2.1 Characters and blocks of finite groups

We recall some standard notation from representation theory of finite groups as can be found in [Reference Isaacs32] and [Reference Navarro46], for instance. Let $\mathrm{Irr}(G)$ be the set of ordinary irreducible characters. If $N\unlhd G$ and $\vartheta \in \mathrm{Irr}(N)$ , then we denote by $\mathrm{Irr}(G\mid \vartheta )$ the set of irreducible characters of G that lie above $\vartheta $ . More generally, if $\mathcal {S}$ is a subset of irreducible characters of N, then we denote by $\mathrm{Irr}(G\mid \mathcal {S})$ the union of the sets $\mathrm{Irr}(G\mid \vartheta )$ for $\vartheta \in \mathcal {S}$ , that is, the set of irreducible characters of G that lie above some character in the set $\mathcal {S}$ .

Next, we denote by $G_\vartheta $ the stabiliser of the irreducible character $\vartheta \in \mathrm{Irr}(N)$ under the conjugacy action of G and say that $\vartheta $ is G-invariant if $G=G_\vartheta $ . In this case, we say that $(G,N,\vartheta )$ is a character triple. These objects provide important information in the study of Clifford theory and play a crucial role in many aspects of the local-global conjectures. Of paramount importance is the introduction of certain binary relations on the set of character triples. We refer the reader to [Reference Navarro47, Chapter 5 and 10] and [Reference Späth65] for a more detailed introduction to these ideas and for the necessary background on projective representations. The binary relation considered here was introduced in [Reference Späth64, Definition 3.6] and is known as N-block isomorphism of character triples, denoted by . This equivalence relation has further been studied in [Reference Rossi52].

In order to construct N-block isomorphisms of character triples, it is often useful to prove certain results on the extendibility of characters. Here, we introduce the notion of maximal extendibility (see [Reference Malle and Späth40, Definition 3.5]) that will be considered in the following sections. Let $N\unlhd G$ be finite groups, and consider $\mathcal {S}$ a subset of irreducible characters of N. Then, we say that maximal extendibility holds for the set $\mathcal {S}$ with respect to the inclusion $N\unlhd G$ if every character $\vartheta \in \mathcal {S}$ extends to its stabiliser $G_\vartheta $ . More precisely, we can specify an extension map

(2.1) $$ \begin{align} \Lambda:\mathcal{S}\to\coprod\limits_{N\leq H\leq G}\mathrm{Irr}(H) \end{align} $$

that sends each character $\vartheta \in \mathcal {S}$ to an extension $\Lambda (\vartheta )$ of $\vartheta $ to the stabiliser $G_\vartheta $ .

Next, we consider modular representation theory with respect to a fixed prime number $\ell $ . For $\chi \in \mathrm{Irr}(G)$ , there exist unique nonnegative integers $d(\chi )$ , called the $\ell $ -defect of $\chi $ , such that $\ell ^{d(\chi )}=|G|_\ell /\chi (1)_\ell $ and where for an integer n we denote by $n_\ell $ the largest power of $\ell $ that divides n. For any $d\geq 0$ , let $\mathrm{Irr}^d(G)$ be the set of irreducible characters $\chi $ of G that satisfy $d(\chi )=d$ and denote by ${\mathbf {k}}^d(G)$ its cardinality. Associated to the prime $\ell $ , we also have the set of Brauer $\ell $ -blocks of G. Each block is uniquely determined by the central functions $\lambda _B$ (see [Reference Navarro46, p. 49]). For every $\chi \in \mathrm{Irr}(G)$ , we denote by $\mathrm{bl}(\chi )$ the unique block that satisfies $\chi \in \mathrm{Irr}(\mathrm{ bl}(\chi ))$ . Furthermore, if $H\leq G$ and b is a block of H, then $b^G$ denotes the block of G obtained via Brauer’s induction (when it is defined). If B is a block of G and $d\geq 0$ , then let $\mathrm{Irr}^d(B)$ be the set of irreducible characters belonging to the block B and having defect d. The cardinality of $\mathrm{Irr}^d(B)$ is denoted by ${\mathbf {k}}^d(B)$ .

We conclude this introductory section with an analogue of [Reference Isaacs32, Problem 5.3] for blocks that will be used in the sequel.

Proof. The first point is [Reference Rizo51, Lemma 2.1]. Denote by $\mathcal {R}$ the ring of algebraic integers in $\mathbb {C}$ and by $\overline {\mathcal {R}}$ its quotient modulo a fixed maximal ideal $\mathcal {M}$ of $\mathcal {R}$ containing $\ell \mathcal {R}$ . Let $g\in G$ and denote by ${\mathfrak {Cl}}_G(g)$ the G-conjugacy class of g and by ${\mathfrak {Cl}}_G(g)^+$ the corresponding conjugacy class sum in the group algebra over $\overline {\mathcal {R}}$ . Since the intersection ${\mathfrak {Cl}}_G(g)\cap H$ is a union of H-conjugacy classes, we can find $h_1,\dots ,h_n\in {\mathfrak {Cl}}_G(g)\cap H$ such that

$$\begin{align*}{\mathfrak{Cl}}_G(g)\cap H=\coprod\limits_{i=1}^n{\mathfrak{Cl}}_H(h_i)\end{align*}$$

and where n is zero if ${\mathfrak {Cl}}_G(g)\cap H$ is empty. In particular, observe that $\zeta (h_i)=\zeta (g)$ since $\zeta $ is a class function of G. Notice also that $\zeta $ is a group homomorphism. Now, using the notation of [Reference Navarro46, p.87] and recalling for a block B its central character is denoted by $\lambda _B$ , we obtain

$$ \begin{align*} \lambda_{B\cdot\zeta}\left({\mathfrak{Cl}}_G(g)^+\right)&=\lambda_B\left({\mathfrak{Cl}}_G(g)^+\right)\overline{\zeta(g)} \\ &=\lambda_b^G\left({\mathfrak{Cl}}_G(g)^+\right)\overline{\zeta(g)} \\ &=\sum\limits_{i=1}^n\lambda_b\left({\mathfrak{Cl}}_H(h_i)^+\right)\overline{\zeta(g)} \\ &=\sum\limits_{i=1}^n\lambda_b\left({\mathfrak{Cl}}_H(h_i)^+\right)\overline{\zeta_H(h_i)} \\ &=\sum\limits_{i=1}^n\lambda_{b\cdot\zeta_H}\left({\mathfrak{Cl}}_H(h_i)^+\right)=\lambda_{b\cdot\zeta_H}^G\left({\mathfrak{Cl}}_G(g)^+\right), \end{align*} $$

where for every algebraic integer $\alpha \in \mathcal {R}$ we denote by $\overline {\alpha }$ its reduction modulo $\mathcal {M}$ . This shows that $B\cdot \zeta =(b\cdot \zeta _H)^G$ and we are done.

2.2 Finite reductive groups and unipotent characters

Let ${\mathbf {G}}$ be a connected reductive group defined over an algebraic closure of a field of positive characteristic p different from $\ell $ , and consider a Frobenius endomorphism $F:{\mathbf {G}}\to {\mathbf {G}}$ associated with an $\mathbb {F}_q$ -structure for a power q of p. The set of $\mathbb {F}_q$ -rational points on the variety ${\mathbf {G}}$ is denoted by ${\mathbf {G}}^F$ and is called a finite reductive group. By abuse of notation, we also refer to the pair $({\mathbf {G}},F)$ as a finite reductive group. In this paper, we say that the algebraic group ${\mathbf {G}}$ is simply connected, if its derived subgroup $[{\mathbf {G}},{\mathbf {G}}]$ is simply connected.

Let ${\mathbf {L}}$ be a Levi subgroup of a parabolic subgroup ${\mathbf {P}}$ of ${\mathbf {G}}$ , and assume that ${\mathbf {L}}$ (but not necessarily ${\mathbf {P}}$ ) is F-stable. Using $\ell $ -adic cohomology, Deligne–Lusztig [Reference Deligne and Lusztig24] and Lusztig [Reference Lusztig37] defined a $\mathbb {Z}$ -linear map

$$\begin{align*}{\mathbf{R}}_{{\mathbf{L}}\leq {\mathbf{P}}}^{\mathbf{G}}:\mathbb{Z}\mathrm{Irr}\ \left({\mathbf{L}}^F\right)\to\mathbb{Z}\mathrm{Irr}\ \left({\mathbf{G}}^F\right)\end{align*}$$

with adjoint

$$\begin{align*}^*{\mathbf{R}}_{{\mathbf{L}}\leq {\mathbf{P}}}^{\mathbf{G}}:\mathbb{Z}\mathrm{Irr}\ \left({\mathbf{G}}^F\right)\to\mathbb{Z}\mathrm{Irr}\ \left({\mathbf{L}}^F\right).\end{align*}$$

The exact definition can be found in [Reference Cabanes and Enguehard17, Section 8.3]. These maps are known to be independent of the choice of the parabolic subgroup ${\mathbf {P}}$ in almost all cases (see [Reference Bonnafé and Michel3] and [Reference Taylor66]) and, in particular, in those considered in this paper. Therefore, we will always omit ${\mathbf {P}}$ and denote ${\mathbf {R}}_{{\mathbf {L}}\leq {\mathbf {P}}}^{\mathbf {G}}$ simply by ${\mathbf {R}}_{\mathbf {L}}^{\mathbf {G}}$ . Next, using Deligne–Lusztig induction we define the unipotent characters of ${\mathbf {G}}^F$ . These are the irreducible characters $\chi $ of ${\mathbf {G}}^F$ that appear as an irreducible constituent of the virtual character ${\mathbf {R}}_{{\mathbf {T}}}^{{\mathbf {G}}}(1_{\mathbf {T}})$ for some F-stable maximal torus ${\mathbf {T}}$ of ${\mathbf {G}}$ . The set of unipotent characters of ${\mathbf {G}}^F$ is denoted by $\mathrm{Uch}({\mathbf {G}}^F)$ and its cardinality by ${\mathbf {k}}_{\mathrm{u}}({\mathbf {G}}^F)$ . Similarly, if B is a block of ${\mathbf {G}}^F$ and d a nonnegative integer, then ${\mathbf {k}}_{\mathrm{u}}^d(B)$ denotes the cardinality of the intersection $\mathrm{Uch}({\mathbf {G}}^F)\cap \mathrm{Irr}^d(B)$ .

2.3 e-Harish-Chandra theory for unipotent characters

Denote by e the multiplicative order of q modulo $\ell $ , if $\ell $ is odd, or modulo $4$ , if $\ell =2$ . In this section, we collect the main results of e-Harish-Chandra theory for unipotent characters. This was first introduced by Fong and Srinivasan [Reference Fong and Srinivasan28] for classical groups and then further developed by Broué, Malle and Michel [Reference Broué, Malle and Michel9] for unipotent characters. The compatibility of this theory with Brauer $\ell $ -blocks was first described by Fong and Srinivasan for classical groups [Reference Fong and Srinivasan27], [Reference Fong and Srinivasan29] and then completed by Broué, Malle and Michel for large primes [Reference Broué, Malle and Michel9], by Cabanes and Enguehard for all good primes [Reference Cabanes and Enguehard15] and by Enguehard for the remaining bad primes [Reference Enguehard26]. These results also provide a description of the characters belonging to unipotent blocks (see [Reference Cabanes and Enguehard15, Theorem (iii)]). Another description of these characters was provided by the author in [Reference Rossi57] under certain restrictions on the prime $\ell $ (see also [Reference Rossi57, Remark 4.14] for a comparison between the two descriptions). We refer the reader to the monographs [Reference Cabanes and Enguehard17] and [Reference Geck and Malle30] for a more complete account of e-Harish-Chandra theory and to the papers [Reference Cabanes and Enguehard16] and [Reference Kessar and Malle34] for results on nonunipotent blocks.

