Appendix: Adequate subgroups

Note that for any $g\in\Gamma$, $\Gamma$ contains both its semisimple and unipotent parts $g_{s}$ and $g_{u}$, respectively. (They are powers of $g$, as we work over $\overline{{\mathbb{F}}}_{l}$.) Since $e_{g,\alpha}=e_{g_{s},\alpha}$ for all $g\in\Gamma$, the first two conditions are equivalent.

To show that the last two conditions are equivalent, let $Z\subset\operatorname{ad}V$ be the span of the semisimple elements in $\Gamma$. Let $U$ denote the annihilator of $Z$ under the (non-degenerate, $\Gamma$-invariant) trace pairing:

where we used that $e_{g,\alpha}$ is a polynomial in $g$ and that $g=\sum\alpha e_{g,\alpha}$ for $g$ semisimple.

Note that $U\subset\operatorname{ad}^{0}V$ by taking $g=1$ in (1). From (2) it thus follows that the second condition is equivalent to $U=0$. Equivalently, $Z=\operatorname{ad}V$, which is the third condition. ∎

By Burnside’s theorem, $\Gamma$ spans $\operatorname{ad}V$. If $\Gamma$ has order prime to $l$, then every element is semisimple, so the lemma above applies.

The second part of the proposition follows on noting that if $g$, $h$ are semisimple elements then $g\otimes h$ is semisimple, and appealing to the third characterization of condition (C) in the lemma above. ∎

We can identify $\operatorname{Hom}(T({\mathbb{F}}_{l}),\overline{{\mathbb{F}}}_{l}^{\times})$ with $X^{*}/(l-\operatorname{Fr})X^{*}$. To prove the first part, suppose that $|\langle\mu,\delta\rangle|<l-1$ for $\delta\in\Delta_{*}$ and that $\mu(T({\mathbb{F}}_{l}))$ is trivial, so $\mu=(l-\operatorname{Fr})\lambda$. Choose $\delta_{1}$ in $\Delta_{*}$ with $|\langle\lambda,\delta_{1}\rangle|$ maximal. If $\langle\lambda,\delta_{1}\rangle\neq 0$ then

a contradiction. Therefore $\langle\lambda,\delta_{1}\rangle=0$, so $\lambda=0$ and $\mu=0$. In particular we see that if $\mu_{1}$ and $\mu_{2}$ are two elements of $X^{*}$ with $|\langle\mu_{i},\delta\rangle|<(l-1)/2$ for $\delta\in\Delta_{*}$ and $i=1$, $2$ then $\mu_{1}|_{T({\mathbb{F}}_{l})}=\mu_{2}|_{T({\mathbb{F}}_{l})}$ iff $\mu_{1}=\mu_{2}$. The second part now follows since both subspaces of $\operatorname{ad}V$ equal the $\overline{{\mathbb{F}}}_{l}$-linear span of the $T_{/\overline{{\mathbb{F}}}_{l}}$-equivariant projectors onto the weight spaces of $T_{/\overline{{\mathbb{F}}}_{l}}$ in $V$. ∎

