Orbit Closures and Invariants

Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let Y denote the set of fixed points of H in X, and N(H) the normalizer of H in G. In this paper we study the natural map from the quotient of Y by N(H) to the quotient of X by G induced by the inclusion of Y in X. We show that, given G and H, this map is a finite morphism for all G-varieties X if and only if H is G-completely reducible (in the sense defined by J-P. Serre); this was proved in characteristic zero by Luna in the 1970s. We discuss some applications and give a criterion for the map of quotients to be an isomorphism. We show how to extend some other results in Luna's paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic g in G if and only if the intersection of generic conjugates of H and K is reductive.


Introduction
The purpose of this paper is to establish some results in geometric invariant theory over fields of positive characteristic, where tools from characteristic 0-such as Luna's Étale Slice Theorem-are not available. In particular, we prove the following theorem and give some applications (see Sect. 2 for precise definitions of terms). Let k be an algebraically closed field of characteristic p ≥ 0.

Theorem 1.1 Suppose G is a reductive linear algebraic group over k and H is a reductive subgroup of G. Then the following are equivalent:
(i) H is G-completely reducible; (ii) N G (H ) is reductive and, for every affine G-variety X , the natural map of quotients ψ X ,H : X H /N G (H ) → X /G is a finite morphism (here X H denotes the H -fixed points in X ).
The study of closed orbits is central in geometric invariant theory-the closed orbits for G in X parametrise the points of the quotient variety X /G. An important piece of the proof of Theorem 1.1 is Proposition 4.1, which gives a connection between the closed G-orbits in X and the closed H -orbits in X ;cf. [2,33,48]and [7], for example. Theorem 1.1 reduces to the main result in Luna's paper [33]whenk has characteristic 0, because condition (i) and the first hypothesis of (ii) are automatic in characteristic 0 if H is already assumed to be reductive. Luna's methods use the powerful machinery of étale slices, based on his celebrated "Étale Slice Theorem" [32]; see Sect. 3.1 below for more on étale slices. Many useful consequences flow from the existence of an étale slice (see Proposition 3.1 below, for example). Although étale slices sometimes exist in positive characteristic [1], in general they do not. Our methods differ from Luna's in that they apply equally well in all characteristics. These methods also allow us to provide extensions to positive characteristic of other results from [33] (see Proposition 3.10, Remark 4.2(i) and Proposition 4.7).
The work in this paper fits into a broad continuing programme of taking results about algebraic groups and related structures from characteristic zero and proving analogues in positive characteristic. A basic problem with this process is that results-such as the existence of an étale slice-that are true when p = 0 may simply fail when p > 0(cf.Examples3.2, 8.1, 8.2 and 8.3); a further illustration of this in the context of this paper is that a reductive group may fail to be linearly reductive (recall that a linear algebraic group is called reductive if it has trivial unipotent radical, and linearly reductive if all its rational representations are semisimple). When p = 0, a connected group is linearly reductive if and only if it is reductive, whereas if p > 0 a connected group is linearly reductive if and only if it is a torus [42]. Even if a result remains true in positive characteristic, it may be much harder to prove, an example here being the problem of showing that the ring of invariants R G is finitely generated, where R is a finitely-generated k-algebra and G ⊆ Aut(R) is reductive. This was resolved in characteristic 0 in the 1950s, but not in positive characteristic until the 1970s (see the introduction to Haboush's paper [22]).
In some contexts in positive characteristic where the hypothesis of reductivity is too weak and linear reductivity is too strong, it has been found that a third notion, that of G-complete reducibility, provides a good balance (cf. [37,Cor. 1.5]) and our main theorem is another example of this phenomenon. See Sect. 2.4 for the definition. The idea is that when p = 0 there is no distinction between demanding that a subgroup H of a reductive group G is reductive or linearly reductive or G-completely reducible, but there is a huge difference in positive characteristic. The notion of complete reducibility was introduced by Serre [52]and Richardson 1 [49], and over the past twenty years or so has found many applications in the theory of algebraic groups, their subgroup structure and representation theory, geometric invariant theory, and the theory of buildings: for examples, see [2,5,9,30,35,36,39,[56][57][58].
The paper is set out as follows. Section 2 contains preparatory material from geometric invariant theory and the theory of complete reducibility. The proof of Theorem 1.1 contains three main ingredients, each dealt with in a separate section. In Sect. 3 we build on work of Bardsley and Richardson to establish the important technical result Proposition 3.10,which gives a criterion for a map of quotient varieties to be finite. In Sect. 4 we carry out our analysis of the closed G-a n dH -orbits and show that ψ X ,H is quasi-finite if H is G-completely reducible (Theorem 4.4). In Sect. 5 we show that the image of ψ X ,H is closed (Theorem 5.1).
The key idea here is to consider the map of projectivisations P(X H ) → P(X ) induced by the inclusion of X H in X when X is a G-module; the G-complete reducibility of H guarantees that we get a well-defined map of quotient varieties P(X H )/G → P(X )/G. Section 6 draws these strands together and completes the proof of Theorem 1.1 using Proposition 3.8 (a variation on Zariski's Main Theorem). Section 7 gives a criterion for ψ X ,H to be an isomorphism onto its image (Theorem 7.2). In Sect. 8 we use representation theory to construct some examples relevant to Theorem 1.1. In Sect. 9 we give a criterion (Theorem 9.1) for generic double cosets HgK of G to be closed, where H and K are reductive subgroups of G. Luna proved a stronger result [31]in characteristic 0 using étale slice methods, but our techniques work when étale slices are not available. We give some applications of Theorem 9.1 (Examples 9.11 and 9.12); these serve as applications of Theorem 1.1 as well. We finish in Sect. 10 by using the theory we have developed to prove some results on complete reducibility.

Notation
Our basic references for the theory of linear algebraic groups are the books [10]a n d [ 54]. Unless otherwise stated, we work over a fixed algebraically closed field k with no restriction on the characteristic. By a variety we mean a quasi-projective variety over k, and we identify a variety X with its set of k-points. For a linear algebraic group G over k,weletG 0 denote the connected component of G containing the identity element 1 and R u (G) G 0 denote the unipotent radical of G.WesaythatG is reductive if R u (G) ={1}; note that we do not require a reductive group to be connected. When we discuss subgroups of G, we really mean closed subgroups; for two such subgroups H and K of G,wesetHK := {hk | h ∈ H , k ∈ K }.W e denote the centralizer of a subgroup H of G by C G (H ), and the normalizer by N G (H ). All group actions are left actions unless otherwise indicated.
We make repeated use of the following result [35,Lemma 6.8]: if G is reductive and if H is a reductive subgroup of G then N G (H ) 0 = H 0 C G (H ) 0 .
Given a linear algebraic group G,letY (G) denote the set of cocharacters of G,wherea cocharacter is a homomorphism of algebraic groups λ : k * → G. Note that since the image of a cocharacter is connected, we have Y (G) = Y (G 0 ). A linear algebraic group G acts on its set of cocharacters: for g ∈ G, λ ∈ Y (G) and a ∈ k * ,weset(g · λ)(a) = gλ(a)g −1 .
G i v e na na f fi n ev a r i e t yX over k, we denote the coordinate ring of X by k[X ] and the functionfieldofX (when X is irreducible) by k(X ).Givenx ∈ X ,weletT x (X ) denote the tangent space to X at x. Recall that for a linear algebraic group G, T 1 (G) has the structure of a Lie algebra, which we also denote by Lie(G) or g. Given a morphism φ : X → Y of affine varieties X and Y and a point x ∈ X ,weletd x φ : T x (X ) → T φ(x) (Y ) denote the differential of φ at x. We say that X is a G-variety if the linear algebraic group G acts morphically on X .IfX is affine then the action of G on X gives a linear action of G on k[X ],definedby (g · f )(x) = f (g −1 · x) for all g ∈ G, f ∈ k[X ] and x ∈ X .GivenaG-variety X and x ∈ X , we denote the G-orbit through x by G · x and the stabilizer of x in G by G x .Ifx, y ∈ X are two points on the same G-orbit, then we sometimes say x and y are G-conjugate.Forx ∈ X , we denote the orbit map G → G · x, g → g · x by κ x ;wesaytheorbitG · x is separable if κ x is separable. We denote by X G the set of G-fixed points in X , and by k[X ] G the ring of G-invariant functions in k [X ].
Given a morphism of varieties f : V → W ,definee(v) for v ∈ V to be max(dim(Z )), where Z ranges over the irreducible components of f −1 ( f (v)) that contain v.B y[ 10, AG. 10.3], e(v) is an upper semi-continuous function of v. This implies the following useful result about dimensions of stabilizers for a G-variety X [44,Lemma 3.7(c)]: for any r ∈ N ∪{0},theset{x ∈ X | dim(G x ) ≥ r } is closed. We deduce the lower semi-continuity of orbit dimension: that is, for any r ∈ N ∪{0},theset{x ∈ X | dim(G · x) ≤ r } is closed. In particular, the set {x ∈ X | dim(G · x) is maximal} is open. We also need an infinitesimal version of these results. Given a variety Z , we denote the (reduced) tangent bundle of Z by TZ; we may identify TZ with the set of pairs {(z,v) | z ∈ Z ,v ∈ T z (Z )}, and we have a canonical embedding from Z to TZgiven by z → (z, 0). (The tangent bundle is constructed in [10, AG.16] as a possibly non-reduced scheme over k; here we take the tangent bundle to be the corresponding reduced scheme.) If ψ : Z → W is a morphism of varieties then we have a map dψ : TZ → TW given by dψ(z,v) = (ψ(z), d z ψ(v)).

