ROOT DATA WITH GROUP ACTIONS

arXiv:1707.01935v1 [math.RT] 6 Jul 2017

JEFFREY D. ADLER AND JOSHUA M. LANSKY
Abstract. Suppose k is a field, G is a connected reductive algebraic
k-group, T is a maximal k-torus in G, and Γ is a finite group that acts on
(G, T ). From the above, one obtains a root datum Ψ on which Gal(k)×Γ
acts. Provided that Γ preserves a positive system in Ψ, not necessarily
invariant under Gal(k), we construct an inverse to this process. That is,
given a root datum on which Gal(k) × Γ acts, we show how to construct
a pair (G, T ), on which Γ acts as above.
Although the pair (G, T ) and the action of Γ are canonical only up
to an equivalence relation, we construct a particular pair for which G
is k-quasisplit and Γ fixes a Gal(k)-stable pinning of G. Using these
choices, we can define a notion of taking “Γ-fixed points” at the level
of equivalence classes, and this process is compatible with a general
“restriction” process for root data with Γ-action.

1. Introduction
Let k be a field with separable closure ksep . Let Γ be a finite group.
Suppose Ψ is a (reduced) based root datum on which the absolute Galois
group Gal(k) acts. Then it is well known ([7, Theorem 6.2.7]) that there
exists a connected, reductive, k-quasisplit k-group G, uniquely determined
up to k-isomorphism, such that the root datum of G (with respect to a
maximal k-torus contained in a Borel k-subgroup) is isomorphic to Ψ and
carries the same action of Gal(k). We generalize this result in two directions.
(A) Suppose G is a connected reductive k-group, and T is an arbitrary
maximal torus. Then the root datum Ψ(G, T ) carries an action of
Gal(k). We show that one can reverse this process. That is, given a
root datum Ψ with an action of Gal(k), one can obtain a pair (G, T )
that gives rise to Ψ. In general, the pair (G, T ) need not be uniquely
determined up to k-isomorphism. However, we can always choose G to
be k-quasisplit, and all possibilities for G must be k-inner forms of each
other.
(B) Now suppose that Γ acts on a pair (G, T ) as above via k-automorphisms.
Then Γ acts on the root datum Ψ(G, T ), and the actions of Γ and
Gal(k) commute. We show that one can reverse this process under mild
Date: February 22, 2018.
2010 Mathematics Subject Classification. Primary 20G15, 20G40. Secondary 20C33.
Key words and phrases. Reductive algebraic group, root datum, quasi-semisimple
automorphisms.
1

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JEFFREY D. ADLER AND JOSHUA M. LANSKY

conditions. That is, suppose that Ψ is a root datum with an action of
Gal(k) × Γ. Assume that Γ (but not necessarily Gal(k)) preserves a
base. Then one can obtain a pair (G, T ) as above, carrying an action
of Γ. That is, under appropriate conditions, we can lift an action of Γ
from a root datum to a reductive group. Moreover, one can choose G
to be k-quasisplit, and can choose the action of Γ to preserve a pinning.
The above are all contained in our main result, Theorem 1. In order to
state it more precisely, let us consider the collection of abstract root data Ψ
that carry an action of Gal(k) × Γ such that the action of Γ stabilizes a base
for Ψ. We consider two data Ψ and Ψ′ with such actions to be equivalent
if there is a Gal(k) × Γ-equivariant isomorphism Ψ −→ Ψ′ . Let RΓ denote
the set of equivalence classes of reduced data with such actions.
Let G be a connected reductive k-group and T ⊆ G a maximal k-torus.
Suppose there exists some Borel subgroup B ⊆ G (not necessarily defined over k) containing T , and a homomorphism ϕ from Γ to the group
Autk (G, B, T ) of k-automorphisms of G stabilizing T and B. Suppose G′ , T ′ ,
and ϕ′ are defined similarly. We say that the triples (G, T, ϕ) and (G′ , T ′ , ϕ′ )
are equivalent if there exists an isomorphism ν : G −→ G′ whose restriction
gives a Γ-equivariant k-isomorphism T −→ T ′ . (In this situation, ν must be
an inner twisting by [5, §3.2].) Let TΓ be the set of equivalence classes of
such triples (G, T, ϕ).
A triple (G, T, ϕ) as above naturally determines a root datum with appropriate actions of Gal(k) and Γ, hence an element of RΓ . It is easily seen
that if (G′ , T ′ , ϕ′ ) and (G, T, ϕ) are equivalent, then they determine the same
class in RΓ . Hence we have a natural map rΓ : TΓ −→ RΓ .
Our main result is the following:
Theorem 1. The map rΓ : TΓ −→ RΓ is a bijection.
We also prove a generalization of a second well-known result. Suppose
char k = 0, and Γ is a finite, cyclic group acting algebraically on a connected
reductive group G. Then Γ must fix a Borel-torus pair (B, T ) in G. If Γ
fixes a pinning, then the root system of the connected part Ḡ := (GΓ )◦ of
the group of fixed points is obtained as follows. The set of restrictions of
roots of G from T to T̄ := (T Γ )◦ is a root system, not necessarily reduced,
but there is a preferred way to choose a maximal reduced subsystem.
We generalize the above result in several directions.
(C) Instead of assuming char k = 0, we impose the weaker condition that Γ
fix a Borel-torus pair.
(D) The group Γ need not be cyclic.
(E) We describe the root datum, not just the root system, of Ḡ with respect
to T̄ .
The above are all contained in Theorem 2. To state this result more
precisely, suppose that the triple (G, T, ϕ) represents an element of TΓ .
Then we know [1, Proposition 3.5] that Ḡ is a connected reductive k-group,