The theory of $\Phi _e$ -subgroups that constitutes the foundation of e-Harish-Chandra theory was introduced in [Reference Broué and Malle8]. Following their terminology, we say that an F-stable torus ${\mathbf {S}}$ of ${\mathbf {G}}$ is a $\Phi _e$ -torus if its order polynomial $P_{({\mathbf {S}},F)}$ is a power of the e-th cyclotomic polynomial, that is, if $P_{({\mathbf {S}},F)}=\Phi _e^{n}$ for some integer n and where $\Phi _e$ denotes the e-th cyclotomic polynomial (see [Reference Cabanes and Enguehard17, Definition 13.3]). Then, we say that a Levi subgroup ${\mathbf {L}}$ of ${\mathbf {G}}$ is an e-split Levi subgroup if there exists a $\Phi _e$ -torus ${\mathbf {S}}$ such that ${\mathbf {L}}={\mathbf {C}}_{\mathbf {G}}({\mathbf {S}})$ . More precisely, we say that ${\mathbf {L}}$ is an e-split Levi subgroup of $({\mathbf {G}},F)$ to emphasise the role of the Frobenius endomorphism F. Observe that, for any F-stable torus ${\mathbf {T}}$ , there exists a unique maximal $\Phi _e$ -torus of ${\mathbf {T}}$ denoted by ${\mathbf {T}}_{\Phi _e}$ (see [Reference Cabanes and Enguehard17, Proposition 13.5]). Then, it can be shown that an F-stable Levi subgroup ${\mathbf {L}}$ of ${\mathbf {G}}$ is e-split if and only if ${\mathbf {L}}={\mathbf {C}}_{\mathbf {G}}({\mathbf {Z}}^\circ ({\mathbf {L}})_{\Phi _e})$ (see, for instance, [Reference Geck and Malle30, Proposition 3.5.5]).

Next, recall that $({\mathbf {L}},\lambda )$ is an e-cuspidal pair of $({\mathbf {G}},F)$ if ${\mathbf {L}}$ is an e-split Levi subgroup of $({\mathbf {G}},F)$ and $\lambda \in \mathrm{Irr}({\mathbf {L}}^F)$ satisfies $^*{\mathbf {R}}_{{\mathbf {M}}}^{\mathbf {L}}(\lambda )=0$ for every e-split Levi subgroup ${\mathbf {M}}<{\mathbf {L}}$ . A character $\lambda $ with the property above is said to be an e-cuspidal character of ${\mathbf {L}}^F$ . If in addition the character $\lambda $ is unipotent, then we say that $({\mathbf {L}},\lambda )$ is a unipotent e-cuspidal pair and that $\lambda $ is a unipotent e-cuspidal character. We denote by ${\mathcal {C} \mathcal {P}_{\mathrm{u}}}({\mathbf {G}},F)$ the set of unipotent e-cuspidal pairs of $({\mathbf {G}},F)$ and by ${\mathbf {k}}_{\mathrm{c,u}}({\mathbf {G}}^F)$ the number of unipotent e-cuspidal characters of ${\mathbf {G}}^F$ . Moreover, we define the e-Harish-Chandra series associated to the e-cuspidal pair $({\mathbf {L}},\lambda )$ to be the set of irreducible constituents of the virtual character ${\mathbf {R}}_{\mathbf {L}}^{\mathbf {G}}(\lambda )$ , denoted by ${\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ . As before, when $\lambda $ is unipotent we say that ${\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ is a unipotent e-Harish-Chandra series.

Unipotent characters were parametrised by Broué, Malle and Michel [Reference Broué, Malle and Michel9, Theorem 3.2] by using e-Harish-Chandra theory. Their description can be divided into two parts. First, each unipotent character lies in a unique e-Harish-Chandra series, that is,

$$\begin{align*}\mathrm{Uch}\left({\mathbf{G}}^F\right)=\coprod\limits_{({\mathbf{L}},\lambda)}{\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right)\end{align*}$$

where $({\mathbf {L}},\lambda )$ runs over a set of representatives for the action of ${\mathbf {G}}^F$ on the set of unipotent e-cuspidal pairs of $({\mathbf {G}},F)$ as explained in [Reference Broué, Malle and Michel9, Theorem 3.2 (1)]. This is a well-known fact and will be used throughout the paper without further reference. As a consequence of the partition above, it now remains to parametrise the unipotent e-Harish-Chandra series. If $({\mathbf {L}},\lambda )$ is a unipotent e-cuspidal pair, we denote by $W_{\mathbf {G}}({\mathbf {L}},\lambda )^F:={\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F_\lambda /{\mathbf {L}}^F$ the corresponding relative Weyl group. Then, [Reference Broué, Malle and Michel9, Theorem 3.2 (2)] parametrises the characters in an e-Harish-Chandra series in terms of the characters in the relative Weyl group by showing the existence of a bijection

(2.2) $$ \begin{align} \mathrm{Irr}\ \left(W_{\mathbf{G}}({\mathbf{L}},\lambda)^F\right)\to {\mathcal{E}}({\mathbf{G}}^F,({\mathbf{L}},\lambda)). \end{align} $$

In Section 3, we reformulate (2.2) in order to obtain Theorem C.

Unipotent e-Harish-Chandra series are also used to parametrise the so-called unipotent blocks, that is, those blocks that contain unipotent characters. This was proved in [Reference Cabanes and Enguehard15] (see also [Reference Broué, Malle and Michel9] for the case of large primes). More precisely, if $\ell $ is odd and good for ${\mathbf {G}}$ , with $\ell \neq 3$ if ${^3\mathbf {{D}}_4}$ is an irreducible rational component of $({\mathbf {G}},F)$ , then for every $\ell $ -block B of ${\mathbf {G}}^F$ there exists a unipotent e-cuspidal pair $({\mathbf {L}},\lambda )$ , with $({\mathbf {L}},\lambda )$ unique up to ${\mathbf {G}}^F$ -conjugation such that all the irreducible constituents of ${\mathbf {R}}_{\mathbf {L}}^{\mathbf {G}}(\lambda )$ belongs to the block B. In this case, we write $B=b_{{\mathbf {G}}^F}({\mathbf {L}},\lambda )$ and we also have

$$\begin{align*}\mathrm{Uch}({\mathbf{G}}^F)\cap \mathrm{Irr}\ \left(b_{{\mathbf{G}}^F}({\mathbf{L}},\lambda)\right)={\mathcal{E}}({\mathbf{G}}^F,({\mathbf{L}},\lambda)).\end{align*}$$

Moreover, [Reference Cabanes and Enguehard15, Proposition 3.3 (ii) and Proposition 4.2] imply that $\mathrm{bl}(\lambda )^{{\mathbf {G}}^F}=B$ .

2.4 Pseudo-unipotent characters

We denote by $({\mathbf {G}}^*,F^*)$ a group in duality with $({\mathbf {G}},F)$ with respect to a choice of an F-stable maximal torus ${\mathbf {T}}$ of ${\mathbf {G}}$ and an $F^*$ -stable maximal torus ${\mathbf {T}}^*$ of ${\mathbf {G}}^*$ . If $\tau :{\mathbf {G}}_{\mathrm{sc}}\to [{\mathbf {G}},{\mathbf {G}}]$ is a simply connected covering (see [Reference Geck and Malle30, Remark 1.5.13]), then there exists an isomorphisms between the abelian groups

$$ \begin{align*} {\mathbf{Z}}\left({\mathbf{G}}^*\right)^{F^*}&\to\mathrm{Irr}\ \left({\mathbf{G}}^F/\tau({\mathbf{G}}_{\mathrm{sc}})^F\right) \\ z &\mapsto \hat{z}_{{\mathbf{G}}} \end{align*} $$

according to [Reference Cabanes and Enguehard17, (8.19)]. Notice that, if ${\mathbf {L}}$ is an F-stable Levi subgroup of ${\mathbf {G}}$ , then its dual ${\mathbf {L}}^*$ is an $F^*$ -stable Levi subgroup of ${\mathbf {G}}^*$ and we have ${\mathbf {Z}}({\mathbf {G}}^*)^{F^*}\leq {\mathbf {Z}}({\mathbf {L}}^*)^{F^*}$ . In particular, every element $z\in {\mathbf {Z}}({\mathbf {G}}^*)^{F^*}$ defines a linear characters of $\hat {z}_{{\mathbf {L}}}$ and restriction of characters yields the equality

$$\begin{align*}(\hat{z}_{\mathbf{G}})_{{\mathbf{L}}^F}=\hat{z}_{\mathbf{L}}.\end{align*}$$

In the next definition, we consider characters that are obtained by multiplying these linear characters with unipotent characters.

In accordance with the terminology introduced above, we say that an e-Harish-Chandra series of $({\mathbf {K}},F)$ is pseudo-unipotent if it is of the form ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\nu ))$ for some $\nu \in \mathrm{ps}_{\mathbf {K}}(\lambda )$ and where $({\mathbf {L}},\lambda )$ is a unipotent e-cuspidal pair of $({\mathbf {K}},F)$ . In this case, we also say that $({\mathbf {L}},\nu )$ is a pseudo-unipotent e-cuspidal pair. We define the union of all the series associated to characters in $\mathrm{ps}_{\mathbf {K}}(\lambda )$ by ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\mathrm{ps}_{\mathbf {K}}(\lambda )))$ . Since

$$\begin{align*}{\mathbf{R}}_{\mathbf{L}}^{\mathbf{K}}(\lambda\hat{z}_{\mathbf{L}})={\mathbf{R}}_{\mathbf{L}}^{\mathbf{K}}(\lambda)\hat{z}_{\mathbf{K}}\end{align*}$$

for every $z\in {\mathbf {Z}}({\mathbf {K}}^*)^{F^*}$ by [Reference Cabanes and Enguehard17, (8.20)], we deduce that the elements of the pseudo-unipotent e-Harish-Chandra series ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda \hat {z}))$ are exactly the irreducible characters of the form $\varphi \hat {z}_{{\mathbf {K}}}$ for some unipotent character $\varphi \in {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda ))$ . Moreover, we point out that $\lambda $ is the unique unipotent character in the set $\mathrm{ps}_{\mathbf {K}}(\lambda )$ according to [Reference Cabanes and Enguehard17, Proposition 8.26]. Similarly, the unipotent characters in the set ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\mathrm{ps}_{\mathbf {K}}(\lambda )))$ are those in the series ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda ))$ .

Our next lemma shows that blocks covering pseudo-unipotent characters are regular as defined in [Reference Navarro46, p.210].

3.1 Equivariance and maximal extendibility

In this section, we consider some equivariance properties for the parametrisation (3.1) which are related to maximal extendibility (see (2.1)) of unipotent characters.

As in the previous sections, consider a connected reductive group ${\mathbf {G}}$ with a Frobenius endomorphism $F:{\mathbf {G}}\to {\mathbf {G}}$ defining an $\mathbb {F}_q$ -structure on ${\mathbf {G}}$ . We denote by $\mathrm{ Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ the set of those automorphisms of ${\mathbf {G}}^F$ obtained by restricting some bijective morphism of algebraic groups $\sigma :{\mathbf {G}}\to {\mathbf {G}}$ that commutes with F to the set of $\mathbb {F}_q$ -rational points ${\mathbf {G}}^F$ . Notice that the restriction of such a morphism $\sigma $ to ${\mathbf {G}}^F$ , which by abuse of notation we denote again by $\sigma $ , is an automorphism of the finite group ${\mathbf {G}}^F$ . We refer the reader to [Reference Cabanes and Späth18, Section 2.4] for further details. In particular, observe that any morphism $\sigma $ with the properties above is determined by its restriction to ${\mathbf {G}}^F$ up to a power of F (this follows from [Reference Gorenstein, Lyons and Solomon31, Lemma 2.5.7]) and hence it follows that $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ acts on the set of F-stable closed connected subgroups of ${\mathbf {G}}$ . Then, given an F-stable closed connected subgroup ${\mathbf {H}}$ of ${\mathbf {G}}$ , we can define the set $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{\mathbf {H}}$ consisting of those automorphisms $\sigma $ as above that stabilise the algebraic group ${\mathbf {H}}$ .

