On Semisimple Classes and Component Groups in Type $\textsf{D}$

Abstract

In adjoint simple algebraic groups $\textbf{H}$ of type $\textsf{D}$ we show that for every semisimple element s, its centralizer splits over its identity component, i.e. $C _{\textbf{H}}(s)= C _{\textbf{H}}(s)^{\circ }\rtimes \check{A}$ for some complement $\check{A}$ with strong stability properties. We derive several consequences about the action of automorphisms on semisimple conjugacy classes. This helps to parametrize characters of the finite groups $\textsf{D}_{l,\text {sc}}(q)$ and $^2\textsf{D}_{l,\text {sc}}(q)$ and describe the action of automorphisms on them. It is also a contribution to the final proof of the McKay conjecture for the prime 3, see (B. Späth: 2023).

1 Introduction

One of the many outstanding ideas in Deligne–Lusztig theory was to point out the relevance of a dual group $\textbf{G}^*$ to the study of representations of a finite group of Lie type $G=\textbf{G}^F$ (see for instance [4, Chapters 11 and 12]). In particular, the set $\textrm{Irr}(G)$ of irreducible (complex) characters is partitioned according to semisimple elements of $G^*=\textbf{G}^*{}^F$ up to $G^*$-conjugacy (“rational series”) or $\textbf{G}^*$-conjugacy (“geometric series”). This explains how the study of $\textrm{Irr}(G)$, for instance as an $\textrm{Out}(G)$-set, may lead to purely group-theoretic questions about $G^*$ and its semisimple classes.

Together with Lusztig’s Jordan decomposition of characters, rational series help to study characters in terms of unipotent characters of centralizers of semisimple elements $\textrm{C}_{G^*}(s)$. It is a well known fact that the rational classes in a given geometric class are parametrized via the component group $A(s):=\textrm{C}_{\textbf{G}^*}(s)/\textrm{C}_{\textbf{G}^*}^\circ (s)$ of the algebraic centralizer $\textrm{C}_{\textbf{G}^*}(s)$. The various lifts in $\textrm{C}_{\textbf{G}^*}(s)$ of a fixed element of A(s) provide different rational structures on the centralizer $\textrm{C}_{\textbf{G}^*}(s)$ and different finite groups whose unipotent characters are then in correspondence with the characters of a given rational series. The results presented here are applied in [11] to establish a Jordan decomposition compatible with automorphisms.

The study of $\textrm{Irr}(\textbf{G}^F)$ is probably most difficult when $\textbf{G}$ is a simply connected simple group and its type is $\textsf{D}$. The difficulties come from the graph automorphism and a possibly non-cyclic fundamental group. However the dual group $\textbf{G}^*$ in that type, denoted here as $\textbf{H}$, has a couple of remarkable features with regard to semisimple classes and their centralizers. In particular we single out, for s a semisimple element of $\textbf{H}$, a semidirect decomposition

$$ \textrm{C}_{\textbf{H}}(s)=\textrm{C}_{\textbf{H}}^\circ (s)\rtimes \check{A}, $$

where the complement $\check{A}$ has a strong stability with regard to Frobenius endomorphisms and graph automorphisms. In [11], the elements of $\check{A}$ are crucial to parametrize the different rational structures on $\textrm{C}_{\textbf{G}^*}(s)$ and the action of $\textrm{Out}(\textbf{G}^F)$ on rational series of characters. This then also contributes to the verification of McKay’s conjecture for the prime 3 in [11].

In type different from $\textsf{D}$ our questions in representation theory have been solved by simpler methods not requiring this study of adjoint groups. It could be interesting however to prove results similar to Theorem A below in other types, at least classical. See [6, 3.2.1] for a property a bit weaker than the semidirect product structure. Theorem A also shows that Condition (*) introduced in [5, Definition 2] is satisfied for the groups $\textrm{C}_{\textbf{G}^*}(s)$ and all semisimple elements s, whenever $\textbf{G}$ is of type $\textsf{D}$, thus establishing the first (minor) part of [5, Conjecture 7].

2 Notations and Main Theorems

Let $\textbf{H}_0$ be a reductive group over the algebraically closed field $\mathbb {F}$ of positive characteristic p. Assume $\textbf{H}_0$ is simple simply connected. Let $\textbf{B}_0\ge \textbf{T}_0$ some Borel subgroup and maximal torus of $\textbf{H}_0$. We denote by $X(\textbf{T}_0)\supseteq \Phi \supseteq \Delta$ the associated roots and basis of the root system. Let $x_\alpha :\mathbb {F}\rightarrow \textbf{H}_0$ be the unipotent 1-parameter subgroup associated with $\alpha \in \Phi$. We set $F_p:\textbf{H}_0\rightarrow \textbf{H}_0$ defined by $F_p(x_\alpha (t))=x_\alpha (t^p)$ for any $\alpha \in \Phi$ and $t\in \mathbb {F}$. Let $\Gamma$ be the group of algebraic automorphisms of $\textbf{H}_0$ that satisfy $x_{\epsilon \delta }(t)\mapsto x_{\epsilon \delta '}(t)$ for any $t\in \mathbb {F}$, $\epsilon \in \{1,-1\}$, $\delta \in \Delta$ and where $\delta \mapsto \delta '$ is an automorphism of the Dynkin diagram. Note that $\Gamma$ is cyclic of order 2 whenever $\textbf{H}_0$ is of type $\textsf{A}_l$ ($l\ge 2$), $\textsf{D}_l$ ($l\ge 5$) or $\textsf{E}_6$, dihedral of order 6 for type $\textsf{D}_4$ and trivial for other types. Let $E :=\langle F_p\rangle \Gamma$ be the group of abstract automorphisms of $\textbf{H}_0$ generated by $\Gamma$ and $F_p$.

Let $\textbf{H}:=\textbf{H}_0/\textrm{Z}(\textbf{H}_0)$ and

$$ \pi :~ \textbf{H}_0\rightarrow \textbf{H} $$

the quotient map. Then E clearly acts on $\textbf{H}$ and we can form the semidirect product $\textbf{H}\rtimes E$.

Let s be a semisimple element of $\textbf{H}$. Then $\textrm{C}_{\textbf{H}}(s)$ is reductive but may not be connected. The component group $A_\textbf{H}(s):=\textrm{C}_{\textbf{H}}(s)/\textrm{C}_{\textbf{H}}^\circ (s)$ is abelian and can be seen as a subgroup of $Z(\textbf{H}_0)$ via the homomorphism

$$ \omega _s:\textrm{C}_{ \textbf{H}}(s)\rightarrow \textrm{Z}(\textbf{H}_0) \text { defined by }g\mapsto [\dot{g},\dot{s}]\text { for }\dot{g}\in \pi ^{-1}(g),~\dot{s}\in \pi ^{-1}(s) $$

(see [2, 8.2]).

