Abstract
ABSTRACT. Let G be a reductive group over an algebraically closed field of characteristic p, and let u ∈ G be a unipotent element of order p. Suppose that p is a good prime for G. We show in this paper that there is a homomorphism φ : SL 2/k → G whose image contains u. This result was first obtained by D. Testerman (J. Algebra, 1995) using case considerations for each type of simple group (and using, in some cases, computer calculations with explicit representatives for the unipotent orbits).
Abstract
The proof we give is free of case considerations (except in its dependence on the Bala-Carter theorem). Our construction of φ generalizes the construction of a principal homomorphism made by J.-P. Serre in (Invent. Math. 1996); in particular, φ is obtained by reduction modulo p from a homomorphism of group schemes over a valuation ring A in a number field. This permits us to show moreover that the weight spaces of a maximal torus of φ(SL 2/k ) on Lie(G) are "the same as in characteristic 0"; the existence of a φ with this property was previously obtained, again using case considerations, by Lawther and Testerman (Memoirs AMS, 1999) and has been applied in some recent work of G. Seitz (Invent. Math. 2000).
INTRODUCTION
Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic p > 0. It is the main goal of this note to give another proof of the following theorem: Theorem 1. (Testerman [Tes95] ) Suppose that p is a good prime for G. If u ∈ G is unipotent and has order p, then there is a homomorphism φ : SL 2/k → G with u in its image.
INTRODUCTION
One might regard Theorem 1 as a group analogue of the Jacobson-Morozov theorem for Lie algebras. If one considers instead any field E of characteristic 0, a reductive group G /E over E, and u ∈ G /E an E-rational unipotent element, one may write u = exp(X) for a nilpotent X ∈ Lie(G /E ); from the Jacobson-Morozov theorem for Lie(G /E ) one deduces a homomorphism SL 2/E → G /E over E with u in its image.
INTRODUCTION
Testerman's original proof of Theorem 1 used case considerations for each type of simple group (and used, in some cases, computer calculations with explicit unipotent class representatives known from the work of Mizuno). Our proof for the most part avoids case considerations (except that it depends on Pommerening's proof of the Bala-Carter theorem in good characteristic). We will exploit a weak Jacobson-Morozov-type result for an integral form of the Lie algebra of G. We obtain φ as a suitable G-conjugate of the reduction mod p of a homomorphism of group schemes φ /A : SL 2/A → G /A , where A is a valuation ring in a number field.
INTRODUCTION
When u is regular unipotent and has order p, the theorem yields a principal homomorphism; see [Ser96, §2] . The argument we give specializes in the regular case to the proof of loc. cit. Proposition 2. To prove Theorem 1 in general, one needs to consider only distinguished u. To such a u, one can associate a distinguished parabolic subgroup P with unipotent radical U . Since u has order p, the author's recent proof of Testerman's "order formula" [McN, Theorem 1] shows that both the nilpotent group U and the nilpotent Lie algebra Lie(U ) have nilpotence class ≤ p. Since one may as well suppose that P is a standard parabolic subgroup, this permits one to deduce that the P /Q equivariant isomorphism exp : Lie(U ) /Q → U /Q is defined over Z (p) ; the construction of a suitable homomorphism over integers is then the same as in [Ser96] .
INTRODUCTION
Actually, the fact that we obtain a homomorphism of group schemes over A permits us to prove a more precise version of Theorem 1, which was first obtained by Lawther and Testerman. To explain this, we need a few remarks. Recall that the Bala-Carter classification (due in all good characteristics to Pommerening) puts both the unipotent classes of G and the nilpotent orbits of g = Lie(G) in bijection with G-conjugacy classes of pairs (L, Q) where L is a "Levi subgroup" of G and Q is a distinguished parabolic subgroup of L. The group Q has a dense orbit on its unipotent radical which is called the Richardson orbit; the G-orbit of this Richardson orbit is the unipotent class corresponding to (L, Q). Similarly, Q has a dense (Richardson) orbit on the nilradical of its Lie algebra, leading to the analogous nilpotent G-orbit. [Note that in this paper, the nilradical of a parabolic subalgebra means the Lie algebra of the unipotent radical of the corresponding parabolic subgroup.]