ROOT DATA WITH GROUP ACTIONS

3

and T̄ is a maximal k-torus in Ḡ. Thus, if we let “1” represent the map
from the trivial group 1 to Aut(G), then the triple (Ḡ, T̄ , 1) represents an
element of T1 . The equivalence class of (G, T, ϕ) does not determine that
of (Ḡ, T̄ , 1), or even the ksep -isomorphism class of Ḡ. Nonetheless, we can
obtain a well-defined map TΓ −→ T1 as follows: From Remark 25, we will
see that every class in TΓ contains a triple (G, T, ϕ) such G is k-quasisplit
and ϕ fixes a Gal(k)-invariant pinning. Use this choice of triple to define Ḡ
and T̄ , and it is straightforward to show that our choices determine Ḡ and
T̄ up to k-isomorphism.
Suppose that the root datum Ψ represents an element of RΓ . We will see
in §2 that the action of Γ on Ψ allows us to construct a “restricted” root
datum Ψ̄ that has a preferred choice of reduced subdatum. We thus obtain
a map RΓ −→ R1 .
Theorem 2. Our maps TΓ −→ T1 and RΓ −→ R1 above are compatible with the maps of Theorem 1, in the sense that the following diagram
commutes:
rΓ
// RΓ
TΓ


T1

r1


// R1

We prove both theorems in §4.
2. Restrictions of root data
Let Ψ = (X ∗ , Φ, X∗ , Φ∨ ) be a root datum. (We do not assume that Φ is
reduced.) Let Γ denote a finite group of automorphisms of Ψ. We assume
that there exists a Γ-stable set ∆ of simple roots in Φ. Let V ∗ = X ∗ ⊗ Q and
V∗ = X∗ ⊗ Q. Let i∗ denote the quotient map from V ∗ to its space V̄ ∗ := VΓ∗
of Γ-coinvariants. From [1, §2], there is an embedding ι : V̄ ∗ −→ V ∗ with
image V ∗ Γ given by
1 X
ι(v̄) =
γv,
|Γ|
∗

γ∈Γ
V̄ ∗ . Let

where v is any preimage in V of v̄ ∈
X̄ ∗ and Φ̄ denote the images
∗
∗
∗
of X and Φ under i . Then X̄ is the torsion-free part of the module XΓ∗
of Γ-coinvariants of X ∗ . It is straightforward to see that X̄ ∗ and X̄∗ := X∗Γ ,
and thus V̄ ∗ and V̄∗ := V∗Γ , are in duality via the pairing given by hx̄, λ̄i :=
hιx̄, i∗ λ̄i, where i∗ : X̄∗ −→ X∗ is the inclusion map. With respect to these
pairings, i∗ is the transpose of i∗ .
For each β ∈ Φ, let wβ denote the automorphism of X ∗ defined by
wβ (x) = x − hx, β ∨ iβ.
Let W denote the Weyl group of Ψ, i.e., the (finite) subgroup of Aut(X ∗ )
generated by the wβ . Then Γ acts naturally on W , and W acts on X ∗ . The
group W Γ of Γ-fixed elements of W acts on on both V̄ ∗ and X̄ ∗ via the rule

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JEFFREY D. ADLER AND JOSHUA M. LANSKY

w(i∗ x) := i∗ (w(x)) for w ∈ W Γ and x ∈ X ∗ . Equivalently, for x̄ ∈ X̄ ∗ and
w ∈ W Γ , we have ι(w(x̄)) = w(ιx̄).
Lemma 3 (cf. [8, §1.32(a)]). The natural action of W Γ on X̄ ∗ is faithful.
Proof. Let w be a nontrivial element of W Γ . Then there exists a positive
root β ∈ Φ such that w(β) is negative. Since Γ stabilizes ∆, it follows that
w(γ ·β) = γ ·(wβ) is also negative for every γ ∈ Γ. Thus ι(w(i∗ β)) is a linear
combination of roots in ∆ in which all of the coefficients are nonpositive, so
w(i∗ β) 6= i∗ β.

Notation 4. For each root β ∈ Φ, define a Γ-orbit Ξβ in Φ as in [1, §5].
That is, let Ξβ = Γ · β if this is an orthogonal set. Otherwise, for each
θ ∈ Γ · β, there exists a unique root θ ′ 6= θ in Γ · β such that θ and θ ′ are not
orthogonal. Moreover, θ + θ ′ is a root in Φ and does not belong to Γ · β. In
this case, let Ξβ = {θ + θ ′ | θ ∈ Γ · β}.
Remark 5. Thus, in all cases, Ξβ is an orthogonal Γ-orbit of roots.
Lemma 6. If α ∈ Φ̄, then i∗−1 (α) is a Γ-orbit of roots in Φ.
Proof. This argument is similar to but more general than that given in the
proof of [6, Lemma 10.3.2(ii)]. Suppose β ∈ Φ and i∗ (β) = α. Then clearly
i∗ θ = α for any θ ∈ Γ · β.
Now suppose β ′ ∈ Φ, β ′ 6= β, and i∗ β ′ = α. Since ι(i∗ (β ′ − β)) = 0 and
since Γ preserves ∆, when β ′ − β is written as a linear combination of simple
roots, the coefficients must sum to 0. In particular, β ′ − β ∈
/ Φ. Since β ′
′
∨
cannot be multiple of β, we have that hβ , β i ≤ 0 by standard results about
root systems. Similarly, hβ ′ , θ ∨ i ≤ 0 for all θ 6= β ′ in Γ · β. Therefore,
D
E D
E
X
X
X
E D
β ′ − β,
θ ∨ = i∗ (β ′ − β),
θ ∨ = β ′ − β, i∗
θ∨ ,
θ∈Γ·β