In type $\textsf{D}$, we denote by

$$ \underline{E} =\langle F_p, \gamma \rangle $$

the subgroup of E generated by $F_p$ and some $\gamma \in \Gamma$ of order 2 (essentially in the case of type $\textsf{D}_4$ a choice is made and the graph automorphism of order 3 is left out). Note that $\underline{E}$ acts on both $\textbf{H}_0$ and $\textbf{H}$. We constantly consider the semidirect product $\textbf{H}\rtimes \underline{E}=\textbf{H}\underline{E}$. For $\sigma \in \textbf{H}\underline{E} \setminus \textbf{H}$ and $h\in \textbf{H}$ we occasionally write $\sigma (h)$ for $\sigma h\sigma ^{-1}\in \textbf{H}$. We also define the submonoid

$$ \underline{E}^+:=\{F_p^i\circ \gamma ^j~|~ i\ge 1,~j\in \{0,1\}\} $$

whose elements can be considered as Frobenius endomorphisms $F:\textbf{H}\rightarrow \textbf{H}$ with fixed points forming the finite group $\textbf{H}^F:=\{h\in \textbf{H}~|~ F(h)=h\}$. For $x\in \textbf{H}^F$ ($F\in \underline{E}^+$) we denote by $[x]_{\textbf{H}^F}$ its conjugacy class in $\textbf{H}^F$.

In the following theorems, $\textbf{H}=\textsf{D}_{l,\textrm{ad}}(\mathbb {F})$ and $\textbf{H}_0=\textsf{D}_{l,\textrm{sc}}(\mathbb {F})$ are groups of rank $l\ge 4$ over an algebraically closed field $\mathbb {F}$ of characteristic $p\ne 2$. The odd characteristic ensures that $\textsf{D}_{l,\textrm{ad}}(\mathbb {F})$ is actually the quotient $\textbf{H}_0/\textrm{Z}(\textbf{H}_0)$ (see [5, 2.4.4]).

Theorem A Let $F_0\in \underline{E}^+$, $s\in \textbf{H}^{F_0}$ a semisimple element. Then there is a semidirect product decomposition

$$ \textrm{C}_\textbf{H}(s)=\textrm{C}_\textbf{H}^\circ (s)\rtimes \check{A} $$

with $F_0(\check{A})=\check{A}$.

Moreover, if $\sigma '\in \underline{E}$, $F:=F_0^k$ for some $k\ge 1$ and $\sigma '(s)\in [s]_{\textbf{H}^F}$, then there exists $\sigma \in \textbf{H}^F\sigma '\cap \textrm{C}_{\textbf{H}\underline{E}}(s)$ such that $\sigma (\check{A})=\check{A}$ and $[{F_0,\sigma }]\in \check{A}$.

The considerations to prove the following are more classical, using only the simply connected covering $\pi :\textbf{H}_0\rightarrow \textbf{H}$.

Theorem B Let $\mathcal {C}$ be the $\textbf{H}$-conjugacy class of a semisimple element $t\in \textbf{H}$ such that $|A_\textbf{H}(t)|=4$. Let $F_0\in \underline{E}^+$ and assume $F_0(\mathcal {C})=\gamma (\mathcal {C})=\mathcal {C}$. Then $\pi (\textbf{H}_0^{F_0})\cap \mathcal {C}\ne \emptyset$ and there exists $s\in \pi (\textbf{H}_0^{F_0})\cap \mathcal {C}$ such that $\gamma (s)\in [s]_{\textbf{H}^{F_0}}$.

Some more technical corollaries will be given in our last section.

3 Lifting of Component Groups and Automorphisms

We continue with the same notations about $\textbf{H}$, the adjoint group of type $\textsf{D}_l$ ($l\ge 4$) over an algebraically closed field $\mathbb {F}$ of odd characteristic, its maximal torus $\textbf{T}=\pi (\textbf{T}_0)$ and the abstract automorphism group $\underline{E}$.

Proposition 3.1

There exists an $\underline{E}$-stable subgroup $\check{W}\le \textbf{H}$ such that $\textrm{N}_\textbf{H}(\textbf{T})=\textbf{T}\rtimes \check{W}$.

Proof

The statement is probably known to experts familiar with adjoint groups (see [7, Theorem 1]). We recall an elementary construction of $\check{W}$ and emphasize the stability property. For $d\ge 1$ let $J_d\in \textrm{GL}_d(\mathbb {F})$ be the permutation matrix corresponding to the product of transpositions $(1,d)(2,d-1)\dots \in \mathcal {S}_d$.

Let $\textbf{G}:=\textrm{SO}_{2l}(\mathbb {F})=\textrm{O}_{2l}(\mathbb {F})\cap \textrm{SL}_{2l}(\mathbb {F})$, where $\textrm{O}_{2l}(\mathbb {F}):=\{x\in \textrm{GL}_{2l}(\mathbb {F})~|~ {}^txJ_{2l}x=J_{2l} \}$. Then $\textbf{G}$ is connected simple of type $\textsf{D}_l$ (see for instance [4, 1.5.5]) with center $\{\textrm{Id}_{2l}, -\textrm{Id}_{2l}\}$ and $\textbf{H} =\textbf{G}/\{\textrm{Id}_{2l}, -\textrm{Id}_{2l}\}$. A maximal torus of $\textbf{G}$ is the group $\overline{\textbf{T}}$ of diagonal matrices whose diagonal is $(t_1,\dots , t_l,t_l^{-1},\dots , t_1^{-1})$ with $t_1,\dots , t_l\in \mathbb {F}^\times$; a Borel subgroup $\overline{\textbf{B}}$ containing it and consisting of upper diagonal matrices is also described in [4, 1.5.5]. Numbering the basis in dimension 2l by $(1,2,\dots , l,-l,-l+1,\dots ,-2,-1)$ the normalizer of $\overline{\textbf{T}}$ in $\textrm{GL}_{2l}(\mathbb {F})$ is generated by $\overline{\textbf{T}}$ and the permutation matrices corresponding to $(l,-l)$ and the various involutions $(i,j)(-i,-j)$ ($1\le i,j\le l$). All those permutation matrices clearly belong to $\textrm{O}_{2l}(\mathbb {F})$ and generate a subgroup $\overline{V}\le \textrm{O}_{2l}(\mathbb {F})$ with $\overline{V}\cong \mathbb {Z}/2\mathbb {Z}\wr \mathcal {S}_l$. The intersection $\overline{W}:=\overline{V}\cap \textrm{SL}_{2l}(\mathbb {F})$ corresponds to elements of the wreath product whose number of non-trivial coordinates in the base group is even, thus forming a Coxeter group of type $\textsf{D}_l$.