INTRODUCTION
If X ∈ g is in the nilpotent orbit corresponding to the pair (L, Q), then after replacing (L, Q) with a conjugate we may suppose that X is in the Richardson Qorbit on the nilradical of Lie(Q). A co-character ν of G is then said to be associated to X if ν takes values in the derived group of L and if Ad(ν(t))X = t 2 X for all t ∈ G m = k × . Now, any X has associated cocharacters, and any two co-characters associated to X are conjugate by C This show that to the G-class of a pair (L, Q), and so to the corresponding nilpotent orbit, one can associate a co-character which is defined up to G conjugacy. Since the unipotent classes also correspond to pairs (L, Q), one can associate a co-character (up to G-conjugacy) to a unipotent class as well.
INTRODUCTION
We will prove Theorem 1 by proving the following stronger result: Following the terminology of Serre [Ser96, §2.4], we may call a homomorphism φ : SL 2/k → G a sub-principal homomorphism if it satisfies the conclusion of Theorem 2 for some unipotent u ∈ G of order p.
INTRODUCTION
In the language used by Lawther and Testerman [LT99] , the theorem yields an A 1 subgroup of G whose "labeled Dynkin diagram" is the same as the labeled diagram of u. Indeed, the labeled diagram of the A 1 subgroup is obtained by choosing a maximal torus T 0 of SL 2/k and maximal torus T of G containing φ(T 0 ). The homomorphism µ = φ |T0 : T 0 ≃ G m → G is then a cocharacter. For a suitable choice of Borel subgroup B containing T (equivalently: a suitable choice of positive roots) the values α, µ at the simple roots in X * (T ) are non-negative and constitute the labels on the Dynkin diagram. One checks that these labels are independent of the choices made; see [Hum95, §7.6] . Similarly, the labels on the diagram of u are the non-negative integers α, ν where ν is a co-character associated to u (where again T and B are suitably chosen).
INTRODUCTION
Now let E be an algebraically closed field of characteristic 0, and let G /E be a reductive group over E with the same root datum as G. There is a bijection between G-conjugacy classes of pairs (L, Q) (as above) for G and G /E -conjugacy classes of pairs (L /E , Q /E ) for G /E which preserves labeled diagrams. In particular, there is a bijection between the unipotent classes; we writeû for a representative in G /E for the class corresponding to u ∈ G. The Dynkin-Kostant classification of nilpotent orbits in characteristic 0 implies: If φ /E : SL 2/E → G /E is any homomorphism withû in its image, the labeled diagram of φ /E coincides with that ofû. Thus Theorem 2 asserts that one may find a homomorphism φ : SL 2/k → G whose image contains u and for which the weights of a maximal torus of SL 2/k on Lie(G) are "the same as in characteristic 0." (We refer the reader to the extensive tables in [LT99] to see that for some u there are homomorphisms φ : SL 2/k → G whose image contains u, but whose labeled diagram differs from that of u).
INTRODUCTION
We mention that Theorem 2 was used by Seitz in [Sei00] . In loc. cit., Seitz introduced the notion of a "good A 1 ". In the language above, he calls a homomorphism φ : SL 2 (k) → G good if each weight λ ∈ Z of the representation (Ad •φ, g) for a maximal torus of SL 2 (k) satisfies λ ≤ 2p − 2, where g = Lie(G). Seitz proves that for each unipotent u of order p, there is a good homomorphism φ with u in its image; his proof of the existence of such a φ depends in a crucial way on Theorem 2 (when u is distinguished, the existence of a good φ is an immediate consequence of Theorem 2 combined with the "order formula" of Testerman which one will find in [Tes95] or [McN, Theorem 1 
INTRODUCTION
We also obtain a refinement of Theorem 2 for finite fields: if G is defined over a finite field F q of good characteristic and u is F q -rational, we show that in 5.3 φ may be defined over F q as well. In the course of our proof, we establish the following result which may be of independent interest. Let G be defined over an arbitrary field F (of good characteristic) and let u be an F -rational distinguished unipotent element. Suppose that either p is "very good" for G, or that F is perfect. Then the canonical parabolic subgroup attached to u is defined over F . (The same statement holds for F -rational distinguished nilpotent elements).
INTRODUCTION
Finally, we present in an appendix a proof that in good characteristic, there is always a G-equivariant homeomorphism between the nilpotent variety and the unipotent variety of a reductive group. Of course, in "very good" characteristic, there is an isomorphism of varieties due to Springer; however, our argument handles e.g. the group PGL p/k in characteristic p.
INTRODUCTION
The author would like to thank the Mathematics Institute at the University of Arhus for its hospitality during the academic year 2000/2001. Thanks also to Jens C. Jantzen for some suggestions on this manuscript.