Lemma 2.1 For any r
Define a function s : X → TG × TX by s(x) = ((1, 0), (x, 0)). We identify X with a closed subset of TX× TX via the embedding x → ((x, 0), (x, 0)).Sinces is a morphism, we deduce from the upper semi-continuity of the function e(v)-taking (V , W , f ) = (TG× TX, TX× TX, dα)-that the function e ′ (y) is also upper semi-continuous. The result now follows.
⊓ ⊔  [40,Proposition I.7.3(ii)], but the converse is not true. A dominant morphism φ : X → Y of irreducible varieties is called birational if the comorphism induces an isomorphism of function fields k(X ) ∼ = k(Y ). Given an irreducible affine variety X , we can form the normalization of X by considering the normal affine variety X whose coordinate ring is the integral closure of k[X ] in the function field k(X ). The normalization map ν X : X → X is, by construction, finite, birational and surjective.

Remark 2.2
We record an observation which we use several times in the sequel. Let φ : X → Y and ψ : Y → Z be morphisms of affine varieties with ψ • φ finite. Then it is easy to see that: We say that a property P(x) holds for generic x ∈ X if there is an open dense subset U of X such that P(x) holds for all x ∈ U .
For the remainder of the paper, we fix the convention that G denotes a reductive linear algebraic group over k.

Group actions and quotients
The main result of this paper, Theorem 1.1, concerns quotients of affine varieties by reductive algebraic group actions. Let X be an affine G-variety. As noted above, G acts on k[X ],and we can form the subring k[X ] G ⊆ k[X ] of G-invariant functions on X . It follows from [43] and [22]thatk[X ] G is finitely generated, and hence we can form an affine variety denoted gives rise to a morphism from X to X /G, which we shall denote by π X ,G : X → X /G.Themap π X ,G has the following properties [ (iv) each fibre of π X ,G contains a unique closed G-orbit, and π X ,G determines a bijective map from the set of closed G-orbits in X to X /G; (v) X /G is a categorical quotient of X : that is, for every variety V and every morphism ψ : (This means π X ,G is a good quotient in the sense of [44,Chapter 3,Sect. 4,p 57]. More generally, if X is a quasi-projective G-variety and π is a map from X to another quasiprojective variety Y then we call π a good quotient if it is an affine map and satisfies (i)-(v) above.) We say that π X ,G : X → X /G is a geometric quotient if the fibres of π X ,G are precisely the G-orbits. This is the case if and only if every G-orbit is closed (for instance, if every G-orbit has the same dimension-e.g., if G is finite). If φ : Y → X is a G-equivariant morphism of affine G-varieties, then the restriction of the comorphism to k[X ] G induces a natural morphism from Y /G to X /G, which we shall denote by φ G . In a special case of this construction, we have the following result, which follows from [44, Theorem 3.5, Lemma 3.4.1].

Lemma 2.3
Let X be an affine G-variety and let i : Y → X be an embedding of a closed G-stable subvariety Y in X . Then π X ,G (Y ) is closed in X /G. Moreover, the induced map i G : Y /G → X /G is injective and finite.

Remark 2.4
If char(k) = 0theni G is an isomorphism onto its image. This need not be the case in positive characteristic: see Example 3.2.
We record some other useful results. First, note that if G is a finite group, then the map π G above is a finite morphism. To see this, let f ∈ k[X ] and let T be an indeterminate. Then the polynomial F(T ) : For technical reasons, we sometimes need to work with affine G-varieties satisfying an extra property. Definition 2.5 Let X be an affine G-variety. We denote by X cl the closure of the set {x ∈ X | G · x is closed}. Following Luna [32,Sec. 4], we say that X has good dimension where m is the maximal orbit dimension.

Lemma 2.7 Let X be an irreducible affine G-variety with good dimension. Then k(X
Then U is a nonempty open subset of X , and clearly U is G-stable. Hence C := X \U is closed and G-stable. As X has good dimension, there exists 0 = h ∈ k[X ] G such that h| C = 0. Now f is a globally defined regular function on the corresponding principal open The second assertion is [1, 2.1.9 (b)]. Note that separability can fail if X does not have good dimension: see [38].
⊓ ⊔ Lemma 2.8 Let φ : X → Y be a finite surjective G-equivariant map of affine G-varieties.
(i) Fo r a l l x ∈ X, G · x is closed if and only if G · φ(x) is closed. Moreover, if y ∈ Y and G · yi sc l o s e dt h e nφ −1 (G · y) is a finite union of G-orbits, each of which is closed and has the same dimension as G · y.

has good dimension if and only if Y does.
Proof By Lemma 2.8(iii), if one of X or Y has good dimension then they both do. It follows from Lemma 2.
Later we also need some material on constructing quotients of projective varieties by actions of reductive groups, but we delay this until Sect. 5.
Suppose H is a subgroup of G. Recall that the quotient G/H (which as a set is just the coset space) has the structure of a quasi-projective homogeneous G-variety, and H is the stabilizer of the image of 1 ∈ G under the natural map π G,H : G → G/H .Richardsonhas proved the following in this situation ([47, Theorem A]; see also [23]).

Theorem 2.10 Suppose H is a subgroup of G. Then G/H is an affine variety if and only if H is reductive.
Recall also that the Zariski topology on G/H is the quotient topology: that is, a subset Proof Since H is reductive, G/H is affine. The group G acts on G/H by left multiplication. In general, however, we need a slightly more complicated construction. Let Y be the G-module X ⊕ X . Note that Y H = X H ⊕ X H and for any (y 1 , y 2 ) ∈ Y H , G (y 1 ,y 2 ) = G y 1 ∩ G y 2 . We show that Y has the desired properties. For each r ≥ 0, define Then C r is empty for all but finitely many r by [ then C r is the image of C r under projection onto the first factor. Set D r = C r \C r +1 . Then the nonempty D r form a finite collection of disjoint constructible sets that cover the irreducible set U 1 × U 1 ,soD s contains a nonempty open subset U 2 of U 1 × U 1 for precisely one value of s.
We show that s = 1. Suppose not. Choose y = (x 1 , x 2 ) ∈ U 2 .L e tg 1 , g 2 ,...,g r be coset representatives for G x 1 /H with g 1 ∈ H .N o t et h a tU 3 is an open dense subset of X H .L e tz = (x 1 , x) ∈ U 2 . Then our hypothesis means that g · (x 1 , x) = (x 1 , x) for some g / ∈ H .No wg must fix x 1 ,sog ∈ g i H for some i ≥ 1; in fact, i ≥ 2sinceg / ∈ H . It follows that g i fixes (x 1 , x) since H fixes (x 1 , x),sog i fixes x.
But r j=2 (X g j ∩ X H ) is a proper closed subset of X H as none of the g j for j ≥ 2fi x e s x 0 , so we have a contradiction. We conclude that s = 1 after all. Hence G y = H for all y ∈ U 2 .
Set y 0 = (x 0 , 0). The orbit N G (H ) · x 0 is closed in G/H , so the orbit N G (H ) · y 0 is closed in Y H . Moreover, N G (H ) y 0 = H ,s oN G (H ) · y 0 has maximal dimension among the N G (H )-orbits on Y H . Hence y 0 is a stable point of Y H for the N G (H )-action. A similar argument shows that y 0 is a stable point of G · Y H for the G-action. Since the set of stable points is open in each case, we can find a nonempty open subset U of U 2 such that (ii) and (iii) hold for U ; then (i) holds for U by construction. This completes the proof. ⊓ ⊔

Cocharacters, G-actions and R-parabolic subgroups
Suppose that X is a G-variety. For any cocharacter λ ∈ Y (G) and x ∈ X we can define a morphism ψ = ψ x,λ : k * → X by ψ(a) = λ(a) · x for each a ∈ k * .W esaythatthelimit lim a→0 λ(a) · xe x i s t sif ψ extends to a morphism ψ : k → X . If the limit exists, then the extension ψ is unique, and we set lim a→0 λ(a) · x = ψ(0). It is clear that, for any G and X , if there exists λ ∈ Y (G) such that lim a→0 λ(a) · x exists but lies outside G · x,thenG · x is not closed in X . A subgroup P of G is called a parabolic subgroup if the quotient G/P is complete; this is the case if and only if G/P is projective. If G is connected and reductive, then all parabolic subgroups of G have a Levi decomposition P = R u (P) ⋊ L, where the reductive subgroup L is called a Levi subgroup of P. In this case, the unipotent radical R u (P) acts simply transitively on the set of Levi subgroups of P, and given a maximal torus T of P there exists a unique Levi subgroup of P containing T . For these standard results see [10,11]o r [ 54] for example. It is possible to extend these ideas to a non-connected reductive group using the formalism of R-parabolic subgroups described in [5,Sec. 6]. We give a brief summary; see loc. cit. for further details. Given a cocharacter λ ∈ Y (G),w e have: (i) P λ := {g ∈ G | lim a→0 λ(a)gλ(a) −1 exists} is a parabolic subgroup of G;wecalla parabolic subgroup arising in this way an R-parabolic subgroup of G. (ii) L λ := C G (λ) ={ g ∈ G | lim a→0 λ(a)gλ(a) −1 = g} is a Levi subgroup of P λ ;w e call a Levi subgroup arising in this way an R-Levi subgroup of G.
The R-parabolic (resp. R-Levi) subgroups of a connected reductive group G are the same as the parabolic and Levi subgroups of G. Moreover, the results listed above for parabolic and Levi subgroups of connected reductive algebraic groups also hold for R-parabolic and R-Levi subgroups of non-connected reductive groups; that is, the unipotent radical R u (P) acts simply transitively on the set of R-Levi subgroups of an R-parabolic subgroup P,and given a maximal torus T of P there exists a unique R-Levi subgroup of P containing T . Now, if H is a reductive subgroup of G and λ ∈ Y (H ),thenλ gives rise in a natural way to R-parabolic and R-Levi subgroups of both G and H . In such a situation, we reserve the notation P λ (resp. L λ ) for R-parabolic (resp. R-Levi) subgroups of G, and use the notation