Now, let $\ell $ be a prime number not dividing q and denote by e the order of q modulo $\ell $ or q modulo $4$ if $\ell =2$ . In order to control the action of automorphisms on unipotent e-Harish-Chandra series, we exploit a result of Cabanes and Späth. More precisely, in [Reference Cabanes and Späth18, Theorem 3.4] it was shown that the parametrisation given by Broué, Malle and Michel in [Reference Broué, Malle and Michel9, Theorem 3.2 (2)] commutes with the action of those automorphisms in the set $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ . Notice that the statement of [Reference Cabanes and Späth18, Theorem 3.4] only considers unipotent e-cuspidal pairs $({\mathbf {L}},\lambda )$ , where ${\mathbf {L}}$ is a minimal e-split Levi subgroup (which is enough for the purpose of dealing with the McKay Conjecture). However, their proof works for the general case as well.

Proposition 3.1. For every unipotent e-cuspidal pair $({\mathbf {L}},\lambda )$ of the group $({\mathbf {G}},F)$ , there exists an $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -equivariant bijection

$$\begin{align*}I_{({\mathbf{L}},\lambda)}^{\mathbf{G}}:\mathrm{Irr}\ \left(W_{\mathbf{G}}({\mathbf{L}},\lambda)^F\right)\to{\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right)\end{align*}$$

such that

$$\begin{align*}I^{\mathbf{G}}_{({\mathbf{L}},\lambda)}(\eta)(1)_\ell=\left|\hspace{1pt}{\mathbf{G}}^F:{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}},\lambda)^F\hspace{1pt}\right|{}_\ell\cdot\lambda(1)_\ell\cdot\eta(1)_\ell\end{align*}$$

for every $\eta \in \mathrm{Irr}(W_{\mathbf {G}}({\mathbf {L}},\lambda )^F)$ .

As explained at the beginning of this section, if $\lambda $ extends to the stabiliser ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F_\lambda $ , then we can use the bijection (2.2) to obtain (3.1). A similar argument can be used to include the equivariance property described above and obtain an equivariant version of (3.1). Observe that, by the discussion on automorphisms above, it follows that the group $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ acts on the set of e-cuspidal pairs $({\mathbf {L}},\lambda )$ , and therefore, we can define the stabiliser $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ . Furthermore, recall that we denote by $d(\chi )$ the $\ell $ -defect of an irreducible character $\chi $ .

Corollary 3.2. Let $({\mathbf {L}},\lambda )$ be a unipotent e-cuspidal pair of $({\mathbf {G}},F)$ , and suppose that $\lambda $ has an extension $\lambda ^\diamond \in \mathrm{Irr}({\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})_\lambda ^F)$ . Then there exists a bijection

$$\begin{align*}\Omega_{({\mathbf{L}},\lambda)}^{\mathbf{G}}:{\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\enspace\middle|\enspace\lambda\right)\end{align*}$$

such that

$$\begin{align*}d(\chi)=d\left(\Omega^{\mathbf{G}}_{({\mathbf{L}},\lambda)}(\chi)\right)\end{align*}$$

for every $\chi \in {\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ . Furthermore, the bijection $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is $\mathrm{ Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -equivariant whenever the extension $\lambda ^\diamond $ is $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -invariant.

Proof. Consider the bijection $I^{{\mathbf {G}}}_{({\mathbf {L}},\lambda )}$ given by Proposition 3.1, and define the map

$$ \begin{align*} \Omega_{({\mathbf{L}},\lambda)}^{\mathbf{G}}:{\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right)&\to\mathrm{Irr}\ \left({\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\enspace\middle|\enspace \lambda\right) \\ I_{({\mathbf{L}},\lambda)}^{\mathbf{G}}(\eta)&\mapsto\left(\lambda^\diamond\eta\right)^{{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F} \end{align*} $$

for every $\eta \in \mathrm{Irr}(W_{\mathbf {G}}({\mathbf {L}},\lambda )^F)$ and where $\lambda ^\diamond $ is the extension of $\lambda $ to ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})_\lambda ^F$ given in the statement. This is a well defined bijection by the Clifford correspondence [Reference Isaacs32, Theorem 6.11] and Gallagher’s theorem [Reference Isaacs32, Corollary 6.17]. Moreover, for every $\alpha \in \mathrm{ Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ such that $({\mathbf {L}},\lambda )^\alpha =({\mathbf {L}},\lambda )$ and every $\eta \in \mathrm{Irr}(W_{\mathbf {G}}({\mathbf {L}},\lambda )^F)$ we have

$$ \begin{align*} \left(\left(\lambda^\diamond\eta\right)^{{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F}\right)^\alpha&=\left(\left(\lambda^\diamond\eta\right)^\alpha\right)^{{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F} \\ &=\left(\lambda^\diamond\eta^\alpha\right)^{{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F} \end{align*} $$

whenever $\alpha $ stabilises $\lambda ^\diamond $ . On the other hand,

$$\begin{align*}I_{({\mathbf{L}},\lambda)}^{\mathbf{G}}(\eta)^\alpha=I_{({\mathbf{L}},\lambda)}^{\mathbf{G}}\left(\eta^\alpha\right)\end{align*}$$

by Proposition 3.1 and hence we conclude that $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -equivariant provided that $\lambda ^\diamond $ is $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -invariant. Furthermore, if we consider $\eta \in \mathrm{Irr}(W_{\mathbf {G}}({\mathbf {L}},\lambda )^F)$ and define the characters $\chi :=I_{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\eta )$ and $\psi :=(\lambda ^\diamond \eta )^{{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F}$ , then the degree formula from Proposition 3.1 implies that

$$ \begin{align*} \ell^{d(\chi)}=\dfrac{\left|\hspace{1pt}{\mathbf{G}}^F\hspace{1pt}\right|{}_\ell}{\chi(1)_\ell}=\dfrac{\left|\hspace{1pt}{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}},\lambda)^F\hspace{1pt}\right|{}_\ell}{\lambda(1)_\ell\cdot\eta(1)_\ell}=\dfrac{\left|\hspace{1pt}{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\hspace{1pt}\right|{}_\ell}{\psi(1)_\ell}=\ell^{d(\psi)} \end{align*} $$

and hence we deduce that $d(\chi )=d(\psi )$ as required.

Next, we consider a regular embedding ${\mathbf {G}}\leq \widetilde {{\mathbf {G}}}$ as defined in [Reference Cabanes and Enguehard17, (15.1)]. Then, $\widetilde {{\mathbf {G}}}$ is a connected reductive group with connected centre and whose derived subgroup coincides with that of ${\mathbf {G}}$ , that is, $[\widetilde {{\mathbf {G}}},\widetilde {{\mathbf {G}}}]=[{\mathbf {G}},{\mathbf {G}}]$ . In particular, observe that $\widetilde {{\mathbf {G}}}={\mathbf {Z}}(\widetilde {{\mathbf {G}}}){\mathbf {G}}$ , that ${\mathbf {G}}$ is normal in $\widetilde {{\mathbf {G}}}$ and that the quotient $\widetilde {{\mathbf {G}}}/{\mathbf {G}}$ is an abelian group. For every Levi subgroup ${\mathbf {L}}$ of ${\mathbf {G}}$ , and recalling that Levi subgroups are exactly the centralisers of tori, we deduce that $\widetilde {{\mathbf {L}}}:={\mathbf {Z}}(\widetilde {{\mathbf {G}}}){\mathbf {L}}$ is a Levi subgroup of $\widetilde {{\mathbf {G}}}$ and that ${\mathbf {L}}\leq \widetilde {{\mathbf {L}}}$ is again a regular embedding. Notice also that ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}(\widetilde {{\mathbf {L}}})$ . These observations will be used throughout this paper without further reference.

We also recall that, according to [Reference Digne and Michel25, Proposition 13.20], restriction of characters yields a bijection between the unipotent characters of $\widetilde {{\mathbf {G}}}^F$ and those of ${\mathbf {G}}^F$ . In particular, every unipotent character of ${\mathbf {G}}^F$ is $\widetilde {{\mathbf {G}}}^F$ -invariant. Using this observation, we can compare the relative Weyl groups in $\widetilde {{\mathbf {G}}}^F$ with those in ${\mathbf {G}}^F$ .

As a consequence of the lemma above, we show that when $\lambda $ extends to its stabiliser ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F_\lambda $ , then every irreducible character of ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})$ that lies above $\lambda $ is ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ -invariant and extends to ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F={\mathbf {N}}_{\widetilde {{\mathbf {G}}}}(\widetilde {{\mathbf {L}}})^F$ .

We can now construct a parametrisation of unipotent e-Harish-Chandra series in the group $\widetilde {{\mathbf {G}}}^F$ which agrees with the bijection $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ from Corollary 3.2 via restriction of characters.

Proposition 3.5. Let $({\mathbf {L}},\lambda )$ be a unipotent e-cuspidal pair of $({\mathbf {G}},F)$ , and suppose that $\lambda $ has an extension $\lambda ^\diamond \in \mathrm{Irr}({\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})_\lambda ^F)$ . If $\widetilde {\lambda }$ is the unipotent extension of $\lambda $ to $\widetilde {{\mathbf {L}}}^F$ , then there exists a bijection $\widetilde {\Omega }_{(\widetilde {{\mathbf {L}}},\widetilde {\lambda })}^{\widetilde {{\mathbf {G}}}}$ making the following diagram commute

and where $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ is the bijection given by Corollary 3.2.

Proof. First, observe that $\lambda $ has a unique unipotent extension $\widetilde {\lambda }$ to $\widetilde {{\mathbf {L}}}^F$ according to [Reference Digne and Michel25, Proposition 13.20]. Moreover, restriction from $\widetilde {{\mathbf {G}}}^F$ to ${\mathbf {G}}^F$ induces a bijection from the set ${\mathcal {E}}(\widetilde {{\mathbf {G}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))$ to ${\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ according to [Reference Cabanes and Enguehard15, Proposition 3.1]. Consider a character $\psi \in \mathrm{Irr}({\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F)$ lying above $\lambda $ and observe that $\psi $ admits an extension $\widetilde {\psi }_0\in \mathrm{Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F)$ by Corollary 3.4. Let $\widetilde {\lambda }_0$ be an irreducible constituent of the restriction $\widetilde {\psi }_{0,\widetilde {{\mathbf {L}}}^F}$ , and notice that $\widetilde {\lambda _0}$ is an extension of $\lambda $ since $\widetilde {{\mathbf {L}}}^F/{\mathbf {L}}^F$ is abelian. Now, Gallagher’s theorem [Reference Isaacs32, Corollary 6.17] implies that there exists a linear character $\nu \in \mathrm{ Irr}(\widetilde {{\mathbf {L}}}^F/{\mathbf {L}}^F)$ such that $\widetilde {\lambda }_0\nu =\widetilde {\lambda }$ . Since ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F/{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F\simeq \widetilde {{\mathbf {L}}}^F/{\mathbf {L}}^F$ , we can identify $\nu $ with a character of ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ . Then, it follows that the character $\widetilde {\psi }:=\widetilde {\psi }_0\nu $ is an extension of $\psi $ to ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ lying above $\widetilde {\lambda }$ . Then the assignment $\psi \mapsto \widetilde {\psi }$ defines a bijection between $\mathrm{ Irr}({\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F\mid \lambda )$ and $\mathrm{Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F\mid \widetilde {\lambda })$ whose inverse is given by restriction of characters. We can now define

$$\begin{align*}\widetilde{\Omega}_{(\widetilde{{\mathbf{L}}},\widetilde{\lambda})}^{\widetilde{{\mathbf{G}}}}\left(\widetilde{\chi} \right):=\widetilde{\psi}\end{align*}$$

for every $\widetilde {\chi }\in {\mathcal {E}}(\widetilde {{\mathbf {G}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))$ and $\widetilde {\psi }\in \mathrm{Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F\mid \widetilde {\lambda })$ whenever $\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\widetilde {\chi }_{{\mathbf {G}}^F})=\widetilde {\psi }_{{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F}$ .

