I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.

Let me first recall the classical set-up. According to Steinberg's lecture notes, every simple untwisted Chevalley group $G$ can be obtained as follows. Consdider a finite-dimensional, complex simple Lie algebra $L$ together with a Chevalley basis and a finite field $\mathbb{F}$. Associate to each root $\alpha$ a one-parameter subgroup $x_\alpha : \mathbb{F} \longrightarrow \operatorname{GL}(L) : t \mapsto x_\alpha(t)$, called the root subgroup of $\alpha$, in the usual way (using the exponential of the ad-nilpotent matrices, interpreted over the field $\mathbb{F}$). Then each such $x_\alpha$ is an injective homomorphism from the additive group $(\mathbb{F},+)$ of the field $\mathbb{F}$ to the group $\operatorname{GL}(L)$ and $G$ is (isomorphic to) the group that is generated by the elements $x_\alpha(t)$ (where $\alpha$ ranges over the roots and where $t$ ranges over the elements of the field). These root subgroups are moreover compatible with field automorphisms in the obvious way.

The case of twisted Chevalley groups is more complicated to describe. We assume that the Dynkin diagram and the field allow us to define a distinguished automorphism $\sigma = \gamma \cdot \phi$ of the group $G$ in the usual way (with $\gamma$ a symmetry of the Dynkin diagram and with $\phi$ a suitable field automorphism), so that we may define the twisted Chevalley group associated to $G$ and $\sigma$ as $G^\sigma$, the subgroup of $G$ that is fixed elementwise by $\sigma$.

I would now like to have a collection of injective homomorphisms $y_B : (\mathbb{F},+) \longrightarrow G^\sigma : t \mapsto y_B(t)$ that are compatible with field automorphisms and that generate $G^\sigma$, as in the classical case. Here, the field $\mathbb{F}$ is the defining field, (or possibly the subfield $\mathbb{F}_0$ of $\mathbb{F}$ that is fixed by the distinguished field automorphism $\phi$).

The obvious choice of $x_\alpha$ does not work, since the $x_\alpha$ need not be contained in $G^\sigma$. A natural, but slightly naive, approach would be to consider a $\langle \gamma \rangle$-orbit B and to then define

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(t)$ (for an orbit $B = \{ \alpha \}$ of length $1$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) \}$ of length $2$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(\phi^0(t)) \cdot x_{\gamma^1(\alpha)}(\phi^1(t)) \cdot x_{\gamma^2(\alpha)}(\phi^2(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) , \gamma^2(\alpha) \}$ of length 3).

Question: Is there a standard text that constructs such one-parameter subgroups of $G^\sigma$ and that then proves the following properties?

• Property 1: Each $y_B$ is an injective homomorphism.

• Property 2: The group $G^\sigma$ is generated by the elements $y_B(t)$.

• Property 3: The one-parameter groups $y_B$ are compatible with field automorphisms, in the sense that if the field automorphism $A$ is induced by the automorphism $a$ of the field $\mathbb{F}$, then $A(y_B(t)) = y_B(a(t))$ for all $t \in \mathbb{F}$.

Assuming that the $y_B$ are indeed defined using the above $\langle \gamma \rangle$, Property 1 would seem to follow from the "uniqueness of expression" in the product formulae for the classical case, while Property 3 would follow immediately.

Gorenstein and Lyons make a vague reference to such a construction in [The local structure of finite groups of characteristic 2 type: Part I, Chapter I, Paragraph 4]. It also looks like E. Abe has indeed tried "something along those lines" in [Coverings of twisted Chevalley groups over commutative rings. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1977), no. 366-382, 194–218.] But the text is difficult to understand and I am still unsure about the conventions / assumptions / generality of it.

Any help would be appreciated.

I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.

Let me first recall the classical set-up. According to Steinberg's lecture notes, every simple untwisted Chevalley group $G$ can be obtained as follows. Consdider a finite-dimensional, complex simple Lie algebra $L$ together with a Chevalley basis and a finite field $\mathbb{F}$. Associate to each root $\alpha$ a one-parameter subgroup $x_\alpha : \mathbb{F} \longrightarrow \operatorname{GL}(L) : t \mapsto x_\alpha(t)$, called the root subgroup of $\alpha$, in the usual way (using the exponential of the ad-nilpotent matrices, interpreted over the field $\mathbb{F}$). Then each such $x_\alpha$ is an injective homomorphism from the additive group $(\mathbb{F},+)$ of the field $\mathbb{F}$ to the group $\operatorname{GL}(L)$ and $G$ is (isomorphic to) the group that is generated by the elements $x_\alpha(t)$ (where $\alpha$ ranges over the roots and where $t$ ranges over the elements of the field). These root subgroups are moreover compatible with field automorphisms in the obvious way.

The case of twisted Chevalley groups is more complicated to describe. We assume that the Dynkin diagram and the field allow us to define a distinguished automorphism $\sigma = \gamma \cdot \phi$ of the group $G$ in the usual way (with $\gamma$ a symmetry of the Dynkin diagram and with $\phi$ a suitable field automorphism), so that we may define the twisted Chevalley group associated to $G$ and $\sigma$ as $G^\sigma$, the subgroup of $G$ that is fixed elementwise by $\sigma$.