Let $J$ denote the connected component of the kernel of $\phi$ with its reduced scheme structure. Then $J$ is smooth ([Mil], Proposition I.5.18). By Theorem 8.1.5 of [Spr09] and its proof, $J$ is semisimple and there is a second semisimple algebraic group $I\subset G$ which commutes with $J$ and such that $I\times J\rightarrow G$ is a central isogeny. It follows from the simply-connectedness of $G$ that it is an isomorphism of $I\times J$ onto $G$. In particular, $I$ and $J$ are simply connected. Note that $T=T_{I}\times T_{J}$ and that $B=B_{I}\times B_{J}$ where $(B_{I},T_{I})$ (resp. $(B_{J},T_{J})$) is a Borel and maximal torus in $I$ (resp. $J$). (This follows from the fact that any smooth connected soluble subgroup of (resp. torus in) $G$ is conjugate to a subgroup of $B$ (resp. $T$).) Moreover $U=U_{I}\times U_{J}$, where $U$ denotes the unipotent radical of $B$. Let $\overline{I}$ denote the image of $I$ under $\phi$. Then $\overline{I}$ is again reduced and connected and hence also smooth. In fact it is semisimple. (See Proposition 14.10(1)(c) of [Bor91].) The map $\phi$ factors through an isogeny $I\rightarrow\overline{I}\subset\operatorname{GL}(V)$. Let $\overline{B}$, $\overline{T}$, $\overline{U}$ denote the images of $B_{I}$, $T_{I}$, $U_{I}$ in $\overline{I}$. Then these are all reduced and hence smooth. Moreover $\overline{T}$ is a torus, $\overline{B}$ is connected and soluble, $\overline{U}$ is connected unipotent and $\overline{B}=\overline{T}\overline{U}$. As $\dim\overline{I}=\dim I=\dim T_{I}+2\dim U_{I}=\dim\overline{T}+2\dim\overline{U}$ we see that $\overline{B}$ must be a Borel subgroup of $\overline{I}$ with unipotent radical $\overline{U}$ and that $\overline{T}$ is a maximal torus in $\overline{I}$. The isogeny $I\rightarrow\overline{I}$ induces an $l$-morphism from the root datum of $\overline{I}$ to the root datum of $I$. (See section 9.6.3 of [Spr09].) Then $I\rightarrow\overline{I}$ is a central isogeny, as otherwise $T$ would have a weight occurring in $\operatorname{Lie}\overline{I}\subset\operatorname{ad}V$ of the form $l\mu$ with $\mu$ non-zero and this would contradict our assumption on the weights of $T$ on $V$. ∎

By the Lie–Kolchin theorem we may suppose $U$ is contained in the group $U^{\prime}=\prod U_{i}^{\prime}$, where $U^{\prime}_{i}$ denotes the unipotent radical of a Borel subgroup of $\operatorname{GL}(W_{i})$. The maps $\exp$ and $\log$ provide mutually inverse isomorphisms of varieties between $U^{\prime}$ and $\operatorname{Lie}U^{\prime}$. It remains to show that $\exp\operatorname{Lie}U=U$. Note that the product of any $l$ elements of $\operatorname{Lie}U^{\prime}$ is zero. Thus the Zassenhaus formula (see [Mag54], section IV) tells us that to check that $\exp\operatorname{Lie}U\subset U$ it suffices to check that for any root $\alpha$ we have $\exp(\operatorname{Lie}U_{\alpha})\subset U$. Let $x_{\alpha}:{\mathbb{G}}_{a}\to U_{\alpha}$ be the root homomorphism corresponding to $\alpha$ and let $X_{\alpha}=dx_{\alpha}(1)\in\operatorname{Lie}U_{\alpha}$. Then formula II.1.19(6) of [Jan03] shows that for $a\in\overline{{\mathbb{F}}}_{l}$,

in $\operatorname{GL}(V)$, on noting that for $n<l$ we have $X_{\alpha,n}=X_{\alpha}^{n}/n!$ while $X_{\alpha,n}$ acts trivially on $V$ for $n\geq l$. (This latter assertion follows from formula II.1.19(5) of [Jan03] because $V_{\lambda}$ and $V_{\lambda+n\alpha}$ cannot both be non-zero.) Now by the Baker–Campbell–Hausdorff formula (see section IV.8 in part I of [Ser92]) and the fact that the product of any $l$ elements of $\operatorname{Lie}U^{\prime}$ is zero we see that $\exp\operatorname{Lie}U$ is a subgroup of $U$. As $U$ is connected and smooth and $\dim\operatorname{Lie}U\geq\dim U$ we deduce that $\exp\operatorname{Lie}U=U$. This proves the first two parts.

The third part follows inductively from equation (3) and the Zassenhaus formula: fix a total order $<$ on the set of positive roots such that if $\alpha$, $\beta$, $\alpha+\beta$ are positive roots, then $\max(\alpha,\beta)<\alpha+\beta$. We induct on the positive root $\gamma$. Suppose that we know that $\exp$ depends only on $G$ and $U$ on the subspace $\bigoplus_{\alpha>\gamma}\operatorname{Lie}U_{\alpha}$. Then the same is true for $\exp(X+Y)$ for any $X\in\operatorname{Lie}U_{\gamma}$ and $Y\in\bigoplus_{\alpha>\gamma}\operatorname{Lie}U_{\alpha}$ by the Zassenhaus formula. (Note that $[\operatorname{Lie}U_{\alpha},\operatorname{Lie}U_{\beta}]\subset\operatorname{Lie}U_{\alpha+\beta}$ whenever $\alpha$, $\beta$ are positive roots.) This completes the proof of the third part.

