . Then the result follows from Proposition 7.11.
Next, we consider the connection between Conjecture 5.4, the Character Triple Conjecture and the inductive condition for Dade's Conjecture.
Proposition 7.13. Assume Hypothesis 4.1 with ℓ large, (G, F, e)-adapted and such that O ℓ (G F ) = 1 = Z(G * ) F * ℓ . Consider an ℓ-block B of G F with non-trivial defect and fix d ≥ 0. If Conjecture 5.4 holds for B and d, then the Character Triple Conjecture (Conjecture 7.5) holds for B and d.
Proof. Consider (E, ϑ) ∈ C d (B) + . By [Spä17, Proposition 6.10] we may assume that E is an ℓelementary abelian chain. If E is a bad ℓ-elementary abelian chain (see Definition 7.7), then we define where E ′ is the chain corresponding to E via the bijection given by Lemma 7.9. Notice in this case that G F E = G F E ′ and therefore that (E ′ , ϑ) ∈ C d (B) − . Assume that E is a good ℓ-elementary abelian chain and consider the corresponding chain of e-split Levi subgroups σ given by Lemma 7.8. Since the map D ↦ σ is Aut F (G F )-equivariant, we know that G F E = G F σ . Recall that L(σ) denotes the final term of σ. By Lemma 5.5 (iii), there exists an e-cuspidal pair (M, µ) of L, unique up to G F σ -conjugation, such that ϑ ∈ Irr d (G F σ E(L F , (M, µ)). Then, as bl(ϑ) G F = B, we have ϑ ∈ Irr d (B σ E(L(σ) F , (M, Ab(µ)))).
Since (M, µ) is unique up to G F σ -conjugation while Λ and the bijections given by Lemma 7.8 and Lemma 7.9 are equivariant, we conclude that Ω is a well defined Aut F (G F )-equivariant bijection. Moreover, using the property on character triples of Λ it is immediate to show that Ω satisfies the analogous properties required by Conjecture 7.5 with respect to X ∶= G F ⋊ Aut F (G F ). This completes the proof.
To obtain the inductive condition for Dade's Conjecture we apply Lemma 7.6. Recall that, apart from a few exceptions, the universal covering group of a (non-abelian) simple group of Lie type is a finite reductive group G F with G simple and simply connected. Moreover, in this case, Aut F (G F ) = Aut(G F ) (see [GLS98,1.15] and [CS13, 2.4]).
Corollary 7.14. Consider the set up of Proposition 7.13 and suppose that G F Z(G F ) is a non-abelian simple group with universal covering group G F where G is simple and simply connected. If Conjecture 5.4 holds for B and d, then the inductive condition for Dade's Conjecture holds for B and d.
Proof. Consider the bijection Ω ∶ C d (B) + G F → C d (B) − G F constructed in the proof of Proposition 7.13 and fix (D, ϑ) ∈ C d (B) + and (E, χ) ∈ Ω((D, ϑ)). Since Ω satisfies the requirements of Conjecture 7.5, we know that with X ∶= G F ⋊ Aut F (G F ) and applying [Spä17, Lemma 3.4] we obtain Z ∶= Ker(ϑ Z(G F ) ) = Ker(χ Z(G F ) ). Because, under our assumption, Z(G F ) has order coprime to ℓ, it follows from [Spä17, Corollary 4.5] (see also Lemma 2.1) that where ϑ and χ correspond to ϑ and χ respectively via inflation of characters. Now, the result follows from Lemma 7.6 after noticing that Aut(G F ) = Aut F (G F ) under our hypothesis.
To conclude we show that Parametrisation 5.8 implies the inductive Alperin-McKay condition (Conjecture 7.2).
Proposition 7.15. Assume Hypothesis 4.1 with ℓ large and (G, F, e)-adapted and consider an ℓblock B of G F . If Parametrisation 5.8 holds with respect to every (L, λ) ∈ CP e (B), then the inductive Alperin-McKay condition (Conjecture 7.2) holds for B with respect to G F ⊴ G F ⋊ Aut F (G F ).
for every (L, λ) ∈ T , χ ∈ T (L,λ) B and x ∈ Aut F (G F ) L,B . Recalling that N G (L) F = N G F (D) and Aut F (G F ) L,B = Aut F (G F ) D,B , we deduce that the equivalence of character triples required by Conjecture 7.2 follow from the analogue properties given by Parametrisation 5.8.