On the Smoothness of Centralizers in Reductive Groups

Let G be a connected reductive algebraic group over an algebraically closed field k. In a recent paper, Bate, Martin, R\"ohrle and Tange show that every (smooth) subgroup of G is separable provided that the characteristic of k is very good for G. Here separability of a subgroup means that its scheme-theoretic centralizer in G is smooth. Serre suggested extending this result to arbitrary, possibly non-smooth, subgroup schemes of G. The aim of this note is to prove this more general result. Moreover, we provide a condition on the characteristic of k that is necessary and sufficient for the smoothness of all centralizers in G. We finally relate this condition to other standard hypotheses on connected reductive groups.


Introduction
Let G be a connected reductive algebraic group over an algebraically closed field k. A closed subgroup H ⊆ G is called separable in G if the Lie algebra of its centralizer coincides with the infinitesimal fixed points of H in Lie (G). Here, in contrast to the later sections of the paper, we do not consider scheme-theoretic centralizers. A similar notion exists for subalgebras of Lie (G). See [BMR05] and [BMRT] for these concepts and their importance in the context of Serre's notion of G-complete reducibility. The starting point of this note is the following result of Bate Theorem. Suppose that char k is very good for G. Then any subgroup of G is separable in G and any subalgebra of Lie (G) is separable in Lie (G).
Separability of a subgroup H of G can also be characterized as the smoothness of the schemetheoretic centralizer of H in G, see Section 3.1. In particular, all subgroups are separable if the characteristic of the ground field is zero (all algebraic group schemes in characteristic zero are smooth, due to a theorem of Cartier). The above theorem is therefore an instance of the general principle that a characteristic zero result becomes true in positive characteristic, provided that the characteristic is "big enough". More precisely, the assertion of the theorem then can be interpreted in the following way: If H is an arbitrary closed subgroup scheme of G and if char k is very good for G, then the scheme-theoretic centralizer of H in G is smooth provided that either H is smooth or H is infinitesimal of height one ([BMRT, Rem. 3.5(vi)], see Lemma 3.1 below).
We are going to extend this result to arbitrary closed subgroup schemes, and to the weaker condition that the characteristic is only "pretty good for G" (Definition 2.11); see Theorem 3.3. We are also going to exhibit examples of non-smooth centralizers in small characteristic. In [BMRT,Ex. 3.11], the authors construct non-separable subgroups of simple groups of type G 2 and F 4 in characteristic two. We observe that non-separable subgroups may be found in any reductive group in non pretty good characteristic; see Example 4.2.
Combining these results, the following is our main theorem: Theorem 1.1. Let G be a connected reductive algebraic group. Then the characteristic of k is zero or pretty good for G if and only if all centralizers of closed subgroup schemes in G are smooth.
Finally, we show that the universal smoothness of centralizers holds in many classes of "standard" reductive groups, as introduced by Jantzen and McNinch-Testerman; see Corollary 5.3. In fact, the different notions of standardness almost coincide (Theorem 5.2). This demonstrates that requiring the characteristic of the ground field to be pretty good for G is a natural assumption for connected reductive groups.
The paper is organized as follows. In Section 2, we recall general facts about group schemes. We also recall the notions of good and very good primes. In Section 2.10, we introduce the notion of a pretty good prime.
In Section 3, we investigate conditions for the smoothness of centralizers. First, in Section 3.1, we relate the above notion of separability to the smoothness of certain scheme-theoretic centralizers. In Section 3.2, it is our goal to prove the forward implication of Theorem 1.1. In fact, we prove a slightly more general result in Theorem 3.3.
In Section 4, we provide methods to construct non-smooth centralizers outside of pretty good characteristic. This allows us to give a proof of Theorem 1.1 at the end of this section.
Finally, in Section 5, we recall other notions of standard reductive groups, and prove that these amount to essentially the same as requiring the group to be defined in pretty good characteristic.

Preliminaries
In this section, let k be a field and k a fixed algebraic closure of k. Let S be a commutative ring.
2.1. Basic definitions. We call a functor from the category of S-algebras to the category of sets (resp. groups) an S-functor (resp. S-group functor ). We adopt the point of view that (group) schemes over S are certain S-(group) functors. See [DG70] and [Ja03] for this approach. In particular, morphisms of (group) schemes are just morphisms of S-(group) functors. If X is an S-(group)-functor and if r : R → R ′ is a morphism of S-algebras, we write x r for the image of an element x ∈ X(R) under the induced map X(R) → X(R ′ ). Since we are mainly interested in affine objects, let us recall the basic definitions: An affine S-(group) scheme is an S-(group) functor X of the form X = Hom(A, ), where A is an S-(Hopf) algebra (and where Hom denotes homomorphisms of S-algebras). We also write S[X] instead of A. We call X of finite type or algebraic if S[X] is a finitely generated S-algebra. If R is an S-algebra, every affine S-(group) scheme X gives rise to an affine R-(group) scheme X R , with R[X R ] = S[X] ⊗ S R. If M is an S-module, we define the S-group functor M a via M a (R) = M ⊗ S R, see [DG70, II, §1, 2.1]. If M is projective and finite, then M a is an affine S-group scheme represented by the symmetric algebra S(M * ).