GENERALITIES ON REDUCTIVE GROUPS
Let G /Z be a split reductive group scheme over Z. If G /Z is moreover semisimple, one may regard G /Z as a "Chevalley group scheme" as in [Bor70] . Let g Z be the Lie algebra. For any commutative ring Λ, we put G /Λ = G /Z × Z Λ, and
GENERALITIES ON REDUCTIVE GROUPS
Let (X, Y, R, R ∨ ) be the root datum of G /Z with respect to a fixed maximal torus T /Z . Fix a Z-basis γ 1 , . . . , γ r for Y , the co-character group of T /Z . Let H γi = dγ i (1) ∈ h Z , the Lie algebra of T /Z . The algebra g Z has a Chevalley basis {E α | α ∈ R} ∪ {H γ1 , . . . , H γr }. We have h Z = i ZH γi , and b Z = h Z ⊕ α∈R + ZE α is a Borel subalgebra of g Z .
GENERALITIES ON REDUCTIVE GROUPS
2.1. Good primes. Recall the notion of a good prime for the root system R (or for the root datum (X, Y, R, R ∨ ), it is the same). For the indecomposable root systems, a prime is bad (= not good) only in the following situations: 2 is bad unless R is of type A, 3 is bad if R is of type E, F or G, and 5 is bad if R = E 8 .
GENERALITIES ON REDUCTIVE GROUPS
For general R, p is good for R provided that it is good for each indecomposable component of R.
Parabolic subalgebras.
If S denotes the simple roots in R + , any subset I ⊂ S determines a subroot system R I in a well-known way, and hence a standard parabolic subalgebra
Parabolic subalgebras.
Consider the function f : ZR → Z which satisfies
Parabolic subalgebras.
We may regard f as a co-character of the adjoint group, so that g Z becomes a module via f for G m/Z ; as such, it is the direct sum of its weight spaces. Thus, we have g Z = i∈Z g Z (i), where
Parabolic subalgebras.
We obtain the original parabolic algebra as
Parabolic subalgebras.
There are "group scheme versions" of each of these constructions: i.e. there are parabolic subgroup schemes P (I) /Z and P (I) − /Z with respective subgroup schemes U (I) /Z and U (I)
Parabolic subalgebras.
2.3. Distinguished nilpotents and parabolics. Let F be an algebraically closed field. If X ∈ g F is nilpotent, we say that X is distinguished if the connected center of G /F is a maximal torus of
Parabolic subalgebras.
where u is the nilradical of p.
Parabolic subalgebras.
Let p Z = p(I) Z be a standard parabolic subalgebra of g Z , and let u Z = u(I) Z . For any algebraically closed field F , u F = u Z ⊗ Z F is then the nilradical of p F . If moreover the characteristic of F is good for the root system R, Thus the condition that p F = p Z ⊗ Z F is distinguished is independent of F so long as the characteristic of F is good; we say that p Z is a distinguished standard parabolic subalgebra if p F is distinguished for some (hence any) algebraically closed field F of good characteristic.
Parabolic subalgebras.
When p Z is distinguished, the map f : ZR → Z of (2.2.1) extends uniquely to a linear function X * (T /Z ) → Z and hence determines a cocharacter τ of T /Z satisfying:
Parabolic subalgebras.
For this see [Jantzen, Lemma 5.2] . Note that the argument in loc. cit. applies to semisimple G, which we may reduce to by considering the derived group of G.
2.4.
Richardson orbits and the Bala-Carter Theorem. Let again F be an algebraically closed field. Let p ⊂ g F be any parabolic subalgebra, with nilradical u. There is a unique parabolic subgroup P ≤ G /F with Lie(P ) = p. Moreover, u is the Lie algebra of the unipotent radical U of P . A theorem of R. Richardson [Hum95, Theorem 5.3] says that there is a nilpotent G-orbit O ⊂ g with the property that O ∩ u is an open P -orbit. Similarly, there is a unipotent class C in G with the property that C ∩ U is an open P -orbit. By a Richardson element, we mean an orbit representative for C or O lying in U respectively u. The orbits O and C are known as the Richardson orbits associated with p (or with P ).
2.4.
By a Levi subgroup of G, we mean a Levi factor of a parabolic subgroup. 
Suppose that the characteristic of F is good for the root system R. Associate to each distinguished parabolic subalgebra its nilpotent Richardson orbit. Then this map defines a bijection between the conjugacy classes of distinguished parabolic subalgebras and the distinguished nilpotent orbits in g.