3.2 Construction of ${\mathbf {G}}^F$ -block isomorphisms of character triples

For the rest of this section, we assume that ${\mathbf {G}}$ is simple, simply connected and of type $\mathbf {A}$ , $\mathbf {B}$ or $\mathbf {C}$ .

We now give a more explicit construction of the group of automorphisms $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ . Fix a maximally split torus ${\mathbf {T}}_0$ contained in an F-stable Borel subgroup ${\mathbf {B}}_0$ of ${\mathbf {G}}$ . This choice corresponds to a set of graph automorphisms $\gamma :{\mathbf {G}}\to {\mathbf {G}}$ and a field endomorphism $F_0:{\mathbf {G}}\to {\mathbf {G}}$ . More precisely, if we consider the set of simple roots $\Delta \subseteq \Phi ({\mathbf {G}},{\mathbf {T}}_0)$ corresponding to the choice ${\mathbf {T}}_0\subseteq {\mathbf {B}}_0$ , then we have an automorphism $\gamma :{\mathbf {G}}\to {\mathbf {G}}$ given by $\gamma (x_\alpha (t)):=x_{\gamma (\alpha )}(t)$ for every $t\in \mathbb {G}_{\mathrm{a}}$ and $\alpha \in \pm \Delta $ and where $\gamma $ is a symmetry of the Dynkin diagram of $\Delta $ , while $F_0(x_\alpha (t)):=x_{\alpha }(t^p)$ for every $t\in \mathbb {G}_{\mathrm{a}}$ and $\alpha \in \Phi ({\mathbf {G}},{\mathbf {T}}_0)$ . Here, we denote by $x_\alpha :\mathbb {G}_{\mathrm{a}}\to {\mathbf {G}}$ the homomorphism corresponding to $\alpha \in \Phi ({\mathbf {G}},{\mathbf {T}}_0)$ . We define the subgroup $\mathcal {A}$ of $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ generated by the graph and field automorphisms described above.

In addition, we choose our regular embedding ${\mathbf {G}}\leq \widetilde {{\mathbf {G}}}$ to be defined in such a way that the graph and field automorphisms extend to $\widetilde {{\mathbf {G}}}$ (see, for instance, [Reference Malle and Späth40, Section 2B]). In particular, the group $\mathcal {A}$ acts via automorphisms on $\widetilde {{\mathbf {G}}}^F$ and we can form the external semidirect product $\widetilde {{\mathbf {G}}}^F\rtimes \mathcal {A}$ which acts on ${\mathbf {G}}^F$ . It turns out that $\widetilde {{\mathbf {G}}}^F\rtimes \mathcal {A}$ and $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ induce the same set of automorphisms on the finite group ${\mathbf {G}}^F$ (see, for instance, [Reference Gorenstein, Lyons and Solomon31, Section 2.5]).

Throughout this section, we consider a fixed unipotent e-cuspidal pair $({\mathbf {L}},\lambda )$ of $({\mathbf {G}},F)$ and a unipotent extension $\widetilde {\lambda }$ of $\lambda $ to $\widetilde {{\mathbf {L}}}^F$ (whose existence is ensured by [Reference Digne and Michel25, Proposition 13.20]) where, as always, we define $\widetilde {{\mathbf {L}}}:={\mathbf {L}}{\mathbf {Z}}(\widetilde {{\mathbf {G}}})$ . In the next lemma, we show that the hypothesis of Corollary 3.2 is satisfied under our assumptions.

As an immediate consequence of the lemma above, we deduce that every character of ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ lying above $\lambda $ extends to ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ . This can be considered as a local analogue of [Reference Digne and Michel25, Proposition 13.20].

We point out that, under our assumptions, every irreducible character of ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ lying above $\lambda $ extends to its stabiliser in ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ because the quotient ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F/{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ is cyclic according to [Reference Geck and Malle30, Proposition 1.7.5]. However, in the lemma above we are also showing, using independent methods, that each such character is ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ -invariant.

Using Lemma 3.6, we can define bijections $\Omega :=\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}$ and $\widetilde {\Omega }_{(\widetilde {{\mathbf {L}}},\widetilde {\lambda })}^{\widetilde {{\mathbf {G}}}}$ as described in Corollary 3.2 and Proposition 3.5 respectively. In what follows, we consider the sets of characters $\mathcal {G}:={\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ , $\mathcal {L}:=\mathrm{Irr}({\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F\mid \lambda )$ , $\widetilde {\mathcal {G}}:={\mathcal {E}}(\widetilde {{\mathbf {G}}}^F,(\widetilde {{\mathbf {L}}},\widetilde {\lambda }))$ and $\widetilde {\mathcal {L}}:=\mathrm{Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F\mid \widetilde {\lambda })$ . Our next aim is to show that the parametrisation $\Omega $ is compatible with ${\mathbf {G}}^F$ -block isomorphisms of character triples. We start by checking the group theoretic properties required for the existence of such isomorphisms (see [Reference Späth64, Remark 3.7 (i)]).

We now apply Lemma 3.8 to show that the map $\widetilde {\Omega }$ satisfies some useful equivariance properties. Before doing so, we need to introduce some notation. For this purpose, consider a pair $({\mathbf {G}}^*,F^*)$ dual to $({\mathbf {G}},F)$ and a pair $(\widetilde {{\mathbf {G}}}^*,F^*)$ dual to $(\widetilde {{\mathbf {G}}},F)$ . Let $i^*:\widetilde {{\mathbf {G}}}^*\to {\mathbf {G}}^*$ be the surjection induced by duality from the inclusion ${\mathbf {G}}\leq \widetilde {{\mathbf {G}}}$ , and observe that $\mathrm{Ker}(i^*)={\mathbf {Z}}(\widetilde {{\mathbf {G}}}^*)$ since ${\mathbf {G}}$ is simply connected (see [Reference Cabanes and Enguehard17, Section 15.1]). As shown in [Reference Cabanes and Enguehard17, (15.2)], there exists an isomorphism

(3.2) $$ \begin{align} \mathrm{Ker}(i^*)^F&\to \mathrm{Irr}\ \left(\widetilde{{\mathbf{G}}}^F/{\mathbf{G}}^F\right) \\ z&\mapsto \widehat{z}_{\widetilde{{\mathbf{G}}}}\nonumber. \end{align} $$

Furthermore, if ${\mathbf {L}}$ is an F-stable Levi subgroup of ${\mathbf {G}}$ and $z\in \mathrm{Ker}(i^*)$ , then we define $\widehat {z}_{\widetilde {{\mathbf {L}}}}$ to be the restriction of $\widehat {z}_{\widetilde {{\mathbf {G}}}}$ to $\widetilde {{\mathbf {L}}}^F$ and $\widehat {z}_{{\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})}$ to be the restriction of $\widehat {z}_{\widetilde {{\mathbf {G}}}}$ to ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ . We set $\mathcal {K}:=\mathrm{Ker}(i^*)$ and obtain an action of the group $\mathcal {K}$ on the characters of $\widetilde {{\mathbf {G}}}^F$ , $\widetilde {{\mathbf {L}}}^F$ and ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ as defined in [Reference Rossi54, Definition 2.1]. Moreover, we consider the external semidirect product $(\widetilde {{\mathbf {G}}}^F\mathcal {A})\ltimes \mathcal {K}$ given by defining $z^x$ as the unique element of $\mathcal {K}$ corresponding to the character $(\widehat {z}_{\widetilde {{\mathbf {G}}}})^x$ of the quotient $\widetilde {{\mathbf {G}}}^F/{\mathbf {G}}^F$ via the isomorphism specified in (3.2), whenever $x\in \widetilde {{\mathbf {G}}}^F\mathcal {A}$ and $z\in \mathcal {K}$ . Then, for every F-stable Levi subgroup ${\mathbf {L}}$ of ${\mathbf {G}}$ , we obtain an action of $(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\mathbf {L}}\ltimes \mathcal {K}$ on the irreducible characters of $\widetilde {{\mathbf {L}}}^F$ and ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ . We denote by $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\mathbf {L}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ the stabiliser of $\widetilde {\lambda }\in \mathrm{Irr}(\widetilde {{\mathbf {L}}}^F)$ . In particular, it follows that $((\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\mathbf {L}}\ltimes \mathcal {K})_{\widetilde {\lambda }}$ acts on the sets of characters $\widetilde {\mathcal {G}}$ and $\widetilde {\mathcal {L}}$ . Next, we show that the bijection $\widetilde {\Omega }$ is compatible with this action.

Proof. Let $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ and $\widetilde {\psi }\in \widetilde {\mathcal {L}}$ . By the definition of $\widetilde {\Omega }$ , we have $\widetilde {\Omega }(\widetilde {\chi })=\widetilde {\psi }$ if and only if $\Omega (\chi )=\psi $ , where $\chi :=\widetilde {\chi }_{{\mathbf {G}}^F}$ and $\psi :=\widetilde {\psi }_{{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F}$ . Now, if we consider $g\in {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ , $x\in (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{({\mathbf {L}},\lambda )}$ and $z\in \mathcal {K}$ such that $(gx,z)$ stabilises ${\widetilde {\lambda }}$ , then we obtain

$$\begin{align*}\widetilde{\Omega}\left(\widetilde{\chi}^{(gx,z)}\right)=\widetilde{\psi}^{(gx,z)}\end{align*}$$

if and only if

(3.3) $$ \begin{align} \Omega\left(\left(\widetilde{\chi}^{(gx,z)}\right)_{{\mathbf{G}}^F}\right)=\left(\widetilde{\psi}^{(gx,z)}\right)_{{\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F}. \end{align} $$

However, since the restriction of $\widetilde {\chi }^{(gx,z)}$ to ${\mathbf {G}}^F$ coincides with $\chi ^x$ and the restriction of $\widetilde {\psi }^{(gx,z)}$ to ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ coincides with $\psi ^x$ , we deduce that the equality in (3.3) holds by the equivariance properties of $\Omega $ as described in Corollary 3.2.

One of the main ingredients for the construction of the projective representations needed to obtain ${\mathbf {G}}^F$ -block isomorphisms of character triples is given by the following two lemmas on maximal extendibility.

The local version of the lemma above is a consequence of the results obtained in [Reference Brough and Späth13].

Finally, we can start constructing isomorphisms of character triples for the bijection $\Omega $ . As a first step, we obtain a weaker isomorphism, known as ${\mathbf {G}}^F$ -central isomorphism of character triples and denoted by , whose requirements are given by [Reference Späth64, Remark 3.7 (i)-(iii)] and replacing the condition on defect groups by imposing that ${\mathbf {C}}_G(N)\leq H_1\cap H_2$ with the notations used there. We refer the reader to [Reference Rossi53, Definition 3.3.4] for a precise definition.

