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arXiv:math/0510377v1 [math.RT] 18 Oct 2005

COMPLETELY REDUCIBLE SL(2)-HOMOMORPHISMS

GEORGE J. MCNINCH AND DONNA M. TESTERMAN

Abstract. Let K be any field, and let G be a semisimple group over K. Suppose the

characteristic of K is positive and is very good for G.

homomorphisms φ : SL2 → G whose image is geometrically G-completely reducible – or

G-cr – in the sense of Serre; the description resembles that of irreducible modules given by

Steinberg’s tensor product theorem. In case K is algebraically closed and G is simple, the

result proved here was previously obtained by Liebeck and Seitz using different methods.

A recent result shows the Lie algebra of the image of φ to be geometrically G-cr; this plays

an important role in our proof.

We describe all group scheme

1. Introduction

Let K be an arbitrary field of characteristic p > 0.

rated K-scheme of finite type. An algebraic group will mean a smooth and affine K-group

scheme; a subgroup will mean a K-subgroup scheme, and a homomorphism will mean a K-

homomorphism. A smooth group scheme G is said to be reductive if G/Kalg is reductive in

the usual sense – i.e. it has trivial unipotent radical – where Kalgis an algebraic closure of

K. The Lie algebra g = Lie(G) may be regarded as a scheme over K; we permit ourselves to

write g for the set of K-points g(K).

For G a reductive group, a subgroup H ⊂ G is said to be geometrically G-completely

reducible – or G-cr – if whenever k is an algebraically closed field containing K and H/kis

contained in a parabolic k-subgroup P of G/k, then H/k⊂ L for some Levi k-subgroup L of

P; see §2.3 for more details. The notion of G-cr was introduced by J-P. Serre; see e.g. [Ser

05] for more on this notion. It is our goal here to describe all homomorphisms φ : SL2→ G

whose image is geometrically G-cr; this we achieve under some assumptions on G which are

described in §2.4. For the purposes of this introduction, let us suppose that G is semisimple.

Then our assumption is: the characteristic of K is very good for G (again see §2.4 for the

precise definition of a very good prime).

Let F : SL2 → SL2 be the Frobenius endomorphism obtained by base change from the

Frobenius endomorphism of SL2/Fp; cf. §2.8 below. We say that a collection of homomor-

phisms φ0,φ1,...,φr: SL2→ G is commuting if

By a scheme we mean a sepa-

imφi⊂ CG(imφj)for all 0 ≤ i ?= j ≤ r.

Let?φ = (φ0,...,φr) where the φi are commuting homomorphisms SL2 → G, and let ? n =

(n0 < ··· < nr) where the ni are non-negative integers. Then the data (?φ,? n) determines

a homomorphism Φ?φ,? n: SL2 → G given for every commutative K-algebra Λ and every

g ∈ SL2(Λ) by the rule

g ?→ φ0(Fn0(g)) · φ1(Fn1(g))···φr(Fnr(g)).

Date: October 18, 2005.

Research of McNinch supported in part by the US National Science Foundation through DMS-0437482.

Research of Testerman supported in part by the Swiss National Science Foundation grant PP002-68710.

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2GEORGE J. MCNINCH AND DONNA M. TESTERMAN

We say that Φ = Φ?φ,? nis the twisted-product homomorphism determined by (?φ,? n).

A notion of optimal homomorphisms SL2→ G was introduced in [Mc 05]; see §2.7 for the

precise definition. When G is a K-form of GL(V ) or SL(V ), a homomorphism f : SL2→ G is

optimal just in case the representation (f/Ksep,V ) is restricted and semisimple, where Ksepis a

separable closure of K; see Remark 18. We will say that the list of commuting homomorphisms

?φ = (φ0,φ1,...,φr) is optimal if each φiis an optimal homomorphism.

Theorem 1. Let G be a semisimple group for which the characteristic is very good, and let

Φ : SL2→ G be a homomorphism. If the image of Φ is geometrically G-cr, then there are

commuting optimal homomorphisms?φ = (φ0,...,φr) and non-negative integers ? n = (n0 <

n1 < ··· < nd) such that Φ is the twisted-product homomorphism determined by (?φ,? n).

Moreover,?φ and ? n are uniquely determined by Φ.

We actually prove the theorem for the strongly standard reductive groups described below

in 2.4; see Theorem 39.