In $\textbf{G}=\textrm{SO}_{2l}(\mathbb {F})$ we indeed get $\textrm{N}_\textbf{G}(\overline{\textbf{T}})=\overline{\textbf{T}} \rtimes \overline{W}$. The elements of $\overline{V}$ are clearly fixed by $F_p$ which here is just the raising of matrix entries to the p-th power. On the other hand, conjugation by the permutation matrix associated to $(l,-l)$ induces the graph automorphism of $\textbf{G}$ of order 2 associated to $\overline{\textbf{T}}$ and $\overline{\textbf{B}}$, and swapping the fundamental reflections corresponding to $(l-1,l)(-l+1,-l)$ and $(l-1,-l)(-l+1,l)$ (see [4, 4.3.6]). This too preserves $\overline{W}$, since the permutation matrix associated to $(l,-l)$ belongs to $\overline{V}$. We get our Proposition by taking for $\check{W}$ the image of $\overline{W}$ in $\textbf{H} =\textbf{G}/\{\textrm{Id}_{2l}, -\textrm{Id}_{2l}\}$.$\square$

Remark 3.2

In type $\textsf{B}_l$ the same statement is proved similarly in $\textbf{H} =\textrm{SO}_{2l+1}(\mathbb {F})$ seen as $\textrm{O}_{2l+1}(\mathbb {F})/\{\textrm{Id}_{2l+1}, -\textrm{Id}_{2l+1}\}$ (see again [7]). In type $\textsf{A}_{l-1}$ with $\textbf{H}=\textrm{GL}_{l}(\mathbb {F})/\{ \lambda \textrm{Id}_{l}~|~\lambda \in \mathbb {F}^\times \}$, the permutation matrices also provide a complement whose elements are fixed under $F_p$. Note that in that type the non-trivial element of $\Gamma$ sends our group of permutation matrices to a distinct $\textrm{N}_{\textbf{H} }(\textbf{T})$-conjugate in odd characteristic. Note also that permutation matrices have a somewhat cumbersome expression in terms of Chevalley generators. For instance the permutation matrix for the transposition (1, 2) has a class $\textrm{mod}\,\textrm{Z}(\textrm{GL}_l(\mathbb {F}))$ corresponding to $n_{\alpha _1}(1)h_{\alpha _1}(-\omega )h_{\alpha _2}(-\omega ^2)\dots h_{\alpha _{l-1}}(-\omega ^{l-1})\in \textrm{SL}_l(\mathbb {F})$ for some $\omega$ with $\omega ^l=-1$ in the notations of [9, 1.12.1], a similarly complex formula being necessary in type $\textsf{D}$.

In type $\textsf{C}_l$, or exceptional types $\ne \textsf{G}_2$ it is known that $\textbf{T}$ has no complement in $\textrm{N}_{\textbf{H}}(\textbf{T})$ (see [1, 4.11]).

We now consider the centralizer of some $x\in \textbf{T}$, where $\textbf{T}$ is as before a maximal torus of an adjoint group that is maximally split for any Frobenius endomorphism belonging to $\underline{E}^+$.

Recall that for any $x\in \textbf{T}$ the connected centralizer $\textrm{C}_{ \textbf{H} }^\circ (x)$ is a reductive group containing $\textbf{T}$ as maximal torus (see for instance [3, 3.5.4]).

Proposition 3.3

Let $\textbf{T}$ and $\check{W}$ be as in Proposition 3.1, noting that $\check{W}$ acts on $\textbf{T}$ hence naturally on the set $\Phi$ of $\ \textbf{T}$-roots of $\ \textbf{H}$. Let $x\in \textbf{T}$, $\Phi _x\subseteq X(\textbf{T})$ the root system of $\ \textrm{C}_\textbf{H}^\circ (x)$ with respect to $\textbf{T}$ and choose $\Delta _x$ a basis of $\Phi _x$. We set $\check{B}:=\textrm{C}_{\textbf{H}}(x)\cap \check{W}_{\Delta _x}=\{w\in \check{W}~|~ {}^wx=x, w(\Delta _x)=\Delta _x\}$ and similarly $\check{B}_{\underline{E}}:=\textrm{C}_{\textbf{H}\underline{E}}(x)\cap {(\check{W}\rtimes \underline{E} )}_{\Delta _x}$. Then

(a):

$\textrm{C}_{\textbf{H}}(x)= \textrm{C}_{\textbf{H}}^\circ (x) \rtimes \check{B}$, $\textrm{C}_{\textbf{H} \underline{E} }(x)= \textrm{C}_{\textbf{H}}^\circ (x) \rtimes \check{B}_{\underline{E}}$,

(b):

$\check{B} \lhd \check{B}_{\underline{E}}$ with abelian quotient.

Proof

Note first that $\underline{E}$ stabilizes $\textbf{T}$ and sends maximal tori of $\textbf{H}$ to maximal tori since $\sigma (^g\textbf{T})={}^{\sigma (g)}\textbf{T}$ for any $g\in \textbf{H}$ and $\sigma \in \underline{E}$. So $\textbf{H}\underline{E}$ permutes the maximal tori of $\textbf{H}$. The connected centralizer $\textbf{C}:=\textrm{C}_{\textbf{H}}^\circ (x)$ is the reductive group containing $\textbf{T}$ with root system $\Phi _x$, so

$$\begin{aligned} \begin{array}{lllr} \textrm{N}_{\textbf{H} \underline{E} }(\textbf{C})&{}=&{}\textbf{C}.\textrm{N}_{\textbf{H} \underline{E} }(\textbf{T},\textbf{C})\,\,\qquad \text { by conjugacy of maximal tori in}\ \textbf{C} \\ &{}=&{}\textbf{C}.\textrm{N}_{\check{W}\underline{E}}(\textbf{C})\,\,\qquad \text {by Proposition~3.1} \\ &{}=&{}\textbf{C}\rtimes (\check{W}\underline{E})_{\Delta _x}\,\,\qquad \text {by the regular action of}\ \textrm{N}_{\textbf{C} }(\textbf{T})/\textbf{T}\ \text {on the bases of}\ \Phi _{x}. \end{array} \end{aligned}$$

On the other hand, $\textbf{C}$ is obviously normal in $\textrm{C}_{\textbf{H} \underline{E} }(x)$, so $\textrm{C}_{ \textbf{H} \underline{E} }(x)$ is the centralizer of x in the above $\textbf{C}\rtimes (\check{W}\underline{E})_{\Delta _x}$. This gives the second claim of (a) with $\check{B}_{\underline{E}}:=\textrm{C}_{\check{W} \underline{E} }(x)_{\Delta _x}$. The first is clear by taking the intersection with $\textbf{H}$, which also gives the first part of (b). The quotient $\check{B}_{\underline{E}}/\check{B}$ is abelian, since it injects into $\underline{E}$.$\square$

Remark 3.4

It could be interesting to classify those groups $\textrm{C}_{ \textbf{H} \underline{E} }(x)/\textrm{C}_{ \textbf{H} }^\circ (x)$ or equivalently $\check{B}_{\underline{E}}$.

Using the simply connected covering $\textbf{H}_0$ we can embed $\textrm{C}_{\textbf{H} \underline{E}}(x)/\textrm{C}_{ \textbf{H} }^\circ (x)$ into $\mathbb {Z}\times V\times \mathbb {Z}/2\mathbb {Z}$, where V is the dihedral group of order 8. Let us explain briefly how this can be seen.