Proposition 3.12. For every $\chi \in \mathcal {G}$ and $\psi :=\Omega (\chi )\in \mathcal {L}$ , we have

Proof. First, notice that $(\widetilde {{\mathbf {G}}}^F\mathcal {A})_\chi =\widetilde {{\mathbf {G}}}^F\mathcal {A}_\chi $ since $\chi $ is $\widetilde {{\mathbf {G}}}^F$ -invariant. We start by constructing projective representations associated with $\chi $ and $\psi $ . According to Proposition 3.5, we can find a unipotent extension $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ of $\chi $ to $\widetilde {{\mathbf {G}}}^F$ . Furthermore, by Lemma 3.10, there exists an extension $\chi '$ of $\chi $ to ${\mathbf {G}}^F\mathcal {A}_\chi $ . Let $\widetilde {\mathcal {D}}_{\mathrm{glo}}$ be a representation of $\widetilde {{\mathbf {G}}}^F$ affording $\widetilde {\chi }$ and $\mathcal {D}^{\prime }_{\mathrm{glo}}$ a representation of ${\mathbf {G}}^F\mathcal {A}_\chi $ affording $\chi '$ . Now, [Reference Späth63, Lemma 2.11] implies that

$$\begin{align*}{\mathcal{P}}_{\mathrm{glo}}:\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_\chi\to \mathrm{GL}_{\chi(1)}(\mathbb{C})\end{align*}$$

defined by ${\mathcal {P}}_{\mathrm{glo}}(x_1x_2):=\widetilde {\mathcal {D}}_{\mathrm{glo}}(x_1)\mathcal {D}^{\prime }_{\mathrm{glo}}(x_2)$ for every $x_1\in \widetilde {{\mathbf {G}}}^F$ and $x_2\in {\mathbf {G}}^F\mathcal {A}_\chi $ is a projective representation associated with $\chi $ . Next, observe that $\widetilde {\psi }:=\widetilde {\Omega }(\widetilde {\chi })\in \widetilde {\mathcal {L}}$ is an extension of $\psi $ to ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ , and consider an extension $\psi '$ of $\psi $ to $({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\psi }$ given by Lemma 3.11. Let $\widetilde {\mathcal {D}}_{\mathrm{loc}}$ be a representation of ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ affording $\widetilde {\psi }$ and $\mathcal {D}^{\prime }_{\mathrm{loc}}$ a representation of $({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\psi }$ affording $\psi '$ . Once again, [Reference Späth63, Lemma 2.11] shows that the map

$$\begin{align*}{\mathcal{P}}_{\mathrm{loc}}:\left(\widetilde{{\mathbf{G}}}^F\mathcal{A}\right)_{{\mathbf{L}},\psi}\to \mathrm{GL}_{\psi(1)}(\mathbb{C})\end{align*}$$

given by ${\mathcal {P}}_{\mathrm{loc}}(x_1x_2):=\widetilde {\mathcal {D}}_{\mathrm{loc}}(x_1)\mathcal {D}^{\prime }_{\mathrm{loc}}(x_2)$ for every $x_1\in {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ and $x_2\in ({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\psi }$ is a projective representation associated with $\psi $ . We denote by $\alpha _{\mathrm{glo}}$ and $\alpha _{\mathrm{loc}}$ the factor set of ${\mathcal {P}}_{\mathrm{glo}}$ and ${\mathcal {P}}_{\mathrm{loc}}$ , respectively. As explained in the proof of [Reference Rossi54, Theorem 4.3], in order to prove that $\alpha _{\mathrm{glo}}$ coincides with $\alpha _{\mathrm{loc}}$ via the isomorphism $\widetilde {{\mathbf {G}}}^F\mathcal {A}_\chi /{\mathbf {G}}^F\simeq (\widetilde {{\mathbf {G}}}^F\mathcal {A})_{{\mathbf {L}},\psi }/{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F$ , it suffices to show that

(3.4) $$ \begin{align} (\mu_x^{\mathrm{glo}})_{{\mathbf{N}}_{\widetilde{{\mathbf{G}}}}({\mathbf{L}})^F}=\mu_x^{\mathrm{loc}} \end{align} $$

for every $x\in ({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }$ and where $\mu _x^{\mathrm{glo}}\in \mathrm{Irr}(\widetilde {{\mathbf {G}}}^F/{\mathbf {G}}^F)$ and $\mu _x^{\mathrm{loc}}\in \mathrm{ Irr}({\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F/{\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F)$ are determined by Gallagher’s theorem (see [Reference Isaacs32, Corollary 6.17]) via the equalities $\widetilde {\chi }=\mu _x^{\mathrm{glo}}\widetilde {\chi }^x$ and $\widetilde {\psi }=\mu _x^{\mathrm{loc}}\widetilde {\psi }^x$ respectively. Because $({\mathbf {G}}^F\mathcal {A})_{{\mathbf {L}},\chi }={\mathbf {N}}_{\mathbf {G}}({\mathbf {L}})^F({\mathbf {G}}^F\mathcal {A})_{({\mathbf {L}},\lambda ),\chi }$ , we may assume that x stabilises $\lambda $ . Let $z\in \mathcal {K}$ such that $\mu _x^{\mathrm{glo}}=\hat {z}_{\widetilde {{\mathbf {G}}}}$ , and observe that $(x,z)$ is an element of $({\mathbf {G}}^F\mathcal {A})_{({\mathbf {L}},\lambda ),\chi }\rtimes \mathcal {K}$ that stabilises $\widetilde {\chi }$ . Then, applying [Reference Broué, Malle and Michel9, Theorem 3.2 (1)], we deduce that $\widetilde {\lambda }$ and $\widetilde {\lambda }^{(x,z)}$ are ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ -conjugate and we may choose $g\in {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ such that $\widetilde {\lambda }=(\widetilde {\lambda }^{(x,z)})^g=\widetilde {\lambda }^{(xg,z)}$ . In other words,

$$\begin{align*}(xg,z)\in\left({\mathbf{N}}_{\widetilde{{\mathbf{G}}}}({\mathbf{L}})^F(\widetilde{{\mathbf{G}}}^F\mathcal{A})_{({\mathbf{L}},\lambda)}\rtimes \mathcal{K}\right)_{\widetilde{\lambda}}\end{align*}$$

and thus Lemma 3.9 implies that the equality $\widetilde {\chi }=\widetilde {\chi }^{(xg,z)}$ holds if and only if $\widetilde {\psi }=\widetilde {\psi }^{(xg,z)}$ . From this, we immediately deduce the equality required in (3.4).

Next, denote by $\zeta _{\mathrm{glo}}$ and $\zeta _{\mathrm{loc }}$ the scalar functions associated to ${\mathcal {P}}_{\mathrm{glo}}$ and ${\mathcal {P}}_{\mathrm{loc}}$ , respectively. To conclude the proof, it remains to show that the central functions $\zeta _{\mathrm{glo}}$ and $\zeta _{\mathrm{loc}}$ coincide on ${\mathbf {C}}_{(\widetilde {{\mathbf {G}}}^F\mathcal {A})_{\chi }}({\mathbf {G}}^F)={\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)$ . As in the proof of [Reference Rossi54, Theorem 4.3], it is enough to show that the restrictions of $\widetilde {\chi }$ and $\widetilde {\psi }$ to ${\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)$ are multiples of a common irreducible constituent. This follows from the fact that unipotent characters contain the center in their kernel. In fact, on one hand, $1_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}$ is the unique irreducible constituent of $\widetilde {\chi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}$ because $\widetilde {\chi }$ is unipotent. On the other hand, $\widetilde {\psi }$ lies above $\widetilde {\lambda }$ and, since ${\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)\leq {\mathbf {Z}}(\widetilde {{\mathbf {L}}}^F)$ and $\widetilde {\lambda }$ is unipotent, we deduce that $1_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}$ is the unique irreducible constituent of $\widetilde {\psi }_{{\mathbf {Z}}(\widetilde {{\mathbf {G}}}^F)}$ . This completes the proof.

We conclude this section by verifying the remaining condition [Reference Späth64, Remark 3.7 (iv)] and obtain the required ${\mathbf {G}}^F$ -block isomorphisms of character triples for the map $\Omega $ .

Proposition 3.13. If the prime $\ell $ is odd, then we have

for every $\chi \in \mathcal {G}$ and where $\psi :=\Omega (\chi )$ .

Proof. By Proposition 3.12, it is enough to check the block theoretic requirement given by [Reference Späth64, Remark 3.7 (ii) and (iv)]. First, observe that under our assumption [Reference Cabanes and Enguehard15, Proposition 3.3 (ii)] shows that ${{\mathbf {L}}^F={\mathbf {C}}_{{\mathbf {G}}^F}(E)}$ where $E:={\mathbf {Z}}({\mathbf {L}})^F_\ell $ . In particular, ${\mathbf {N}}_J({\mathbf {L}})={\mathbf {N}}_J(E)$ for every ${\mathbf {G}}^F\leq J\leq \widetilde {{\mathbf {G}}}^F$ . Furthermore, for every block $C_0$ of ${\mathbf {N}}_J({\mathbf {L}})$ and every defect group D of $C_0$ , we have $E\leq {\mathbf {O}}_\ell ({\mathbf {N}}_J({\mathbf {L}}))\leq D$ and hence ${\mathbf {C}}_{\widetilde {{\mathbf {G}}}^F}(D)\leq {\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ . Now, [Reference Koshitani and Späth36, Theorem B] implies that for every block C of ${\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F$ covering $C_0$ , the induced blocks $B:=C^{\widetilde {{\mathbf {G}}}^F}$ and $B_0:=C_0^J$ are well defined and B covers $B_0$ .

Let $\widetilde {\chi }\in \widetilde {\mathcal {G}}$ be an extension of $\chi $ , and set $\widetilde {\psi }:=\widetilde {\Omega }(\widetilde {\chi })$ . By Lemma 2.3, the block of $\widetilde {C}$ of $\widetilde {\psi }$ coincides with the induced block $\mathrm{bl}(\widetilde {\lambda })^{{\mathbf {N}}_{\widetilde {{\mathbf {G}}}}({\mathbf {L}})^F}$ . Furthermore, by [Reference Cabanes and Enguehard15, Proposition 4.2] we know that the block $\widetilde {B}$ of $\widetilde {\chi }$ coincides with $b_{\widetilde {{\mathbf {G}}}^F}(\widetilde {{\mathbf {L}}},\widetilde {\lambda })=\mathrm{bl}(\widetilde {\lambda })^{\widetilde {{\mathbf {G}}}^F}$ . Then, by the transitivity of block induction we get $\widetilde {B}=\widetilde {C}^{\widetilde {{\mathbf {G}}}^F}$ . Consider now ${\mathbf {G}}^F\leq J\leq \widetilde {{\mathbf {G}}}^F$ as in the previous paragraph and notice that $\mathrm{ bl}(\widetilde {\chi }_J)$ is the unique block of J covered by $\widetilde {B}$ . Now, since $\mathrm{bl}(\widetilde {\psi }_{{\mathbf {N}}_J({\mathbf {L}})})$ is covered by $\widetilde {C}$ , we deduce that $\mathrm{ bl}(\widetilde {\psi }_{{\mathbf {N}}_J({\mathbf {L}})})^J$ is covered by $\widetilde {B}$ and therefore

(3.5) $$ \begin{align} \mathrm{bl}\left(\widetilde{\chi}_J\right)=\mathrm{bl}\left(\widetilde{\psi}_{{\mathbf{N}}_J({\mathbf{L}})}\right)^J. \end{align} $$

As explained in the proof of [Reference Rossi54, Theorem 4.8], we can now use (3.5) together with Proposition 3.12 to conclude the proof via an application of [Reference Späth64, Theorem 4.1 (i)].

3.3 Proof of Theorem C

Proof of Theorem C.

The hypothesis of Corollary 3.2 is satisfied under our restrictions on ${\mathbf {G}}$ according to Lemma 3.6, and therefore, we obtain an $\mathrm{ Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{({\mathbf {L}},\lambda )}$ -equivariant bijection

$$\begin{align*}\Omega^{\mathbf{G}}_{({\mathbf{L}},\lambda)}:{\mathcal{E}}\left({\mathbf{G}}^F,({\mathbf{L}},\lambda)\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_{\mathbf{G}}({\mathbf{L}})^F\enspace\middle|\enspace\lambda\right)\end{align*}$$

that, furthermore, preserves the $\ell $ -defect of characters. Next, observe that the groups $\widetilde {{\mathbf {G}}}^F\mathcal {A}$ and $X:={\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ induce the same automorphisms on ${\mathbf {G}}^F$ according to the description given in [Reference Gorenstein, Lyons and Solomon31, Section 2.5]. Then, by applying [Reference Späth64, Theorem 5.3] and Proposition 3.13, we conclude that

for every $\chi \in {\mathcal {E}}({\mathbf {G}}^F,({\mathbf {L}},\lambda ))$ and where $\psi :=\Omega _{({\mathbf {L}},\lambda )}^{\mathbf {G}}(\chi )$ and the proof is now complete.