The last part follows from the third part, by considering the representation $G\xrightarrow{\theta}G\hookrightarrow\operatorname{GL}(V)$. ∎

The universal Chevalley group over $\overline{{\mathbb{F}}}_{l}$ constructed using the complex semisimple Lie algebra ${\mathcal{L}}$ of the same root system as $G$ is an algebraic group isomorphic to $G$ (see [Ste68b], §5). (In the notation of [Ste68b], we can let $V$ be any representation whose weights span the weight lattice, so that ${\mathcal{L}}_{\mathbb{Z}}\subset{\mathcal{L}}$ is the ${\mathbb{Z}}$-lattice spanned by the fixed Chevalley basis $H_{i}$, $X_{\alpha}$; see Cor. 2 on p. 18 of [Ste68b].) In particular, ${\mathfrak{g}}\cong{\mathcal{L}}_{\mathbb{Z}}\otimes\overline{{\mathbb{F}}}_{l}$ (by the remark on p. 64 of [Ste68b]). Write $G=\prod G_{i}$ as a product of almost simple simply connected algebraic groups and correspondingly ${\mathfrak{g}}=\bigoplus{\mathfrak{g}}_{i}$. Then $Z({\mathfrak{g}}_{i})=0$ by our assumption on $l$ and $G$ (see Theorem 2.3 in [Hur82]) and hence all ${\mathfrak{g}}_{i}$ are simple ([Ste61], 2.6(5)). Moreover ${\mathfrak{g}}_{i}\cong{\mathfrak{g}}_{j}$ implies $G_{i}\cong G_{j}$ ([Ste61], 8.1). The $G_{i}$ (resp., ${\mathfrak{g}}_{i}$) are uniquely characterised as the minimal non-trivial connected normal subgroups of $G$ (resp., minimal non-trivial ideals of ${\mathfrak{g}}$), so they are permuted by automorphisms. Therefore if $\operatorname{Aut}(G_{i})\to\operatorname{Aut}({\mathfrak{g}}_{i})$ is a bijection for all $i$, then so is $\operatorname{Aut}(G)\to\operatorname{Aut}({\mathfrak{g}})$, and also the final claim of the proposition follows. (Note that any connected normal subgroup is a product of some of the $G_{i}$.) We can thus assume, without loss of generality, that $G$ is almost simple.

Let $G^{\operatorname{ad}}$ denote the adjoint form of $G$. As $G$ is the universal cover of $G^{\operatorname{ad}}$ and as $G^{\operatorname{ad}}=G/Z(G)$, we have $\operatorname{Aut}(G)=\operatorname{Aut}(G^{\operatorname{ad}})$. As $Z({\mathfrak{g}})=0$ we see that the natural map ${\mathfrak{g}}\to\operatorname{Lie}G^{\operatorname{ad}}$ is an isomorphism. Thus it suffices to show that $\operatorname{Aut}(G)=\operatorname{Aut}({\mathfrak{g}})$ whenever $G$ is simple of adjoint type and ${\mathfrak{g}}=\operatorname{Lie}G$. Thus we write $G$ for $G^{\operatorname{ad}}$ from now on.

As an algebraic group $G$ is isomorphic to the adjoint Chevalley group over $\overline{{\mathbb{F}}}_{l}$ (again by [Ste68b], §5). (In the notation of [Ste68b], we take $V$ to be the adjoint representation ${\mathfrak{g}}$.) Thus we can identify $G(\overline{{\mathbb{F}}}_{l})$ with the subgroup of $\operatorname{GL}({\mathfrak{g}})$ generated by the elements $x_{\alpha}(t):=\exp(\operatorname{ad}(tX_{\alpha}))$, where $t\in\overline{{\mathbb{F}}}_{l}$ and $\alpha$ is any root. As each $\operatorname{ad}(tX_{\alpha})$ is a derivation of ${\mathfrak{g}}$, the group $G(\overline{{\mathbb{F}}}_{l})$ is actually contained in $\operatorname{Aut}({\mathfrak{g}})$. For any $\eta\in\operatorname{Aut}({\mathfrak{g}})$, we have $\eta\circ\operatorname{ad}X\circ\eta^{-1}=\operatorname{ad}(\eta X)$ in $\operatorname{GL}({\mathfrak{g}})$. It follows that the natural action of $G(\overline{{\mathbb{F}}}_{l})\subset\operatorname{GL}({\mathfrak{g}})$ on ${\mathfrak{g}}$ agrees with the adjoint action of $G(\overline{{\mathbb{F}}}_{l})$ on ${\mathfrak{g}}\subset\operatorname{End}({\mathfrak{g}})$.