In case K is algebraically closed and G is simple, this theorem was obtained by Liebeck

and Seitz [LS 03, Theorem 1]; cf. Remark 17 to see that the notion of restricted – or good –

A1-subgroup used in [LS 03] is“the same”as the notion of optimal homomorphism used here.

Note that Liebeck and Seitz prove a version of Theorem 1 where SL2is replaced by any

quasisimple group H. If G is a split classical group over K in good characteristic, the more

general form of Theorem 1 found in [LS 03] is a consequence of Steinberg’s tensor product

theorem [Jan 87, Cor. II.3.17]; cf. [LS 03, Lemma 4.1]. The proof given by Liebeck and

Seitz of Theorem 1 for a quasisimple group G of exceptional type relies instead on detailed

knowledge of the subgroup structure – in particular, of the maximal subgroups – of G; see

e.g. [LS 03, Theorem 2.1, Proposition 2.2, and §4.1] for the case H = SL2. In contrast, when

p > 2, our proof uses in an essential way the complete reducibility of the Lie algebra of a G-cr

subgroup of G [Mc 05a]; cf. the proofs of Lemma 24, Proposition 25, and Lemma 29 [when

p = 2, we have essentially just used the proof of Liebeck and Seitz].

We obtain also the converse to Theorem 1, though we do so only under a restriction on

p. Write h(G) for the maximum value of the Coxeter number of a simple k-quotient of G/k,

where k is an algebraically closed field containing K.

Theorem 2. Let G be semisimple in very good characteristic, and suppose that p > 2h(G)−2,

let?φ = (φ0,...,φd) be commuting optimal homomorphisms SL2→ G, and let ? n = (n0< n1<

··· < nd) be non-negative integers. Then the image of the twisted-product homomorphism

Φ : SL2→ G determined by (?φ,? n) is geometrically G-cr.

Again, this result is proved for a more general class of reductive groups; see Theorem 43.

The assumption on p made in the last theorem is unnecessary if G is a classical group – or

a group of type G2– in good characteristic; see Remark 44. However, it is not clear to the

authors how to eliminate the prime restriction in general.

The first named author would like to acknowledge the hospitality of the Centre Interfacul-

taire Bernoulli at the´Ecole Polytechnique F´ ed´ erale de Lausanne during a visit in June 2005;

this visit permitted much of the collaboration which led to the present manuscript.

2. Preliminaries

2.1. Reduced subgroups. Let k be a perfect field – in the application we take k to be

algebraically closed. Let B be a group scheme of finite type over k.

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Lemma 3. There is a unique smooth subgroup Bred ⊂ B which has the same underlying

topological space as B. If A is any smooth group scheme over k and f : A → B is a k-

morphism, then f factors in a unique way into a k-morphism A → Bred followed by the

inclusion Bred→ B.

Proof. Use [Li 02, Prop. 2.4.2] to find the reduced k-scheme Bredwith the same underlying

topological space as B; the result just quoted then yields the uniqueness of Bred. It is clear

that Bredis a k-group scheme, and the assertion about A and f follows from loc. cit. Prop

2.4.2(d). Since k is perfect, apply [KMRT, Prop. 21.9] to see that a k-group is smooth if and

only if it is geometrically reduced if and only if it is reduced. Thus Bredis indeed smooth

.

?

We are going to consider later some group schemes which we do not a priori know to be

smooth, and we want to choose maximal tori in these group schemes. The following example

explains why in those cases we first extend scalars to an algebraically closed field (see e.g.

§3.2 below).

Example 4. If B is a group scheme over an imperfect field K, and if k is a perfect field

containing K, then a maximal torus of B/k,redneed not arise by base-change from a K-

subgroup of B. Let us give an example.

Let A = Gm⋉ Ga where Gm acts on Ga“with weight one”; i.e. K[A] = K[T,T−1,U]

where the comultiplication µ∗is given by

µ∗(T±1) = (T ⊗ T)±1

and

Suppose that K is not perfect, and let L = K(β) where βp= α ∈ K but β ?∈ K. Consider

the subgroup scheme B ⊂ A defined by the ideal I = (αTp− Up− α) ⊳ K[A].

If k is a perfect field containing K, notice that the image¯f ∈ k[B] of f = βT −U−β ∈ k[A]

satisfies¯fp= 0 but¯f ?= 0; thus B/kis not reduced. The subgroup B/k,red⊂ A/kis defined by

J = (βT −U−β), so that B/k,red≃ Gm/kis a torus. The group of k-points B/k,red(k) ⊂ A(k)

may be described as:

B/k,red(k) = {(t,βt − β) ∈ Gm(k) ⋉ Ga(k) | t ∈ k×}.