G-complete reducibility
Our main result, and many of the intermediate ones, uses the framework of G-complete reducibility introduced by Serre [52], which has been shown to have geometric implications in [5] and subsequent papers. We give a short recap of some of the key ideas concerning complete reducibility.
Let H be a subgroup of G. Following Serre (see, for example, [52]), we say that H is G-completely reducible (G-cr for short) if whenever H ⊆ P for an R-parabolic subgroup P of G, there exists an R-Levi subgroup L of P such that H ⊆ L. For example, if G = SL n (k) or GL n (k) then H is G-cr if and only if the inclusion of H is completely reducible in the usual sense of representation theory. If H is G-cr then H is reductive, while if H is linearly reductive then H is G-cr (see [5,Sects. 2.4,6]). Hence in characteristic 0, H is G-cr if and only if H is reductive.
In [2]and [37] it was shown that the notion of complete reducibility is useful when one considers G-varieties and, as explained in the introduction, one of the purposes of this paper is to expand upon this theme.
The geometric approach to complete reducibility outlined in [5] rests on the following construction, which was first given in this form in [9]. Given a subgroup H of a reductive group G and a positive integer n,w ec a l lat u p l eo fe l e m e n t sh ∈ H n a generic tuple for H if there exists a closed embedding of G in some GL m (k) such that h generates the associative subalgebra of m × m matrices spanned by H [9,Defn. 5.4]. A generic tuple for H always exists for sufficiently large n. Suppose h ∈ H n is a generic tuple for H ;thenin [9, Theorem 5.8(iii)] it is shown that H is G-completely reducible if and only if the G-orbit of h in G n is closed, where G acts on G n by simultaneous conjugation.

Optimal cocharacters
Let X be an affine G-variety. The classic Hilbert-Mumford Theorem [28, Theorem 1.4] says that via the process of taking limits, the cocharacters of G can be used to detect whether or not the G-orbit of a point in X is closed. Kempf strengthened the Hilbert-Mumford Theorem in [28](seealso [24,41,51]), by developing a theory of "optimal cocharacters" for non-closed G-orbits. We give an amalgam of some results from Kempf's paper; see [28,Theorem 3.4,Cor. 3.5] (and see also [9,Sect. 4] for the case of non-connected G). Theorem 2.12 Let x ∈ X be such that G · x is not closed, and let S be a closed G-stable subset of X which meets G · x. Then there exists an R-parabolic subgroup P(x) of G and a nonempty subset (x) ⊆ Y (G) such that: (i) for all λ ∈ (x), lim a→0 λ(a) · x exists, lies in S, and is not G-conjugate to x; (ii) for all λ ∈ (x),P λ = P(x); (iii) R u (P(x)) acts simply transitively on (x); (iv) G x ⊆ P(x).

Preparatory results
In this section we collect some results concerning algebraic group actions on varieties which will be useful in the rest of the paper. Recall our standing assumption that G is a reductive group.

Étale slices
Étale slices are a powerful tool in geometric invariant theory. Let X be an affine G-variety and let x ∈ X such that G · x is closed. Luna introduced the notion of an étale slice through x [32, III.1]: this is a locally closed affine subvariety S of X with x ∈ S satisfying certain properties. He proved that an étale slice through x always exists when the ground field has characteristic 0. Bardsley and Richardson later defined étale slices in arbitrary characteristic [1, Defn. 7.1] and gave some sufficient conditions for an étale slice to exist [1,]. If an étale slice exists through x, the orbit G · x must be separable. We record an important consequence of the étale slice theory [1, Proposition 8.6]. Proposition 3.1 Let X be an affine G-variety and let x ∈ X such that G · x is closed and there is an étale slice through x. Then there is an open neighbourhood U of x such that for all u ∈ U, G u is conjugate to a subgroup of G x .
The following example, based on a construction from [37,Example 8.3], shows that this result need not hold when there is no étale slice.
.S e tK x = im( f x ). Note that for each x ∈ k, there are only finitely . These actions commute with each other, so we have an action of G on the quotient space Since H is finite, ϕ is a geometric quotient. A straightforward calculation shows that for any (x, g) ∈ k × G, the stabilizer G ϕ(x,g) is precisely gK x g −1 . It follows that the G-orbits on V are all closed, but the assertion of Proposition 3.1 cannot hold for any v ∈ V . Hence no v ∈ V admits an étale slice. Note that generic stabilizers are nontrivial, but there do exist orbits with trivial stabilizer (take x = 0). Nonetheless we can even show (using étale slice methods!) that generic G-orbits in V are An easy computation shows that the map (ψ x ) H : G/H → V induced by ψ x is bijective and separable when regarded as a map onto its image, so (ψ x ) H gives by Zariski's Main Theorem In contrast, consider the orbit G · ϕ(0, g). This cannot be separable: for otherwise ϕ(0, g) admits an étale slice by [1, Proposition 7.6], since the stabilizer G ϕ(0,g) is trivial, and we know already that this is impossible. It follows easily that (ψ 0 ) H : G/H → V is not an isomorphism onto its image. We see from this that if i is the obvious inclusion of Y : The failure of Proposition 3.1 and other consequences of the machinery of étale slices when slices do not exist is behind many of the technical difficulties we need to overcome in order to prove Theorem 1.1.

Some results on closed orbits
We first need a technical lemma which collects together various properties of orbits and quotients and the associated morphisms. For more details, see the proofs of [48,Lemmas 4.2,10.1.3] or the discussion in [27, Sect. 2.1], for example; the extension to non-connected G is immediate. Note that if G acts on a variety X then for any x ∈ X , G · x is locally closed [10, Proposition 1.8], so it has the structure of a quasi-affine variety. Lemma 3.3 Let X be a G-variety. Suppose x ∈ X , and let ψ x : G/G x → G · x be the natural map. Then:

is affine if and only if G/G x is affine if and only if G x is reductive; (iii) ψ x is an isomorphism of varieties if and only if the orbit G · x is separable.
Remark 3.4 All the subtleties here are only really important in positive characteristic since in characteristic 0 the orbit map is always separable, so the morphism ψ x is always an isomorphism. The result shows that even in bad cases where the orbit map is not separable we can reasonably compare the quotient G/G x with the orbit G · x, as one might hope.

is closed in X if and only if H K is closed in G.
Proof Part (ii) follows immediately from part (i). For part (i), since the map ψ x : G/K → G · x from Lemma 3.3 is a homeomorphism, H · x is closed in G · x if and only if the corresponding subset H · π G,K (1) is closed in G/K (recall that π G,K : G → G/K is the canonical projection). Since G/K has the quotient topology, this is the case if and only if the preimage of this orbit is closed in G. But the preimage is precisely the subset HK.
⊓ ⊔ Our next result involves the following set-up: Suppose Y is another G-variety. Then G ×G acts on the product X × Y via (g 1 , g 2 ) · (x, y) = (g 1 · x, g 2 · y), and identifying G with its diagonal embedding (G) in G × G, we can also get the diagonal action of G on X × Y : Lemma 3.6 With the notation just introduced, let x ∈ X, y ∈ Y and set K = G x ,H= G y . Then: Proof (i). The first equivalence follows from Lemma 3.5 since KH = (HK) −1 is closed in G if and only if HK is closed in G (note that this argument is based on the one in the proof of [48,Lemma 10.1.4]). For the second equivalence, consider the orbit map κ 1 : G × G → G associated to the orbit of 1 ∈ G for the double coset action of G × G on G (cf. Sect. 9); so κ 1 is given by κ 1 (g 1 , g 2 ) = g 1 g −1 2 .Thenκ 1 is surjective and open. Now, since the (G × G)-orbit of (x, y) is (G · x) × (G · y) and the stabilizer of (x, which happens if and only if K · y is closed in G · y, by Lemma 3.5(i) again.
(ii). This chain of equivalences follows quickly from part (i). ⊓ ⊔ Remark 3.7 The results above give criteria for a result of the form "G · x closed implies H · x closed" for a point x in a G-variety X . We can't hope for a general converse to this. For example, let G be any connected reductive group and, in the language of Sect. 2.4,let x ∈ X = G n be a generic tuple for a Borel subgroup of G and y ∈ Y = G n be a generic tuple for G itself. Then, G x = G y = Z (G),theG-orbits of y and (x, y) are closed, but the G-orbit of x is not closed.

Finite morphisms and quotients
In this section we provide some general results on finite morphisms and quotients by reductive group actions. We begin with an extension of Zariski's Main Theorem which deals with nonseparable morphisms. Recall that if X is an irreducible affine variety then ν X : X → X denotes the normalization of X .
by construction, so ψ is finite and surjective, and hence α is quasi-finite and has the same image as φ.Butα is birational by construction, so α is an isomorphism from the affine variety Z onto an open subvariety of Y by Zariski's Main Theorem (since Y is normal). To complete the proof of the first assertion, it is enough to show that ψ is injective. This follows because any k-algebra homomorphism k[X ]→k is completely determined by its values on f Because ν X is finite and birational, the map φ • ν X : X → Y satisfies the hypotheses of the proposition. Hence φ • ν X is injective. This forces ν X to be injective also. But ν X is also surjective, and we are done.