Recall from the introduction that a cocharacter ν : G m → G is said to be associated to a nilpotent element X ∈ g provided that Ad(ν(t))X = t 2 X for all t ∈ G m , and that ν takes values in the derived group of some Levi subgroup L for which X ∈ Lie(L) is distinguished. We record: Proposition 5. Suppose the characteristic is good for G. Let X be nilpotent, and let ν be a cocharacter associated to X. Consider the parabolic subalgebra p = i≥0 g(i), and let P be the corresponding parabolic subgroup of G.
P and p depend only on X.
Proof. [Jantzen, Prop. 5 .9]
P and p depend only on X.
The subalgebra p is known as the canonical (or Jacobson-Morozov) parabolic determined by X.
P and p depend only on X.
Remark 6. If L is a Levi subgroup, and X is a distinguished nilpotent element in Lie(L). Let ι : L → G be the inclusion map. Then a cocharacter τ of L is associated to X (with respect to L) if and only if ι • τ is associated to X (with respect to G). 
P and p depend only on X.
Then dρ restricts to a representation of the sl 2 triple. The Chevalley group construction [Ste68] applied to the Lie algebra sl 2/F and the representation (dρ, V ) gives a homomorphism φ : SL 2/F → GL(V ) which maps the upper triangular subgroup to the image of t → exp(tdρX) and the lower triangular subgroup to the image of s → exp(sdρY ). Now [Bor91, Prop. 6.12] implies that φ takes values in G /F and is unique. It is clear by construction that dφ has the desired form.
THE MAIN RESULT
In this section, we suppose that k is an algebraic closure of the finite field F p . The split reductive group scheme G /Z is as in the previous section; we now suppose that G /Z is (split) semisimple. We fix an algebraic closure E = Q of the rational field; in what follows, we regard all finite extensions of Q as subextensions of E/Q. Recall that we suppose p to be good for G.
THE MAIN RESULT
Let F be a finite extension of Q, A ⊂ F a valuation ring whose residue field has characteristic p, and X ∈ g A . We write
THE MAIN RESULT
and X k for "the" analogous element in g k (note that we must choose some embedding of the residue field of A in k; the particular choice is not important for us).
THE MAIN RESULT
3.1. sl 2 -triples over integers. Lemma 8. Let p Z be a distinguished standard parabolic subalgebra of g Z , and let τ ∈ X * (T /Z ) be the corresponding co-character as in (2.3.1).
THE MAIN RESULT
1. There is a finite field extension F ⊃ Q, a valuation ring A ⊂ F (whose residue field we embed in k), and an element
THE MAIN RESULT
Proof. The assertion (1) is elementary; a proof is written down in [McN, Lemma 5.2] . Since H = dτ (1), (2) will follow provided that we find Y ∈ g A (−2) with [X, Y ] = H. Since p k is distinguished and since G /Z is semisimple, we have rankg Z (−2) = rankg Z (0). It follows from the main result of [Spa84] that the centralizer in g k of X k is contained in p. This implies that ad(X k ) :
THE MAIN RESULT
is injective, and is therefore a linear isomorphism. Thus ad(X) : 
THE MAIN RESULT
Proof. Recall that u Z is the Lie algebra of the group scheme U Z ; U Q is obtained by base change.
THE MAIN RESULT
Using the Hausdorff series, we may regard the nilpotent Lie algebra u Q as an algebraic group over Q, and the exponential map exp : u Q → U Q is an isomorphism of algebraic groups; for this see [Bou89, ch. II, §6].
THE MAIN RESULT
The p-nilpotence of the Richardson element implies, by [McN, Theorem 1] , that the nilpotence class of the Lie algebra u k is ≤ p, or equivalently, that g Z (i) = 0 for all i ≥ 2p (for the grading induced by the cocharacter τ in (2.3.1)).
THE MAIN RESULT
The lemma now follows from [McN, §8] (the case p Z = b Z was described in [Ser96, §2]).
THE MAIN RESULT
We record for later use the following improvement of the lemma.
THE MAIN RESULT
Lemma 10. Let F be an arbitrary subfield of the algebraically closed field k, and suppose that G is defined (not necessarily split) over F . Let p be a parabolic subalgebra defined over F , and suppose that a Richardson element X of its nilradical u satisfies X
THE MAIN RESULT
[p] = 0. Then the exponential isomorphism exp : u → U obtained from the previous lemma is defined over F .
THE MAIN RESULT
Proof. This follows from [Sei00, Proposition 5.2].