θ∈Γ·β

i∗ (β ′

θ∈Γ·β

′ ∨
and since
− β) = 0, this pairing vanishes. Thus
θ∈Γ·β hβ , θ i =
P
∨
θ∈Γ·β hβ, θ i = 2 or 1, depending on whether or not Γ · β is orthogonal.
(This follows from the properties of root orbits discussed in [1, §5].) Since
hβ ′ , θ ∨ i ≤ 0 for all θ 6= β ′ in Γ · β, it follows that β ′ ∈ Γ · β.


P

For each α ∈ Φ̄, define
α∨ =

|Γ · β| X ∨
ξ ∈ X̄∗ ,
|Ξβ |
ξ∈Ξβ

where β is any element of Φ such that i∗ β = α. The element of X̄∗ defined by
the above formula is independent of the particular choice of β by Lemma 6.
Note that α∨ does indeed lie in X̄∗ since |Γ · β|/|Ξβ | = 1 or 2. Let Φ̄∨ =
{α∨ | α ∈ Φ̄}.
Theorem 7. With the above notation, Ψ̄ := (X̄ ∗ , Φ̄, X̄∗ , Φ̄∨ ) is a root datum.

ROOT DATA WITH GROUP ACTIONS

5

Remark 8. If Ψ comes equipped with an action of Gal(k), and the action of
Γ commutes with that of Gal(k), then it is clear that the action of Gal(k)
preserves Ψ̄.
Proof of Theorem 7. According to [5, §1.1], it suffices to show that
• X̄ ∗ and X̄∗ are in duality (which we have already observed),
• hα, α∨ i = 2 for all α ∈ Φ̄, and
• The automorphisms wα of X̄ ∗ of the form
wα (x̄) = x̄ − hx̄, α∨ iα

(for α ∈ Φ̄)

stabilize Φ̄ and generate a finite subgroup of Aut(X̄ ∗ ).
Let α ∈ Φ̄. Choose β ∈ Φ such that i∗ β = α, and choose ξ0 ∈ Ξβ . Then
we have
hα, α∨ i = hια, i∗ α∨ i
D 1
X
|Γ · β| X ∨ E
ξ
θ,
=
|Γ · β|
|Ξβ |
θ∈Γ·β
ξ∈Ξβ
E
D
X
X
1
=
ξ∨
θ,
|Ξβ |
θ∈Γ·β
ξ∈Ξβ
1 D X ′ X ∨E
=
ξ,
ξ
|Ξβ | ′
ξ ∈Ξβ

(by the definition of Ξβ )

θ∈Ξβ

hξ0 , ξ0∨ i

=
= 2,

(by Remark 5)

as desired.
Now let x̄ ∈ X̄ ∗ , and choose x ∈ X ∗ such that i∗ x = x̄. Then

(9)

hx̄, α∨ i = hx, i∗ α∨ i
D |Γ · β| X E
(by Remark 5)
ξ∨
= x,
|Ξβ |
ξ∈Ξβ

|Γ · β| X
hx, ξ ∨ i.
=
|Ξβ |
ξ∈Ξβ

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JEFFREY D. ADLER AND JOSHUA M. LANSKY

It follows that
wα (x̄) = x̄ − hx̄, α∨ iα
= i∗ x − hx̄, α∨ ii∗ β
hx̄, α∨ i ∗  X 
i
θ
= i∗ x −
|Γ · β|

(by Lemma 6)

θ∈Γ·β

hx̄, α∨ i ∗  X ′ 
ξ
i
|Γ · β|
ξ ′ ∈Ξβ
X 
1 X
= i∗ x −
hx, ξ ∨ ii∗
ξ′
|Ξβ |
ξ∈Ξβ
ξ ′ ∈Ξβ
 1 X 
X
ξ′ .
hx, ξ ∨ ii∗
= i∗ x −
|Ξβ | ′
= i∗ x −

ξ∈Ξβ

(by the definition of Ξβ )
(by (9))

ξ ∈Ξβ

But by Remark 5, for any ξ ∈ Ξβ ,
 1 X 
i∗
ξ ′ = i∗ ξ,
|Ξβ | ′
ξ ∈Ξβ

so we have
(10)



X
wα (x̄) = i∗ x −
x, ξ ∨ ξ .
ξ∈Ξβ

Also by Remark 5, the reflections wξ ∈ W for ξ ∈ Ξβ all commute with one
another. If w denotes their product, then
X
hx, ξ ∨ iξ = w(x),
x−
ξ∈Ξβ

so by (10), we have
(11)

wα (x̄) = i∗ (w(x)).