4.1 Parametrisation of pseudo-unipotent characters of Levi subgroups

In this section, we assume that ${\mathbf {G}}$ is a simply connected reductive group whose irreducible components are of type $\mathbf {A}$ , $\mathbf {B}$ or $\mathbf {C}$ . Recall that, by abuse of terminology, we say that ${\mathbf {G}}$ is simply connected if so is its derived subgroup $[{\mathbf {G}},{\mathbf {G}}]$ (a semisimple group). Furthermore, we assume that the prime $\ell $ is odd.

Let ${\mathbf {K}}$ be an F-stable Levi subgroup of ${\mathbf {G}}$ , and set ${\mathbf {K}}_0:=[{\mathbf {K}},{\mathbf {K}}]$ . Observe that since the group ${\mathbf {G}}$ is simply connected, the subgroup ${\mathbf {K}}_0$ is also simply connected according to [Reference Malle and Testerman41, Proposition 12.14]. In addition, under our assumption on the type of ${\mathbf {G}}$ , we deduce that the simple components of ${\mathbf {K}}_0$ can only be of some of the types $\mathbf {A}$ , $\mathbf {B}$ or $\mathbf {C}$ .

Proposition 4.1. For every unipotent e-cuspidal pair $({\mathbf {L}}_0,\lambda _0)$ of $({\mathbf {K}}_0,F)$ , there exists a defect preserving $\mathrm{Aut}_{\mathbb {F}}({\mathbf {K}}_0^F)_{({\mathbf {L}}_0,\lambda _0)}$ -equivariant bijection

$$\begin{align*}\Omega^{{\mathbf{K}}_0}_{({\mathbf{L}}_0,\lambda_0)}:{\mathcal{E}}\left({\mathbf{K}}_0^F,({\mathbf{L}}_0,\lambda_0)\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_{{\mathbf{K}}_0}({\mathbf{L}}_0)^F\hspace{1pt}\middle|\hspace{1pt} \lambda_0\right)\end{align*}$$

such that

for every $\vartheta \in {\mathcal {E}}({\mathbf {K}}_0^F,({\mathbf {L}}_0,\lambda _0))$ and where $Y:={\mathbf {K}}_0^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {K}}_0^F)$ .

Proof. Notice that ${\mathbf {K}}_0$ is the direct product of simple algebraic groups ${\mathbf {K}}_1,\dots , {\mathbf {K}}_n$ and that the action of F permutes the simple components ${\mathbf {K}}_i$ . Denote the direct product of the simple components in each F-orbit by ${\mathbf {H}}_j$ for $j=1,\dots , t$ . The $({\mathbf {H}}_j,F)$ are the irreducible rational components of $({\mathbf {K}}_0,F)$ , and we have ${\mathbf {K}}_0^F={\mathbf {H}}_1^F\times \cdots \times {\mathbf {H}}_t^F$ . Similarly, if we define the intersections ${\mathbf {M}}_j:={\mathbf {L}}_0\cap {\mathbf {H}}_j$ , then we have a decomposition ${\mathbf {L}}_0^F={\mathbf {M}}_1^F\times \cdots \times {\mathbf {M}}_t^F$ . In particular, we can write $\lambda _0=\mu _1\times \cdots \times \mu _t$ with $\mu _j\in \mathrm{Irr}({\mathbf {M}}_j^F)$ . In this case, notice that $({\mathbf {M}}_j,\mu _j)$ is a unipotent e-cuspidal pair of $({\mathbf {H}}_j,F)$ . Next, suppose that ${\mathbf {H}}_j={\mathbf {H}}_{j,1}\times \dots \times {\mathbf {H}}_{j,m_j}$ , and observe that ${\mathbf {H}}_j^F\simeq {\mathbf {H}}_{j,1}^{F^{m_j}}$ . By the discussion at the beginning of this section, we know that ${\mathbf {H}}_{j,1}$ is a simple, simply connected group of type $\mathbf {A}$ , $\mathbf {B}$ or $\mathbf {C}$ and hence it satisfies the assumptions of Theorem C. Then, via the isomorphism ${\mathbf {H}}_j^F\simeq {\mathbf {H}}_{j,1}^{F^{m_j}}$ , we obtain an $\mathrm{Aut}_{\mathbb {F}}({\mathbf {H}}_j^F)_{({\mathbf {M}}_j,\mu _j)}$ -equivariant bijection

$$\begin{align*}\Omega^{{\mathbf{H}}_j}_{({\mathbf{M}}_j,\mu_j)}:{\mathcal{E}}\left({\mathbf{H}}_j^F,({\mathbf{M}}_j,\mu_j)\right)\to\mathrm{ Irr}\left({\mathbf{N}}_{{\mathbf{H}}_j}({\mathbf{M}}_j)^F\hspace{1pt}\middle|\hspace{1pt}\mu_j\right)\end{align*}$$

that preserves the defect of characters and such that

(4.1)

for every $\vartheta \in {\mathcal {E}}({\mathbf {H}}_j^F,({\mathbf {M}}_j,\mu _j))$ and where $Y_j:={\mathbf {H}}_j^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {H}}_j^F)$ . Since the characters in the sets ${\mathcal {E}}({\mathbf {K}}_0^F,({\mathbf {L}}_0,\lambda _0))$ and $\mathrm{Irr}({\mathbf {N}}_{{\mathbf {K}}_0}({\mathbf {L}}_0)^F\mid \lambda _0)$ are products of characters belonging to the sets ${\mathcal {E}}({\mathbf {H}}_j^F,({\mathbf {M}}_j,\mu _j))$ and $\mathrm{Irr}({\mathbf {N}}_{{\mathbf {H}}_j}({\mathbf {M}}_j)^F\mid \mu _j)$ , respectively, we obtain a bijection

$$\begin{align*}\Omega_{({\mathbf{L}}_0,\lambda_0)}^{{\mathbf{K}}_0}:{\mathcal{E}}\left({\mathbf{K}}_0^F,({\mathbf{L}}_0,\lambda_0)\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_{{\mathbf{K}}_0}({\mathbf{L}}_0)^F\hspace{1pt}\middle|\hspace{1pt} \lambda_0\right)\end{align*}$$

by setting

$$\begin{align*}\Omega_{({\mathbf{L}}_0,\lambda_0)}^{{\mathbf{K}}_0}\left(\vartheta_1\times\dots\times \vartheta_t\right):=\Omega_{({\mathbf{M}}_1,\mu_1)}^{{\mathbf{H}}_1}(\vartheta_1)\times \dots\times \Omega_{({\mathbf{M}}_t,\mu_t)}^{{\mathbf{H}}_t}(\vartheta_t)\end{align*}$$

for every $\vartheta _j\in {\mathcal {E}}({\mathbf {H}}_j^F,({\mathbf {M}}_j,\mu _j))$ . Finally, arguing as in the proof of [Reference Rossi57, Proposition 6.5], we deduce that the bijection $\Omega ^{{\mathbf {K}}_0}_{({\mathbf {L}}_0,\lambda _0)}$ preserves the defect of characters, is $\mathrm{Aut}_{\mathbb {F}}({\mathbf {K}}_0^F)_{({\mathbf {L}}_0,\lambda _0)}$ -equivariant, and, using (4.1), it induces the ${\mathbf {K}}_0^F$ -block isomorphisms of character triples required in the statement.

In our next result, we replace the automorphism group $Y:={\mathbf {K}}_0^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {K}}_0^F)$ with the group of automorphisms of ${\mathbf {G}}^F$ stabilising ${\mathbf {K}}$ , that is, $X:=({\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{\mathbf {K}}$ . To do so, we apply the so-called Butterfly Theorem [Reference Späth64, Theorem 5.3] which basically states that, for any finite group G, the notion of G-block isomorphism of character triples only depends on the automorphisms induced on G.

Corollary 4.2. If $({\mathbf {L}}_0,\lambda _0)$ is a unipotent e-cuspidal pair of $({\mathbf {K}}_0,F)$ , then the map $\Omega _{({\mathbf {L}}_0,\lambda _0)}^{{\mathbf {K}}_0}$ given by Proposition 4.1 is $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{{\mathbf {K}},({\mathbf {L}}_0,\lambda _0)}$ -equivariant and satisfies

(4.2)

for every $\vartheta \in {\mathcal {E}}({\mathbf {K}}_0^F,({\mathbf {L}}_0,\lambda _0))$ and where $X:=({\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{\mathbf {K}}$ .

The notion of isomorphism of character triples, introduced by Isaacs, plays a fundamental role in representation theory of finite groups and in the study of the local-global conjectures. One of the most important consequences of the existence of isomorphisms of character triples is the possibility to lift character bijections. For instance, the main result of [Reference Navarro and Späth48] shows how to apply this technique to construct bijections above characters of height zero in the context of the Alperin–McKay Conjecture [Reference Navarro and Späth48, Theorem B]. The main consequence of this result, which follows from an argument introduced by Murai [Reference Murai45], is a reduction theorem for the celebrated Brauer’s Height Zero Conjecture [Reference Navarro and Späth48, Theorem A]. This strategy ultimately lead to the solution of Brauer’s conjecture (see [Reference Malle, Navarro, Schaffer Fry and Tiep39] and [Reference Ruhstorfer60]). For other applications of isomorphisms of character triples, see [Reference Martínez and Rossi42, Proposition 1.1], [Reference Martínez, Rizo and Rossi43], [Reference Navarro, Späth and Vallejo49], [Reference Rossi52], [Reference Rossi55], [Reference Rossi59], [Reference Ruhstorfer61] and [Reference Turull68].

In our next result, we exploit this idea in order to lift the bijections given by Proposition 4.1 to the Levi subgroup ${\mathbf {K}}$ . Consequently, we extend the parametrisation of unipotent e-Harish-Chandra series given by Theorem C for the simple group ${\mathbf {G}}$ to a parametrisation of e-Harish-Chandra series associated to $({\mathbf {K}},F)$ -pseudo-unipotent characters for every F-stable Levi subgroup ${\mathbf {K}}$ of ${\mathbf {G}}$ . First, we need a preliminary lemma.

Theorem 4.4. Suppose that ${\mathbf {G}}$ is a simply connected group whose irreducible components are of type $\mathbf {A}$ , $\mathbf {B}$ or $\mathbf {C}$ , and assume that $\ell $ is odd. For every F-stable Levi subgroup ${\mathbf {K}}\leq {\mathbf {G}}$ and every unipotent e-cuspidal pair $({\mathbf {L}},\lambda )$ of $({\mathbf {K}},F)$ , there exists a defect preserving $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{{\mathbf {K}},({\mathbf {L}},\lambda )}$ -equivariant bijection

$$\begin{align*}\Omega_{({\mathbf{L}},\lambda)}^{\mathbf{K}}:{\mathcal{E}}\left({\mathbf{K}}^F,({\mathbf{L}},{\mathrm{ps}_{\mathbf{K}}(\lambda)})\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_{\mathbf{K}}({\mathbf{L}})^F\enspace\middle|\enspace {\mathrm{ps}_{\mathbf{K}}(\lambda)}\right)\end{align*}$$

such that

for every $\chi \in {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\mathrm{ps}_{\mathbf {K}}(\lambda )))$ and where $X:=({\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{\mathbf {K}}$ .