The main theorem. It is:
Theorem 11. Let p Z be a distinguished standard parabolic subalgebra of g Z , and let τ ∈ X * (T /Z ) be the corresponding co-character as in (2.3.1).
The main theorem. It is:
Choose the number field F , valuation ring A , and element X ∈ g A (2) as in Lemma 8(1), and let (X, Y, H) be an sl 2 -triple over A as in part (2) of that lemma. Proof. Lemma 9 implies that the exponential isomorphism exp : u Q → U Q is defined over Z (p) and hence over A . Applying this lemma to the opposite parabolic, the isomorphism exp : u − F → U − /F is defined over A as well. Consider the subgroup schemes of SL 2 = SL 2/Z :
The main theorem. It is:
The "big cell" of SL 2 is the subscheme
The main theorem. It is:
Let φ /F : SL 2/F → G /F be the homomorphism as in (1). Then the restriction of φ /F to Ω /F is (s, t, u) → exp(sY F )·τ /F (t)·exp(uX F ). Since τ is defined over A , and since we have just seen that the relevant exponential maps are defined over A , it follows that the restriction of φ /F to Ω /F is defined over A . The proof of [Ser96, Prop. 2] now implies that φ is defined over A which proves (1). (The argument of loc. cit. uses that SL 2/A is covered by Ω /A and wΩ /A , for a suitable w ∈ SL 2 (Z)).
The main theorem. It is:
One knows that the isomorphism of varieties u k → U /k is P /k -equivariant. Since X k is in the dense P /k orbit on u k , the element exp(X k ) is in the dense
The main theorem. It is:
is now immediate.
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
We return to the assumptions of the introduction; thus k is algebraically closed of characteristic p > 0 (but not necessarily an algebraic closure of a finite field), G is reductive, and the characteristic of k is good for G.
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
Since Theorem 2 is a stronger form of Theorem 1; we just prove this latter result using Theorem 11.
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
Proof of Theorem 2. First, we may replace the reductive group G by its derived group, so that we may suppose G is semisimple.
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
We next note that it suffices to prove the theorem when k is an algebraic closure of a finite field. Indeed, Let k 0 ⊂ k denote the algebraic closure of the prime field F p . Then by [McN, Cor. 7 .3] each nilpotent orbit in g k has a point rational over k 0 .
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
Since p is good, there is a G-equivariant homeomorphism ε : N → U; see Proposition 23 of the appendix. Moreover, by Remark 24 (in the appendix), ε restricts to a G-equivariant homeomorphism N /k0 → U /k0 . This shows that also each unipotent class has a point rational over k 0 . Thus u = gu ′ g −1 for some u ′ rational over k 0 , where g is over k. The theorem for k 0 and u ′ yields a suitable homomorphism φ : SL 2/k0 → G /k0 , and Int(g) • φ then works for k and u.
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
We suppose first that u is a distinguished unipotent element. Suppose that P is the distinguished parabolic subgroup with u ∈ U = U P a Richardson element. Replacing P by a G-conjugate, we may suppose that Lie(P ) = p Z ⊗ Z k where p Z is a distinguished parabolic subalgebra of g Z .
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
Since u has order p, [McN, Theorem 1] shows that a Richardson element X in Lie(U ) satisfies X
WHY THEOREM 11 IMPLIES THEOREMS 1 AND 2
[p] = 0. Theorem 11 now gives us a homomorphism φ : SL 2/k → G /k whose image meets the Richardson class determined by P and which has the same labeled diagram as u. Replacing φ by Int(g) • φ for a suitable g ∈ G, the proof is complete for distinguished u.
When u is not distinguished, it is distinguished in a proper Levi subgroup L.
We may then apply the result in the distinguished case to (the derived group of) L. Now Remark 6 shows that the resulting homomorphism has the desired properties.
RATIONALITY PROPERTIES
Let F be a ground field, suppose that G /F is a reductive group over F . Recall that the characteristic p of F is very good for G if p is good for G, and moreover if the root system of G has an irreducible component of type A r , then r ≡ −1 (mod p).
Some rational Levi and parabolic subgroups.
Theorem 12. Suppose that p is a good prime for G, and that moreover either p is very good for G, or F is perfect. Let X ∈ g(F ) be an F -rational nilpotent element.
Some rational Levi and parabolic subgroups.
1. Then X is distinguished in the Lie algebra of a Levi subgroup which is defined over F . 2. If X is distinguished in g, then the canonical parabolic subgroup attached to X is defined over F .
Some rational Levi and parabolic subgroups.