In particular, if α′ ∈ Φ̄, and β ′ ∈ Φ satisfies i∗ β ′ = α′ , then
wα (α′ ) = i∗ (w(β ′ )) ∈ i∗ (Φ) = Φ̄,
so Φ̄ is stable under the action of wα , as desired.
It remains to show that the group W̄ := hwα | α ∈ Φ̄i ⊂ Aut(X̄ ∗ ) is finite.
To accomplish this, we show that W̄ embeds naturally in the finite group
W Γ . By Lemma 3, there is a natural injection
W Γ −→ Aut(X̄ ∗ ).
To construct an embedding W̄ −→ W Γ , it is therefore enough to show that
the image of this injection contains W̄ . Thus, given w̄ ∈ W̄ , we will show
that there exists w ∈ W Γ whose action on X̄ ∗ coincides with that of w̄. It
suffices to prove the existence of w only in the caseQin which w̄ is a reflection
wα through a root α ∈ Φ̄. In this case, let w = ξ∈Ξβ wξ , where β ∈ Φ is

ROOT DATA WITH GROUP ACTIONS

7

such that i∗ β = α. It follows from Remark 5 that w ∈ W Γ , and it follows
from (11) that for any x ∈ X ∗ ,
wα (i∗ x) = i∗ (w(x)) = w(i∗ x).
This establishes the existence of the desired embedding.



Remark 12. If Φ is reduced, then so is the root system Φ̄ constructed above,
unless Φ has an irreducible factor of type A2n whose stabilizer in Γ acts
upon it nontrivially. To see this, it is easy to reduce to the case where Φ is
irreducible and Γ is cyclic (see [1, Proposition 3.5]). The result then follows
from [3, §1.3].
Remark 13. There is a way to choose a maximal reduced subsystem that
we will later see is preferred. Specficially, take only the nondivisible (resp.
nonmultipliable) roots of Φ̄ according as char k is not two (resp. two).
Lemma 14. The map i∗ induces a bijection between the set of Γ-invariant
positive systems in Φ and the set of positive systems in Φ̄.
Proof. Let Π ⊆ Φ be a Γ-invariant positive system. Let Π̄ = i∗ (Π) ⊆ Φ̄.
Then there is some vector v ∈ V∗ such that for every root β ∈ Φ, we have
that hβ, vi =
6 0, and P
hβ, vi > 0 if and only if β ∈ Π. Since Π is Γ-invariant,
we may replace v by γ∈Γ γv, and thus assume that v is Γ-invariant, and so
lies in V̄∗ . Suppose α ∈ Φ̄. Then α = i∗ β for some β ∈ Φ, so hα, vi = hβ, vi.
Thus, hα, vi =
6 0, and hα, vi > 0 if and only if α ∈ Π̄. This shows that Π̄ is
a positive system in Φ̄.
Conversely, suppose that Π̄ ⊆ Φ̄ is a positive system, and let Π = i∗ −1 Π̄.
Then there is some vector v̄ ∈ V̄∗ such that for every root α ∈ Φ̄, we have
that hα, v̄i =
6 0, and hα, v̄i > 0 if and only if α ∈ Π̄. For every root β ∈ Φ,
we have hβ, i∗ vi = hi∗ β, vi, which is never zero, and is positive if and only
if β ∈ Π. Thus, Π ⊂ Φ is a positive system. Since i∗ v is Γ-invariant, so is
Π.

Corollary 15. Let W̄ be the Weyl group of Ψ̄. Then the embedding of W̄
into W Γ in the proof of Theorem 7 is an isomorphism.
Proof. Since W̄ acts simply transitively on the set of positive systems in Φ̄,
and W Γ acts simply transitively on the set of Γ-invariant positive systems
in Φ, the result follows from Lemma 14.

3. From automorphisms of root data to automorphisms of
reductive groups
Let Ψ = (X ∗ , Φ, X∗ , Φ∨ ) be a root datum on which a group Λ acts via
automorphisms. Choosing a root basis ∆ of Φ, we obtain a corresponding
based root datum Ψ̇. Then Ψ̇ also carries an action of Λ. Namely, for σ ∈ Λ,
there exists a unique element c(σ) in the Weyl group W (Ψ) of Ψ such that
σ(∆) = c(σ)(∆). If we define σ ⋆ to be the automorphism of Ψ̇ given by
(16)

σ ⋆ χ = c(σ)−1 (σχ)

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JEFFREY D. ADLER AND JOSHUA M. LANSKY

for χ ∈ X ∗ , then the action of Λ on Ψ̇ is given by σ 7→ σ ⋆ .
Since Λ acts on Ψ and Ψ̇, it acts naturally on Aut(Ψ) and Aut(Ψ̇), as
well as on the Weyl groups W (Ψ) ⊂ Aut(Ψ) and W (Ψ̇) ⊂ Aut(Ψ̇). Just as
the actions of Λ on Ψ̇ and on Ψ differ, so the actions of Λ on W (Ψ̇) and on
W (Ψ) differ, even though the Weyl groups themselves are equal. For σ ∈ Λ
and w ∈ W (Ψ̇), let (σ, w) 7→ σ ⋆ (w) denote the action of Λ on W (Ψ̇). Then
we have
σw = c(σ) (σ ⋆ w) c(σ)−1 .
One can check readily that map c : Λ −→ W (Ψ̇) is a cocycle in Z 1 (k, W (Ψ̇)).
We now turn our attention to based root data arising from reductive
algebraic groups. Let G be a connected reductive k-group, B a Borel subgroup of G, and T ⊆ B a maximal torus of G. Let Ψ̇(G, B, T ) denote the
corresponding based root datum.
Any map ϑ ∈ Aut(G) determines an obvious isomorphism
ϑ∗ : Ψ̇(G, B, T ) −→ Ψ̇(G, ϑ(B), ϑ(T )).
There is a natural homomorphism
π : Aut(G) −→ Aut(Ψ̇(G, B, T ))
defined as follows. For ϑ ∈ Aut(G), choose gϑ ∈ G(ksep ) such that Int(gϑ )
takes ϑ(B) to B, and ϑ(T ) to T . Then Int(gϑ ) ◦ ϑ stabilizes B and T , and
we let π(ϑ) be the automorphism (Int(gϑ ) ◦ ϑ)∗ of Ψ̇(G, B, T ) (which is, in
fact, independent of the choice of gϑ ).
Now suppose that T is defined over k. Then an element σ ∈ Gal(k)
naturally determines an automorphism of Ψ(G, T ) hence an automorphism
σ ⋆ of Ψ̇(G, B, T ) as defined in (16). We thus obtain an action of Gal(k)
on Ψ̇(G, B, T ), hence one on Aut(Ψ̇(G, B, T )) as above. These actions are
independent of the particular choice of B and T in the sense that if g ∈
G(ksep ) and σ ∈ Gal(k), then we have
(17)