The simply connected covering $\textbf{H}_0\xrightarrow {\pi } \textbf{H}$ has kernel $\textrm{Z}(\textbf{H}_0)$ of order 4 while the covering $\textbf{H}_0\rightarrow \textrm{SO}_{2l}(\mathbb {F})$ has a kernel $\{1,h_0\}$ of order 2 with $F_p(h_0)=\gamma (h_0)=h_0$ (see [9, p. 70], where $h_0$ is called $z_c$). The group $\textbf{H}$ acts on $\textbf{H}_0$ by conjugacy, with $\underline{E}$ also acting in a compatible way, so we get an action of $\textbf{H}\underline{E}$ on $\textbf{H}_0$ and therefore an action of $\textrm{C}_{\textbf{H} \underline{E}}(s)$ on $\pi ^{-1}(s)$. Denote $\textrm{pr}_1:\textrm{C}_{\textbf{H} \underline{E} }(s)\rightarrow \mathcal {S}_{\pi ^{-1} (s)}$ the induced group morphism with values in a permutation group on four elements. Note that the kernel of the action of $\textrm{C}_{\textbf{H}}(s)$ on $\pi ^{-1} (s)$ is $\textrm{C}_{ \textbf{H} }^\circ (s)$, since this is $\pi \big (\textrm{C}_{\textbf{H}_0 }(\pi ^{-1}(s))\big )$ by connectedness of centralizers of semisimple elements in $\textbf{H}_0$ (see [3, 3.5.6]). Moreover since the action of $\textrm{C}_{ \textbf{H} \underline{E} }(s)$ on $\pi ^{-1}(s)$ commutes with translation by $h_0$, the image of $\textrm{pr}_1$ is included in the centralizer of a product of two disjoint transpositions in the symmetric group on four elements. The latter is a dihedral group V of order 8.

Letting $\textrm{pr}_2:\textbf{H}\rtimes \underline{E}\rightarrow \underline{E}$ be the projection on the second term of the semidirect product, we get that the kernel of the map $\textrm{C}_{ \textbf{H} \underline{E} }(s)\rightarrow V \times \underline{E}$ defined by $c\mapsto (\textrm{pr}_1(c),\textrm{pr}_2(c))$ is $\textrm{C}^\circ _{ \textbf{H} }(s)$. Restricting this morphism to $\check{B}_{\underline{E}}$ then makes it injective thanks to Proposition 3.3(a).

Our applications to centralizers of semisimple elements and semisimple conjugacy classes will stem from the omnibus lemma below.

Lemma 3.5

Let $F_0\in \underline{E}^+$. Let $s\in \textbf{H}^{F_0}$ be a semisimple element and let $x\in \textbf{T}$ be an $\textbf{H}$-conjugate of s. Let $\textrm{C}_{ \textbf{H} \underline{E} }(x)=\textrm{C}_{\textbf{H} }^\circ (x)\rtimes \check{B}_{\underline{E}}$ the decomposition of Proposition 3.3. Then there is an inner automorphism $\iota :\textbf{H}\underline{E}\rightarrow \textbf{H}\underline{E}$ induced by an element of $\textbf{H}$ such that

• (a) $\iota (s)=x$,

• (b) $\iota (F_0) \in \check{B}_{\underline{E}}$ and

• (c) if $F=F_0^i$ for some $i\ge 1$, $\tau \in \underline{E}$ and $h\in \textbf{H}^F$ are such that $\tau (s)=s^h$, then

  $$ \iota (h\tau ) \in \textrm{C}_{\textbf{H}}^\circ (x)\textrm{C}_{\check{B}_{\underline{E}}}(\iota (F)). $$

Proof

We denote $x:=s^g$ for some $g\in \textbf{H}$ and $\iota ':\textbf{H} \underline{E} \longrightarrow \textbf{H} \underline{E}$ given by $y\mapsto y^g$. Note that $\iota '$ preserves cosets $\textbf{H} e$ ($e\in \textbf{H}\underline{E}$). We have $\iota '(F_0)\in \textbf{H} {F_0}\cap \textrm{C}_{\textbf{H} \underline{E}}(x)$ since s is $F_0$-invariant. Recalling $\check{B}_{\underline{E}}$ the group associated to x from Proposition 3.3 and the decomposition $\textrm{C}_{\textbf{H} \underline{E}}(x)= \textrm{C}^\circ _\textbf{H}(x)\check{B}_{\underline{E}}$ we can write $\iota '({F_0})$ as cb with $c\in \textrm{C}^\circ _\textbf{H}(x)$ and $b \in \check{B}_{\underline{E}}$. Lang’s theorem (see for instance [8, 4.1.2]) applied to $\textrm{C}^\circ _\textbf{H}(x)$ and $\iota '(F_0)$ allows us to write $c^{-1} =c'{}^{-1} \iota '(F_0){c'}\iota '(F_0)^{-1}$ in $\textbf{H}\underline{E}$ for some $c'\in \textrm{C}^\circ _\textbf{H}(x)$. Calling now $\iota : \textbf{H} \underline{E} \longrightarrow \textbf{H} \underline{E}$ the isomorphism given by $y\mapsto y^{gc'}$, we get

$$ b=c^{-1} \iota '(F_0)=c'{}^{-1} \iota '(F_0){c'}=\iota '(F_0)^{c'}=\iota (F_0) $$

and $\iota (s)=s^{gc'}=x^{c'}=x$.

It remains to check (c). We have $\tau \in \underline{E}$ and $h\in \textbf{H}^{F}$ such that $h\tau \in \textrm{C}_{\textbf{H}\underline{E}}(s)$. Then $\iota (h\tau )\in \textrm{C}_{\textbf{H}\underline{E}}(\iota (s)) = \textrm{C}_{\textbf{H}\underline{E}}(x)= \textrm{C}^\circ _{\textbf{H} }(x)\rtimes \check{B}_{\underline{E}}$ by Proposition 3.3(a). So

$$ \iota (h\tau )\in \textrm{C}^\circ _{\textbf{H} }(x)b' $$

for some $b'\in \check{B}_{\underline{E}}$. Note that $h \tau$ commutes with F and accordingly $\iota (h \tau )$ commutes with $\iota (F)$. Because of the semidirect product structure $\textrm{C}_{\textbf{H}\underline{E}}(x)=\textrm{C}_\textbf{H}^\circ (x) \rtimes \check{B}_{\underline{E}}$ and since $\iota (F )=\iota (F_0)^i\in \check{B}_{\underline{E}}$ thanks to (b), we see that $b'$ commutes with $\iota (F)$ as well. This gives our point (c).$\square$

Proof of Theorem A

By Lemma 3.5, we have an inner automorphism $\iota$ of $\textbf{H}\underline{E}$ such that $x:=\iota (s)\in \textbf{T}$, and $\textrm{C}_{ \textbf{H} \underline{E} }(x)=\textrm{C}_{\textbf{H}}^\circ (x)\rtimes \check{B}_{\underline{E}}$ with $\iota (F_0)\in \check{B}_{\underline{E}}$. We define $\check{A}:=\iota ^{-1}(\check{B})$, recalling $\check{B} =\check{B}_{\underline{E}}\cap \textbf{H}$ from Proposition 3.3. It is $F_0$-stable because $\check{B}_{\underline{E}}\cap \textbf{H}$ is stable under $\iota (F_0)$ since the latter belongs to $\check{B}_{\underline{E}}$.