2.2.
Notions for group morphisms. Let ϕ : G → G ′ be a morphism of affine algebraic k-group schemes. It is called surjective if the comorphism ϕ * : k[G ′ ] → k [G] is injective (which usually does not imply surjectivity of all induced maps G(R) → G ′ (R)). Surjectivity can be characterized by the following property (see [Wat79,15.5]): For every k-algebra R and every g ′ ∈ G ′ (R) there is a faithfully flat extension s : R → R and an element g ∈ G(R) such that ϕ(R)(g) = g ′ s . Let H ⊆ G and H ′ ⊆ G ′ be closed subgroup schemes. The functor ϕ −1 H ′ : R → ϕ(R) −1 (H ′ (R)) is a closed subgroup scheme of G, defined by the ideal generated by ϕ * [KMRT,VI,22.4]). The kernel of ϕ is the closed subgroup scheme ker ϕ := ϕ −1 {e}. Let ϕH be the closed subgroup scheme of G ′ defined by the ideal (ϕ * ) −1 (I), if I ⊆ k [G] defines H. Note that ϕ(R)(H(R)) ⊆ (ϕH)(R) for all k-algebras R. The induced map H → ϕH then is surjective in the above sense, since its comorphism is injective. In particular, where the forward containment follows from the above characterization of surjectivity; the reverse inclusion follows from the observation that x s (ϕ * ) −1 (I) = 0 implies x(ϕ * ) −1 (I) = 0.
A sequence of morphisms of affine algebraic group schemes 1 → N ϕ − → G ψ − → H → 1 is called exact provided that ψ is surjective and ϕ induces an isomorphism of N with ker(ψ). A morphism π : G → G ′ of affine algebraic k-group schemes is called an isogeny if it is surjective and if its kernel is a finite group scheme.
2.3. Fixed points and centralizers. If G is a k-group functor and X is a k-functor, there is an obvious notion of a natural action α : G × X → X. Suppose such an action is given. Then we can define the fixed point functor X G . It associates to each k-algebra R the set We also define the centralizer of a subfunctor Y of X in G as the functor Cent G (Y ) that associates to each k-algebra R the group It is well known that if G, X and Y are k-schemes and if X is separated, then X G is closed in X, and Cent G (Y ) is closed in G (see [Ja03,I,2.6]).
We are interested in two special cases of the above situation. If H is a closed subgroup scheme of an affine algebraic k-group scheme G, it acts on G by conjugation. We denote the corresponding fixed point functor by C G (H). It is the same as the centralizer Cent G (H), where G acts on itself via conjugation. We call C G (H) the centralizer of H in G. We have the equality If M ⊆ N are two k-vector spaces, we can consider M a as a subfunctor of N a . If G acts linearly on N a (i.e. R-linearly on each N ⊗R), then N is called a G-module. In this case, we write Cent G (M ) for the centralizer Cent G (M a ).
and we can endow T x X with the S-module structure induced by the bijection with the space of all such point derivations. If ϕ : X → Y is a morphism of affine schemes that sends x ∈ X(S) to y ∈ Y (S), then it induces an S-linear map dϕ If G is an affine algebraic S-group scheme, we write Lie(G) = g = T e (G), where e ∈ G(S) is the identity element. Suppose that char(S) = p. Then the S-module g is finite and has the structure of a p-Lie algebra over S, see [DG70, II, §7, 3.4], and we call it the Lie algebra of G. This construction has the following converse (see Ibid., II, §7, 3.5): If l is a finite projective S-module with a p-Lie algebra structure over S, then there is an affine algebraic S-group scheme G(l) with Lie algebra isomorphic to l such that taking differentials yields a bijection Hom(G(l), G) ∼ = Hom(l, g), where on the right hand side we consider homomorphisms of p-Lie algebras over S. Moreover, G commutes with changing the base ring. If S = k is a field, G induces an equivalence between the category of finite-dimensional p-Lie algebras and the category of algebraic k-group schemes of height one; then for sub-p-Lie algebras l ⊆ Lie(G) (where G is an affine algebraic k-group scheme) the group G(l) can be identified with a closed subgroup scheme of G (Ibid., II, §7, 4).
If G is an affine algebraic k-group scheme and if S is a k-algebra, then Lie(G) ⊗ S ∼ = Lie(G S ) (see Ibid., II, §4, 4.8). In the same way we can prove part (a) of the following lemma. The other parts then follow from the definitions and an argument similar to Ibid., II, §4, 2.5. Lemma 2.3. Let X be an affine algebraic k-scheme. Let x ∈ X(k). Let G be an affine algebraic k-group scheme.