Proof. For the first assertion, let C = C G (X) be the centralizer of X. Recall that if the characteristic is very good, the orbit map G → Ad(G)X ⊂ g is separable [Jantzen, Theorem 2.5, §2.6]. With our assumption on F , we see that either F is perfect, or the orbit map is separable. According to [Spr98, Prop. 12.1.2], the group C is then defined over F . Let T ≤ C be a maximal torus which is defined over F (such a torus exists by [Spr98, Theorem 13.3.6 and Remark 13. Since T is central in L, we see that the connected center of L is a maximal torus of C L (X) hence that X is distinguished in Lie(L); this proves the first assertion of (1).
Some rational Levi and parabolic subgroups.
Let O = Ad(G)X ⊂ g, let P be the canonical parabolic subgroup associated to X, and let p = Lie(P ); see Proposition 5. We will show that P (and hence also p) is defined over F .
Some rational Levi and parabolic subgroups.
Fix an algebraic closure F of F , let F s be the separable closure of F in F , and let Γ be the Galois group of F s /F . Since X is F -rational, it is stable under the action of Γ; since P is canonically attached to X, it is also stable under the action of Γ. If P is defined over F s , (2) now follows by [Spr98, Proposition 11.2.8(i)]. This completes the proof in case F is perfect. In the case where F is not perfect, this shows that we may now suppose F to be separably closed, and we may moreover suppose that the characteristic is very good, so that the orbit map G → O is separable.
Some rational Levi and parabolic subgroups.
Let P be the G-conjugacy class of the parabolic subgroup P . Since F is separably closed, G contains a maximal torus which is defined over F and is F -split. It then follows from [Spr98, Lemma 15.4.5] that P contains an F -rational point; i.e. there is a standard parabolic subgroup P o which is conjugate via G to P . Since F is separably closed, B is split over F , and so P o is defined over F . Thus also P is defined over F , and we may identify P with G/P o as F -varieties.
Some rational Levi and parabolic subgroups.
Since the quotient map G → G/P o is defined over F and has "local sections" (see [Spr98, Lemma 8.5 .2]) we may define the fiber space We now complete the proof of Theorem 12. ConsiderX = p −1 2 (X) ∈ U. Theñ X is a simple point of Y (since Y is smooth). Since X ∈ U, (5.1.1) implies that dp 2 : TXY → T X O is an isomorphism. We now apply the condition [Spr98, Corollary 11.2.14]
Some rational Levi and parabolic subgroups.
1 to see that theX is rational over F . ThusX is represented by a pair (g, Z) where g ∈ G(F ) and Z ∈ u o (F ), so that P = gP o g −1 is rational over F .
Theorem 13.
Suppose that p is a good prime for G. Moreover, suppose that either p is very good for G, or F is perfect. Let u ∈ G(F ) be an F -rational unipotent element.
u is distinguished in a Levi subgroup which is defined over F .

If u is distinguished in G, then the canonical parabolic subgroup associated to u is defined over F .
Proof. Part (1) is proved mutatis mutandum as in part (1) of the previous theorem. For part (2), one must instead replace the vector bundle Y by the "affine-space bundle" G × Po U over P. The remainder of the argument is the same.
If u is distinguished in G, then the canonical parabolic subgroup associated to u is defined over F .
Remark 14. In the case where F = F q is a finite field of order q, we can give a different proof of part (2) of Theorem 12 which shows moreover that the grading of 2.2 is F q -rational. Indeed, let φ : G m → G be a cocharacter associated to X. Then φ determines a subgroup
If u is distinguished in G, then the canonical parabolic subgroup associated to u is defined over F .
Since X is F qrational, the variety of all such subgroups T φ (where φ ranges over all cocharacters associated to X) is stable by the action of Frobenius, and is thus rational over F q . Moreover, this variety is a homogeneous space for the connected group C o G (X) by Proposition 4. Thus an application of Lang's Theorem [DM91, Cor. 3.12] shows that there is an F q rational point T φ ′ . The rationality of T φ ′ is equivalent to that of φ ′ ; thus the weight spaces of φ ′ on g are F q -rational.
Conjugacy of nice homomorphisms.
In this section, G is a reductive group over the algebraically closed field k. Fix a distinguished parabolic subgroup P of G, and let ε : u → U be a P -equivariant homeomorphism, where U is the unipotent radical of P and u is its Lie algebra (in good characteristic, such a homeomorphism always exists; see §6 below.) Fix a Richardson element u ∈ U , and let X = ε −1 (u). We shall say that a homomorphism φ : SL 2/k → G is nice for u with respect to ε if the following property is satisfied:
Conjugacy of nice homomorphisms.