I would now like to have a collection of injective homomorphisms $y_B : (\mathbb{F},+) \longrightarrow G^\sigma : t \mapsto y_B(t)$ that are compatible with field automorphisms and that generate $G^\sigma$, as in the classical case. Here, the field $\mathbb{F}$ is the defining field, (or possibly the subfield $\mathbb{F}_0$ of $\mathbb{F}$ that is fixed by the distinguished field automorphism $\phi$).

The obvious choice of $x_\alpha$ does not work, since the $x_\alpha$ need not be contained in $G^\sigma$. A natural, but slightly naive, approach would be to consider a $\langle \gamma \rangle$-orbit B and to then define

$y_B : \mathbb{F}_0 \longrightarrow G^\sigma : t \mapsto x_\alpha(t)$ (for an orbit $B = \{ \alpha \}$ of length $1$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : t \mapsto x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) \}$ of length $2$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : t \mapsto x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t)) \cdot x_{\gamma^2(\alpha)}(\phi^2(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) , \gamma^2(\alpha) \}$ of length 3).

Question: Is there a standard text that constructs a collection of one-parameter subgroups $t \mapsto y_B(t)$ of $G^\sigma$ satisfying the following properties?

• Property 1: Each $y_B$ is an injective homomorphism.

• Property 2: The group $G^\sigma$ is generated by the elements $y_B(t)$.

• Property 3: The one-parameter groups $y_B$ are compatible with field automorphisms, in the sense that if the field automorphism $A$ is induced by the automorphism $a$ of the field $\mathbb{F}$, then $A(y_B(t)) = y_B(a(t))$ for all $t \in \mathbb{F}$.

Assuming that the $y_B$ are indeed defined using the above $\langle \sigma \rangle$-orbits, Property 1 would seem to follow from the "uniqueness of expression" in the product formulae for the classical case, while Property 3 would follow immediately.

Gorenstein and Lyons make a vague reference to such a construction in [The local structure of finite groups of characteristic 2 type: Part I, Chapter I, Paragraph 4]. It also looks like E. Abe has indeed tried "something along those lines" in [Coverings of twisted Chevalley groups over commutative rings. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1977), no. 366-382, 194–218.] But the text is difficult to understand and I am still unsure about the conventions / assumptions / generality of it.

Any help would be appreciated.

I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.

Let me first recall the classical set-up. According to Steinberg's lecture notes, every simple untwisted Chevalley group $G$ can be obtained as follows. Consdider a finite-dimensional, complex simple Lie algebra $L$ together with a Chevalley basis and a finite field $\mathbb{F}$. Associate to each root $\alpha$ a one-parameter subgroup $x_\alpha : \mathbb{F} \longrightarrow \operatorname{GL}(L) : t \mapsto x_\alpha(t)$, called the root subgroup of $\alpha$, in the usual way (using the exponential of the ad-nilpotent matrices, interpreted over the field $\mathbb{F}$). Then each such $x_\alpha$ is an injective homomorphism from the additive group $(\mathbb{F},+)$ of the field $\mathbb{F}$ to the group $\operatorname{GL}(L)$ and $G$ is (isomorphic to) the group that is generated by the elements $x_\alpha(t)$ (where $\alpha$ ranges over the roots and where $t$ ranges over the elements of the field). These root subgroups are moreover compatible with field automorphisms in the obvious way.

The case of twisted Chevalley groups is more complicated to describe. We assume that the Dynkin diagram and the field allow us to define a distinguished automorphism $\sigma = \gamma \cdot \phi$ of the group $G$ in the usual way (with $\gamma$ a symmetry of the Dynkin diagram and with $\phi$ a suitable field automorphism), so that we may define the twisted Chevalley group associated to $G$ and $\sigma$ as $G^\sigma$, the subgroup of $G$ that is fixed elementwise by $\sigma$.

I would now like to have a collection of injective homomorphisms $y_B : (\mathbb{F},+) \longrightarrow G^\sigma : t \mapsto y_B(t)$ that are compatible with field automorphisms and that generate $G$ in the above sense. Here, the field $\mathbb{F}$ is the defining field, (or possibly the subfield $\mathbb{F}_0$ of $\mathbb{F}$ that is fixed by the distinguished field automorphism $\phi$).

The obvious choice of $x_\alpha$ does not work, since the $x_\alpha$ need not be contained in $G^\sigma$. A natural, but slightly naive, approach would be to consider a $\langle \gamma \rangle$-orbit B and to then define

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(t)$ (for an orbit $B = \{ \alpha \}$ of length $1$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) \}$ of length $2$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(\phi^0(t)) \cdot x_{\gamma^1(\alpha)}(\phi^1(t)) \cdot x_{\gamma^2(\alpha)}(\phi^2(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) , \gamma^2(\alpha) \}$ of length 3).