The choice of Chevalley basis gives rise to a maximal torus $T$ and a Borel $B$ that contains it ([Ste68b], §5). From Theorem 9.6.2 in [Spr09] we deduce the following, using that $G$ is adjoint. For each symmetry $\pi$ of the Dynkin diagram ${\mathcal{D}}$ there is a unique $\pi^{\prime}\in\operatorname{Aut}(G)$ that preserves $(B,T)$ and that permutes the $x_{\alpha_{i}}(1)\in B$ according to $\pi$ (where $\alpha_{i}$ are the simple roots). Moreover, $\operatorname{Aut}(G)$ is the semidirect product of $G$ (acting by inner automorphisms) and $\operatorname{Aut}({\mathcal{D}})$. Also, the elements of $\operatorname{Aut}({\mathcal{D}})$ biject with the “graph automorphisms” of ${\mathfrak{g}}$ ([Ste61], §3).

The result now follows from ([Ste61], 4.2 and 4.5), as the group $\mathfrak{H}$ in [Ste61] is actually contained in $G(\overline{{\mathbb{F}}}_{l})$ since $\overline{{\mathbb{F}}}_{l}$ is algebraically closed (see Lemma 19 on p. 27 of [Ste68b]). (Note that the uniqueness statement in ([Ste61], 4.2) is incorrect and seems to be a typo.) ∎

Write $V=\bigoplus_{i}{W_{i}}$ as a direct sum of irreducible $\Gamma^{0}$-modules. Since $\dim W_{i}\leq l$ for all $i$, we see that every element of $l$-power order in the image of $\Gamma^{0}\rightarrow\operatorname{GL}(W_{i})$ actually has order dividing $l$. Since $\Gamma^{0}\hookrightarrow\prod\operatorname{GL}(W_{i})$, we deduce that every element of $\Gamma^{0}$ of $l$-power order actually has order dividing $l$. Note that $\Gamma/\Gamma^{0}$ has order prime to $l$.

Step 1. We show that there exists a connected simply connected semisimple algebraic group $G^{0}$ over ${\mathbb{F}}_{l}$ and a finite central subgroup $Z_{0}\subset G^{0}({\mathbb{F}}_{l})$ with $G^{0}({\mathbb{F}}_{l})/Z_{0}\cong\Gamma^{0}$. Let $\Gamma_{i}$ denote the image of $\Gamma^{0}$ in $\operatorname{GL}(W_{i})$. Note that $\Gamma_{i}$ has no non-trivial normal subgroup of $l$-power order (since $\Gamma_{i}$ acts faithfully on $W_{i}$, and an $l$-group acting on a non-zero $\overline{{\mathbb{F}}}_{l}$-vector space has non-zero fixed points). So by Theorem B of [Gur99], $\Gamma_{i}$ is a central product of quasisimple Chevalley groups. (Note that if $l=11$ then $\dim W_{i}<7$.) Now $\Gamma^{0}$ is a subgroup of $\prod\Gamma_{i}$ that surjects onto each factor, so $Z(\Gamma^{0})=\Gamma^{0}\cap\prod Z(\Gamma_{i})$. Thus $\Gamma^{0}/Z(\Gamma^{0})$ is a subgroup of $\prod\Gamma_{i}/Z(\Gamma_{i})$, a product of simple Chevalley groups, that surjects onto each factor. By a theorem of Hall (Lemma 3.5 in [Kup]), $\Gamma^{0}/Z(\Gamma^{0})$ is itself isomorphic to a direct product of simple Chevalley groups. It follows that $\Gamma^{0}=[\Gamma^{0},\Gamma^{0}]Z(\Gamma^{0})$. Since $\Gamma^{0}$ is generated by elements of order $l$ and $Z(\Gamma^{0})$ is of order prime to $l$, it follows moreover that $\Gamma^{0}$ is perfect. Therefore $\Gamma^{0}$ is a perfect central extension of a product $\prod H_{j}$ of simple Chevalley groups $H_{j}$, so there exists a surjective homomorphism $\pi:\prod\widetilde{H}_{j}\to\Gamma^{0}$ with central kernel, where $\widetilde{H}_{j}$ is the universal perfect central extension of $H_{j}$.