We now consider some $\sigma '\in \underline{E}$ such that $\sigma '(s)\in [s]_{\textbf{H}^{F}}$ for some $F=F_0^k$, $k\ge 1$. Let $h\in \textbf{H}^F$ such that $\sigma '(s)=s^h$. By Lemma 3.5(c) applied to $\tau :=\sigma '$, we get some $b\in \textrm{C}_{\check{B}_{\underline{E}}}(\iota (F))$ such that $\iota (h\tau ) \in \textrm{C}_{\textbf{H} }^\circ (x)b$. We show that $\sigma :=\iota ^{-1}(b)$ satisfies our claims. First $\sigma$ centralizes s since b centralizes x. Also $\sigma$ belongs to $\textbf{H}\tau$ since $\iota$ and its inverse, being conjugations, are the identity on $\textbf{H}\underline{E}/\textbf{H}$ and therefore $\textbf{H} \sigma =\textbf{H} b=\textbf{H}\tau$. Moreover $\sigma$ commutes with F since b commutes with $\iota (F)$, so $\sigma \in (\textbf{H}\tau )^F=\textbf{H}^F\tau$. The group $\check{A}$ is $\sigma$-stable by the same argument used for $F_0$, namely $b\in \check{B}_{\underline{E}}$ and therefore $\iota ^{-1}(b)$ normalizes $\check{A}=\iota ^{-1}(\check{B})$. That $b\in \check{B}_{\underline{E}}$ also implies that $[\iota (F_0),b]\in \check{B}$, since $\check{B}_{\underline{E}}/\check{B}$ is abelian by Proposition 3.3(b). This in turn tells us that $[F_0,\sigma ]\in \check{A}$, thus completing the proof of Theorem A.$\square$

In the next section, Theorem A will be applied mostly with $\sigma '$ being a graph automorphism. But here is a case reminiscent of Shintani descent with two commuting Frobenius endomorphisms.

Corollary 3.6

Let $F,F_0\in \underline{E}^+$, let $\mathcal {C}$ be a semisimple conjugacy class of $\textbf{H}$ and assume we have $s_0\in \mathcal {C}\cap \textbf{H}^{F_0}$ such that $F(s_0)\in [s_0]_{\textbf{H}^{F_0}}$. Applying Theorem A with $(F_0,s,\sigma ',k)$ being here $(F_0,s_0,F,1)$, we get a group $\check{A}$ and some $\sigma \in \textbf{H}\underline{E}$ that we denote as $\check{A}(s_0)$ and $F'$. Then there exist $s\in \mathcal {C}^F$, $F_0'\in \textbf{H}^F F_0$ centralizing s, and a group $\check{A}(s)$ such that

• (i) $\textrm{C}_{\textbf{H}}(s)=\textrm{C}_{\textbf{H}}^\circ (s) \rtimes \check{A}(s)$;

• (ii) $\check{A}(s)$ is $\langle F,F_0'\rangle$-stable; and there exists an inner isomorphism $\iota _{s_0,s}: \textbf{H} \underline{E} \rightarrow \textbf{H} \underline{E}$ such that

  • $\iota _{s_0,s}(s_0)=s$ and $\iota _{s_0,s}(\check{A}(s_0))=\check{A}(s)$;

  • $\iota _{s_0,s}(F_0)=F_0'$ and $\iota _{s_0,s}(F')=F$.

Proof

Let us recall how Theorem A is proved by application of Lemma 3.5. Choosing $x\in \textbf{T}\cap \mathcal {C}$, we have $\textrm{C}_{\textbf{H}}(x)=\textrm{C}_{\textbf{H}}^\circ (x)\rtimes \check{B}$ from Proposition 3.3. Lemma 3.5 gives us an inner automorphism $\iota :\textbf{H}\underline{E}\rightarrow \textbf{H}\underline{E}$ with $\iota (s_0)=x$, $\iota (F_0):=f_0\in \check{B}$, and we define $\check{A}(s_0):=\iota ^{-1}(\check{B})$.

Having $F(s_0)\in [s_0]_{\textbf{H}^{F_0}}$ implies that ${[hF,s_0]}=1$ for some $h\in \textbf{H}^{F_0}$, so that $\iota (hF)\in \textrm{C}_{\textbf{H}}^\circ (x)f$ with $f\in \textrm{C}_{\check{B}_{\underline{E}}}(f_0)$ thanks to Lemma 3.5(c) for $i=1$ and $\tau =F$. Following the proof of Theorem A, we define $F':=\iota ^{-1}(f)\in \textrm{C}_{\textbf{H}^{F_0}F}(s_0)$ and get $F'(\check{A}(s_0))=\check{A}(s_0)$.

Recalling that $\iota$ is an inner automorphism and $\textbf{H}\underline{E}/\textbf{H}$ is abelian, we have $\textbf{H} f=\textbf{H} F'=\textbf{H} F$ and Lang’s theorem ensures the existence of some inner automorphism $\iota '$ of $\textbf{H}\underline{E}$ induced by an element of $\textbf{H}$ such that $\iota '(f)=F$. We define $\iota _{s_0,s}:=\iota '\iota$, $F'_0:=\iota _{s_0,s}(F_0)=\iota '(f_0)$, $s:=\iota _{s_0,s}(s_0)=\iota '(x)$. The latter belongs to $\mathcal {C}^F$, since $\iota$ and $\iota '$ are induced by elements of $\textbf{H}$ and $[F,s]=\iota _{s_0,s}([\iota ^{-1}(f),s_0])=1$. We have $\iota _{s_0,s}(F')=\iota '(f)=F$. We get $\textrm{C}_{\textbf{H}}(s)=\textrm{C}_{\textbf{H}}^\circ (s)\rtimes \check{A}(s)$ for $\check{A}(s):=\iota _{s_0,s}(\check{A}(s_0))=\iota '(\check{B})$ as a consequence of the same property of $\check{A}(s_0)$ for $s_0$. Moreover $\check{A}(s)$ is $\langle F,F'_0\rangle$-stable, since $\check{A}(s_0)$ has been seen to be $F_0$-stable and $\check{B}$ is f-stable since $f\in \check{B}_{\underline{E}}$. We also have $F'_0\in \textbf{H} F_0$ while $[F'_0,F]=\iota '([f_0, f])=1$ as seen above. So $F'_0\in \textbf{H}^F F_0$ as claimed. We now have all our claims.$\square$

4 Proof of Theorem B and Some Corollaries

We prove Theorem B but also some more technical statements, that can be seen as strengthenings of Theorem A, about semisimple classes of $\textbf{H}$ and the action of elements of $\underline{E}^{+}$. Recall $\gamma \in \underline{E}$ (see Section 2) the graph automorphism stabilizing our choice of a maximal torus and Borel subgroup.

In the following $F_0\in \underline{E}^{+}$ and $\mathcal {C}$ is the $\textbf{H}$-conjugacy class of a semisimple element.

Corollary 4.1

Assume $F_0(\mathcal {C})=\gamma (\mathcal {C})=\mathcal {C}$ and there is some $s\in \mathcal {C}^{F_0}$ such that $\gamma (s) \in [s]_{\textbf{H}^{F_0}}$. Then there is $\gamma ' \in \textbf{H}^{F_0}\gamma \cap \textrm{C}_{\textbf{H}\underline{E}}(s)$ and a $\langle F_0, \gamma '\rangle$-invariant subgroup $\check{A}$ with $\textrm{C}_\textbf{H}(s)=\textrm{C}_\textbf{H}^\circ (s)\rtimes \check{A}$.