σ ⋆ ◦ Int(g)∗ = Int(g)∗ ◦ σ ⋆ ,

where we use the notation σ ⋆ to denote both the action of σ on Ψ̇(G, B, T )
and on Ψ̇(G, g B, g T ).
There is a well-known exact sequence
(18)

π

1 −→ Inn(G) −→ Aut(G) −→ Aut(Ψ̇(G, B, T )) −→ 1.

We note that the homomorphisms in (18) are Gal(k)-equivariant.
Remark 19. Let ∆ be the set of simple roots for (G, B, T ). Let {Xα }α∈∆ ⊂
Lie(G)(ksep ) be a pinning. It is well known [5, Cor. 2.14] that {Xα } determines a unique splitting ψ of (18). Namely, if f ∈ Aut(Ψ̇(G, B, T )), define
ψ(f ) to be the automorphism of G such that
• ψ(f ) stabilizes B and T ,
• the restriction of ψ(f ) to T is determined by the automorphism of
X ∗ (T ) given by f , and

ROOT DATA WITH GROUP ACTIONS

9

• ψ(f )(Xα ) = Xf (α) .
Thus im ψ lies in the subgroup Aut(G, B, T, {Xα }) of Aut(G) consisting of
automorphisms that stabilize B, T , and the pinning {Xα }. If B and T are
defined over k, and {Xα } is Gal(k)-stable, it follows from [2, §3.10] that ψ
is Gal(k)-equivariant.
Lemma 20. Retain the notation of the previous remark, and assume that B
and T are defined over k, and {Xα } is Gal(k)-stable. Suppose a group Γ acts
on G via k-automorphisms, preserving B, T , and {Xα }. Then Ḡ = (GΓ )◦
is a reductive k-group, B̄ = (B Γ )◦ is a Borel k-subgroup of G, T̄ = (T Γ )◦ is
a maximal k-torus in B, and W (G, T )Γ = W (Ḡ, T̄ ).
We prove this result by reducing to the well-known case where Γ is cyclic.
Proof. The statements about Ḡ and T̄ follow from [1, Proposition 3.5]. The
lemma follows for G if it holds for a central quotient of G. Therefore, we
may assume that (over ksep ) G is a direct product of almost simple groups.
We can also reduce to the case where Γ acts transitively on the factors of
G. As in the proof loc. cit., we may identify the factors of G with each
other, and replace Γ by a group S × Γ1 such that Ḡ = (GS×Γ1 )◦ , where S
acts by permuting the factors in our product decomposition of G, and Γ1
preserves each factor and acts in the same way on each. It is clear from the
construction that S × Γ1 preserves {Xα }.
Working in stages, we may assume that Γ is simple. Thus, either Γ acts
by permutation of the factors of G, or G is simple. In the former case,
our result is trivial, so assume that G is simple. Then G has a connected
Dynkin diagram, whose automorphism group is solvable. Since Γ embeds in
this automorphism group, it must be cyclic.
We let Ψ = (X ∗ (T ), Φ(G, T ), X∗ (T ), Φ∨ (G, T )) and will freely use the
notation of §2. We may identify X̄ ∗ with X ∗ (T̄ ). Under this identification,
the restriction βres of a root β ∈ Φ(G, T ) to T̄ corresponds to i∗ β. It follows
from [8, §8.2(2)] that since Γ fixes a pinning (i.e., cβ = 1 for each β ∈ ∆,
in the terminology loc. cit.), then for each β ∈ Φ(G, T ), there exists a root
α ∈ Φ(Ḡ, T̄ ) proportional to βres . Meanwhile, it follows from [1, Proposition
3.5(iv)] that every root in Φ(Ḡ, T̄ ) is the restriction of a root in Φ(G, T ).
It follows that the Weyl group W̄ of Ψ̄ is equal to W (Ḡ, T̄ ). But W̄ is
canonically isomorphic to W (G, T )Γ by Corollary 15.

4. Proofs of Theorems
Proof of Theorem 1. Consider an abstract root datum Ψ = (X ∗ , Φ, X∗ , Φ∨ )
with an action of Gal(k) × Γ. Suppose that ∆ is a Γ-stable base for Ψ. Let
Ψ̇ be the corresponding based root datum. As discussed in §3, the action of
Gal(k) × Γ on Ψ determines one of Gal(k) × Γ on Ψ̇. Since ∆ is Γ-stable, the
actions of Γ on Ψ and Ψ̇ coincide. In the notation of (16) with Λ = Gal(k),
the elements c(σ) ∈ W (Ψ̇) that arise from the action of Gal(k) on Ψ̇ must
lie in W (Ψ̇)Γ since this action commutes with that of Γ. Therefore, the