As $l>3$ (to rule out Suzuki and Ree groups) there exist connected simply connected algebraic groups ${G_{j}}$ over ${\mathbb{F}}_{l}$ such that $H_{j}\cong G_{j}({\mathbb{F}}_{l})/Z(G_{j}({\mathbb{F}}_{l}))$. (Note that $G_{j}$ is the restriction of scalars of an absolutely almost simple algebraic group over a finite extension of ${\mathbb{F}}_{l}$.) Since $l>3$ it is known that $\widetilde{H}_{j}\cong G_{j}({\mathbb{F}}_{l})$ (see section 6.1 in [GLS98], particularly table 6.1.3). So we can take $G^{0}=\prod G_{j}$ and $Z_{0}=\ker\pi$.

Since $\Gamma^{0}/Z(\Gamma^{0})$ is a product of nonabelian simple groups and since $Z(\Gamma^{0})$ and $\Gamma/\Gamma^{0}$ are of order prime to $l$, it follows that $\Gamma$ does not have any composition factor of order $l$.

Let $G^{0}\supset B\supset T$ denote a Borel and maximal torus defined over ${\mathbb{F}}_{l}$.

Step 2. We lift $V$ to a $G^{0}_{/\overline{{\mathbb{F}}}_{l}}$-module and compare the actions of $T({\mathbb{F}}_{l})$ and $T(\overline{{\mathbb{F}}}_{l})$ on $V$. Let $U$ denote the unipotent radical of $B$ and set $N=N_{G^{0}}(T)$. Let $B^{\mathrm{op}}$ denote the opposite Borel subgroup to $B$ containing $T$ and let $U^{\mathrm{op}}$ denote its unipotent radical. (See Theorem 14.1 of [Bor91]. By uniqueness we see it is defined over ${\mathbb{F}}_{l}$.) Let $X=X^{*}(T_{/\overline{{\mathbb{F}}}_{l}})$ with its subset $\Phi$ of roots and $\Phi^{+}$ (resp. $\Delta$) the set of positive (resp. simple) roots corresponding to $B$. Let $X^{+}\subset X$ be the subset of dominant weights. There is a semisimple algebraic action of $G^{0}_{/\overline{{\mathbb{F}}}_{l}}$ on $V$, say $\phi:G^{0}_{/\overline{{\mathbb{F}}}_{l}}\rightarrow\operatorname{GL}(V)$, such that:

1. (i)

   the highest weight $\lambda$ of a simple submodule is restricted (i.e. $0\leq\langle\lambda,\alpha^{\vee}\rangle<l$ for all $\alpha\in\Delta$),

2. (ii)

   the action of $G^{0}({\mathbb{F}}_{l})$ is the one induced by the map $G^{0}({\mathbb{F}}_{l})\rightarrow\Gamma^{0}$,

3. (iii)

   the subspaces $W_{i}$ are $G^{0}_{/\overline{{\mathbb{F}}}_{l}}$-stable.

(This follows from a result of Steinberg: see Theorem 2.11 in [Hum06]. Note that [Hum06] works with an algebraic group $\mathbf{G}$ that is simple, but the proof given does not depend on that assumption.) By Proposition 3 of [Ser94] we see that if $\lambda$ in $X^{+}$ is a weight of $T_{/\overline{{\mathbb{F}}}_{l}}$ on $V$ then $\sum_{\alpha\in\Phi^{+}}\langle\lambda,\alpha^{\vee}\rangle<d$; in particular, $\langle\lambda,\alpha^{\vee}\rangle<(l-1)/2$ for all $\alpha\in\Phi^{+}$. (Note that $\dim W_{i}\leq(l-1)/2$ and that the proof of that proposition does not require that $G^{0}_{/\overline{{\mathbb{F}}}_{l}}$ be almost simple.) If $\mu$ is a weight of $T_{/\overline{{\mathbb{F}}}_{l}}$ on $V$ then we see that there is $w$ in the Weyl group with $w\mu\in X^{+}$ and $0\leq\langle w\mu,\alpha^{\vee}\rangle<(l-1)/2$ for all $\alpha\in\Phi^{+}$, and we deduce that $|\langle\mu,\alpha^{\vee}\rangle|<(l-1)/2$ for all $\alpha\in\Phi$. We also deduce that if $\mu$ is a weight of $T_{/\overline{{\mathbb{F}}}_{l}}$ on $\operatorname{ad}V$ then $|\langle\mu,\alpha^{\vee}\rangle|<l-1$ for all $\alpha\in\Delta$.