Recall that $\pi :\textbf{H}_0\rightarrow \textbf{H}$ is a composition $\textbf{H}_0\rightarrow \textrm{SO}_{2l}(\mathbb {F})\rightarrow \textbf{H}$. The kernel of the second map is $\{\pm \textrm{Id}_{2l}\}$ while the kernel of the first is $\{1,h_0\}$ where $h_0\in \textrm{Z}(\textbf{H}_0)$ is of order 2.

We keep $\mathcal {C}$ a semisimple conjugacy class of $\textbf{H}$ and $F_0\in \underline{E}^+$. We describe the situation complementary to the one of Corollary 4.1, but while the latter is a direct consequence of Theorem A, the following will require a bit more work.

Corollary 4.2

Assume that $F_0(\mathcal {C})=\gamma (\mathcal {C})=\mathcal {C}$ but $\gamma (x) \notin [x]_{\textbf{H}^{F_0}}$ for every $x\in \mathcal {C}^{F_0}$.

Let $F:=F_0^{2k}$ for some $k\ge 1$ and let $s\in \mathcal {C}^F$.

Then there exist $F_0'\in \textrm{C}_{\textbf{H}^F F_0}(s)$, $\gamma '\in \textrm{C}_{\textbf{H}^F \gamma }(s)$ and an $\langle F,F_0',\gamma '\rangle$-stable group $\check{A}$ such that

(i):

$\textrm{C}_{\textbf{H}}(s)= \textrm{C}^\circ _{\textbf{H}}(s)\rtimes \check{A}$;

(ii):

$\omega _s(\check{A})=\langle h_0\rangle$ and $|\textrm{A}_{\textbf{H}}(s)|=|\check{A}|=2$;

(iii):

$[s_0,F'_0]\in \textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$ for every $s_0\in \pi ^{-1}(s)$ and

(iv):

$[F_0', \gamma ']=\check{a}$, where $\check{a}$ is the generator of $\check{A}$.

Let $\mathcal {C}$ be a semisimple conjugacy class of $\textbf{H}$ and $F'\in \textbf{H}\underline{E}^{+}$ (seen as an endomorphism of $\textbf{H}$) such that $F'(\mathcal {C})=\mathcal {C}$. Thanks to Lang’s Theorem we have $\mathcal {C}^{F'}\ne \emptyset$.

Taking $s\in \mathcal {C}^{F'}$ and recalling that $A_\textbf{H}(s)$ is abelian, we can combine the parametrization of $\textbf{H}^{F'}$-conjugacy classes in $\mathcal {C}^{F'}$ by $A_\textbf{H}(s)_{F'}:=A_\textbf{H}(s)/[A_\textbf{H}(s),F']$ (see [8, 4.3.6]) and the injection $A_\textbf{H}(s)\rightarrow \textrm{Z}(\textbf{H}_0)$ induced by the map $\omega _s$ from our introduction.

$$\begin{aligned} \mathcal {C}^{F'}/\textbf{H}^{F'}\text {-conj}~{\mathop {\longleftrightarrow }\limits ^{\sim }} ~&A_\textbf{H}(s)_{F'}&\\&A_\textbf{H}(s)&\hookrightarrow ~ \textrm{Z}(\textbf{H}_0) \end{aligned}$$

An easy calculation then yields that the resulting map

$$\begin{aligned} \begin{array}{clrl} \Theta _{F^{\prime }} :\mathcal {C}^{F^{\prime }}/\textbf{H}^{F^{\prime }}\text {-conj} \hookrightarrow \textrm{Z}(\textbf{H}_0)_{F^{\prime }}\quad \text { is injective and defined by} \\ \qquad \qquad \qquad \qquad \qquad \quad {[x]}_{\textbf{H}^{F^{\prime }}} \quad \mapsto x_0^{-1} {F^{\prime }}(x_0) {[\textrm{Z}(\textbf{H}_0),F^{\prime }]}\quad \text { for any } x_0\in \pi ^{-1}(x). \end{array} \end{aligned}$$

Note that, in contrast to the other two maps, $\Theta _{F'}$ is independent of the choice of s in $\mathcal {C}^{F'}$. Note also that if moreover $F'\in \underline{E}^+$ and denoting by $\underline{E}_\mathcal {C}$ the stabilizer of $\mathcal {C}$ in $\underline{E}$, then $\Theta _{F'}$ is $\underline{E}_\mathcal {C}$-equivariant by its definition (since $\underline{E}$ commutes with $F'$).

Let us show Theorem B (see also the third paragraph of the proof of [12, 6.17] for related considerations). We consider the above with $F'=F_0\in \underline{E}$ and s being denoted by t. We are assuming that $|A_\textbf{H}(t)|=4=|\textrm{Z}(\textbf{H}_0)|$. The injection $A_\textbf{H}(t)\hookrightarrow \textrm{Z}(\textbf{H}_0)$ is then an $\langle F_0\rangle$-equivariant isomorphism and therefore $\Theta _{F_0}$ is onto. So $1=x_0^{-1} F_0(x_0) [\textrm{Z}(\textbf{H}_0),F_0]$ for some $x_0\in \textbf{H}_0$ with $\pi (x_0)\in \mathcal {C}^{F_0}$. Rewriting this as $x_0^{-1} F_0(x_0)z^{-1} F_0(z)=1$ for some $z\in \textrm{Z}(\textbf{H}_0)$, we get $x'_0:=x_0z\in \textbf{H}_0^{F_0}$ and $\pi (x_0')=\pi (x_0)\in \mathcal {C}$. Note also that the equivariance and injectivity of $\Theta _{F_0}$ imply that $[\pi (x_0)]_{\textbf{H}^{F_0}}$ is $\gamma$-stable. This shows all claims of Theorem B.

We now prove the corollaries. If there is some $s\in \mathcal {C}^{F_0}$ such that ${[s]_{\textbf{H}^{F_0}}}$ is $\gamma$-stable then Theorem A with $k=1$ and $\sigma '$ being here $\gamma$ clearly gives the claims of Corollary 4.1.

Proof of Corollary 4.2

The action of $\underline{E}$ on $\textrm{Z}(\textbf{H}_0)$ is easy to deduce from the description of $\textrm{Z}(\textbf{H}_0)$ (see [9, Table 1.12.6], [10, 2.9]). In particular $\textrm{C}_{\textrm{Z}(\textbf{H}_0)}(\gamma )=\langle h_0\rangle$, a subgroup of $\textrm{Z}(\textbf{H}_0)$ of order 2.

By assumption, no $\textbf{H}^{F_0}$-class contained in $\mathcal {C}^{F_0}$ is $\gamma$-stable, so the $\textbf{H}^{F_0}$-conjugacy classes contained in $\mathcal {C}$ are mapped via $\Theta _{F_0}$ to elements in $\textrm{Z}(\textbf{H}_0)_{F_0}$ not fixed by $\gamma$. Then $\textrm{Z}(\textbf{H}_0)_{F_0}$ can’t have order 1 or 2, so it implies the following

Lemma 4.3

Keeping the assumptions of Corollary 4.2, $F_0$, and therefore also F, act trivially on $\textrm{Z}(\textbf{H}_0)$.