Proof. Recall that ${\mathbf {K}}_0=[{\mathbf {K}},{\mathbf {K}}]$ , and define ${\mathbf {L}}_0:={\mathbf {L}}\cap {\mathbf {K}}_0$ and $\lambda _0$ the restriction of $\lambda $ to ${\mathbf {L}}_0^F$ . Observe that $({\mathbf {L}}_0,\lambda _0)$ is a unipotent e-cuspidal pair of $({\mathbf {K}}_0,F)$ . Let $z\in {\mathbf {Z}}({\mathbf {K}}^*)^{F^*}$ , and consider a character $\chi $ belonging to ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda \hat {z}_{\mathbf {L}}))$ . Since the restriction of $\lambda \hat {z}_{\mathbf {L}}$ to ${\mathbf {L}}_0^F$ coincides with $\lambda _0$ , [Reference Geck and Malle30, Corollary 3.3.25] implies that $\chi $ lies above some character in ${\mathcal {E}}({\mathbf {K}}^F_0({\mathbf {L}}_0,\lambda _0))$ . On the other hand, assume that $\chi \in \mathrm{Irr}({\mathbf {K}}^F)$ lies above $\chi _0\in {\mathcal {E}}({\mathbf {K}}^F_0,({\mathbf {L}}_0,\lambda _0))$ . By [Reference Cabanes and Enguehard15, Proposition 3.1], the character $\chi _0$ has an extension $\chi '\in {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda ))$ and hence, using Gallagher’s theorem [Reference Isaacs32, Corollary 6.17] and [Reference Cabanes and Enguehard17, (8.19)], we can find $z\in {\mathbf {Z}}({\mathbf {K}}^*)^{F^*}$ such that $\chi =\chi '\hat {z}_{\mathbf {K}}$ . Since $\chi '\hat {z}_{\mathbf {K}}$ is a character of ${\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\lambda \hat {z}_{\mathbf {L}}))$ according to [Reference Cabanes and Enguehard17, (8.20)], we conclude that

(4.3) $$ \begin{align} {\mathcal{E}}\left({\mathbf{K}}^F,({\mathbf{L}},\mathrm{ps}_{\mathbf{K}}(\lambda))\right)=\mathrm{Irr}\ \left({\mathbf{K}}^F\enspace\middle|\enspace {\mathcal{E}}\left({\mathbf{K}}^F_0,({\mathbf{L}}_0,\lambda_0)\right)\right). \end{align} $$

Next, suppose that $\psi \in \mathrm{Irr}({\mathbf {N}}_{\mathbf {K}}({\mathbf {L}})^F\mid \lambda \hat {z}_{\mathbf {L}})$ . In this case, $\psi $ lies above the restriction of $\lambda \hat {z}_{\mathbf {L}}$ to ${\mathbf {L}}_0^F$ which coincides with $\lambda _0$ . In particular, there exists some $\varphi \in \mathrm{Irr}({\mathbf {N}}_{{\mathbf {K}}_0}({\mathbf {L}}_0)^F\mid \lambda _0)$ such that $\psi $ lies above $\varphi $ . On the other, if $\chi $ lies above such a character $\varphi \in \mathrm{Irr}({\mathbf {N}}_{{\mathbf {K}}_0}({\mathbf {L}}_0)^F\mid \lambda _0)$ , then it lies above $\lambda _0$ , and therefore, we can find $z\in {\mathbf {Z}}({\mathbf {K}}^*)^{F^*}$ such that $\psi \in \mathrm{Irr}({\mathbf {N}}_{\mathbf {K}}({\mathbf {L}})^F\mid \lambda \hat {z}_{\mathbf {L}})$ . This shows that

(4.4) $$ \begin{align} \mathrm{Irr}\ \left({\mathbf{N}}_{\mathbf{K}}({\mathbf{L}})^F\enspace\middle|\enspace \mathrm{ps}_{\mathbf{K}}(\lambda)\right)=\mathrm{Irr}\ \left({\mathbf{N}}_{{\mathbf{K}}}({\mathbf{L}})^F\enspace\middle|\enspace \mathrm{ Irr}\left({\mathbf{N}}_{{\mathbf{K}}_0}({\mathbf{L}}_0)^F\enspace\middle|\enspace \lambda_0\right)\right). \end{align} $$

Finally, consider the map $\Omega _{({\mathbf {L}}_0,\lambda _0)}^{{\mathbf {K}}_0}$ given by Proposition 4.1. Then, the result follows from (4.3) and (4.4) by applying [Reference Rossi57, Proposition 6.1 and Remark 6.2] as explained in the proof of [Reference Rossi57, Corollary 6.10] and using the ${\mathbf {K}}^F$ -block isomorphisms of character triples obtained in Corollary 4.2. Here, we consider $A:={\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ , $A_0:={\mathbf {N}}_A({\mathbf {L}})$ , $K:={\mathbf {K}}_0^F$ , $K_0={\mathbf {N}}_{{\mathbf {K}}_0}({\mathbf {L}})^F={\mathbf {N}}_{{\mathbf {K}}_0}({\mathbf {L}}_0)^F$ , $G:={\mathbf {G}}^F$ , $X:=({\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{\mathbf {K}}$ , $\mathcal {S}:={\mathcal {E}}({\mathbf {K}}_0^F,({\mathbf {L}}_0,\lambda _0))$ , $\mathcal {S}_0:=\mathrm{Irr}({\mathbf {N}}_{{\mathbf {K}}_0}({\mathbf {L}}_0)^F\mid \lambda _0)$ , $V:=({\mathbf {G}}^F\rtimes \mathrm{ Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{{\mathbf {K}},\mathcal {S}}$ and $U:=({\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{{\mathbf {K}},{\mathbf {L}},\lambda _0}$ . Observe that the condition on defect groups required by [Reference Rossi57, Proposition 6.1] is satisfied by Lemma 4.3.

4.2 Above e-Harish-Chandra series

We now further extend Theorem C by lifting the character bijections from Theorem 4.4 with respect to normal inclusions.

Proposition 4.5. Consider the setup of Theorem 4.4, and let ${\mathbf {K}}^F\leq H\leq {\mathbf {N}}_{\mathbf {G}}({\mathbf {K}})^F$ . Then, there exists a defect preserving $\mathrm{ Aut}_{\mathbb {F}}({\mathbf {G}}^F)_{H,{\mathbf {K}},({\mathbf {L}},\lambda )}$ -equivariant bijection

$$\begin{align*}\Omega_{({\mathbf{L}},\lambda)}^{{\mathbf{K}},H}:\mathrm{Irr}\ \left(H\hspace{1pt}\middle|\hspace{1pt} {\mathcal{E}}\left({\mathbf{K}}^F,({\mathbf{L}},\mathrm{ps}_{\mathbf{K}}(\lambda))\right)\right)\to\mathrm{ Irr}\left({\mathbf{N}}_H({\mathbf{L}})\hspace{1pt}\middle|\hspace{1pt} \mathrm{ps}_{\mathbf{K}}(\lambda)\right)\end{align*}$$

such that

for every $\chi \in \mathrm{Irr}(H\mid {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\mathrm{ps}_{\mathbf {K}}(\lambda ))))$ and where $X:=({\mathbf {G}}^F\rtimes \mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F))_{\mathbf {K}}$ .

Before proceeding further, we point out an interesting analogy with another important character correspondence. The Glauberman correspondence plays a fundamental role in the study of the local-global counting conjectures and lies at the heart of most reduction theorems. In its most basic form, it states that for every finite $\ell $ -group L acting on a finite $\ell '$ -group K, there exists a bijection

$$\begin{align*}f_L:\mathrm{Irr}_L(K)\to\mathrm{Irr}({\mathbf{N}}_K(L))\end{align*}$$

between the set of L-invariant characters of K and the characters of the normaliser ${\mathbf {N}}_K(L)$ (see, for instance, [Reference Navarro47, Section 2.3]). A deep result due to Dade [Reference Dade20], and recently reproved by Turull [Reference Turull67], shows that, if K and L are subgroups of a finite group G and $KL\leq H\leq K{\mathbf {N}}_G(L)$ , then the Glauberman correspondence $f_L$ can be lifted to a character correspondence for H, that is, there exists a bijection

(4.5) $$ \begin{align} f_L^H:\mathrm{Irr}\ \left(H\enspace\middle|\enspace\chi\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_H(L)\enspace\middle|\enspace f_L(\chi)\right) \end{align} $$

for every $\chi \in \mathrm{Irr}_L(K)$ . On the other hand, the parametrisation of unipotent e-Harish-Chandra series obtained by Broué, Malle and Michel [Reference Broué, Malle and Michel9, Theorem 3.2] lies at the centre of the proofs of the local-global counting conjectures for finite reductive groups. It is interesting to note that our methods yield a character bijection above e-Harish-Chandra series which is analogous to (4.5) in the context of the Glauberman correspondence. This is an immediate consequence of Proposition 4.5.

Corollary 4.6. Consider the setup of Theorem 4.4, and let ${\mathbf {K}}^F\leq H\leq {\mathbf {N}}_{\mathbf {G}}({\mathbf {K}})^F$ . Then, there exists a bijection

$$\begin{align*}\Psi_\chi^H:\mathrm{Irr}\ \left(H\enspace\middle|\enspace \chi\right)\to\mathrm{Irr}\ \left({\mathbf{N}}_H({\mathbf{L}})\enspace\middle|\enspace\Omega_{({\mathbf{L}},\lambda)}^{\mathbf{K}}(\chi)\right)\end{align*}$$

for every $\chi \in {\mathcal {E}}({\mathbf {K}}^F,({\mathbf {L}},\mathrm{ps}_{\mathbf {K}}(\lambda )))$ .

5.1 Preliminaries on e-chains

Our first aim is to define e-local structures for finite reductive groups that play a role analogous to that of $\ell $ -chains in the context of Dade’s Conjecture and the Character Triple Conjecture. The connection between the set of e-chains and that of $\ell $ -chains has already been studied in [Reference Rossi57, Section 7.2]. These results provide a way to obtain Dade’s Conjecture and the Character Triple Conjecture as a consequence of [Reference Rossi57, Conjecture C and Conjecture D]. The possibility to use different types of chains is crucial in the study of Dade’s Conjecture and has been introduced by Knörr and Robinson [Reference Knörr and Robinson35]. Their results were insipred by previous studies conducted by many authors including Brown [Reference Brown14] and Quillen [Reference Quillen50] who analysed the homotopy theory of associated simplicial complexes. As in the previous section, we assume that ${\mathbf {G}}$ is a simply connected reductive group whose irreducible components are of type $\mathbf {A}$ , $\mathbf {B}$ or $\mathbf {C}$ and that $\ell $ is an odd prime.

Definition 5.1. We denote by ${\mathcal {L}}_e({\mathbf {G}},F)$ the set of e -chains of the finite reductive group $({\mathbf {G}},F)$ , that is, chains of the form

$$\begin{align*}\sigma=\left\lbrace{\mathbf{G}}={\mathbf{L}}_0>{\mathbf{L}}_1>\dots>{\mathbf{L}}_n\right\rbrace,\end{align*}$$

where n is a nonnegative integer and each ${\mathbf {L}}_i$ is an e-split Levi subgroup of $({\mathbf {G}},F)$ . We denote by $|\sigma |:=n$ the length of the e-chain $\sigma $ and by ${\mathbf {L}}(\sigma )$ its last term. Furthermore, we define ${\mathcal {L}}_e({\mathbf {G}},F)_{>0}$ to be the set of e-chains having length strictly larger than $0$ .

Observe that the notion of length defined above induces a partition of the set ${\mathcal {L}}_e({\mathbf {G}},F)$ into e-chains of even and odd length. More precisely, we denote by ${\mathcal {L}}_e({\mathbf {G}},F)_\pm $ the subset of those e-chains $\sigma \in {\mathcal {L}}_e({\mathbf {G}},F)$ that satisfy $(-1)^{|\sigma |}=\pm 1$ .