Step 3. The semisimple group $\overline{I}\subset\operatorname{GL}(V)$ and its simply connected cover $I\subset G^{0}_{/\overline{{\mathbb{F}}}_{l}}$. Since $|\langle\mu,\alpha^{\vee}\rangle|<l/2$ for all weights $\mu$ of $T_{/\overline{{\mathbb{F}}}_{l}}$ on $V$ and all $\alpha\in\Delta$ we may apply Lemma 4 to $\phi:G^{0}_{/\overline{{\mathbb{F}}}_{l}}\rightarrow\operatorname{GL}(V)$. We obtain connected simply connected semisimple algebraic subgroups $I$, $J$ of $G^{0}_{/\overline{{\mathbb{F}}}_{l}}$ such that $G^{0}_{/\overline{{\mathbb{F}}}_{l}}=I\times J$, $\phi(J)=1$, and $\phi$ induces a central isogeny of $I$ onto its image $\overline{I}$, which is a semisimple algebraic group. Note that $T_{/\overline{{\mathbb{F}}}_{l}}=T_{I}\times T_{J}$ and that $B_{/\overline{{\mathbb{F}}}_{l}}=B_{I}\times B_{J}$ where $(B_{I},T_{I})$ (resp. $(B_{J},T_{J})$) is a Borel and maximal torus in $I$ (resp. $J$). Moreover $U_{/\overline{{\mathbb{F}}}_{l}}=U_{I}\times U_{J}$. Let $\overline{B}$, $\overline{T}$, $\overline{U}$, $\overline{B}^{\mathrm{op}}$, $\overline{U}^{\mathrm{op}}$ denote the images of $B_{I}$, $T_{I}$, $U_{I}$, $B_{I}^{\mathrm{op}}$, $U_{I}^{\mathrm{op}}$ in $\overline{I}$. Then $\overline{T}$ is a maximal torus of $\overline{I}$, and $\overline{B}$, $\overline{B}^{\mathrm{op}}$ are opposite Borel subgroups containing it. Also $\overline{U}$, $\overline{U}^{\mathrm{op}}$ are the unipotent radicals of $\overline{B}$, $\overline{B}^{\mathrm{op}}$. Since $I\to\overline{I}$ is a central isogeny, $U_{I}\rightarrow\overline{U}$ and $U_{I}^{\mathrm{op}}\rightarrow\overline{U}^{\mathrm{op}}$ are isomorphisms.

Step 4. The maps $\log$ and $\exp$ provide inverse isomorphisms of varieties between $\overline{U}\subset\operatorname{GL}(V)$ and $\operatorname{Lie}\overline{U}\subset\operatorname{ad}V$. This follows from Lemma 5 applied to $\overline{I}\subset\operatorname{GL}(V)$ since $\dim W_{i}\leq l$ for all $i$ and $|\langle\mu,\alpha^{\vee}\rangle|<l/2$ for all weights $\mu$ of $T_{/\overline{{\mathbb{F}}}_{l}}$ on $V$ and all $\alpha\in\Delta$. (Note that $T_{I}\to\overline{T}$ induces a bijection on coroots since $I\to\overline{I}$ is a central isogeny; thus $T\to\overline{T}$ induces a surjection on coroots.)