We also get that the image of $\Theta _{F_0}$, being $\gamma$-stable, is indeed equal to the whole $\textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$. This is of order 2, so $|A_\textbf{H}(s)_{F_0}|=|A_\textbf{H}(s)|=2$ for any $s\in \mathcal {C}^{F_0}$ by the diagram defining $\Theta _{F_0}$ from $\omega _s$. The equality $|A_\textbf{H}(s)|=2$ of course holds for any $s\in \mathcal {C}$. We have $\gamma \omega _s\gamma =\omega _{\gamma (s)}$ and the image of $\omega _s$ depends only on the conjugacy class of s (by the same formula for inner automorphisms), so $\gamma (\mathcal {C})=\mathcal {C}$ implies that $\omega _s(A_\textbf{H}(s))$ is $\gamma$-stable of order 2 for any $s\in \mathcal {C}$. Then indeed $\omega _s(A_\textbf{H}(s))=\langle h_0\rangle$, which amounts to part (ii) once the other parts are checked.

We prove the other statements (i), (iii) and (iv) in several steps, first for some $s\in \mathcal {C}^{F_0}$ and then for $t\in \mathcal {C}^F$ via an application of Theorem A and Lemma 3.5.

For the following we fix some $x\in \mathcal {C}\cap \textbf{T}$ and the groups $\check{B}_{\underline{E}}\le \textbf{H} \underline{E}$ and $\hat{B}:=\check{B}_{\underline{E}} \cap \textbf{H}$ with $\textrm{C}_{ \textbf{H} \underline{E} }(x)=\textrm{C}_\textbf{H}^\circ (x)\rtimes \check{B}_{\underline{E}}$ from Proposition 3.3.

Let us take $s\in \mathcal {C}^{F_0}$. Let $s_0\in \pi ^{-1}(s)$ and $h':=s_0^{-1} F_0(s_0)$. Note that $\Theta _{F_0}([s]_{\textbf{H}^{F_0}})=h'\in \textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$ by the above. This ensures $[s_0,F_0]\in \textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$ for every $s_0\in \pi ^{-1}(s)$. Defining $F'_0:=F_0$ in that case, we get part (iii) for that s.

We turn to the question of actually finding $\gamma '$ in $\textrm{C}_{ \textbf{H} \underline{E} }(s)^F$ and proving (i) and (iv). In order to apply Theorem A with $\sigma '=\gamma$, we need to check that $[s]_{\textbf{H}^F}$ is $\gamma$-stable.

Since $F=F_0^{2k}$ and $F_0(s_0)=s_0h'$ with $h'\in \textrm{Z}(\textbf{H}_0)$, we have

$$ \Theta _F([s]_{\textbf{H}^F})=s_0^{-1} F_0^{2k}(s_0)=s_0^{-1} F_0(s_0) F_0(s_0^{-1} F_0(s_0))\cdots F_0^{2k-1}(s_0^{-1} F_0(s_0))=(h')^{2k} $$

thanks to Lemma 4.3. Squares in $\textrm{Z}(\textbf{H}_0)$ are $\underline{E}$-fixed since they form a stable subgroup of order 1 or 2. So $\Theta _F([s]_{\textbf{H}^F})$ is $\gamma$-fixed and by equivariance of $\Theta _F$, this implies that $[s]_{\textbf{H}^F}$ is $\gamma$-stable. Note that it is also $F_0$-stable since $F_0(s)=s$. On the other hand $\mathcal {C}^F/{\textbf{H}^F}\text {-conj}$ is of cardinality $|A(s)_F|=|A(s)|=2$ as seen before and using Lemma 4.3 again. Since $\mathcal {C}^F$ is $\langle F_0,\gamma \rangle$-stable and $[s]_{\textbf{H}^F}$ is $\langle F_0,\gamma \rangle$-stable it implies the following

Lemma 4.4

Under the assumptions of Corollary 4.2$\langle F_0,\gamma \rangle$ acts trivially on $\mathcal {C}^{F}/\textbf{H}^{F}\text {-conj}$.

We can now apply Theorem A to s with $i=2k$, $\sigma '=\gamma$. Taking also $F'_0=F_0$, this gives us a $\sigma$ that we call $\gamma '$ and a decomposition $\textrm{C}_\textbf{H}(s)=\textrm{C}_\textbf{H}^\circ (s)\rtimes \check{A}(s)$ satisfying our part (i) and $[F_0,\gamma ']\in \check{A}(s)$.

Let us check that $[F_0,\gamma ']\ne 1$. Remembering that $\textbf{H}\underline{E}$ acts on $\textbf{H}_0$, it suffices to show that ${[\gamma ',F_0]}(s_0)\ne s_0$. Recall that $F_0(s_0)=s_0h'$ for some $h'\in \textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$ so that $\gamma '(h')=\gamma '{}^{-1} (h')=\gamma (h')=h_0h'$ since $\textrm{Z}(\textbf{H}_0)$ is a group of order 4 in which the centralizer of $\gamma$ is the subgroup of order 2 generated by $h_0$. Let $z\in \textrm{Z}(\textbf{H}_0)$ such that $\gamma '(s_0)=s_0z$. Note that z exists since $\gamma '(s)=s$. Using also Lemma 4.3 we get

$$\begin{aligned} {[\gamma ',F_0]}(s_0)= & {} \gamma 'F_0\gamma '{}^{-1}(s_0h'{} ^{-1})=\gamma 'F_0(s_0\gamma '{}^{-1}(z^{-1})h_0h'{}^{-1})=\\= & {} \gamma '(s_0\gamma '{}^{-1}(z^{-1})h_0)=s_0h_0\ne s_0. \end{aligned}$$

This implies our claim that $[F_0,\gamma ']\ne 1$. (Another proof is possible with the three subgroups Lemma.) Since $\check{A}(s)$ is a group of order 2 and $[F_0,\gamma ']\in \check{A}(s)$, $[F_0,\gamma ']$ generates $\check{A}(s)$. This gives (iv) in that case.

Let us recall however how Theorem A is deduced from Lemma 3.5 in our case (with $\sigma '$ being $\gamma$). We first get an inner automorphism of $\textbf{H}\underline{E}$ induced by an element of $\textbf{H}$, namely $\iota _s:\textbf{H}\underline{E}\rightarrow \textbf{H}\underline{E}$, such that $\iota _s(s)=x$, $f_s:=\iota _s(F_0)\in \check{B}_{\underline{E}}$ and then $\check{A}(s):=\iota _s^{-1}(\check{B})$ satisfies $\textrm{C}_{\textbf{H}}(s)=\textrm{C}_\textbf{H}^\circ (s)\rtimes \check{A}(s)$. Finally we take some $g_s\in \check{B}_{\underline{E}}\cap \textbf{H}^F\gamma$ and define $\gamma ':= \iota _s^{-1}(g_s)$.