⊓ ⊔
We need some further results about the behaviour of affine G-varieties under normalization. If X is an affine G-variety then X inherits a unique structure of a G-variety such that ν X is G-equivariant (cf. [1,Sect. 3]). This gives a map of quotients (ν X ) G : X /G → X /G. Lemma 3.9 Let X be an irreducible affine G-variety with good dimension and let (ν X ) G be as above. Then (ν X ) G is finite and X /G is the normalization of X /G.

Proof
The natural map of quotients X /G 0 → X /G can be viewed as the quotient map by the finite group G/G 0 and is therefore finite. The same is true for X /G 0 → X /G,s ob y The coordinate ring k[ X ] of the normalization of X is the integral closure of k[X ] in the function field k(X ).L e tS be the integral closure of k[X ] G in k(X ).T h e nS is finitely generated as a k-algebra [1, 2.4.3], and birational and quasi-finite by Lemmas 2.9 and 2.8(ii), so α is also birational and quasi-finite. It follows from Zariski's Main Theorem that α is an open embedding.
The map β is finite by construction, so to complete the proof that where pr 1 is projection onto the first factor. The composition X → C → X is finite, where the second map is projection onto the second factor, so θ is a finite map from X to C;i n particular, C = θ( X ).
Let G act on Z × X trivially on the first factor, and by the given action on the second. It is immediate that θ is G-equivariant, so C is G-stable and we have an induced map θ G : X /G → (Z × X )/G.TheimageD of θ G is π Z ×X ,G (C), and this is closed in (Z × X )/G as C is closed and G-stable. There is an obvious map ξ : (Z × X )/G → Z × X /G, and it is easily checked that ξ is an isomorphism; hence ξ(D) is closed. Untangling the definitions, we find that α factors as To finish the proof, we note that for any G (connected or otherwise), the variety X /G is normal since X is normal, and the considerations above show that (ν X ) G : X /G → X /G is finite. Moreover, (ν X ) G is birational by Lemma 2.9 since X has good dimension. The result now follows from another application of Zariski's Main Theorem.
⊓ ⊔ Next we extend a result of Bardsley and Richardson [1, 2.4.2], which they prove in the special case when X and Y are normal and φ is dominant. It provides an extension to positive characteristic of a result used freely in [33].
Proof As at the start of the proof of Lemma 3.9, we can immediately reduce to the case when G is connected, since the natural maps X /G 0 → X /G and Y /G 0 → Y /G arefinite.The map X cl /G → X /G is surjective, and Lemma 2.3 implies it is finite. We may also assume, therefore, that X has good dimension. Since a morphism is finite if and only if its restriction to every irreducible component of the domain is finite, we can assume X is irreducible. By the proof of Lemma 2.8, φ(X cl ) ⊆ Y cl , so after replacing Y with φ(X ) if necessary, we may assume by Lemma 2.3 that φ is dominant and Y is irreducible and has good dimension.
The map φ : X → Y gives rise to a map φ : X → Y ,and φ is finite as φ is. We have a commutative diagram where the vertical arrows are the normalization maps. Taking quotients by G,w eo b t a i na commutative diagram Since φ is finite and dominant and X and Y are irreducible and normal, the map φ G : ⊓ ⊔

Proof of Theorem 1.1, Part 1: quasi-finiteness
In this section we provide the first step towards our proof of Theorem 1.1, showing that the map ψ X ,H in question is quasi-finite. We are also able to retrieve other results from [33] which follow from the main theorem, but in arbitrary characteristic. Our first result is a generalization of [2,Theorem 4.4]; see also [7,Theorem 5.4].

Proposition 4.1 Suppose that G is a reductive group and X is an affine G-variety. Let H be a G-completely reducible subgroup of G and let
x ∈ X H . Then the following are equivalent: and (x) be the R-parabolic subgroup and class of cocharacters given by Theorem 2.12.
) acts simply transitively on (x) and on the set of R-Levi subgroups of is not closed. This shows that if (i) holds then G · x must be closed. Therefore, in order to finish the proof, we need to show that N G (H ) · x is closed if and only if H is G x -cr under the assumption that G · x is closed (note that since G · x is closed, G x is reductive (Lemma 3.3(ii)), and hence it makes sense to ask whether or not H is G x -cr).
To see this equivalence, let h ∈ G n for some n be a generic tuple for the subgroup H and consider the diagonal action of G on G n × X .T h e nC G (H ) = G h . Now, by Lemma 3.6, The latter condition is equivalent to requiring that H is G x -cr, and since x is H -fixed and

Lemma 4.3 Suppose H is a reductive subgroup of G such that H is not G-cr. Then:
(i) There exists an affine G-variety X and a point x ∈ X H such that G · x is not closed.
(ii) There exists a rational G-module V and a nonzero subspace W ⊆ V H such that: Proof Choose a closed embedding G ֒→ SL m (k) for some m and think of H and G as closed subgroups of SL m (k). Let Mat m denote the algebra of all m × m matrices. Let x = (x 1 ,...,x n ) ∈ H n be a basis for the associative subalgebra of Mat m spanned by H ;thenx is a generic tuple for H (see Sect. 2.4). This means that if we let SL m (k) act on Y := (Mat m ) n by simultaneous conjugation, then There is also a right action of GL n (k) on Y , which we denote by * . Given a matrix A = (a ij ) ∈ GL n (k) and an element y = (y 1 ,...,y n ) ∈ Y ,wecanset This is the action obtained by thinking of the tuple y as a row vector of length n and letting the n × n matrix A act on the right in the obvious way. Note that the SL m (k)-andGL n (k)-actions commute.
Given any h ∈ H ,sincex is a basis for the associative algebra generated by H ,wehave that h · x is also a basis for this algebra, and hence there exists a unique A(h) ∈ GL n (k) such that h · x = x * A(h). Note also that and hence the map A : H → GL n (k) is a group homomorphism. This map is in fact a rational representation of H since it arises from the morphic action of H on the vector space spanned by the entries of x.LetK denote the image of H in GL n (k);thenK is a reductive group and x * K = H · x is closed. Moreover, since the elements of the tuple x are linearly independent, the stabilizer of x in K is trivial. Hence x is a stable point for the action of K on Y .NowletX = Y /K and set x := π Y ,K (x). Since the SL m (k)-andGL n (k)-actions on Y commute, we obtain an action of SL m (k) on X . It is immediate that x ∈ X H .
We know that G · x is not closed in Y , so there exists a cocharacter λ ∈ Y (G) such that lim a→0 λ(a)·x = y exists and is not G-conjugate to x.Sinceπ Y ,K is G-equivariant, it is easy to see that lim a→0 λ(a) · x = π Y ,K (y) (and in particular this limit exists). Suppose π Y ,K (y) is G-conjugate to x. Then there exists g ∈ G such that g · π Y ,K (y) = π Y ,K (g · y) = x, ) is precisely K · x, which coincides with H · x by construction. Hence g · y = h · x for some h ∈ H and we see that y and x are G-conjugate, which is a contradiction. Hence π(y) and x are not conjugate, and the G-orbit of x ∈ X H is not closed, which proves (i).
To prove (ii), let S denote the unique closed G-orbit in the closure of G · x. Then, following [28, Lemma 1.1(b)], we can find a rational G-module V with a G-equivariant morphism φ : X → V such that φ −1 (0) = S.SinceG ·x is not closed, it does not meet S, and hence v := φ(x) = 0. However, by Theorem 2.12, there exists μ ∈ Y (G) such that lim a→0 μ(a) · x ∈ S, and since the morphism φ is G-equivariant, we have that {0} is the unique closed G-orbit in the closure of G ·v. Note also that v is H -fixed since x is. Now the tuple x consists of elements of H ,soisC G (H )-fixed, and hence  (i) N G (H ) is reductive and for every affine G-variety X , the natural morphism ψ X ,H : Proof Suppose H is not G-cr. Then either N G (H ) is not reductive, in which case the first part of condition (i) fails, or else N G (H ) is reductive but the second part of condition (i) fails by Lemma 4.3(ii). Hence (i) implies (ii).
Conversely, suppose H is G-cr, and let X be any affine G-variety. Since H is G-cr, and hence H is reductive, we have N G (H ) 0 = H 0 C G (H ) 0 .ThatN G (H ) is reductive is shown in [5,Proposition 3.12], and hence it always makes sense to take the quotient X H /N G (H ).
Suppose x ∈ X H . We first claim that the unique closed G-orbit S in G · x meets X H . Indeed, either G · x is already closed, in which case S = G · x, or we can find the optimal parabolic P(x) and optimal class (x) as given in Kempf's Theorem 2.12.SinceH ≤ G x ≤ P(x) and H is G-cr, there is a Levi subgroup L of P(x) containing H . Since the unipotent radical acts simply transitively on (x) and on the set of Levi subgroups of P(x),thereis precisely one element λ ∈ (x) with L = L λ , and this choice of λ commutes with H .But then y := lim a→0 λ(a) · x ∈ S ∩ X H , which proves the claim. Now any point of X /G has the form π X ,G (x),whereG · x is closed in X .Soletx ∈ X such that G · x is closed. For any y ∈ π −1 X ,G (π X ,G (x))∩ X H , G · x is the unique closed G-orbit in G · y. Hence, if π −1 X ,G (π X ,G (x)) ∩ X H is nonempty, G · x must meet X H , by the claim in the previous paragraph. It follows from the definitions that π −1 X H ,N G (H ) (ψ −1 X ,H (π X ,G (x))) = π −1 X ,G (π X ,G (x)) ∩ X H , so to show that ψ X ,H is quasi-finite, we need to show that for each such x there are only finitely many closed N G (H )-orbits in π −1 X ,G (π X ,G (x)) ∩ X H .Butany y ∈ X H with a closed N G (H )-orbit has a closed G-orbit, by Proposition 4.1, and hence any y ∈ π −1 X ,G (π X ,G (x)) ∩ X H with a closed N G (H )-orbit is already G-conjugate to x.Sowe must show that there are only finitely many closed N G (H )-orbits in G · x ∩ X H . Fix x ∈ X H with G · x closed, and recall that G x is reductive since G · x is closed. Let y ∈ G · x ∩ X H , and write y = g · x for some g ∈ G.SinceG · y is closed, Proposition 4.1 says that N G (H ) · y is closed if and only if H is G y -cr, which is the case if and only if g −1 Hg is G x -cr. Suppose g −1 Hg and H are G x -conjugate: say H = g −1 1 (g −1 Hg)g 1 for some g 1 ∈ G x .T h e ngg 1 ∈ N G (H ) and y = g · x = (gg 1 ) · x,s ow es e et h a tx and y are N G (H )-conjugate. Conversely, suppose x and y are N G (H )-conjugate: say y = m · x for some m ∈ N G (H ).T h e nm −1 g ∈ G x and m −1 g(g −1 Hg)g −1 m = H ,s og −1 Hg and H are G x -conjugate. Hence the distinct closed N G (H )-orbits in G · x ∩ X H correspond to the distinct G x -conjugacy classes of G x -cr subgroups of the form g −1 Hg inside G x .I ti s therefore enough to show that there are only finitely many such conjugacy classes.
Let h ∈ H n be a generic tuple for H in G x for some n and let g ∈ G such that g −1 Hg is a G x -cr subgroup of G x .T h e ng −1 · h is a generic tuple for g −1 Hg.S i n c eg −1 Hg is both G-cr and G x -cr, the G-andG x -orbits of h in G n are both closed. It follows from [35, Theorem 1.1] that the natural map of quotients G n x /G x → G n /G is finite, and hence there are only finitely many closed G x -orbits contained in G · h ∩ G n x . This proves the result. ⊓ ⊔ Remark 4.5 Note that if G x = H and G · x is closed then the argument in the proof above shows that there is precisely one closed N G (H )-orbit inside G · x ∩ X H (namely, N G (H ) · x), and therefore ψ −1 X ,H (π X ,G (x)) is a singleton. We will use this observation in Sects. 6 and 7.
The third paragraph of the proof above shows that for any x ∈ X H , the unique closed orbit contained in G · x also meets X H . This allows us to prove the following: Proof Since G · X H is closed and G-stable, we may replace X with G · X H ;t h e ns a y i n g ψ X ,H has closed image is the same as saying that ψ X ,H is surjective. But this is equivalent to saying that the fibre above every point of X /G meets X H . Since each fibre contains a unique closed orbit, the observation before the Lemma gives the result.
,andletG act by left multiplication on the first factor and trivially on the second factor. Now let X = Y /Z ; this is a special case of a construction described in [32,I.3].SinceZ is reductive and Y is affine, X is affine, and since Z acts freely on Y ,the fibres of π Y ,Z are precisely the Z -orbits in Y . Moreover, since the G-andZ -actions on Y commute, X is naturally a G-variety. Let 0 = v ∈ V and choose a cocharacter λ of Z such that m :