(a) The k-functors R → T x R X R and (T x X) a are isomorphic. Here x R denotes the image of x in X(R). (b) If G acts naturally on X and if x ∈ X G (k), then G acts linearly on T x X and there is an If G is an affine algebraic k-group scheme and if g ∈ G(R), then conjugation with g induces an automorphism Int(g) : G R → G R , sending h ∈ G(R ′ ) to g r hg −1 r for each morphism r : R → R ′ of k-algebras. This in turn induces an automorphism Ad(g) = d Int(g) : Lie(G R ) → Lie(G R ). Using the above isomorphism of k-functors Lie(G ? ) ∼ = g a , we see that G acts linearly on g a via Ad. Now suppose that H is a closed subgroup scheme of G, and that i : H → G is the inclusion. With the notation of Section 2.3 we find that 2.5. The identity component. Let G be an affine algebraic k-group scheme. There is a finite group scheme π 0 G which is represented by the largest separable subalgebra of k [G], see [Wat79, Thm. 6.7]. Let G • denote the kernel of the corresponding morphism G → π 0 G. Then G • is a connected closed normal subgroup scheme of G, called the identity component of G.
Now let G be an affine algebraic S-group scheme. For a prime ideal p ⊆ S, let κ(p) denote the residue field of the local ring S p . Then we can construct the identity component of the affine ) for all prime ideals p ⊆ S}, and we call G • the identity component of G. This construction commutes with base change. In particular, for an affine algebraic k-group scheme G and a k-algebra R we have In fact, our definition coincides with the one in [SGA3, VI B, 3.1]. If ϕ : G → G ′ is a morphism of affine algebraic S-group schemes, then it induces a morphism of S-group functors G • → (G ′ ) • .

2.6.
Smoothness and algebraic groups. Let X be an affine algebraic k-scheme and consider a point x ∈ X(k). Let dim x X be the dimension of the local ring k[X] x obtained by localizing k[X] at the kernel of x. We say that x ∈ X(k) is a regular point if dim x X = dim k T x X. This is equivalent to the regularity of k[X] x . Using the dimension formula for flat morphisms (see [DG70, I, §3, 6.3]), we get the following lemma. Lemma 2.8. Let ϕ : X → Y be a flat morphism of affine algebraic k-schemes. Suppose that x ∈ X(k) and y = ϕ(x) ∈ Y (k) are regular points. Suppose that x is also regular in the fibre ).
An affine algebraic k-scheme X is called smooth provided that X k is regular, i.e. all prime ideals in k[X] ⊗ k induce regular local rings. For affine algebraic k-group schemes, this property can be characterized in the following way (see [KMRT,21.8,21.9], and [DG70, II, §5, 2.1]): Proposition 2.9. An affine algebraic k-group scheme G is smooth if and only if any of the following statements hold: By a theorem of Cartier (see [DG70, II, §6, 1.1]), every affine algebraic k-group scheme G is smooth if the characteristic of k is zero. We reserve the term algebraic group over k for a smooth affine algebraic k-group scheme. There is an equivalence of categories between the category of algebraic groups over k and the category of linear algebraic groups over k that are equipped with a k-structure (see [Bo91] and [Sp98] for this language). This equivalence can be realized as the functor G → G(k), if we regard G(k) as an affine variety with coordinate ring k[G] ⊗ k; it gives a correspondence between the smooth closed algebraic subgroups of G and the closed subgroups of G(k) that are defined over k. Moreover, the Lie algebras g ⊗ k and the Lie algebra associated to G(k) coincide.
If 1 → N → G → H → 1 is an exact sequence of affine algebraic k-group schemes, we have that dim G = dim N + dim H and that G is smooth if H and N are smooth (see [KMRT,).
An isogeny G → G ′ is called separable if its kernel is smooth. Two affine algebraic k-group schemes are called separably isogenous if there exists a separable isogeny between them. 2.7. Reductive groups. We say that an affine algebraic k-group scheme G is a reductive algebraic group over k, if it is smooth and if G(k) corresponds to a reductive linear algebraic group defined over k via the equivalence given in Section 2.6. Equivalently, G k has no non-trivial closed smooth connected normal unipotent subgroup schemes.
If G is a reductive algebraic group, we denote by Rad(G) its radical (i.e. the unique maximal torus of the center of G), and by Der(G) the derived group of G (see [DG70,II,§5,4.8]).
An affine algebraic k-group scheme G is called linearly reductive provided that all G-modules are semisimple. This is equivalent to the vanishing of , where X and Y are free abelian groups of finite rank that are in duality relative to a pairing , : X × Y → Z, and where Φ ⊆ X and Φ ∨ ⊆ Y are finite sets. The following additional axioms are required: This means that the root systems Φ ⊆ RΦ and Φ ∨ ⊆ RΦ ∨ are reduced. A root datum is called semisimple provided that the ranks of X and ZΦ coincide. SupposeR = (X,Φ,Ỹ ,Φ ∨ ) is another root datum, f : X →X is a Z-linear map, and f ∨ is the induced mapỸ → Y . Then f is called an isogeny of root data, and denoted f : R →R, if f is injective with finite cokernel, f maps Φ ontoΦ and f ∨ mapsΦ ∨ onto Φ ∨ . See [SGA3,XXI] for more generalities on root data.