µ∗(U) = U ⊗ T + 1 ⊗ U.

Note that B/k,reddoes not arise by base change from a K-subgroup of A, e.g. since the

intersection B/k,red(k) ∩ A(K) consists only in the identity element [where the intersection

takes place in the group A(k)].

2.2. Cocharacters and parabolic subgroups. A cocharacter of an algebraic group A is a

homomorphism γ : Gm→ A. We write X∗(A) for the set of cocharacters of A.

A linear representation (ρ,V ) of A yields a linear representation (ρ◦γ,V ) of Gmwhich in

turn is determined by the morphism

(ρ ◦ γ)∗: V → K[Gm] ⊗KV = K[t,t−1] ⊗KV.

Then V is the direct sum of the weight spaces

(2.2.1)V (γ;i) = {v ∈ V | (ρ ◦ γ)∗v = ti⊗ v}

for i ∈ Z.

Consider now the reductive group G. If γ ∈ X∗(G), then

PG(γ) = P(γ) = {x ∈ G | lim

t→0γ(t)xγ(t−1) exists}

is a parabolic subgroup of G whose Lie algebra is p(γ) =?

for the notion of limit used here. Moreover, each parabolic subgroup of G has the form P(γ)

for some cocharacter γ; for all this cf. [Spr 98, 3.2.15 and 8.4.5].

i≥0g(γ;i); see e.g. [Spr 98, §3.2]

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4GEORGE J. MCNINCH AND DONNA M. TESTERMAN

We note that γ “exhibits” a Levi decomposition of P = P(γ). Indeed, P(γ) is the semi-

direct product CG(γ)·U(γ), where U(γ) = {x ∈ P | limt→0γ(t)xγ(t−1) = 1} is the unipotent

radical of P(γ), and the reductive subgroup CG(γ) = CG(γ(Gm)) is a Levi factor in P(γ); cf.

[Spr 98, 13.4.2].

2.3. Complete reducibility, Lie algebras. Let G be a reductive group, and write g for its

Lie algebra.

A smooth subgroup H ⊂ G is geometrically G-cr if whenever k is an algebraically closed

field containing K and H/k⊂ P for a parabolic k-subgroup P ⊂ G/k, then H/k⊂ L for some

Levi k-subgroup L ⊂ P.

Similarly, if h ⊂ g is a Lie subalgebra, we say that h is geometrically G-cr if whenever

k is an algebraically closed field containing K and P ⊂ G/kis a parabolic k-subgroup with

h/k= h ⊗Kk ⊂ Lie(P), then h/k⊂ Lie(L) for some Levi k-subgroup L ⊂ P.

Lemma 5. Let X and Y be schemes of finite type over K, and let f : X → Y be a (K-)

morphism. The following are equivalent:

i) f is surjective,

ii) f/k: X(k) → Y (k) is surjective for all algebraically closed fields k containing K, and

iii) f/k: X(k) → Y (k) is surjective for some algebraically closed field k containing K.

Proof. This follows from [DG70, I §3.6.10]

?

Lemma 6. Fix an algebraically closed field k containing K. Let G be a reductive group, let

J ⊂ G be a smooth subgroup, and let h ⊂ g be a Lie subalgebra. Then

(1) J is geometrically G-cr if and only if J/kis G/k-cr.

(2) h is geometrically G-cr if and only if h/kis G/k-cr.

Proof. We prove (1); the proof of (2) is essentially the same. We are going to apply the

previous Lemma.

First let P be the scheme of all parabolic subgroups of G, and let Y = PJbe the fixed

point scheme for the action of J; thus Y is the closed subscheme of those parabolic subgroups

containing J.1

Let also PL be the scheme such that for each commutative K-algebra Λ, the Λ-points

PL(Λ) are the pairs P ⊃ L where P is a parabolic of G/Λand L is a Levi subgroup of P;

cf. [SGA3, Exp. XXVI, §3.15]. Let X = (PL)Jbe the scheme of those pairs P ⊃ L where L

contains J.

There is an evident morphism PL → P given by (P ⊃ L) ?→ P; cf. [SGA3, Exp. XXVI,

§3.15]. By restriction one gets a morphism f : X → Y . Then f is surjective if and only if J

is G-cr, and (1) follows from the preceding Lemma.

?