Step 5. The $\overline{{\mathbb{F}}}_{l}$-span of $\log U({\mathbb{F}}_{l})$ is $\operatorname{Lie}\overline{U}$. Since $d\phi:\operatorname{Lie}U\rightarrow\operatorname{Lie}\overline{U}$ is surjective, it suffices to show that there is an isomorphism $\log:U\to\operatorname{Lie}U$ defined over ${\mathbb{F}}_{l}$ such that $d\phi\circ\log=\log\circ\phi$. Pick an ${\mathbb{F}}_{l}$-structure on $V$. The map $G^{0}_{/\overline{{\mathbb{F}}}_{l}}\to\operatorname{GL}(V)$ can be defined over some ${\mathbb{F}}_{l^{s}}$ and so taking restrictions of scalars from ${\mathbb{F}}_{l^{s}}$ to ${\mathbb{F}}_{l}$ we get an ${\mathbb{F}}_{l}$-vector space $V^{\prime}$ and a map $\psi:G^{0}\to\operatorname{GL}(V^{\prime})$. The map $G^{0}_{/\overline{{\mathbb{F}}}_{l}}\to\operatorname{GL}(V)$ is obtained from $\psi$ by extending scalars to $\overline{{\mathbb{F}}}_{l}$ and projecting to a direct summand $V$ of $V^{\prime}\otimes\overline{{\mathbb{F}}}_{l}$. The dimension of all irreducible factors of $V^{\prime}\otimes\overline{{\mathbb{F}}}_{l}$ is at most $l$. Moreover for any weight $\lambda$ of $T_{/\overline{{\mathbb{F}}}_{l}}$ on $V^{\prime}\otimes\overline{{\mathbb{F}}}_{l}$ we have $|\langle\lambda,\alpha^{\vee}\rangle|<(l-1)/2$ for all $\alpha\in\Phi^{+}$.

By Lemma 4 we see that $\psi:G^{0}\to\operatorname{GL}(V^{\prime})$ is a central isogeny onto its image. (By construction we have $(\ker\psi)({\mathbb{F}}_{l})=Z_{0}$. Suppose that $\ker\psi$ is not finite. Then it has to contain one of the ${\mathbb{F}}_{l}$-almost simple factors of $G^{0}=\prod G_{j}$. But $G_{j}({\mathbb{F}}_{l})$ is nonabelian.)

In particular, $\psi$ induces an isomorphism $U\to\psi(U)$. Then Lemma 5 (applied to the image of $\psi_{/\overline{{\mathbb{F}}}_{l}}$) gives the desired map $\log:U\to\operatorname{Lie}U\subset\operatorname{ad}V^{\prime}$.

Step 6: Some properties of $G^{0}({\mathbb{F}}_{l})$. The pair $(B({{\mathbb{F}}_{l}}),N({{\mathbb{F}}_{l}}))$ is a split $BN$ pair in $G^{0}({{\mathbb{F}}_{l}})$ (see section 1.18 of [Car93]). Also $U({{\mathbb{F}}_{l}})$ is a Sylow $l$-subgroup of $G^{0}({{\mathbb{F}}_{l}})$ and $B({{\mathbb{F}}_{l}})=N_{G^{0}({{\mathbb{F}}_{l}})}(U({{\mathbb{F}}_{l}}))=N_{G^{0}({{\mathbb{F}}_{l}})}(B({{\mathbb{F}}_{l}}))$ (see Proposition 2.5.1 of [Car93]).

Moreover $T({{\mathbb{F}}_{l}})$ is a Sylow $l$-complement in $B({{\mathbb{F}}_{l}})$. Note that $U^{\mathrm{op}}({{\mathbb{F}}_{l}})$ is $N({{\mathbb{F}}_{l}})$-conjugate to $U({{\mathbb{F}}_{l}})$. (The longest Weyl element $w_{0}$ is stable under Frobenius, hence represented by an element $n_{0}\in N({\mathbb{F}}_{l})$. Then use that $U^{\mathrm{op}}=n_{0}Un_{0}^{-1}$.) Moreover the second-last displayed equation on page 74 (section 2.9) of [Car93] shows that $U^{\mathrm{op}}({{\mathbb{F}}_{l}})$ is the unique $N({{\mathbb{F}}_{l}})$-conjugate of $U({{\mathbb{F}}_{l}})$ with trivial intersection with $U({{\mathbb{F}}_{l}})$.