and we have our contradiction. Hence N G (H )/H is finite. This completes the proof that (i) implies (ii).
Conversely, suppose (ii) holds and X is any affine G-variety. Let x ∈ X H ,sothatH ≤ G x . Since N G (H )/H is finite, N G (H ) · x is a finite union of H -orbits. But N G (H ) · x ⊆ X H , so each of these H -orbits is a singleton and N G (H ) · x is finite, and therefore closed in X . Now we can apply Proposition 4.1 to deduce that the G-orbit of x is also closed, which gives (i).

Proof of Theorem 1.1,Part2:surjectivity
In this section, we prove the following:

Theorem 5.1 Let X be an affine G-variety and let H be a G-cr subgroup of G. Then the map
The proof of Theorem 5.1 in positive characteristic requires some preparation. Before we begin, we note that if char(k) = 0 then we can give a much quicker proof using the machinery of étale slices, as follows. Let x ∈ G · X H such that G · x is closed. Then there is an étale slice through x for the G-action [32, III.1]. By Proposition 3.1, there is an open G-stable neighbourhood O of x such that G y is conjugate to a subgroup of G x for all y ∈ O.Since O meets G · X H , H must be conjugate to a subgroup of G x . Hence x ∈ G · X H ,andweare done by Lemma 4.6.
We need some material on weighted projective varieties and their quotients by reductive groups (cf. [ is homogeneous if f ∈ k[V ] i for some i; in this case we write deg( f ) = i (so deg( f ) is the weighted degree rather than the usual degree of a polynomial). The action of k * can be diagonalised, so we can choose a basis {v 1 ,...,v n } for V consisting of weight vectors. Then the corresponding elements X 1 ,...,X n of the dual V * are weight vectors and we can write );wecallthistheweighted projectivization of V according to the k * -action [18]. Then W(V ) is a projective variety and we may identify the points of W(V ) with the equivalence classes of V \{0} under the equivalence relation ∼,wherev ∼ w if and only if v = c · w for some c ∈ k * . If the weights of the k * -action on V are all 1-that is, if the action of k * on V is by ordinary scalar multiplication-then W(V ) is just the ordinary projective space P(V ) associated to V , but in Sect. 6 we will need to consider the general weighted case. One can show that the canonical projection Since the G-andk * -actions commute, the ring k[V ] G of invariants also inherits a grading by non-negative integers: It is easily checked that the action of G on V descends to give an action of G on W(V ). We say that We have an analogous notion of semistable points in the affine variety V .W es a yt h a t v ∈ V semistable (or G-semistable)i f f (v) = 0 for some homogeneous f ∈ k[V ] G such that deg( f ) ≥ 1, and we define V ss,G to be the set of semistable points; note that V ss,G = ξ −1 V (W(V ) ss,G ).I fv is not stable then we say that v is unstable (or G-unstable).

Since the homogeneous elements of
. By the Hilbert-Mumford Theorem, this is the case if and only if there exists λ ∈ Y (G) such that lim a→0 λ(a) · v = 0. We denote the composition V ss,G → Now suppose K is a reductive subgroup of G and X is a closed (K × k * )-stable subvariety of V (sothatinparticular0 ∈ X ). Then the vanishing ideal for X in k[V ] is homogeneous with respect to our fixed k * -grading, so k[X ] inherits a grading. The constructions above still go through replacing V and G with X and K .WehaveprojectivevarietiesW(X ) := Proj(k[X ]) and W(X ) ss,K /K := Z := Proj(k[X ] K ),whereW(X ) ss,K is defined analogously to above; the map W(X ) ss,K → Z is a good quotient. Note that the proof of [44, Theorem 3.14] still goes through: all one needs is that k[X ] is graded and the G-action preserves the grading. Since W(V ) and W(X ) are categorical quotients of V \{0} and X \{0} respectively, the inclusion of X in V gives rise to a map from W(X ) to W(V ).

It is clear from the characterisation of semistable points in terms of the Hilbert-Mumford
Theorem that X ∩ V ss,G ⊆ X ss,K . Suppose X ss,K ⊆ V ss,G ;thenX ss,K = X ∩ V ss,G .Since Y and Z are categorical quotients of W(V ) ss,G and W(X ) ss,K , respectively, the inclusion of W(X ) ss,K in W(V ) ss,G gives rise to a map φ : Z → Y . Now we come to the point: because Y and Z are projective, the image of φ is closed.
We can now state and prove the main result of this section.

