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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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COMPLETELY REDUCIBLE SL(2)-HOMOMORPHISMS3
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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18GEORGE J. MCNINCH AND DONNA M. TESTERMAN
For general K, the above argument represents the base-changed morphism Φ/Ksep as the
twisted product homomorphism Φ?φ,? nfor unique commuting optimal Ksep-homomorphisms
?φ = (φ0,...,φt) and unique ? n = (n0 < n1 < ··· < nt). Since?φ and ? n are unique, we
may apply Galois descent to see that each φi arises by base change from an optimal K-
homomorphism, and the proof is complete.
?
5. Proof of a partial converse to the main theorem
In this section, we will prove Theorem 2, which is a geometric statement – it depends only
on G and H over an algebraically closed field. Thus we will suppose in this section that K is
algebraically closed, and we write “G-cr”rather than “geometrically G-cr”.
We begin with a result on G-cr subgroups.
Proposition 40. Let G be reductive, let h(G) be the maximum Coxeter number of a simple
quotient of G, and suppose that p > 2h(G)−2. Let A,B ⊂ G be smooth, connected, and G-cr,
and suppose that B ⊂ CG(A). Then A · B is G-cr.
Proof. Under our assumptions on p, it follows from [Ser 05, Corollaire 5.5] that a subgroup
Γ ⊂ G is G-cr if and only if the representation of Γ on Lie(G) is semisimple.
Since a smooth, connected G-cr subgroup is reductive [Ser 05, Prop. 4.1], the proposition
is now a consequence of the lemma which follows.
?
Lemma 41. Let G1,G2⊂ GL(V ) be connected and reductive, and suppose G2⊂ CGL(V )(G1).
Then V is semisimple for G1· G2.
Proof. Write H = G1·G2. Since H is a quotient of the reductive group G1×G2by a central
subgroup, H is reductive.
Since G1and G2 commute, G2 leaves stable the G1-isotypic components of V . Thus we
may write V as a direct sum of H-submodules which are isotypic for both G1and G2. Thus
we may as well assume that V itself is isotypic for G1and for G2.
Let Bi⊂ Gibe Borel subgroups and let Ti⊂ Bibe maximal tori for i = 1,2. Note that the
choice of a Borel subgroup determines a system of positive roots in each X∗(Ti); the weights
of Ti on Ui= Ru(Bi) are positive. Our hypothesis means that there are dominant weights
λi∈ X∗(Ti) such that each simple Gi-submodule of V is isomorphic to LGi(λi), the simple
Gi-module with highest weight λi.
Now, B = B1·B2is a Borel subgroup of H, and T = T1·T2is a maximal torus of B. Since
T1∩ T2lies in the center of H, one knows that there is a unique character λ ∈ X∗(T) such
that λ|Ti= λifor i = 1,2. Moreover, it is clear that λ is dominant. Put U = U1·U2= Ru(B).
It follows from [Jan 87, II.2.12(1)] that there are no non-trivial self-extensions of simple
H-modules; thus the Lemma will follow if we show that all simple H-submodules of V are
isomorphic to LH(λ).
Let L ⊂ V be a simple H-submodule; we claim that L ≃ LH(λ). Since L is simple, the
fixed point space of U on L satisfies dimKLU= 1 and our claim will follow once we show
that LU⊂ LT;λsince then LU= LT;λand L ≃ LH(λ); for all this, see [Jan 87, Prop. II.2.4]
[we are writing LT;λ for the λ weight space of the torus T on L]. Since L is semisimple
and G1-isotypic, LU1= LT1;λ1. Since G2⊂ CGL(V )(G1), LU1is a G2-submodule. Since LU1
is semisimple and isotypic as G2-module, we know that LU= (LU1)U2= (LU1)T2;λ2. Thus
LU⊂ LT1;λ1∩ LT2;λ2so indeed LU= LT;λas required.
?
Suppose that H is a simple group, and that for each strongly standard group G, one has
a set CGof homomorphisms H → G satisfying (C1)–(C5) of §4.1.
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COMPLETELY REDUCIBLE SL(2)-HOMOMORPHISMS 19
Theorem 42. Let G be strongly standard and assume that p > 2h(G) − 2. Let h0,...,hrbe
commuting CG-homomorphisms, and let n0< n1< ··· < nr be non-negative integers. Then
the image of the twisted-product homomorphism h determined by (?h,? n) is geometrically G-cr.
Proof. Write Sifor the image of hi, 0 ≤ i ≤ r. By (C1), Siis G-cr for 0 ≤ i ≤ r. In view of
our assumption on p, it follows from Proposition 40 that the subgroup A = S0· S1···Sr is
G-cr.
Write X for the building of G. If S = imh, Corollary 33 shows that XS= XA. Since A is
G-cr, XA= XSis not contractible, so that S is G-cr [Ser 05, Th´ eor` eme 2.1].
?
Theorem 43. Let G be a strongly standard reductive group, 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.
Proof. As in the proof of Theorem 1, write CGfor the set of optimal homomorphisms SL2→ G
for a strongly standard group G. Note that the condition p > 2h(G) − 2 implies that p > 2.
Then Theorem 2 is a consequence of Proposition 30 and Theorem 42.
?
Of course, Theorem 2 is a special case of the previous result.
Remark 44. Let G be one of the following groups: (i) GL(V ), (ii) the symplectic group Sp(V ),
(iii) the orthogonal group SO(V ), or (iv) a group of type G2. In cases (ii), (iii) assume p > 2
while in case (iv) assume that p > 3; then p is very good for G. In case (iv), write V for
the 7 dimensional irreducible module for G; thus in each case V is the “natural” module for
G. Then a closed subgroup H ⊂ G is G-cr if and only if V is semisimple as an H-module;
see [Ser 05, 3.2.2]. Thus, the conclusion of Theorem 2 holds for G (with no further prime
restrictions). Indeed, in view of Lemma 41, one finds that the conclusion of Proposition 40
is valid with no further assumption on p by using V rather than the adjoint representation
of G. Now argue as in the proof of Theorem 42 when p > 2, or just use Steinberg’s tensor
product theorem when p = 2 (since we are supposing G = GL(V ) in that case).
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[Mc 05]
[Mc 05a]
[Sei 00]
[Ser 05]
Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, MA 02155, USA
E-mail address: george.mcninch@tufts.edu
Institut de g´ eom´ etrie, alg` ebre et topologie, Bˆ atiment BCH,´Ecole Polytechnique F´ ed´ erale de
Lausanne, CH-1015 Lausanne, Switzerland
E-mail address: donna.testerman@epfl.ch