The significance of these notions is given by the following: Consider the category of pairs (G, T ), where G is a connected reductive group over k and T is a maximal torus in G, and where morphisms are central isogenies respecting the torus. Considering characters, roots, cocharacters and coroots relative to T , one associates a root datum R(G, T ) to (G, T ). In this way one obtains a contravariant functor to the category of reduced root data with isogenies, which is almost an equivalence of categories (the functor is not faithful, but two central isogenies giving the same isogenies of root data differ only by conjugation with a torus element; to fix this one may add more data involving the choice of positive roots, see [SGA3,XXIII,Thm. 4.1]). The reductive group G is semisimple if and only if its root datum is semisimple. Separable isogenies between connected reductive groups are automatically central and correspond to isogenies f of root data such that coker(f ) has order invertible in k.
with the obvious pairing and bijection gives rise to the dual group G ∨ .
2.9. Good and very good primes. Let Φ be a (reduced) root system with irreducible components Φ 1 , . . . , Φ t . Let ∆ = ∆ 1 ∪ · · · ∪ ∆ t be a base of Φ, where each ∆ i is a base of Φ i . For each i, write the highest rootα i of Φ i as a linear combination of simple roots. If a prime number p divides any coefficient occurring among the expressions of theα i , it is said to be bad (for the root system Φ). Otherwise it is called good. By inspecting the coefficients of the highest roots of irreducible root systems, one can verify that a prime p is bad if and only if it occurs among the coefficients of the highest roots, if and only if it is smaller than the largest coefficient. A prime number p is called very good for Φ, if p is good and if p does not divide n + 1 for any irreducible component of type Now let G be a reductive algebraic group over k and let Φ be the root system associated to G k . Then we call a prime number p (very) good for G provided that p is (very) good for the root system Φ.
2.10. Pretty good primes for root data. Let R = (X, Φ, Y, Φ ∨ ) be a (reduced) root datum. Motivated by Lemma 2.10, we introduce the following terminology: Definition 2.11. A prime number p is called pretty good for the root datum R if the groups X/ZΦ ′ and Y /ZΦ ′∨ have no p-torsion, for all subsets Φ ′ ⊆ Φ. If G is a connected reductive algebraic group over k, we say that a prime p is pretty good for G provided that it is pretty good for the root datum associated to G k . This notion is related to the notions of good and very good primes for root systems as follows: Lemma 2.12. Let p be a prime number and R = (X, Φ, Y, Φ ∨ ) as above. Then the following hold: (a) p is pretty good for R if and only if p is good for Φ and the groups X/ZΦ and Y /ZΦ ∨ have no p-torsion; (b) p is very good for Φ ⇒ p is pretty good for R ⇒ p is good for Φ; (c) if R is semisimple, then p is pretty good for R if and only if it is very good for Φ; (d) let f : R →R = (X,Φ,Ỹ ,Φ ∨ ) be an isogeny of root data; if coker(f : X →X) has no p-torsion, then p is pretty good for R if and only if it is pretty good forR; (e) if R = R 1 ⊕ R 2 , then p is pretty good for R if and only if it is so for both R 1 and R 2 .
Proof. (a). For a subset Φ ′ ⊆ Φ we consider the exact sequence If p is pretty good, then the p-torsion of the middle group in this sequence vanishes, so p is good by Lemma 2.10 (a). This proves the forward implication of (a). For the reverse implication, the above exact sequence and its obvious dual version imply that the p-torsion of X/ZΦ ′ (resp. Y /ZΦ ′∨ ) coincide with the p-torsion of ZΦ/ZΦ ′ (resp. ZΦ ∨ /ZΦ ′∨ ). Since p is good for Φ and Φ ∨ , the assertion follows again from Lemma 2.10 (a).
(b). According to (a) it remains to show: if p is very good for Φ, then X/ZΦ and Y /ZΦ ∨ have no p-torsion. For the first group, note that (X ∩ QΦ)/ZΦ has no p-torsion according to Lemma 2.10 (b), since it is a subgroup of Λ/ZΦ. But then the same is true for X/ZΦ, since the quotient X/(X ∩ QΦ) has no torsion at all. Since p is also very good for Φ ∨ , a similar argument proves the assertion for Y /ZΦ ∨ .
(c). Suppose that R is a semisimple root datum. According to (b) it remains to show that a pretty good prime p for R does not divide the order of the fundamental group Λ/ZΦ. In the semisimple setting, the character lattice is a sublattice of the weight lattice, and the inclusion X → Λ is dual to the inclusion ZΦ ∨ → Y . So both Λ/X and X/ZΦ have no p-torsion by assumption, and the assertion of (c) follows. (d).
The condition on f guarantees that the p-torsion of X/ZΦ ′ coincides with the p-torsion of X/Zf (Φ ′ ), and similarly for f ∨ , whose cokernel satisfies the same condition as the one of f .
A similar decomposition holds for Y /ZΦ ′∨ , which finishes the proof.