Question: Is there a standard text that constructs such one-parameter subgroups of $G^\sigma$ and that then proves the following properties?

• Property 1: Each $y_B$ is an injective homomorphism.

• Property 2: The group $G^\sigma$ is generated by the elements $y_B(t)$.

• Property 3: The one-parameter groups $y_B$ are compatible with field automorphisms, in the sense that if the field automorphism $A$ is induced by the automorphism $a$ of the field $\mathbb{F}$, then $A(y_B(t)) = y_B(a(t))$ for all $t \in \mathbb{F}$.

Assuming that the $y_B$ are indeed defined using the above $\langle \gamma \rangle$, Property 1 would seem to follow from the "uniqueness of expression" in the product formulae for the classical case, while Property 3 would follow immediately.

Gorenstein and Lyons make a vague reference to such a construction in [The local structure of finite groups of characteristic 2 type: Part I, Chapter I, Paragraph 4]. It also looks like E. Abe has indeed tried "something along those lines" in [Coverings of twisted Chevalley groups over commutative rings. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1977), no. 366-382, 194–218.] But the text is difficult to understand and I am still unsure about the conventions / assumptions / generality of it.

Any help would be appreciated.

I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.

Let me first recall the classical set-up. According to Steinberg's lecture notes, every simple untwisted Chevalley group $G$ can be obtained as follows. Consdider a finite-dimensional, complex simple Lie algebra $L$ together with a Chevalley basis and a finite field $\mathbb{F}$. Associate to each root $\alpha$ a one-parameter subgroup $x_\alpha : \mathbb{F} \longrightarrow \operatorname{GL}(L) : t \mapsto x_\alpha(t)$, called the root subgroup of $\alpha$, in the usual way (using the exponential of the ad-nilpotent matrices, interpreted over the field $\mathbb{F}$). Then each such $x_\alpha$ is an injective homomorphism from the additive group $(\mathbb{F},+)$ of the field $\mathbb{F}$ to the group $\operatorname{GL}(L)$ and $G$ is (isomorphic to) the group that is generated by the elements $x_\alpha(t)$ (where $\alpha$ ranges over the roots and where $t$ ranges over the elements of the field). These root subgroups are moreover compatible with field automorphisms in the obvious way.

The case of twisted Chevalley groups is more complicated to describe. We assume that the Dynkin diagram and the field allow us to define a distinguished automorphism $\sigma = \gamma \cdot \phi$ of the group $G$ in the usual way (with $\gamma$ a symmetry of the Dynkin diagram and with $\phi$ a suitable field automorphism), so that we may define the twisted Chevalley group associated to $G$ and $\sigma$ as $G^\sigma$, the subgroup of $G$ that is fixed elementwise by $\sigma$.

I would now like to have a collection of injective homomorphisms $y_B : (\mathbb{F},+) \longrightarrow G^\sigma : t \mapsto y_B(t)$ that are compatible with field automorphisms and that generate $G^\sigma$, as in the classical case. Here, the field $\mathbb{F}$ is the defining field, (or possibly the subfield $\mathbb{F}_0$ of $\mathbb{F}$ that is fixed by the distinguished field automorphism $\phi$).

The obvious choice of $x_\alpha$ does not work, since the $x_\alpha$ need not be contained in $G^\sigma$. A natural, but slightly naive, approach would be to consider a $\langle \gamma \rangle$-orbit B and to then define

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(t)$ (for an orbit $B = \{ \alpha \}$ of length $1$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) \}$ of length $2$),

$y_B : \mathbb{F} \longrightarrow G^\sigma : x_{\alpha}(\phi^0(t)) \cdot x_{\gamma^1(\alpha)}(\phi^1(t)) \cdot x_{\gamma^2(\alpha)}(\phi^2(t))$ (for an orbit $B = \{ \alpha , \gamma(\alpha) , \gamma^2(\alpha) \}$ of length 3).

Question: Is there a standard text that constructs such one-parameter subgroups of $G^\sigma$ and that then proves the following properties?

• Property 1: Each $y_B$ is an injective homomorphism.

• Property 2: The group $G^\sigma$ is generated by the elements $y_B(t)$.

• Property 3: The one-parameter groups $y_B$ are compatible with field automorphisms, in the sense that if the field automorphism $A$ is induced by the automorphism $a$ of the field $\mathbb{F}$, then $A(y_B(t)) = y_B(a(t))$ for all $t \in \mathbb{F}$.

Assuming that the $y_B$ are indeed defined using the above $\langle \gamma \rangle$, Property 1 would seem to follow from the "uniqueness of expression" in the product formulae for the classical case, while Property 3 would follow immediately.

Gorenstein and Lyons make a vague reference to such a construction in [The local structure of finite groups of characteristic 2 type: Part I, Chapter I, Paragraph 4]. It also looks like E. Abe has indeed tried "something along those lines" in [Coverings of twisted Chevalley groups over commutative rings. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1977), no. 366-382, 194–218.] But the text is difficult to understand and I am still unsure about the conventions / assumptions / generality of it.

Any help would be appreciated.