Proposition 5.2
Let V be a G-module equipped with a k * -action as above. Let K be a reductive subgroup of G, let X be a closed (K × k * )-stable subset of V and suppose X ss,K ⊆ V ss,G . Then the natural morphism of quotients X /K → V /G has closed image (i.e., π V ,G (X ) is closed in V /G).
Proof For the purposes of the proof, we need to replace G with a slightly larger group to take into account possible effects of passing to the weighted projectivisation. Without loss, we may assume that G is a subgroup of GL(V ).L e tR = k[V ] G and let f 1 ,..., f r be homogeneous generators for R.L e tm be the lowest common multiple of the degrees deg( f 1 ),...,deg( f r ) and write m = p α m ′ for some m ′ coprime to p.L e tF be the finite group of m ′th roots of unity, regarded as a subgroup of k * (equipped with its given action on V ). Now set Ŵ = FG.T h e nŴ inherits an action on V from the commuting actions of G and k * ,andV ss,Ŵ = V ss,G because Ŵ 0 = G 0 .Further ,F acts on the quotient V /G and the quotient map π V /G,F : . Hence, to show the result claimed, we may replace G with Ŵ and show that A homogeneous f ∈ R belongs to S if and only if deg( f ) is divisible by m ′ (since then the action of F is killed by the degree).
To show that π V ,Ŵ (X ) is closed in V /Ŵ, it is enough to show that for every x ∈ Ŵ · X with closed Ŵ-orbit, there exists an x ′ ∈ X with π V ,Ŵ (x ′ ) = π V ,Ŵ (x) (cf. Lemma 4.6). If x ∈ Ŵ · X is unstable and Ŵ · x is closed, then x must actually be 0, so x ∈ X also. Therefore, we may assume that we have x ∈ V ss,Ŵ ∩Ŵ · X such that Ŵ·x is closed (in V ). By the discussion before the proposition, we have a morphism φ : W(X ) ss,K /K → W(V ) ss,Ŵ /Ŵ with closed image C, say. Note that since we are assuming X ss,K = V ss,Ŵ ∩ X ,wehaveŴ · X ss,K = V ss,Ŵ ∩Ŵ · X , and this set is dense in V ss,Ŵ ∩Ŵ · X . The composition V ss,Ŵ −→ W(V ) ss,Ŵ /Ŵ takes Ŵ · X ss,K into C, and since C is closed that means that the composition in fact takes all of V ss,Ŵ ∩ Ŵ · X into C. Therefore, we can find z in W(X ) ss,K /K with φ(z) = η V ,Ŵ (ξ V (x)). Tracing back through the definitions, we see that φ(z) = η V ,Ŵ (ξ V (y)) for some y ∈ X ss,K . It follows that η V ,Ŵ (ξ V (x)) = η V ,Ŵ (ξ V (y)); we claim that in fact π V ,Ŵ (x) = π V ,Ŵ (c · y) for some c ∈ k * . Note that suffices to finish the proof, since in particular y ∈ X , so setting x ′ = c · y ∈ X gives us what we want.
Both points x and y lie in V ss,Ŵ , so there are homogeneous generators f i , f j ∈ R for which f i (x) = 0a n d , and hence f ′ (x) = f ′ (c · y).SinceS is generated by homogeneous elements, we see that π V ,Ŵ (x) = π V ,Ŵ (c · y), as required. This finishes the proof. ⊓ ⊔ Proof of Theorem 5. 1 We can choose a G-equivariant closed embedding of X in a G-module V .Letv ∈ V H such that v is G-unstable. By the argument in the proof of Proposition 4.1,there exists λ ∈ Y (N G (H )) such that lim a→0 λ(a) · v = 0. This shows that (V H ) ss,N G (H ) ⊆ V ss,G . The G-action commutes with the natural k * -action by scalars, and this preserves the subspace V H also, so Proposition 5.2 implies that the map ψ V ,H : x ∈ G · X H and we are done by Lemma 4.6.

Remark 5.3
For the proof of Theorem 5.1, we only need to apply Proposition 5.2 when the k * -action is the standard action by scalars, so the weighted projectivization is the usual projectivization in this case. However, we do need Proposition 5.2 in this more general set-up to complete the proof of Theorem 1.1 in the next section.

Example 5.4
Let G act on X := G by conjugation and let H be a maximal torus of G. Assume G is connected. Then X H = H . Since the closed orbits in X are precisely the semisimple conjugacy classes [55], the map ψ X ,H : X H /N G (H ) → X /G is surjective-in fact, it is well known that ψ X ,H is an isomorphism (cf. Sect. 7). Note, however, that although G · X H is dense in X , not every element of X belongs to G · H (just take x ∈ X not semisimple).

Proof of Theorem 1.1, Part 3: finiteness
We now complete the proof of Theorem 1.1. The implication (ii) ⇒ (i) follows from Theorem 4.4, so it remains to show that if H is G-completely reducible then the morphism ψ X ,H is finite. By Lemma 2.3, we can replace X with a larger affine G-variety, hence without loss we can assume that X is a G-module. Let G 1 be the subgroup of G generated by G 0 and H . The inclusion of X H in X gives rise to a morphism ψ 1 X ,H : We have a commutative diagram where the vertical arrows are the obvious maps. We may identify X /G with the quotient of X /G 0 by the finite group G/G 0 ,sothemapX /G 0 → X /G is finite. This map factorizes as X /G 0 → X /G 1 → X /G,sothemapX /G 1 → X /G is finite by Remark 2.2(ii). Likewise, the map X H /N G 1 (H ) → X H /N G (H ) is finite. Hence both of the vertical maps in (6.1)are finite and surjective. Now using Remark 2.2(ii) we see it is enough to show that ψ 1 X ,H is finite. So it is enough to prove that ψ X ,H is finite under the assumption that G = G 1 .
Let Y and U ⊆ Y H be as in Lemma 2.11 and set V = X ⊕ Y .W ehaveaG-equivariant closed embedding of X in V given by x → (x, 0).LetW = G · V H ;thenW H = V H .Note that G · V H = G 0 · V H by our assumption that G = G 1 ,soW is irreducible. By Lemma 2.3 again, it is enough to prove that ψ W ,H is finite.
The subset X H × U of X H ⊕ Y H = V H = W H is open and dense, and G w = H for w ∈ X H × U . Next we claim that W has good dimension (for the G-action). To see this, let y 0 ∈ U and set w 0 = (0, y 0 ).T h e nG · w 0 is a G-orbit of maximal dimension in W ,a n d G · w 0 is closed (as G · y 0 is,byLemma2.11), so w 0 is a stable point of W for the G-action. The claim now follows from Remark 2.6. By a similar argument, W H has good dimension for the N G (H )-action. Now since the stable points form an open subset, we can conclude that G · w and N G (H ) · w are closed for generic w ∈ W H , and it follows from Remark  carries a vector space structure, and we can identify a point 0 C ∈ C corresponding to the zero 0 V ;wehaveν W (0 C ) = 0 V by construction. Furthermore, the action of k * on W by scalar multiplication lifts to an action of k * on W which preserves the closed subset C.
We want to apply Proposition 5.2 to deduce that the map C/N G (H ) → W /G has closed image (note that we cannot use Theorem 5. Let λ 0 : k * → k * be the identity cocharacter of k * .Now{0 V } is the unique closed k * -orbit in W , and each element of W is destabilized to 0 V by λ 0 . It follows from Lemma 2. We deduce that lim a→0 λ 0 (a) · w = 0 C . Hence, we can conclude that k * acts on M with positive weights. Further, C is a closed (N G (H ) × k * )-stable subset of M. By the same argument as in the proof of Proposition 4.1,ifc ∈ C is G-unstable, then since c is H -fixed and H is G-cr, c is also N G (H )-unstable. Hence C ss,N G (H ) ⊆ M ss,G . Thus we can now apply Proposition 5.2 to deduce that C/N G (H ) has closed image in M/G.S i n c ei G : W /G → M/G is injective (Lemma 2.3), we deduce that C/N G (H ) has closed image in W /G, as we wanted. This allows us to draw the following commutative diagram: where by abuse of notation we denote the restriction of ψ W ,H to C/N G (H ) by the same symbol. The leftmost vertical arrow is the isomorphism induced by the isomorphism C ∼ = W H above. The other vertical map is finite (Proposition 3.10) and birational (Lemma 2.9; recall that W has good dimension). By Theorem 5.1, ψ W ,H has closed image, and we have just argued that ψ W ,H (C/N G (H )) is closed. But W = G · W H ,s oψ W ,H is surjective, and it follows that ψ W ,H (C/N G (H )) = W /G.Sinceψ W ,H is quasi-finite (Theorem 4.4) and has singletons as generic fibres, the same is true of ψ W ,H .As W /G is normal, it follows from Proposition 3.8 that ψ W ,H is finite and bijective. This implies that (ν W ) G • ψ W ,H is finite. Since the leftmost vertical arrow is an isomorphism, we have that ψ W ,H is finite, as required. This completes the proof of Theorem 1.1.

Separability of Ã X,H
We now consider the question of when ψ X ,H is an isomorphism, or close to being one. Before we state our result, we need some terminology.

Definition 7.1
Let H be a subgroup of G. We say that H is a principal stabilizer for the G-variety X if there exists a nonempty open subset U of X such that G x is G-conjugate to H for all x ∈ U . We say that H is a principal connected stabilizer for the G-variety X if H is connected and there exists a nonempty open subset U of X such that G 0 x is G-conjugate to H for all x ∈ U . It is immediate that if G permutes the irreducible components of X transitively then a principal stabilizer (resp., principal connected stabilizer) is unique up to conjugacy, if one exists.
In characteristic 0, principal stabilizers exist under mild hypotheses: for instance, if X is smooth [50,Proposition 5.3] or if X has good dimension [34,Lemma 3.4]. For a counterexample in positive characteristic, see Example 3.2.

Theorem 7.2 Let X be an affine G-variety. Suppose that: (a) H is a principal stabilizer for X cl ; (b) H is G-cr; (c) X /G and X H /N G (H ) are irreducible; and (d) X /G is normal. Then ψ X ,H is finite and bijective. In particular, if ψ X ,H is separable then it is an isomorphism.
Observe that this result extends a theorem of Luna and Richardson [34,Theorem 4.2] to positive characteristic; note that in characteristic 0, a reductive group H is automatically G-cr, ψ X ,H is automatically separable and principal stabilizers exist, as noted above.
Proof By Theorem 1.1, ψ X ,H is finite, so its fibres are finite. To prove the first assertion of the theorem it is enough, therefore, by Proposition 3.8 to show that ψ X ,H is surjective and generic fibres of ψ X ,H are singletons. By hypothesis, G · X H contains a nonempty open subset of X cl . The assumption that X /G is irreducible implies that the action of G is transitive on the irreducible components of X cl so we can conclude that π X ,G (X H ) = π X ,G (G · X H ) contains a nonempty open subset of X /G.AsX /G is irreducible, ψ X ,H (X H /N G (H )) = X /G.Ifx is a stable point of X cl and G x = H then ψ −1 X ,H (ψ X ,H (π X H ,N G (H ) (x))) is a singleton, by Remark 4.5. This proves the first assertion as the set of conjugates of such x is open in X cl .If ψ X ,H is separable then the second assertion follows from Zariski's Main Theorem, as X /G is normal. ⊓ ⊔

Remark 7.3
The assertion of Theorem 7.2 also holds by a similar argument if we replace the hypothesis that H is a principal stabilizer for X cl with the hypothesis that H is a principal connected stabilizer for X cl .
Next we study the separability condition. To simplify the arguments below, we consider only the case when X has good dimension for the G-action.