Example 2.13. The notion of a pretty good prime differs from the notions of good and very good primes. For instance, every prime is pretty good for GL 2 , whereas p = 2 is not very good for GL 2 . On the other hand, all primes are good for SL 2 and PSL 2 , whereas p = 2 is not pretty good for SL 2 or PSL 2 .

Smooth centralizers
Let k be a field and k an algebraic closure of k. Note that all constructions (preimages, centralizers, kernels, etc.) will be taken in the scheme-theoretic sense of Section 2.
3.1. Relation to separability. Suppose for the moment that G is an algebraic group over the algebraically closed field k of positive characteristic with Lie algebra g. This is the situation considered in [BMRT]. There, a smooth closed subgroup H ⊆ G is called separable in G if dim C G(k) (H(k)) = dim k g H(k) , where we consider the centralizer to be taken in the category of linear algebraic groups. Similarly, a Lie subalgebra h ⊆ g is called separable in g if dim Cent G(k) (h) = dim k c g (h). Again we consider here the centralizer as a linear algebraic group, whereas c g (h) is the Lie algebra centralizer of h in g.
The next lemma explains the assertion of [BMRT, Rem. 3.5(vi)], that relates the notion of separability to the smoothness of certain centralizers of subgroup schemes. We use the notation and equivalence of categories of Section 2.6.
Lemma 3.1. Let G be an algebraic group over k with Lie algebra g. Let H be a smooth subgroup scheme of G and let h ⊆ g be a subalgebra. Then the following hold:

(i) The centralizer C G (H) is smooth if and only if H(k) is separable in G(k).
(ii) Let h ′ be the p-envelope of h in g, i.e. the smallest p-subalgebra of g that contains h. Let H ′ = G(h ′ ) be the affine group scheme associated to h ′ as in Section 2.4. Then C G (H ′ ) is smooth if and only if h is separable in g.
Proof. (i). Let i : H → G be the inclusion. The smoothness of H implies that for g ∈ G(k), we have Int(g) = i : H → G if and only if Int(g)(k) = i(k) (see [Liu02, Ex. 2.9 in 3.2]). Therefore, according to (2.4), we get the identity H(k)).
This implies that the dimensions of the scheme C G (H) and the linear algebraic group C G(k) (H(k)) coincide (because the defining ideal of the linear algebraic group now proves to be the radical of the ideal defining C G (H)). By (2.6), we also have that Lie(C G (H)) = g H = g H(k) (for the last equation we note that both subsets of g can be defined in terms of the same comodule map). We conclude by Proposition 2.9 that C G (H) is smooth if and only if dim C G (H) = dim k g H(k) , if and only if dim C G(k) (H(k)) = dim k g H(k) . The last equality is equivalent to the separability of H(k) in G(k).
(ii). As in part (i) we first get the identity Cent G (h)(k) = Cent G(k) (h). We also have the identity Lie(Cent G (h)) = c g (h) and can therefore conclude in the same manner as in part (i) that h is separable in g if and only if Cent G (h) is smooth. Since the actions of G(k) and g on g are compatible with the p-mapping, we get that h is separable in g if and only h ′ is separable in g (in fact, c g (h) = c g (h ′ ) and Cent G(k) 2)). So the desired equality follows from the identities (2.4) and (2.5).

Sufficient conditions for the smoothness of centralizers.
A sufficient condition for the smoothness of fixed points is given by the following theorem (see [DG70, II, §5, 2.8]): Theorem 3.2. Let G and H be affine algebraic k-group schemes. Suppose that G is smooth and that H acts via group automorphisms on G. If H 1 (H, g) = 0 for the corresponding H-module structure on g, then G H is a smooth affine algebraic group scheme. In particular, G H is smooth if H is linearly reductive.
Our goal is to prove the following theorem: Theorem 3.3. Let G be a reductive algebraic group over k. Suppose that the characteristic of k is zero or pretty good for G • . Suppose further that the component group π 0 G(k) has order invertible in k. Then for any closed subgroup scheme H ⊆ G, the centralizer C G (H) is smooth. In particular, all centralizers of closed subgroup schemes are smooth for a connected reductive group G in pretty good characteristic.
We defer the proof of the theorem to the end of this section. It relies on a series of lemmas which generalize ideas of [BMRT] to the scheme-theoretic setup. The main steps of the proof then follow closely the proof of [BMRT, Thm. 1.2], plus a careful type A analysis. for each k-algebra R. Then H 1 (H, gl 2 ) is one-dimensional as a k-vector space (which can be checked by computing the cohomology of the cyclic group of order 3 with the corresponding action on k 4 ). So the cohomology criterion does not apply for GL 2 , whereas all primes are pretty good for GL 2 .
(c). As pointed out to me by Steve Donkin, there is the following more generic way to produce examples of non-vanishing cohomology on some gl n as above. Take any affine algebraic group scheme G that is not linearly reductive. Then there exist G-modules M 1 , M 2 such that Ext 1 If V also happens to be a faithful G-module (this can be arranged for example for any simple adjoint algebraic group G in positive characteristic), then we can identify G with its image in GL(V ). This yields again an example where the theorem implies the smoothness of the centralizer C GL(V ) (G), whereas the cohomology criterion does not apply.