10

JEFFREY D. ADLER AND JOSHUA M. LANSKY

map c : Gal(k) −→ W (Ψ̇)Γ is a cocycle in Z 1 (k, W (Ψ̇)Γ ). We note that the
Gal(k) × Γ-isomorphism class of Ψ̇ depends only on that of Ψ.
By [7, Theorem 6.2.7], there exists a triple (G, B0 , T0 ), unique up to kisomorphism, consisting of a k-quasisplit connected reductive group G, a
Borel k-subgroup B0 of G, and a maximal k-torus T0 of B0 , such that the
associated based root datum Ψ̇(G, B0 , T0 ) is Gal(k)-isomorphic to Ψ̇. We
will identify Ψ̇ and Ψ̇(G, B0 , T0 ) via such an isomorphism.
Let {Xα } be a Gal(k)-stable pinning for G relative to B0 and T0 . The
action of Γ on Ψ̇ determines a homomorphism φ : Γ −→ Aut(Ψ̇). Let ϕ be
the composition
φ

ψ

ϕ : Γ −→ Aut(Ψ̇) = Aut(Ψ̇(G, B0 , T0 )) −→ Aut(G, B0 , T0 , {Xα }),
where ψ : Aut(Ψ̇(G, B0 , T0 )) −→ Aut(G, B0 , T0 , {Xα }) is the homomorphism
from Remark 19. 
◦
ϕ(Γ) ◦
Let Ḡ = Gϕ(Γ) and T̄0 = T0
. By Lemma 20, Ḡ is a k-quasisplit
reductive group, T̄0 a maximal k-torus of Ḡ, and
W (Ψ̇)Γ = W (G, T0 )ϕ(Γ) = W (Ḡ, T̄0 ).
Thus we may view c as a cocycle in Z 1 (k, W (Ḡ, T̄0 )).
By [4, Theorem 1.1], there is some g ∈ Ḡ(ksep ) such that for all σ ∈
Gal(k), g−1 σ(g) lies in the normalizer NḠ (T̄0 )(ksep ), and the image of g −1 σ(g)
in W (Ḡ, T̄0 ) is equal to c(σ). Let T = g T0 and B = g B0 . Since g is ϕ(Γ)fixed, T is a ϕ(Γ)-stable maximal k-torus of G, and B is a ϕ(Γ)-stable Borel
subgroup of G containing T . We have therefore associated to Ψ a triple
(G, T, ϕ) of the required type.
Suppose we vary the arbitrary choices made in the above construction of
(G, T, ϕ). That is, suppose we choose
• another based root datum Ψ̇′ with underlying datum Ψ, and hence
a cocycle c′ in Z 1 (k, W (Ψ̇′ )Γ );
• another triple of k-groups (G′ , B0′ , T0′ ) k-isomorphic to (G, B0 , T0 )
and an identification of Ψ̇(G′ , B0′ , T0′ ) with Ψ̇′ ; and
• a Gal(k)-stable pinning {Xα′ } of G′ relative to B0′ and T0′ , along with
the associated map ψ ′ : Aut(Ψ̇(G′ , B0′ , T0′ )) −→ Aut(G′ , B0′ , T0′ , {Xα′ })
from Remark 19.
We will show that these choices lead to a triple (G′ , T ′ , ϕ′ ) that is equivalent
to (G, T, ϕ). We note that replacing Ψ by another datum in its Gal(k) × Γisomorphism class has no additional effect on the triple arising from the
construction.
Use a particular k-isomorphism between (G′ , B0′ , T0′ ) and (G, B0 , T0 ) to
identify these triples. Following the above construction, we obtain a homomorphism
ϕ′ : Γ −→ Aut(G, B0 , T0 , {Xα′ }), as well as an element g ′ ∈
′
′ (Γ) ◦
ϕ
and a k-torus T ′ = g T0 , analogous to g and T , respectively.
G
There is a unique element w ∈ W (Ψ) mapping Ψ̇ to Ψ̇′ , and by uniqueness,
w must in fact lie in W (Ψ)Γ , and the mapping it induces is equivariant with

ROOT DATA WITH GROUP ACTIONS

11

respect to the actions of Gal(k) on these based root data. Via conjugation,
the element w induces Γ-equivariant isomorphisms W (Ψ̇) −→ W (Ψ̇′ ) and
τ : Z 1 (k, W (Ψ̇)) −→ Z 1 (k, W (Ψ̇′ )).
We have a unique element κ of Autk (Ψ̇(G, B0 , T0 )) that produces a commutative square
(21)

Ψ̇

// Ψ̇(G, B0 , T0 )
κ

w





Ψ̇′

// Ψ̇(G, B0 , T0 )

Here the horizontal arrows are the identifications chosen in the respective
constuctions of ϕ and ϕ′ . (That κ is Gal(k)-equivariant follows from the
equivariance of the other three maps in the square.) We therefore obtain a
diagram
(22)

♥66
♥♥♥
♥
♥
♥♥♥
♥♥♥

Γ

// Aut(Ψ̇) ∩ Aut(Ψ̇′ )
PPP
PPP
PPP
PPP
((

Aut(Ψ̇)

// Aut(Ψ̇(G, B0 , T0 ))




// Aut(Ψ̇(G, B0 , T0 ))