A nice homomorphism determines a cocharacter ψ associated to X by the rule 
Conjugacy of nice homomorphisms.
• φ 1 , we may suppose that ψ 1 = ψ 2 . The proposition is now a consequence of the lemma that follows. Lemma 16. Let φ i : SL 2/k → G, i = 1, 2, be nice homomorphisms for u with respect to ε, and let ψ i be the corresponding cocharacters as in (5.2.2). If
Conjugacy of nice homomorphisms.
Proof. Let Ω ⊂ SL 2/k be the big cell Ω = U − 1 T 1 U 1 as in the proof of Theorem 11.
Conjugacy of nice homomorphisms.
Since Ω is a dense subset of SL 2/k , it suffices to show that the restrictions of φ 1 and φ 2 to Ω coincide.
Conjugacy of nice homomorphisms.
For s ∈ k, one has
Conjugacy of nice homomorphisms.
since ψ 1 = ψ 2 , it follows that the restrictions of the φ i to U 1 coincide. Let g(i) the graded components of g with respect to ψ, and let H = dψ(1) ∈ g(0). Lemma 8 shows that there is a unique Y ∈ g(−2) such that (X, H, Y ) is an sl 2 triple over k; it follows that dφ i ( 0 0 1 0 ) = Y for i = 1, 2. In particular, dφ 1 = dφ 2 .
Conjugacy of nice homomorphisms.
We may find w ∈ SL 2/k with Ad(w) 0 1 0 0 = 0 0 1 0 and Int(w) 1 s 0 1 = 1 0 s 1 for all s ∈ k.
Conjugacy of nice homomorphisms.
Since dφ 1 = dφ 2 , we find that φ 1 (w) −1 φ 2 (w) ∈ C G (X) = C G (u). It then follows that φ 1 ( 1 0 1 1 ) = φ 2 ( 1 0 1 1 ). Arguing as before, one sees that the restrictions of the φ i to U − 1 coincide, and the lemma is proved.
Conjugacy of nice homomorphisms.
Remark 17. Let u ∈ G be a distinguished unipotent element of order p, and suppose that u is rational over a ground field F . We make the same hypothesis as in Theorems 12 and 13; thus p is a good prime, and moreover p is either very good for G, or F is perfect. Let U be the unipotent radical of the canonical parabolic subgroup P associated with u (recall by Theorem 13 that P and hence U are defined over F ). Then Lemma 10 yields a P -equivariant isomorphism exp : Lie(U ) → U which is defined over F . The proof of Theorem 11 shows that there is a subprincipal homomorphism φ : SL 2/k → G which is nice for u with respect to exp (note that there is no a priori reason that φ should be defined over F ).
Finite fields.
We now suppose that G is defined over F q , the finite field with q elements, of characteristic p. (Recall that finite fields are perfect.) We also suppose that p is a good prime for G.
Finite fields.
Theorem 18. Let u ∈ G be an F q -rational unipotent element of order p. Then there is a sub-principal homomorphism φ : SL 2/Fq → G /Fq defined over F q whose image contains u.
Finite fields.
Proof. By Theorem 13(1), we may suppose that u is distinguished. Let P be the canonical parabolic associated to u, and let U be its unipotent radical. By Remark 17, there is a sub-principal homomorphism φ which is nice for u with respect to exp, where exp : Lie(U ) → U is the exponential F q -isomorphism of Lemma 10. For any sub-principal homomorphism φ nice for u with respect to exp, we get a subgroup
Finite fields.
Since exp is defined over F q and u is F q -rational, applying the Frobenius to a nice homomorphism yields again a nice homomorphism; see (5.2.1). Thus the variety of all X φ is defined over F q . By Proposition 15, this variety is a homogeneous space for the connected group C o G (u); an application of Lang's Theorem [DM91, Cor. 3.12] shows that there is an F q rational point X φ ′ . Since X φ ′ is F q -rational, the homomorphism φ ′ is defined over F q , as desired.
Finite fields.
Remark 19. In particular, the theorem yields a homomorphism SL 2 (F q ) → G(F q ) between the groups of rational points.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
Let F be an algebraically closed field. In this section, we denote the reductive group G /F by G. 