In what follows, given an e-chain $\sigma $ and an e-split Levi subgroup ${\mathbf {M}}$ of $({\mathbf {L}}(\sigma ),F)$ , we denote by $\sigma +{\mathbf {M}}$ the e-chain obtained by adding ${\mathbf {M}}$ at the end of $\sigma $ . We also allow the possibility that ${\mathbf {M}}={\mathbf {L}}(\sigma )$ , in which case we define $\sigma +{\mathbf {L}}(\sigma ):=\sigma $ . Vice versa, we denote by $\sigma -{\mathbf {L}}(\sigma )$ the e-chain obtained by removing the last term ${\mathbf {L}}(\sigma )$ from $\sigma $ . Here, we use the convention that $\sigma _0-{\mathbf {L}}(\sigma _0)=\sigma _0=\sigma _0+{\mathbf {G}}$ where $\sigma _0=\{{\mathbf {G}}\}$ is the trivial e-chain.

Next, consider the action of ${\mathbf {G}}^F$ on the set of e-chains ${\mathcal {L}}_e({\mathbf {G}},F)$ induced by conjugation: for every $g\in {\mathbf {G}}^F$ and $\sigma =\{{\mathbf {L}}_i\}_i$ , we define

$$\begin{align*}\sigma^g:=\left\lbrace{\mathbf{G}}={\mathbf{L}}_0>{\mathbf{L}}_1^g>\dots>{\mathbf{L}}_n^g\right\rbrace.\end{align*}$$

It follows from this definition that the stabiliser ${\mathbf {G}}^F_\sigma $ coincides with the intersection of the normalisers ${\mathbf {N}}_{\mathbf {G}}({\mathbf {L}}_i)^F$ for $i=1,\dots , n$ . Similarly, we can define an action of $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)$ on ${\mathcal {L}}_e({\mathbf {G}},F)$ and give an analogous description of the chain stabilisers $\mathrm{Aut}_{\mathbb {F}}({\mathbf {G}}^F)_\sigma $ . In particular, notice that the last term of the chain satisfies ${\mathbf {L}}(\sigma )^F\unlhd {\mathbf {G}}_\sigma ^F$ . Using this observation, we can use the results of Section 4.2 to control the characters of ${\mathbf {G}}^F_\sigma $ that lie above pseudo-unipotent series of ${\mathbf {L}}(\sigma )$ .

Definition 5.2. For every e-chain $\sigma \in {\mathcal {L}}_e({\mathbf {G}},F)$ , we denote by ${\mathcal {C} \mathcal {P}_{\mathrm{u}}}(\sigma )$ the set of unipotent e-cuspidal pairs $({\mathbf {M}},\mu )\in {\mathcal {C} \mathcal {P}_{\mathrm{u}}}({\mathbf {L}}(\sigma ),F)$ that satisfy ${\mathbf {M}}<{\mathbf {G}}$ . Furthermore, for any such pair $({\mathbf {M}},\mu )\in {\mathcal {C} \mathcal {P}_{\mathrm{u}}}(\sigma )$ , we define the character set

$$\begin{align*}\mathrm{Irr}_{\mathrm{ps}}({\mathbf{G}}^F_\sigma,({\mathbf{M}},\mu)):=\begin{cases} \mathrm{Irr}\ \left({\mathbf{G}}_\sigma^F\enspace\middle|\enspace{\mathcal{E}}\left({\mathbf{L}}(\sigma)^F,\left({\mathbf{M}},\mathrm{ps}_{{\mathbf{L}}(\sigma)}(\mu)\right)\right)\right)& {\mathbf{L}}(\sigma)>{\mathbf{M}} \\ \mathrm{Irr}\ \left({\mathbf{G}}_\sigma^F\enspace\middle|\enspace{\mathcal{E}}\left({\mathbf{L}}(\sigma)^F,\left({\mathbf{M}},\mathrm{ps}_{{\mathbf{L}}(\sigma-{\mathbf{L}}(\sigma))}(\mu)\right)\right)\right)& {\mathbf{L}}(\sigma)={\mathbf{M}}. \end{cases}\end{align*}$$

The need to distinguish the cases in the above definition will become apparent in the proofs of Proposition 5.7 and Theorem 5.10 below. Observe that in the definition above, we are excluding the degenerate case where ${\mathbf {G}}={\mathbf {L}}(\sigma )={\mathbf {M}}$ . To understand the reason why we are excluding this case, we can consider an analogy with Dade’s Conjecture. For every finite group G, recall that ${\mathbf {k}}(G)$ denotes the number of its irreducible characters and that, for any nonnegative integer d, the symbol ${\mathbf {k}}^d(G)$ denotes the number of those irreducible characters of $\ell $ -defect d. The local-global counting conjectures provide a way to determine the global invariants ${\mathbf {k}}^d(G)$ in terms of $\ell $ -local structures. This idea was made precise by Isaacs and Navarro [Reference Isaacs and Navarro33]. According to their definitions, the block-free version of Dade’s Conjecture can be stated by saying that the functions ${\mathbf {k}}^d$ are chain local for every $d>0$ . Consequently, and because a sum of chain local functions is chain local, we deduce that the difference ${\mathbf {k}}-{\mathbf {k}}^0=\sum _{d>0}{\mathbf {k}}^d$ is (conjecturally) a chain local function. On the other hand, using the fact that groups admitting a character of $\ell $ -defect zero have trivial $\ell $ -core, it is easy to see that ${\mathbf {k}}^0$ is not chain local. The exclusion of the case ${\mathbf {G}}={\mathbf {L}}(\sigma )={\mathbf {M}}$ can be explained by interpreting these observations in the context of unipotent characters. Recall that ${\mathbf {k}}_{\mathrm{u}}({\mathbf {G}}^F)$ and ${\mathbf {k}}_{\mathrm{ c,u}}({\mathbf {G}}^F)$ denote the number of unipotent characters of ${\mathbf {G}}^F$ and unipotent e-cuspidal characters of ${\mathbf {G}}^F$ respectively. If $\ell $ does not divide the order of ${\mathbf {Z}}({\mathbf {G}}^F)$ , then [Reference Cabanes and Enguehard15] implies that the unipotent e-cuspidal characters of ${\mathbf {G}}^F$ have $\ell $ -defect zero. Therefore, as in the case of Dade’s Conjecture, the global invariant we want to determine e-locally is the difference ${\mathbf {k}}_{\mathrm{u}}({\mathbf {G}}^F)-{\mathbf {k}}_{\mathrm{c,u}}({\mathbf {G}}^F)$ . Finally, notice that ${\mathbf {k}}_{\mathrm{c,u}}({\mathbf {G}}^F)$ is exactly the number of unipotent e-cuspidal pairs $({\mathbf {M}},\mu )$ of ${\mathbf {L}}(\sigma )$ satisfying ${\mathbf {G}}={\mathbf {L}}(\sigma )={\mathbf {M}}$ .

In the following lemma, we show that if the set $\mathrm{Irr}_{\mathrm{ps}}({\mathbf {G}}_\sigma ^F,({\mathbf {M}},\mu ))$ is nonempty then $({\mathbf {M}},\mu )$ is uniquely defined up to ${\mathbf {G}}_\sigma ^F$ -conjugation.

Next, we describe the block theory associated to characters in the sets introduced in Definition 5.2.

Proof. The first point follows from Lemma 2.3 by choosing ${\mathbf {L}}={\mathbf {L}}(\sigma )$ and $H={\mathbf {G}}_\sigma ^F$ . Furthermore, in the second case of Definition 5.2 observe that ${\mathbf {L}}(\sigma )\leq {\mathbf {L}}(\sigma -{\mathbf {L}}(\sigma ))$ from which it follows that ${\mathbf {Z}}({\mathbf {L}}(\sigma -{\mathbf {L}}(\sigma ))^*)\leq {\mathbf {Z}}({\mathbf {L}}(\sigma )^*)$ . Therefore, we can always find $\varphi $ and z as in the statement of (ii). Since $\varphi $ is an irreducible constituent of the virtual character ${\mathbf {R}}_{\mathbf {M}}^{{\mathbf {L}}(\sigma )}(\mu )$ , it follows from [Reference Cabanes and Enguehard15, Proposition 4.2] (whose assumptions are satisfied by [Reference Cabanes and Enguehard15, Proposition 3.3 (ii)]) that $\mathrm{bl}(\varphi )=b_{{\mathbf {L}}(\sigma )^F}({\mathbf {M}},\mu )=\mathrm{bl}(\mu )^{{\mathbf {L}}(\sigma )^F}$ . Then, since $\hat {z}_{\mathbf {M}}$ is the restriction of the linear character $\hat {z}_{{\mathbf {L}}(\sigma )}$ to ${\mathbf {M}}^F$ , we deduce from Lemma 2.1 that

$$\begin{align*}\mathrm{bl}(\varphi\hat{z}_{{\mathbf{L}}(\sigma)})=\mathrm{bl}(\mu\hat{z}_{\mathbf{M}})^{{\mathbf{L}}(\sigma)^F}.\end{align*}$$

Now, [Reference Navarro46, Theorem 9.19] implies that

$$\begin{align*}\mathrm{bl}(\vartheta)=\mathrm{bl}\left(\varphi\hat{z}_{{\mathbf{L}}(\sigma)}\right)^{{\mathbf{G}}_\sigma^F}\end{align*}$$

and the second point follows by the transitivity of block induction. Finally, set $Q:={\mathbf {Z}}({\mathbf {M}})^F_\ell $ and observe that $Q{\mathbf {C}}_{{\mathbf {G}}^F}(Q)={\mathbf {M}}^F\leq {\mathbf {N}}_{{\mathbf {G}}^F}(Q)$ by [Reference Cabanes and Enguehard15, Proposition 3.3(ii)]. Then, [Reference Navarro46, Theorem 4.14] implies that $\mathrm{bl}(\mu \hat {z}_{\mathbf {M}})^{{\mathbf {G}}^F}$ is defined and so is $\mathrm{bl}(\vartheta )^{{\mathbf {G}}^F}$ by (ii) and transitivity of block induction. This concludes the proof.

Using the lemma above, we can now define the following character set. This yields the e-local object through which we can determine the number of (pseudo-)unipotent characters in a given block B of ${\mathbf {G}}^F$ and with a given defect $d\geq 0$ (see Section 5.3).

Definition 5.5. Let B be a block of ${\mathbf {G}}^F$ and d a nonnegative integer. For every e-chain $\sigma \in {\mathcal {L}}_e({\mathbf {G}},F)$ and unipotent e-cuspidal pair $({\mathbf {M}},\mu )\in {\mathcal {C} \mathcal {P}_{\mathrm{ u}}}(\sigma )$ , we define the character set

$$\begin{align*}\mathrm{Irr}_{\mathrm{ps}}^d\left(B_\sigma,({\mathbf{M}},\mu)\right):=\left\lbrace\vartheta\in\mathrm{Irr}_{\mathrm{ps}}\left({\mathbf{G}}_\sigma^F,({\mathbf{M}},\mu)\right)\enspace\middle|\enspace d(\vartheta)=d, \mathrm{ bl}(\vartheta)^{{\mathbf{G}}^F}=B\right\rbrace,\end{align*}$$

where $\mathrm{bl}(\vartheta )^{{\mathbf {G}}^F}$ is defined according to Lemma 5.4 (iii). Furthermore, we denote the cardinality of this set by

$$\begin{align*}{\mathbf{k}}_{\mathrm{ps}}^d\left(B_\sigma,({\mathbf{M}},\mu)\right):=\left|\mathrm{Irr}_{\mathrm{ps}}^d\left(B_\sigma,({\mathbf{M}},\mu)\right)\right|.\end{align*}$$

In the following remark, we explain that the character sets defined above might be nonempty even if B is not a unipotent block.