(d). The following example ([BMRT, Rem. 3.5]) shows that the assertion of the theorem may fail for non-connected reductive groups without the hypothesis on the component group. Take The following three lemmas are used in the proof of Theorem 3.3. The key idea is to transfer the desired property from GL n , where all centralizers are smooth, according to the following first lemma.
Lemma 3.5. Let H be a closed subgroup scheme of GL n . Then the centralizer C GL n (H) is smooth.
Proof. According to (2.1) and Proposition 2.9, we can assume that k = k is algebraically closed. We have the identity (gl H n ) a = ((gl n ) a ) H (see [DG70, II, §2, 1.6]). In particular, ((gl n ) a ) H can be represented by a reduced k-algebra (see Section 2.1). Let R be a k-algebra and let us identify gl n ⊗ R = M n×n (R), where the right hand side denotes n × n-matrices with coefficients in R. We can write So C GLn (H) can be represented by the localization A det , if A represents ((gl n ) a ) H . Since A is reduced, the same is true for A det . Hence C GLn (H) is smooth (Proposition 2.9).
The next lemma shows how the smoothness of centralizers descends from one reductive group to the other, provided that the inclusion induces a reductive pair (see Section 2).
Lemma 3.6. Let (G ′ , G) be a reductive pair of reductive algebraic groups over k and let H ⊆ G be a closed subgroup scheme. If C G ′ (H) is smooth, then so is C G (H).
Proof. We proceed in 3 steps. Let g ′ = g ⊕ m be an H-module decomposition of g ′ .
1. It suffices to show that c : Indeed, by assumption (and using (2.6)) we know that dim C G ′ (H) = dim k g ′H = dim k g H + dim k m H . So the desired inequality dim C G (H) ≥ dim k g H (see Proposition 2.9) is equivalent to c ≤ dim k m H .
2. We show that c ≤ dim k T e ((G ′ /G) H ). Consider the quotient X := C G ′ (H)/C G (H). It is a smooth algebraic scheme of dimension dim X = dim e X = c (see [DG70, III, §3 2.7, 5.4 and 5.5(a)]), which can be formally obtained as the faisceau associated to the functor see [Ja03,I,5.6]. Now if we start with the functor F 2 : R → G ′ (R)/G(R) we get as associated faisceau the smooth algebraic scheme G ′ /G (which is in fact affine, due to the reductivity assumptions, see [Ri77]). Moreover, we have an inclusion of functors F 2 ֒→ G ′ /G ([Ja03, I, 5.6]). There is an obvious action of H on F 2 by conjugation, which extends uniquely to an action of H on G ′ /G (see [DG70, III, §3, 1.3]). We now have F 1 ֒→ F H 2 ֒→ (G ′ /G) H , where the first inclusion follows from the fact that C G ′ (H) ∩ G = C G (H). Therefore, we have a monomorphism X ֒→ (G ′ /G) H (see [Ja03,I,5.4 (4)]). Note that (G ′ /G) H is closed in G ′ /G (Ibid., I, 2.6 (10)), hence also affine. We 3. We show that T e (G ′ /G) H ∼ = m H as k-vector spaces. Let us first note that the natural map π : G ′ → G ′ /G is H-equivariant with respect to the action defined in step 2 (because π is the composite G ′ → F 2 → G ′ /G). It is also flat with smooth fibre π −1 (e) = G (see [ Remark 3.7. Let H ⊆ G be a closed subgroup scheme of a smooth affine algebraic k-group scheme and suppose that H 1 (H, g) = 0. Then C G (H) is smooth, according to Theorem 3.2. We can also use the above two lemmas to obtain a new proof for this fact: Pick a closed embedding G ֒→ GL n for a suitable n (see [Wat79,3.4]). By Lemma 3.5, C GL n (H) is smooth. If H 1 (H, g) = 0, then dim k gl H n = dim k g H + dim k (gl n /g) H . Now the 3 steps of the proof of Lemma 3.6 go through with G ′ = GL n and m replaced by gl n /g (we also need to extend our tangent space arguments to the possibly non-affine variety G ′ /G).
The following final lemma states that the smoothness of a centralizer of a subgroup scheme depends on the ambient group only up to central, separable isogenies.
Lemma 3.8. Let π : G → G ′ be a central, separable isogeny of smooth affine algebraic k-group schemes, and suppose that H ⊆ G is a closed subgroup scheme. Then C G (H) is smooth if and only if C G ′ (πH) is smooth. In particular, all centralizers in G are smooth if and only if the same is true for G ′ .
Proof. We let H act on G ′ via h.x = π(h)xπ(h) −1 for h ∈ H(R), x ∈ G ′ (R). In this way π : G → G ′ becomes an H-equivariant morphism. Since ker π is smooth and finite, the differential dπ : g → g ′ is injective. Moreover, according to our smoothness assumptions and Lemmas 2.3 (c) and 2.8, it is also H-equivariant and surjective. Thus it induces an isomorphism g H ∼ = g ′H .