Before turning to the general case of some $t\in \mathcal {C}^F$, let us apply now $\iota _s$ to $F_0$ and $\gamma '$. From what has been said about $[F_0,\gamma ']$ we see that $[f_s,g_s]=\check{b}$, the generator of $\check{B}$. Since $|\check{B}|=|\textrm{A}_{\textbf{H}}(s)|=2$ and hence $\check{B}\le \textrm{Z}(\check{B}_{\underline{E}})$ this implies

$$\begin{aligned}{}[f,g]=\check{b}~\text { for every }f\in \check{B}_{\underline{E}}\cap \textbf{H} F_0 ~(=\check{B} f_s)\text { and }g\in \check{B}_{\underline{E}}\cap \textbf{H} \gamma ~(=\check{B} g_s). \end{aligned}$$

(1)

This is a general property since such an s always exists.

Let us now look at the general case where $t\in \mathcal {C}^F\setminus \mathcal {C}^{F_0}$. Using again Lemma 3.5 but this time with $i=1$ (so that the $F_0$ of that lemma is our F), we get a conjugation $\iota _t:\textbf{H}\underline{E}\rightarrow \textbf{H}\underline{E}$ such that $\iota _t(t)=x$ and some other properties.

We define $\check{A}(t):=\iota _t^{-1}(\check{B})$ which then satisfies $\textrm{C}_{ \textbf{H} }(t)=\textrm{C}_\textbf{H}^\circ (t)\rtimes \check{A}(t)$. Let $f:=f_s=\iota _s(F_0)\in \check{B}_{\underline{E}}\cap \textbf{H} F_0$ (see above), and take any $g\in \check{B}_{\underline{E}}\cap \textbf{H} \gamma$. Let $F_{0}':=\iota _t^{-1}(f)$ and $\gamma _{t}':=\iota _t^{-1}(g)$. By this construction $\check{A}(t)$ is $\langle F_{0}', \gamma _{t}'\rangle$-stable and $F_{0}'(t)=\gamma _{t}'(t)=t$.

In the next step we show that F centralizes in $\textbf{H}\underline{E}$ both $\gamma '_t$ and $F'_0$. In applying Lemma 3.5, we can take $\tau \in \{F_0,\gamma \}$ and $h\in \textbf{H}^F$ such that $\tau (s)=s^h$ thanks to Lemma 4.4. Lemma 3.5(c) tells us that $\iota _t(h \tau ) =c f'$ with $c \in { \textbf{H} }$, $f' \in {\check{B}_{\underline{E}}}$, and $[f',\iota _t(F)]=1$. Since $\iota _t(h\tau )\in \textbf{H}\tau$, this shows that $\iota _t(F)$ fixes an element of the set with two elements $\check{B}_{\underline{E}}\cap \textbf{H}\tau$. On the other hand $\iota _t(F)\in \check{B}_{\underline{E}}$ by Lemma 3.5(b), so it stabilizes $\check{B}_{\underline{E}}\cap \textbf{H}\tau$ (remember again that cosets mod $\textbf{H}$ are stabilized by inner automorphisms of $\textbf{H}\underline{E}$) hence fixes any of its two elements. Taking the image by $\iota _t^{-1}$, we get that F centralizes both $\iota _t^{-1} (\check{B}_{\underline{E}}\cap \textbf{H} F_0)$ and $\iota _t^{-1} (\check{B}_{\underline{E}}\cap \textbf{H} \gamma )$, hence both $F'_0$ and $\gamma '_t$. So we get that $F'_0\in \textrm{C}_{\textbf{H}^F F_0}(s)$ and $\gamma '_t\in \textrm{C}_{\textbf{H}^F \gamma }(s)$. This ensures part (i) in that case.

Letting $t_0\in \pi ^{-1}(t)$, $x_0\in \pi ^{-1}(x)$, we have $[t_0,F'_0]= \iota _t^{-1}( [x_0,f])= \iota _t^{-1}\iota _s( [s_0,F_0])=[s_0,F_0]$, since $\iota _s$ and $\iota _t$ are conjugations by elements of $\textbf{H}$ hence act trivially on $\textrm{Z}(\textbf{H}_0)$. Then $[t_0,F'_0]= [s_0,F_0]\in \textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$ as seen before. We then get (iii) for t.

Let now $\check{a}$ be the generator of $\check{A}(t)$ and hence $\check{a}=\iota _t^{-1} (\check{b})$. By (1), $[f,g]=\check{b}$ and hence $[F_{0}', \gamma _{t}']=\iota _t^{-1}([f,g])= \iota _t^{-1}(\check{b})= \check{a}$. This ensures part (iv) with $\gamma ':=\gamma '_t$.$\square$

Remark 4.5

Both cases covered by our Corollaries are non empty. It is clear for the first. In order to check the relevance of the second corollary we use the notations of [10, Not. 2.3] to describe elements of the maximal torus $\textbf{T}_0$ of $\textbf{H}_0$. Let q be a power of p with $q\equiv 1~\textrm{mod}~4$. Let $F_q$ be the corresponding power of $F_p$. Let $\zeta , \varpi \in \mathbb {F}^\times$ with $\zeta ^{q^2-1}=\varpi ^2=-1$. Recall that $\textbf{h}_{\underline{l}}(\varpi ):=\textbf{h}_{e_1}(\varpi )\dots \textbf{h}_{e_l}(\varpi )$ is an element of $\textrm{Z}(\textbf{H}_0)\setminus \langle h_0\rangle$, not fixed by $\gamma$. Let $l=4$, $s_0=\textbf{h}_{e_2}(\varpi )\textbf{h}_{e_3}(\zeta )\textbf{h}_{e_4}(\zeta ^q\varpi )\in \textbf{T}_0$. One has $\gamma (s_0)=s_0$. It is not difficult to see that with the equation $\zeta ^{q^2}\varpi ^q\varpi =\zeta$ one gets that $F_q(s_0)$ and $s_0\textbf{h}_{\underline{4}}(\varpi )$ are $\textrm{N}_{\textbf{H} _0}(\textbf{T}_0)$-conjugate, so we can write $F_q(s_0)=s_0^n\textbf{h}_{\underline{4}}(\varpi )$ for some $n\in \textrm{N}_{\textbf{H} _0}(\textbf{T}_0)$. Then $\Theta _{nF_q}([s]_{\textbf{H}^{nF_q}}) =\textbf{h}_{\underline{4}}(\varpi )$ for $s:=\pi (s_0)$.

By Lang’s Theorem we can take $\iota :\textbf{H}\underline{E}\rightarrow \textbf{H}\underline{E}$ an inner automorphism induced by an element of $\textbf{H}$ such that $\iota (nF_q)=F_q$. Then $s':=\iota (s)\in \textbf{H}^{F_q}$ and $\Theta _{F_q}([s']_{\textbf{H}^{F_q}})=\textbf{h}_{\underline{4}}(\varpi )$ which is not $\gamma$-stable. Then $[s']_{\textbf{H}^{F_q}}$ is not $\gamma$-stable while s and $s'$ are in the same $\textbf{H}$-class.