Lemma 7.4 Suppose an affine G-variety X has good dimension and hypotheses (a)-(c) of Theorem 7.2 hold. Then ψ X ,H is separable if and only if for generic
, so the content here is in the reverse inclusion. First we claim that X H has good dimension for the N G (H )-action. To see this, observe that G · X H = X by the surjectivity assertion of Theorem 7.2 (which does not depend on hypothesis (d)), so every closed G-orbit in X meets X H by Lemma 4.6.AsH is a principal stabilizer for X ,wemusthaveG x = H for generic x ∈ X H , and it follows from Proposition 4.1 that generic N G (H )-orbits in X H are closed, as required. We now see from Remark 2.6 that for generic x ∈ X H .No wπ X ,G and π X H ,N G (H ) are separable (Lemma 2.7), and it follows from this and from Eq. (7.5) that for generic x ∈ X H , d x π X ,G is surjective at x with kernel T x (G · x) and d x π X H ,N G (H ) is surjective at x with kernel T x (N G (H ) · x). The map ψ X ,H is surjective and finite (by Theorem 1.1), so it is separable if and only its derivative is an isomorphism for generic points in X H /N G (H ). The result now follows from the argument above.
⊓ ⊔ Recall that a pair (G, H ) of reductive groups with H ≤ G is called a reductive pair if h = Lie(H ) splits off as a direct H -module summand of g = Lie(G),whereH acts via the adjoint action of G on g, and a subgroup A ≤ G is called separable in G if

Proposition 7.6 Suppose an affine G-variety X has good dimension and hypotheses (a)-(c) of Theorem 7.2 hold. Suppose one of the following holds:
(i) there exists x ∈ X such that G x = H and there is an étale slice through x for the G-action; (ii) H is separable in G, (G, H ) is a reductive pair and there exists x ∈ X such that G x = H and G · x is separable.
Then ψ X ,H is separable.
Proof By the argument of Theorem 7.2, ψ X ,H is dominant. Suppose first that (i) holds. Let x ∈ X with G x = H and let S be an étale slice through x for the G-action. By the definition of étale slices and the proof of [1, Proposition 8.6], there exists a G-stable open neigbourhood U of x in X such that G y ≤ G x for all y ∈ S ∩ U and the obvious maps G × (S ∩ U ) → X and (S ∩ U )/H → X /G are étale. As H is a principal stabilizer for X , we can assume after replacing U with a smaller open set that G y is conjugate to H for all y ∈ S ∩ U .W ehave G x = H by hypothesis, so it follows that G y = H for all y ∈ S ∩ U . As the set of stable points of X is G-stable, open and nonempty and the set of smooth points of X /G is open and nonempty, there is a nonempty G-stable open subset U 1 of U such that G · y is closed and π X ,G (y) is a smooth point of X /G for all y ∈ U 1 .
contains a nonempty open subset of X H .L e ty ′ ∈ X H ∩ G · (S ∩ U 1 ):s a y ,y ′ = g · y for some y ∈ S ∩ U 1 , g ∈ G.T h e nG y = H and G y ′ is G-conjugate to H ;b u ty ′ ∈ X H ,s oG y ′ = H .I t follows that g ∈ N G (H ). We deduce that So pick y ∈ S ∩ U 1 such that π X H ,N G (H ) (y) is a smooth point of X H /N G (H ).Themap (S ∩ U )/H → X /G is étale, so its derivative is an isomorphism everywhere. Hence the derivative of the map X H → X /G induced by π X ,G is surjective at y. This in turn implies that the derivative of ψ X ,H is surjective at π X H ,N G (H ) (y).Butπ X H ,N G (H ) (y) and π X ,G (y) are smooth points by construction, so ψ X ,H is separable. Now suppose that (ii) holds. We argue along the lines of the proof of [48, Theorem A]. Let d be an H -module complement to h in g.L e tX 0 ={ x 1 ∈ X | G x 1 = H and G · x 1 is closed and separable}.Letx 1 ∈ X 0 . Then the orbit map κ x 1 : G → G · x 1 gives an isomorphism φ : G/H → G · x 1 . In particular, the derivative d 1 φ at 1 ∈ G gives an isomorphism from g/h to the tangent space T x 1 (G · x 1 ), and it is easily checked that To finish, it is enough by Lemma 7.4 to show that generic elements of X H belong to X 0 . As H is a principal stabilizer for X and X has good dimension for the G-action, G x 1 = H and G · x 1 is closed for generic x 1 ∈ X H .Now for all x 1 ∈ X H . But equality holds in Eq. (7.7)forx 1 = x, so it holds for generic x 1 ∈ X H by Lemma 2.1. This shows that G · x 1 is separable for generic x 1 ∈ X H , so we are done. ⊓ ⊔ The following example shows that separability does not hold automatically under the hypotheses of Theorem 7.2, not even when X has good dimension.
wherek has characteristic p and p > 2. Let e 1 ,...,e p be the standard basis vectors for the vector space V := k p and let B 0 be the standard nondegenerate symmetric bilinear form on k p given by B 0 (e i , e j ) = δ ij .NowletY be S 2 (V ) * , the vector space of symmetric bilinear forms on k p ;t h e nG acts on Y by (g · B)(v, w) = B(g −1 · v, g −1 · w).IfB ∈ Y then B is nondegenerate if and only if the p × p matrix with i, j-entry B(e i , e j ) has nonzero determinant, so the subvariety X of nondegenerate forms is open and affine. Moreover, X has good dimension since the G-orbits on X all have the same dimension.
The stabilizer G B 0 is the special orthogonal group H := SO p (k),a n dH is G-cr as char(k) = 2 (in fact, H is contained in no proper parabolic subgroups of G,s oH is "Girreducible"). It is easily seen that X H ={ cB 0 | c ∈ k * } and N G (H ) = H ; hence N G (H ) acts trivially on X H . Moreover, X = G · X H . Hence H is a principal stabilizer and X has good dimension for the action of G on X .
Let 0 = B = cB 0 ∈ X H .Defineλ ∈ Y (G) by λ(a) = diag(a −1 ,...,a −1 , a p−1 ) (the diagonal matrix with given entries with respect to the basis e 1 ,...,e p ). Let B 1 ∈ Y be the degenerate form given by B 1 (a 1 e 1 + ··· +a p e p , b 1 e 1 + ··· + b p e p ) = ca p b p . Then for all a ∈ k * , λ(a) · B = a 2 B + (a 2−2 p − a 2 )B 1 .AsX is open in Y , we may identify T B X with T B Y . Making the usual identification of the tangent spaces T 1 k * and T B Y with k and Y , respectively, we see that and to T B (X H ), but not to T B (N G (H ) · B) since the latter tangent space is zero. It follows from Lemma 7.4 that ψ X ,H is not separable.

Examples
The for each a ∈ k * ,t h e nλ(a) · h = ah, so lim a→0 λ(a) · h = 0. It is obvious that 0 is not G-conjugate to h. Note that the same reasoning works for any nonzero multiple of h.Onthe other hand, the N G (H )-orbit of any nonzero multiple of h is finite (and hence closed), and there are therefore infinitely many such closed N G (H )-orbits. Hence the fibre of ψ X ,H over π X ,G (0) is infinite. Note that this example only works in characteristic 2 because it relies on the existence of the H -fixed vector h. This is consistent with the results above, since away from characteristic 2 the image of the adjoint representation of SL 2 (k) in SL 3 (k) is completely reducible-actually, it is irreducible-and hence is SL 3 (k)-cr.