We claim that (G ′ ) H = C G ′ (πH). The reverse inclusion follows from the definitions. Conversely, suppose that x ∈ (G ′ ) H (R). Let r : R → R ′ be a k-algebra homomorphism and let h ′ ∈ πH(R ′ ). We have to show that h ′ x r h ′−1 = x r . There is a faithfully flat extension s : R ′ → S and an element h ∈ H(S) such that h ′ s = π(h). By our choice of x we get that x s•r = h.x s•r = h ′ s x s•r h ′−1 s . Now the injectivity of s implies that x r = h ′ x r h ′−1 , as required.
We now have established that Lie(C G (H)) = g H ∼ = g ′H = Lie(C G ′ (πH)). To prove the assertion, it therefore remains to show that dim C G (H) = dim C G ′ (πH). Let M = π −1 C G ′ (πH) ⊆ G. The map π induces an exact sequence It follows that dim M = dim C G ′ (πH). By construction, we have Let m ∈ M • (R) and h ∈ H(R ′ ) for some morphism of k-algebras r : R → R ′ . We need to show that hm r h −1 m −1 r = e R ′ . Consider the natural map of R ′ -functors We first note that ϕ is well-defined. Indeed, if x ∈ M • (R ′′ ), then π(x) commutes with all elements of πH(R ′′ ). In particular, it commutes with π(h s ). But then h s xh −1 s x −1 has to lie in ker π(R ′′ ). In fact, ϕ is even a morphism of affine group schemes over R ′ , because we can compute that , where for the last equation we use that ker(π) ⊆ Z(G). Taking identity components, we get an induced homomorphism ϕ • : where we use that ker π is smooth and finite. So in fact our map ϕ factors over M • R ′ → {e R ′ }. This implies that ϕ(m) = e R ′ , as required. For the statement about all centralizers we just note that ππ −1 H ′ = H ′ for all closed subgroup schemes H ′ ⊆ G ′ .
We are now in a position to prove Theorem 3.3.
Proof of Theorem 3.3. According to (2.1) and Proposition 2.9, we can assume that k = k is algebraically closed. Let H be any closed subgroup scheme of G. Using the following idea of [MT09,2.8.1], we can first reduce to the case that G is connected: Let H act via conjugation on G. Since G • is normal in G, this induces an action of H on G • and (G • ) H is a normal subgroup of C G (H). This allows us to consider the exact sequence of affine group schemes It now remains to show that the left and right terms of this sequence are smooth (see Section 2.2). For the right hand side, note that the closed embedding C G (H) → G together with the fact that acts trivially, so the action factors through Γ and we get the asserted equality. Now, as a closed subgroup scheme of π 0 G, Γ is again a smooth finite group scheme with order invertible in k, hence linearly reductive. We can use Theorem 3.2 to conclude that the fixed points are again a smooth group scheme, provided that C G • (H ∩ G • ) is smooth. This allows us to assume from the outset that G is connected and, by hypothesis, that the characteristic of k is zero or pretty good for G.
If the characteristic is zero, all closed subgroup schemes are smooth. So let us assume that p = char(k) is a pretty good prime for G.
Let G 1 , . . . , G t be the simple components of the derived group of G. We may assume that p is not very good for the root systems of G 1 , . . . , G n , whereas it is for G n+1 , . . . , G t . Since p is good for G and hence for all components of the derived group, this implies that all groups G 1 , . . . , G n are of type A. Let T be the radical of G (which is a torus). Then multiplication in G gives a central isogeny T × G 1 × · · · × G t → G. Denote byG the image of T × G 1 × · · · × G n in G. Let π :G × G n+1 × · · · × G t → G be the induced central isogeny. Due to our characteristic assumption, the Lie algebras Lie(G i ), i > n, are simple (see [Ho82,Cor. 2.7]). Using this, one can show that Lie(ker π) = ker(dπ) = 0, i.e. that π is separable. So, by Lemma 3.8, we may assume that G =G × G n+1 × · · · × G t . Now let H ⊆ G be a closed subgroup scheme. Then we find that where H i = p i H for the projection p i : G → G i , and similarly forH. We claim that the centralizers C G i (H i ), i > n, are smooth. In fact, since every central isogeny between simple algebraic groups with the same root system in very good characteristic is separable, we need to prove the assertion only for one algebraic simple group with a given root system in very good characteristic. According to [Ri67], for a given root system and an algebraically closed field in very good characteristic, we can find a simple algebraic group G ′ i of this type that fits into a reductive pair (GL m , G ′ i ) for a suitable m. We may conclude that the C G i (H i ), i > n, are smooth, by employing Lemmas 3.5, 3.6 and 3.8.