Proposition 7. Let G be reductive, and let M ⊂ G be a Levi subgroup. Suppose that J ⊂ M

is a smooth subgroup, and that h ⊂ Lie(M) is a Lie subalgebra. Then J is geometrically

G-cr if and only if J is geometrically M-cr and h is geometrically G-cr if and only if h is

geometrically M-cr.

1For assertion (2), one should instead regard P as the scheme of parabolic subalgebras of g, which may be

regarded as a closed subscheme of a product of Grassman schemes Grd(g) for various d. Now the subscheme

X ⊂ P of parabolic subalgebras containing h is the intersection of P with the subscheme Z of the product of

Grassman schemes consisting of those subspaces containing h. Since Z is closed in the product, Y is closed in

P. Similar remarks apply to the definition of the subscheme Y ⊂ PL to be given in the next paragraph.

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Proof. For the proof, it is enough to suppose that K is algebraically closed. The proof for J

is found in [BMR 05, Theorem 3.10]. The proof for h is deduced from [Ser 05, 2.1.8]; see [Mc

05a, Lemma 2] for the argument.

?

The following theorem was proved in [Mc 05a].

Theorem 8. Let H ⊂ G be a smooth subgroup which is geometrically G-cr. Then h = Lie(H)

is geometrically G-cr.

We recall a similar result of B. Martin [Ser 05, Th´ eor` eme 3.6].

Theorem 9 (Martin). Let H ⊂ G be a smooth subgroup which is geometrically G-cr, and let

H′⊳ H be a smooth normal subgroup. Then H′is geometrically G-cr as well.

Finally, we note:

Lemma 10. Let π : G → G1 be a central isogeny where G1 is a second reductive group, let

J ⊂ G be a smooth subgroup, and let h ⊂ g = Lie(G) be a Lie subalgebra. Then

(1) J is geometrically G-cr if and only if π(J) is geometrically G1-cr, and

(2) h is geometrically G-cr if and only if dπ(h) is geometrically G1-cr.

Proof. We may and will suppose that K is algebraically closed for the proof. It is clear that

J is contained in a parabolic subgroup P of G if and only if π(J) is contained in the parabolic

subgroup π(P) of G1, and similarly h is contained in Lie(P) if and only if dπ(h) is contained

in dπ(Lie(P)) = Lie(π(P)), the result follows since P ?→ π(P) determines a bijection between

the parabolic subgroups of G and those of G1.

?

2.4. Strongly standard reductive groups. If G is geometrically quasisimple with absolute

root system R2, the characteristic p of K is said to be a bad prime for R in the following

circumstances: p = 2 is bad whenever R ?= Ar, p = 3 is bad if R = G2,F4,Er, and p = 5 is

bad if R = E8. Otherwise, p is good. [Here is a more intrinsic definition of good prime: p is

good just in case it divides no coefficient of the highest root in R].

If p is good, then p is said to be very good provided that either R is not of type Ar, or

that R = Arand r ?≡ −1 (mod p).

There is a possibly inseparable central isogeny3

r?

i=1

for some torus T and some r ≥ 1, where for 1 ≤ i ≤ r there is an isomorphism Gi≃ RLi/KHi

for a finite separable field extension Li/K and a geometrically simple, simply connected Li-

group scheme Hi; here, RLi/KHidenotes the “Weil restriction”of Hito K.

Then p is good, respectively very good, for G if and only if that is so for Hi for every

1 ≤ i ≤ r. Since the Hi are uniquely determined by G up to central isogeny, the notions

of good and very good primes depend only on the central isogeny class of the derived group

(G,G). Moreover, these notions are geometric in the sense that they depend only on the

group G/kfor an algebraically closed field k containing K.

(2.4.1)

Gi× T → G

2The absolute root system of G is the root system of G/Ksep where Ksepis a separable closure of K.

3Indeed, the center of the reductive group G is a smooth subgroup scheme; this follows e.g. from [SGA3, II

Exp. XII Th´ eor` eme 4.1] since for reductive G, the center is the same as the “centre r´ eductif”. The radical

R(G) is the maximal torus of the center of G, so R(G) is a smooth torus, and we take T = R(G) in (2.4.1).

Now, multiplication gives a central isogeny G′× R(G) → G where G′is the derived group of G. So (2.4.1)

follows from the corresponding result for semisimple groups; see e.g. [KMRT, Theorems 26.7 and 26.8] or

[TW 02, Appendix (42.2.7)].