Aut(Ψ̇′ )

in which the square on the right is induced by (21) (and hence commutes),
the vertical maps are given respectively by conjugation by w and κ, the
diagonal maps are given by inclusion, and the map out of Γ is given by the
action of Γ on Ψ.
The map τ (c) : σ 7→ wc(σ)w−1 is a cocycle in Z 1 (k, W (Ψ̇′ )Γ ), cohomologous to c′ ; more precisely, for σ ∈ Gal(k),
c′ (σ) = w−1 (wc(σ)w−1 )σ ⋆′ (w) = w−1 (τ (c)(σ))σ ⋆′ (w),
where σ ⋆′ denotes the result of the action of σ on w, viewed as an element of
W (Ψ̇′ ). Identifying c and c′ respectively with cocycles in Z 1 (k, W (G, T0 )ϕ(Γ) )
′
and Z 1 (k, W (G, T0 )ϕ (Γ) ) as in the above construction, it follows from (22)
that
(23)

c′ (σ) = w−1 (κ ◦ c(σ) ◦ κ−1 )σ(w),

where σ(w) here denotes the result of σ ∈ Gal(k) acting on the element
w ∈ W (Ψ̇′ ) via the identification of this group with the concrete Weyl group
W (G, T0 ) in (22).
Let n ∈ NḠ (T̄0 )(ksep ) be a representative for w and set µ = ψ(κ) ∈
Autk (G, B0 , T0 ). Then by (23), g ′ −1 σ(g′ ) and n−1 µ(g−1 σ(g))σ(n) have the
same image in W (G, T0 ). Rearranging terms and letting h = g′ n−1 µ(g)−1 ,

12

JEFFREY D. ADLER AND JOSHUA M. LANSKY

we obtain that σ(h) and h have the same image modulo
(24)

σ(µ(g)n)

T0 = σ(µ(g)n T0 ) = σ(µ(g) T0 ) = σ(µ(g T0 )) = σ(µ(T )) = µ(T ).

Let ν = Int(h) ◦ µ. Since
−1

ν(T ) = Int(h)(µ(T )) = Int(g ′ n−1 µ(g)−1 )(µ(T )) = Int(g ′ n−1 )(µ(g T ))
′

= Int(g′ n−1 )(µ(T0 )) = Int(g ′ n−1 )(T0 ) = g T0 = T ′ ,
it follows from (24) that ν gives a k-isomorphism T −→ T ′ .
To show that (G′ , T ′ , ϕ′ ) is equivalent to (G, T, ϕ), it remains to show that
ν is Γ-equivariant. It follows from the construction of ϕ that π ◦ ϕ is equal
to the composition Γ −→ Aut(Ψ̇) −→ Aut(Ψ̇(G, B0 , T0 )) appearing in (22).
Similarly, π ◦ ϕ′ is equal to the analogous composition Γ −→ Aut(Ψ̇′ ) −→
Aut(Ψ̇(G, B0 , T0 )). Thus for any γ ∈ Γ,
π(ϕ′ (γ)) = κ ◦ π(ϕ(γ)) ◦ κ−1 .
Applying ψ to this equality and noting that ψ ◦ π ◦ ϕ = ϕ by construction,
we obtain
ψ(π(ϕ′ (γ))) = µ ◦ ϕ(γ) ◦ µ−1 .
Note that by definition, ψ(f ) and ψ ′ (f ) agree on T0 for any f ∈ Aut(Ψ̇(G, B0 , T0 )).
Therefore, as automorphisms of T0 , we have
ϕ′ (γ) = ψ ′ (π(ϕ′ (γ))) = ψ(π(ϕ′ (γ))) = µ ◦ ϕ(γ) ◦ µ−1 .
It follows that, as maps on T ,
ϕ′ (γ) ◦ ν
= ϕ′ (γ) ◦ Int(h) ◦ µ
= ϕ′ (γ) ◦ Int(g ′ n−1 µ(g)−1 ) ◦ µ
= Int(g ′ ) ◦ ϕ′ (γ) ◦ Int(µ(g)n)−1 ◦ µ
= Int(g ′ ) ◦ µ ◦ ϕ(γ) ◦ µ−1 ◦ Int(µ(g)n)−1 ◦ µ
= Int(g ′ ) ◦ µ ◦ ϕ(γ) ◦ Int(gµ−1 (n))−1
= Int(g ′ ) ◦ µ ◦ Int(gµ−1 (n))−1 ◦ ϕ(γ),
where the last equality above comes from the fact that g ∈ Ḡ(ksep ) and
Int(µ−1 (n)) ∈ W (G, T0 )ϕ(Γ) . Thus ϕ′ (γ) ◦ ν is equal to
Int(g ′ n−1 µ(g)−1 ) ◦ µ ◦ ϕ(γ) = ν ◦ ϕ(γ),
showing that ν is Γ-equivariant. Therefore, (G′ , T ′ , ϕ′ ) is equivalent to
(G, T, ϕ), and our construction induces a well-defined map sΓ : RΓ −→ TΓ .
We now show that rΓ ◦sΓ is the identity map on RΓ . Let Ψ be a root datum
representing some class in RΓ , and let (G, T, ϕ) be a triple representing the
image of the class of Ψ under sΓ . We need to show that Ψ(G, T ) is Gal(k)×Γisomorphic to Ψ. We will make free use of the notation developed in the
construction of sΓ .