It remains to show that centralizers inG are smooth. By Lemma 2.12 (d) and (e), p is pretty good forG. By construction ofG, and by replacing G byG, we may assume that G is a connected reductive group with root system Φ = A m 1 ×· · ·×A mn , where p divides all integers m 1 +1, . . . , m n +1. We also assume that the radical of G has at least dimension n (else we replace G by G × S, for some torus S, and exploit the fact that C G×S (H × 1) = C G (H) × S).
Let us consider now the reductive group G C over C defined by the root datum (X, Φ, Y, Φ ∨ ) of G. We identify G C and all other (smooth) complex algebraic groups below with their groups of complex points. The prime p now is just a fixed prime that is pretty good for the root datum of G C ; all we need is that it does not divide the order of the finite groups Z(G C )/ Rad(G C ) and Z(G ∨ C )/ Rad(G ∨ C ), which follows from the fact that the abelian groups X/ZΦ and Y /ZΦ ∨ have no p-torsion.
Let r denote the dimension of the radical of G C . By assumption, r ≥ n. To ease notation we set SL = SL m 1 +1 × · · ·×SL mn+1 , PSL = PSL m 1 +1 × · · ·×PSL mn+1 and GL = GL m 1 +1 × · · ·×GL mn+1 , considered as complex algebraic groups. We now fix some data that determines G C : it is determined by its subgroups Der(G C ), Rad(G C ) and the central isogeny Der ( The assertion on p follows. Let N p denote the (unique) p-Sylow subgroup of N , and similarly Z p ⊆ Z. The last paragraph implies that the composite i : N p → N → N/M is injective. It also implies that N p = Z p . Since Z ∼ = µ m 1 +1 × · · · × µ mn+1 , this gives N p ∼ = µ p s 1 × · · · × µ p sn , where m i + 1 = p s i m ′ i with m ′ i and p coprime. In particular, the inclusion N p → Z corresponds to the standard inclusion µ p s 1 × · · · × µ p sn → µ m 1 +1 × · · · × µ mn+1 . We embed Z into G n m via the componentwise inclusion of µ m 1 +1 × · · · × µ mn+1 into G n m . The injective homomorphism ϕ • i : N p → G r m yields a Z-linear surjection of character groups Z r → Z/p s 1 × · · · × Z/p sn . Choose a matrix A ∈ Z n×r such that the above surjection is induced by Z r A − → Z n → Z/p s 1 × · · · × Z/p sn in the obvious way. Since this concatenation is surjective, and since all s i are positive by assumption, the reduction of A mod p is a surjective linear map F r p → F n p of F p -vector spaces. In particular, the elementary divisors α 1 , . . . , α n of A over Z must all be prime to p. Up to change of basis on both sides over Z, A decomposes as a linear map as Z r → Z n → Z n , where the first map is the projection onto the first n components, and the second map sends (x 1 , . . . , x n ) to (α 1 x 1 , . . . , α n x n ). Switching back to the category of multiplicative algebraic groups, we construct the following commutative diagram: (3.9) Here, ϕ ′ is the map induced by A, the isomorphisms correspond to the changes of bases, j is the inclusion into the first n coordinates of G r m , and π is a surjection with finite kernel of order prime to p (this is due to the analysis of the elementary divisors above). In particular, the order of ker(ψ ′ ) ∼ = ker(π) is prime to p.
To conclude, we express the diagram of connected reductive groups and central isogenies in root data terms. This gives rise to connected reductive groups over k and separable isogenies Remark 5.4. George McNinch suggested to give an easier direct description of essentially standard groups, starting from the easiest examples, and operations like (i) and (ii) above. We can do this as follows. Consider the class of connected reductive groups with the following properties: • it contains all simple groups defined in very good characteristic; • it contains GL n for all n; • it contains all tori; • it is closed under taking products and operations (i), (ii) as above. By Lemma 2.12, all groups in this class are defined in pretty good characteristic (for GL n we use part (a) of the Lemma and the fact that in this case X(T )/ZΦ ∼ = Z ∼ = Y (T )/ZΦ ∨ ). The converse follows from the proof of the implication (a) ⇒ (c) in Theorem 5.2 above.
By the characterization in terms of the smoothness of all centralizers, the class of groups defined in pretty good characteristic is closed under taking subgroups such that we obtain a reductive pair. In particular, it is closed under taking centralizers of diagonalizable subgroup schemes (cf. the proof of (d) ⇒ (a) in Theorem 5.2).
The last remark gives a very easy recipe to prove results for groups defined in pretty good characteristic. We illustrate this with the existence of Springer isomorphisms.
Corollary 5.5. Let G be a connected reductive group defined in pretty good characteristic. Then there is a G-equivariant isomorphism of varieties U → N , where U ⊆ G (resp. N ⊆ g) is the unipotent (resp. nilpotent) variety associated to G.
Proof. The result holds for GL n and for tori. In [McN05,Prop. 9], McNinch shows how to extend Springer's work to T -standard groups, by first extending the result to semisimple groups defined in very good characteristic. He also shows that the existence of Springer isomorphisms is compatible with operation (i). The compatibility with operation (ii) and with taking direct products is obvious.