Step 7. We have $N({{\mathbb{F}}_{l}})=N_{G^{0}({{\mathbb{F}}_{l}})}(T({{\mathbb{F}}_{l}}))$ so that $N_{G^{0}({{\mathbb{F}}_{l}})}(T({{\mathbb{F}}_{l}}))\cap N_{G^{0}({{\mathbb{F}}_{l}})}(B({{\mathbb{F}}_{l}}))=T({{\mathbb{F}}_{l}})$ and $Z_{0}\subset Z(G^{0}({\mathbb{F}}_{l}))\subset T({\mathbb{F}}_{l})$.

Suppose that $g$ is in $N_{G^{0}({{\mathbb{F}}_{l}})}(T({{\mathbb{F}}_{l}}))$. One can write $g$ uniquely as $unu^{\prime}$ where $u\in U({{\mathbb{F}}_{l}}),n\in N({{\mathbb{F}}_{l}})$ maps to $w_{n}$ in the Weyl group and $u^{\prime}\in U_{w_{n}}$ in the notation of Theorem 2.5.14 of [Car93]. Then for any $h$ in $T({{\mathbb{F}}_{l}})$ we can find $h^{\prime}$ and $h^{\prime\prime}$ in $T({{\mathbb{F}}_{l}})$ such that

i.e.,

and

As $T({{\mathbb{F}}_{l}})$ normalizes $U({{\mathbb{F}}_{l}})$ and $U_{w_{n}}$ and as $w_{nh}=w_{n}=w_{hn}$ the uniqueness assertion of Theorem 2.5.14 of [Car93] tells us that $huh^{-1}=u$ and $u^{\prime}=h^{-1}u^{\prime}h$. Thus $u\in Z_{U({{\mathbb{F}}_{l}})}(T({{\mathbb{F}}_{l}}))$ and $u^{\prime}\in Z_{U_{w_{n}}}(T({{\mathbb{F}}_{l}}))\subset Z_{U({{\mathbb{F}}_{l}})}(T({{\mathbb{F}}_{l}}))$. So it suffices to prove that $Z_{U(\overline{{\mathbb{F}}}_{l})}(T({{\mathbb{F}}_{l}}))={1}$. By Proposition 8.2.1 in [Spr09]

it suffices to show that $Z_{U_{\alpha}(\overline{{\mathbb{F}}}_{l})}(T({{\mathbb{F}}_{l}}))={1}$ for all $\alpha\in\Phi^{+}$. By Proposition 8.1.1(i) in [Spr09]

it suffices that $\alpha$ is non-trivial on $T({{\mathbb{F}}_{l}})$ for all $\alpha\in\Phi^{+}$. As $l\geq 5$, this follows from Lemma 3(i) (applied with $\Delta_{*}$ the set of simple coroots).

Step 8. We find a subgroup $H$ of order prime to $l$ such that $\Gamma=\Gamma^{0}H$. Let $H$ denote the subgroup of $h\in\Gamma$ which normalize both the image of $B({{\mathbb{F}}_{l}})$ and the image of $T({{\mathbb{F}}_{l}})$ in $\Gamma^{0}$. Then by the previous paragraph we see that $H\cap\Gamma^{0}$ is $T({{\mathbb{F}}_{l}})/Z_{0}$. Thus $H$ has order prime to $l$.

Moreover if $\gamma\in\Gamma$ we see that $\gamma(B({{\mathbb{F}}_{l}})/Z_{0})\gamma^{-1}$ is the normalizer of a Sylow $l$-subgroup of $G^{0}({{\mathbb{F}}_{l}})/Z_{0}$ and hence $G^{0}({{\mathbb{F}}_{l}})$-conjugate to $B({{\mathbb{F}}_{l}})/Z_{0}$, say $\gamma(B({{\mathbb{F}}_{l}})/Z_{0})\gamma^{-1}=k(B({{\mathbb{F}}_{l}})/Z_{0})k^{-1}$ with $k\in G^{0}({{\mathbb{F}}_{l}})$. Then $k^{-1}\gamma(T({{\mathbb{F}}_{l}})/Z_{0})\gamma^{-1}k$ is a Sylow $l$-complement in $B({{\mathbb{F}}_{l}})/Z_{0}$ and hence (by Hall’s theorem) $B({{\mathbb{F}}_{l}})/Z_{0}$-conjugate to $T({{\mathbb{F}}_{l}})/Z_{0}$, say