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
Let σ : Lie(T ) → Lie(T ) be the map A → A [p] ; for a subspace z ⊂ Lie(T ), we write z σ = {A ∈ z | A = σ(A)}. Since Lie(T ) is an Abelian algebra, σ is additive and "semilinear": σ(αA) = α p σ(A) for α ∈ F and A ∈ Lie(T ). Thus z σ is an F p -vectorspace.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
One knows that the canonical map Lie(T ) σ ⊗ Fp F → Lie(T ) is an isomorphism (indeed: it suffices to observe that this is true when T = G m ).
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
We now claim that an F -subspace z ⊂ Lie(T ) is a p-subalgebra if and only if the canonical map z σ ⊗ Fp F → z is an isomorphism.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
This claim follows from the (apparently) more general statement: suppose that V is a finite dimensional F -vector space, that σ : V → V is a bijective, additive, semilinear map, and that To finish the proof of the lemma, note that the choice of an F p -basis for z σ identifies the map
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
and the lemma follows.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
In the situation of the lemma, we will write Θ n : z → z for the inverse of the homeomorphism A → A [p n ] , for n ≥ 1. Thus Θ n (A) is a sort of "p n -th root" of A ∈ z. Note that Θ n is not a morphism of varieties (since the morphism
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
is purely inseparable of degree p n ). We denote by U(G) = U the variety of unipotent elements in G, and by N (G) = N the variety of nilpotent elements in g. Proof. Let us recall from [Bor91, §22] that "π is a central isogeny" means that Z = ker π is finite and hence central (we will regard this kernel as a (reduced) group variety rather than as a group scheme), and that z = ker dπ is central.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
First, we note that π |Û and dπ |N are bijective; for the latter, this is proved in [Jantzen, Prop. 2.6]; the argument for π |Û is the same.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
If F has characteristic 0, then dπ :ĝ → g is an isomorphism (since the kernel of π is finite). Finally, let M = dπ −1 (N ). Since N ⊂ dπ(ĝ), ( * ) shows that dπ |M : M → N is an open map. We claim that the map Φ : z ×N → M via (Z, X) → Z + X is a homeomorphism. This map is a morphism hence continuous. We will produce an explicit inverse. Let n ≥ 1 have the property that X Proof. Suppose first that P = B is a Borel subgroup with unipotent radical U . Then the proof given in [Car93, Proof of Theorem 5.9.6 (2nd paragraph)] shows that ε induces a homeomorphism u = Lie(U ) → U , where U is the unipotent radical of B (in loc. cit. one is in the situation where ε is an isomorphism of varieties, but the argument depends only on topological properties of ε). Now suppose that P is a parabolic subgroup containing B, and that P = L · U P is a Levi decomposition. Let N (L) and U(L) denote the nilpotent and unipotent varieties of L (regarded as subvarieties of N and U). Let U − denote the unipotent radical of the Borel group opposite to B. Then we have:
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
and U P = {u ∈ U | Int(P )u ⊂ U }, Lie(U P ) = {X ∈ u | Ad(P )X ⊂ u}.
APPENDIX: COMPARING THE UNIPOTENT AND NILPOTENT VARIETIES
The required properties of ε are now immediate from equivariance.
Proposition 23.
Suppose that the characteristic of F is good for G. Then there is a Gequivariant homeomorphism ε : N → U.
Proposition 23.
Proof. Suppose first that G is simply connected and semisimple. Then the result is due to Springer; see [Hum95, Theorem 6 .20]. In fact, one gets in this case an isomorphism of varieties; see [BR85, Cor. 9.3.4]. One now deduces the result when G is the product of a torus and a simply connected semisimple group. Since there is a central isogeny from such a group onto our reductive group G [Spr98, Theorem 9.6.5], the result follows from Lemma 21.
Proposition 23.
Remark 24. Let F 0 ⊂ F be a field extension with both F 0 and F algebraically closed. Then the homeomorphism ε : N /F → U /F of the proposition may be chosen so that its restriction to F 0 points defines a homeomorphism N /F0 → U /F0 . Indeed, in the case where G is simply connected, there is an equivariant isomorphism between the two varieties defined over Z (and hence over F 0 ); see [Hum95, §6.21] . Thus the claim is true in the simply connected case. For the general statement, one observes that in the setting of Lemma 21, the homeomorphisms dπ |N /F :N /F → N /F and π |Û /F :Û /F → U /F are morphisms of varieties defined over F 0 which restrict to homeomorphisms on F 0 points.

{'paper_id': '14369610', 'title': 'Sub-principal homomorphisms in positive characteristic', 's2_url': 'https://api.semanticscholar.org/CorpusID:14369610', 'source': 'S2ORC'}