ROOT DATA WITH GROUP ACTIONS

13

The Gal(k)-equivariant isomorphism of based root data Ψ̇ −→ Ψ̇(G, B0 , T0 )
chosen in the definition of sΓ is Γ-equivariant by construction (where the action of Γ on Ψ̇(G, B0 , T0 ) is induced by ϕ). We may therefore identify Ψ̇ and
Ψ̇(G, B0 , T0 ) as based root data with Gal(k)×Γ-action via this isomorphism.
This allows us to identify Ψ and Ψ(G, T0 ) as root data with Γ-action (but
not necessarily with Gal(k)-action since the actions of Gal(k) on Ψ̇ and Ψ
differ in general).
Recall the element g ∈ Ḡ(ksep ) chosen in the definition of sΓ . The map
Int(g)∗ : Ψ = Ψ(G, T0 ) −→ Ψ(G, T ) is Γ-equivariant since g is ϕ(Γ)-fixed.
Furthermore, Int(g)∗ is Gal(k)-equivariant since for σ ∈ Gal(k) and χ ∈
X ∗ (T0 ),

Int(g)∗ (σχ) = Int(g)∗ c(σ)(σ ⋆ χ)
= gg

−1 σ(g)

(σ ⋆ χ)

= σ(g) (σ ⋆ χ)
= σ(g χ)
= σ(Int(g)∗ (χ)).
Thus Ψ(G, T ) is Gal(k) × Γ-isomorphic to Ψ, as desired.
Finally, we show that sΓ ◦ rΓ is the identity map on TΓ . Let (G, T, ϕ)
represent a class in TΓ , and let (G′ , T ′ , ϕ′ ) represent the image of this class
under sΓ ◦ rΓ . Since rΓ ◦ (sΓ ◦ rΓ ) = (rΓ ◦ sΓ ) ◦ rΓ = rΓ , it follows that there
is a Gal(k) × Γ isomorphism Ψ(G, T ) −→ Ψ(G′ , T ′ ). By [5, Theorem 2.9],
this isomorphism is induced by an isomorphism ν : G −→ G′ that restricts
to a Γ-equivariant k-isomorphism T −→ T ′ . Thus (G, T, ϕ) and (G′ , T ′ , ϕ′ )
are equivalent.

Remark 25. Observe that in the definition of the map sΓ above, the triple
(G, T, ϕ) is constructed in such a way that G is k-quasisplit and ϕ fixes a
Gal(k)-invariant pinning of G. Thus, since sΓ ◦ πΓ is the identity map on
TΓ , we see that every equivalence class in TΓ contains such a triple.
Moreover, suppose that (G, T, ϕ) is a triple of this kind. Applying the
construction of sΓ ◦ rΓ to this triple, we see that the triple we obtain is
precisely (G, T, ϕ), provided that we make appropriate choices.
Remark 26. Recall that in the proof, it is shown that if (G, T, ϕ) and
(G′ , T ′ , ϕ′ ) are two triples that arise by applying the sΓ construction to
a root datum Ψ, then (G, T, ϕ) and (G′ , T ′ , ϕ′ ) are equivalent. We note that
the equivalence ν constructed in this case is of a special kind. Namely, ν is
of the form Int(h) ◦ µ, where h ∈ G(ksep ) and µ ∈ Autk (G, B0 , T0 ).
Now suppose that (G, T, ϕ) and (G′ , T ′ , ϕ′ ) are arbitrary equivalent triples
with the properties that G and G′ are k-quasisplit and ϕ and ϕ′ fix Gal(k)invariant pinnings for G and G′ , respectively. Then combining the first part
of this remark with Remark 25, it follows that there is an equivalence ν
between (G, T, ϕ) and (G′ , T ′ , ϕ′ ) of the above special form.

14

JEFFREY D. ADLER AND JOSHUA M. LANSKY

Remark 27. Suppose that G′ is k-quasisplit and T ′ is a maximal k-torus
of G′ . Suppose that the finite group Γ acts via Gal(k)-equivariant automorphisms on Ψ(G′ , T ′ ) preserving a base. Then the equivalence class of
Ψ(G′ , T ′ ) lies in RΓ . Applying the construction in the definition of sΓ
to Ψ(G′ , T ′ ), we obtain a triple (G, T, ϕ) where G is k-quasisplit. Since
Ψ(G′ , T ′ ) and Ψ(G, T ) are Gal(k)-isomorphic, G′ can be taken to equal G.
Moreover, if g ∈ G(ksep ) is chosen such that T ′ = g T0 , then the cocylcle c used to define T can be taken to be the image of σ 7→ g−1 σ(g) in
Z 1 (k, W (G, T0 )Γ ). In particular, it follows from [1, Proposition 6.1] that T ′
is stably conjugate to T .
Proof of Theorem 2. Consider a class in TΓ . From Remark 25, we can represent this class by a triple (G, T, ϕ), where G is k-quasisplit and the action ϕ of Γ on G fixes a Gal(k)-invariant pinning. Let Ḡ = (Gϕ(Γ) )◦ and
T̄ = (T ϕ(Γ) )◦ . Then Ψ = Ψ(G, T ) comes equipped with an action of Γ.
Consider the restricted root datum Ψ̄ of Theorem 7. Construct a new root
datum Ψ̄′ by replacing the root system Φ̄ of Ψ̄ by a maximal reduced subsystem Φ̄′ as in Remark 13, and do likewise with the coroot system. It is clear
from the constructions that the root data Ψ̄′ and Ψ(Ḡ, T̄ ) are equivalent,
provided that their root systems are equivalent.
As in the proof of Lemma 20, one may reduce to the case where Γ is
cyclic. From the proof of [8, §8.2(2′′′′ )], the root system Φ̄′ must contain the
root system Φ(Ḡ, T̄ ), with equality since Γ fixes a pinning.


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ROOT DATA WITH GROUP ACTIONS

15

Department of Mathematics and Statistics, American University, Washington, DC 20016-8050
E-mail address, Adler: jadler@american.edu
E-mail address, Lansky: lansky@american.edu