Example 8.2 We now provide an infinite family of examples generalizing the previous one.
In these examples, G is SL m (k) acting on its natural module X ,andH is the image of some reductive group under a representation in SL(X ).SinceG has only one closed orbit in X (the orbit {0}), the quotient X /G is just a single point.
First we consider polynomial representations of GL n (k) where k is an algebraically closed field of positive characteristic p. A good reference for the polynomial representation theory of GL n (k) is the monograph [21]. Further details may also be found in the monograph [20]. (To apply this here one should take q = 1 in the set-up considered there.) Let the characteristic be p > 0a n dl e tG = GL n (k) be the group of n × n-invertible matrices. The irreducible polynomial representations of G are parametrized by partitions with at most n parts. More precisely, let + (n) be the set of partitions λ = (λ 1 ,...,λ n ) with λ 1 ≥ ··· ≥ λ n ≥ 0. We may regard λ as a weight of the standard maximal torus of G: we set λ(t) = t λ 1 1 ...t λ n n . Then for each λ ∈ + (n) there exists an irreducible polynomial G-module L(λ) such that L(λ) has unique highest weight λ and λ occurs as a weight with multiplicity one. The modules L(λ), λ ∈ + (n), form a complete set of pairwise nonisomorphic polynomial irreducible G-modules. We write T for the maximal torus of G consisting of diagonal matrices and B for the subgroup of G consisting of all invertible lower triangular matrices. We shall also need modules induced from B to G.W edenotebyk λ the 1-dimensional rational T -module on which t ∈ T acts as multiplication by λ(t). The action of T on k λ extends to an action of B. For each λ ∈ + (n) the induced module ∇(λ) := ind G B k λ is a non-zero polynomial representation of G.Then∇(λ) is finite-dimensional and contains the irreducible module L(λ): in fact the G-socle of ∇(λ) is L(λ).
We consider the induced GL n (k)-module ∇(n( p − 1)).W eh a v et h a t∇(n( p − 1)) = S n( p−1) E,whereS n( p−1) E is the n( p − 1)th symmetric power of the natural GL n (k)-module E.
We consider the matrix representation obtained by the SL n (k)-module (n( p−1)). Hence we have a group homomorphism where m = dim( (n( p − 1))) = np−1 np−n .LetX = (n( p − 1)) and let H be the image of ρ inside G = SL m (k) = SL(X ). The previous reasoning shows that X is an indecomposable H -module and the trivial module appears in the H -socle of X . The group H is reductive but not G-cr since the representation X is not semisimple.
Since H is reductive we have that N G (H ) 0 = H 0 C G (H ) 0 . Moreover, End H (X ) = End SL n (k) (X ) = k (see [26,Proposition 2.8]); this implies that C G (H ) is finite, so N G (H )/H is finite. Now the quotient X H /N G (H ) is infinite since H fixes a full 1-dimensional subspace of X and N G (H )/H is finite. On the other hand, the quotient X /G is a single point and so the morphism is not a finite morphism.
Note that Example 8.1 above is just this one with p = n = 2.
Let X = (0, 1) and let H be the image of Sp 4 (k) in G = SL 5 (k) with natural module X . Then X is an indecomposable H -module and the trivial module appears in the H -socle of X . The group H is reductive but not G-cr since the representation X is not semisimple.
Since H is reductive we have that N G (H ) 0 = H 0 C G (H ) 0 . Moreover, we have that End H (X ) = End Sp 4 (k) (X ) = k (see [26,Proposition 2.8]), so the only endomorphisms of X as an H -module are the scalars. Since G = SL 5 (k),thismeansthatC G (H ) is finite and so N G (H )/H is finite. Now, as in our previous examples, the quotient X H /N G (H ) is infinite since H fixes a full one-dimensional subspace of X and N G (H )/H is finite, whereas the quotient X /G is a single point. Therefore the morphism ψ X ,H : X H /N G (H ) → X /G is not a finite morphism.

Example 8.4
The above examples show that if H is the image of a non-completely reducible representation of a reductive group in G = GL(X ) or SL(X ) then the conclusion of Theorem 1.1 can fail. On the other hand, if H is the image of a completely reducible representation then we get an easy representation-theoretic proof of Theorem 1.1 in this special case, as follows. If the representation is trivial (of any dimension), so that H is the trivial group, then X H = X and N G (H ) = G,s ot h em a pψ X ,H is the identity map. If the representation is non-trivial and irreducible, then X H ={0} and the map ψ X ,H : X H /N G (H ) → X /G is just the map from a singleton set to a singleton set and hence is finite. If the representation is non-trivial and completely reducible but not irreducible then X H has an H -complement in X :s a y ,X = X H ⊕ W . The centre of the Levi subgroup of G corresponding to the given decomposition normalizes H and acts as scalars on X H ,soX H /N G (H ) is again a singleton set and ψ X ,H is finite.
In this section we consider a separate but related problem, using techniques from earlier sections. Fix a reductive group G, and reductive subgroups H and K of G. The group H × K acts on G by the formula (h, k) · g = hgk −1 ; the orbits of the action are the (H , K )-double cosets and we call this action the double coset action. The stabilizer (H × K ) g is given by We are interested in the following question: when does G have good dimension for the double coset action? Note that, again, in characteristic 0 this problem was solved by Luna in [31]; he showed using étale slices that G always has good dimension for the double coset action. The problem of translating Luna's results to positive characteristic was also studied by Brundan [12][13][14]16], who considered in particular the question of when there is a dense double coset in G. Our main result gives a necessary and sufficient condition for G to have good dimension for the double coset action in terms of the stabilizers of the action. Remarks 9.2 (i). It follows from [37, Theorem 1.1] that in order to show that generic stabilizers are reductive, it is enough to show that (H × K ) g has minimal dimension and is reductive for some g ∈ G.
(ii). Work of Popov [45] implies that if a connected semisimple group G acts on a smooth irreducible affine variety V and the divisor class group Cl(V ) has no elements of infinite order then generic orbits of G on V are closed if and only if generic stabilizers of G on V are reductive. By work of Tange [59, Theorem 1.1], if G is connected then Cl(G) has no elements of infinite order, so Theorem 9.1 follows if H and K are connected and semisimple.
⊓ ⊔ In the special case when A is reductive, the next result is [35,Lemma 4.1].W etakethe opportunity to correct the proof given in loc. cit. Lemma 9.4 Let k ′ be an algebraically closed extension field of k. Let A be a linear algebraic group acting on an affine variety X , and let A ′ (resp. X ′ ) be the group (resp. variety) over k ′ obtained from K (resp. X ) by extension of scalars. Let x ∈ X. Then: Proof (i) Let H = HZ(G) 0 and let K = KZ(G) 0 .LetA = (σ × σ) −1 ((H 1 × K 1 ) σ(g) ),a subgroup of H × K .Defineψ : A → G by ψ( h, k) = g −1 hg k −1 . A short calculation shows that ψ gives a homomorphism from A to Z (G) 0 , with kernel (H × K ) g . Moreover, σ × σ gives an epimorphism from A to (H 1 × K 1 ) σ(g) , with kernel Z (G) 0 × Z (G) 0 .Part(i)now follows.
Since g ∈ C was arbitrary, it now follows that r i=1 (H × K ) · (P λ n i ) contains (H × K ) · C and, since G is connected, (H × K ) · (P λ n i ) is dense in G for at least one i.Notealsothat λ = g 2 · μ = ln i · μ,soμ = n −1 i l −1 · λ = n −1 i · λ,soλ = n i · μ ∈ Y (n i Kn −1 i ). Keeping the notation in the previous paragraph, for each i, (H ×K )·(P λ n i ) is constructible, so (H × K ) · (P λ n i ) is either dense or contained in a proper closed subset of G. Thus the union of those subsets (H × K ) · (P λ n i ) that are dense contains an open subset of G; note also that this union is (H × K )-stable. Since (H × K ) · C is open, we can find g ′ ∈ C such that for any i,ifg ′ ∈ (H × K ) · (P λ n i ) then (H × K ) · (P λ n i ) is dense. By the arguments in the paragraph above applied to g ′ , there exists i such that g ′ ∈ (H × K ) · (P λ n i ) and for this i we have λ = n i · μ ∈ Y (n i Kn −1 i ); moreover, (H × K ) · (P λ n i ) is dense by construction. It follows that (H × n i Kn −1 i ) · P λ = r n i ((H × K ) · (P λ n i )) is dense in G, so the first assertion of part (i) follows with g 0 = n i . It is obvious that (λ, λ) destabilizes g for all g ∈ P λ ,sowe have proved part (i).
If g ∈ C and τ g fixes g then (λ, μ) fixes g 1 ,s og 1 = g 2 ∈ L λ n i for some i.T h efi r s t assertion of (ii) follows by a similar argument to that above but applied to r i=1 (H × K ) · (L λ n i ), and the second assertion is again obvious.
⊓ ⊔ Proof of Theorem 9. 1 We have shown already that (ii) and (iii) are equivalent, so it is enough to prove that (i) and (ii) are equivalent. First note that for any g ∈ G, (H ∩ K ) · g is closed if and only if (H ∩ K ) 0 · g = (H 0 ∩ K 0 ) 0 · g is closed, and H ∩ gKg −1 is reductive if and only if (H ∩ gKg −1 ) 0 = (H 0 ∩ gK 0 g −1 ) 0 is reductive, which is the case if and only if H 0 ∩ gK 0 g −1 is reductive. Hence we can assume that H and K are connected. Moreover, we can assume by Lemma 9.5 that k is uncountable. The implication (i) ⇒ (ii) follows immediately from Lemma 3.3(ii). For the reverse implication, we use induction on dim(G). Suppose generic stabilizers are reductive. The result is immediate if dim(G) = 0. If G is not semisimple then let G 1 , σ , H 1 and K 1 be as in Lemma 9.7. Then generic stabilizers of H 1 × K 1 on G 1 are reductive, by Lemma 9.7(i). Since dim(G 1 )<dim(G), it follows by induction that generic orbits of H 1 × K 1 on G 1 are closed. Part (ii) of Lemma 9.7 now implies that generic orbits of H × K on G are closed, so we are done. Hence we can assume that G is semisimple.
First we consider the case when generic stabilizers of H ×K on G are positive-dimensional. Then all stabilizers of H × K on G are positive-dimensional, by semi-continuity of stabilizer dimension. For each g ∈ G such that (H × K ) g is reductive, choose a nontrivial cocharacter τ g ∈ Y ((H × K ) g ). The fixed point set G τ g := G im(τ g ) is closed, so C g := (H × K ) · G τ g is constructible. Since generic stabilizers of H × K on G are reductive, the constructible sets C g for g ∈ G such that (H × K ) g is reductive cover an open dense subset U of G,by [37,Theorem 1.1]. There are only countably many of these sets, as H × K has only countably many conjugacy classes of cocharacters. By [37,Cor.2.5],C g is dense in G for some g ∈ G.
Hence there exists τ = (λ, μ) ∈ Y (H × K ) such that for generic g ∈ G, g is fixed by an (H × K )-conjugate of τ . It follows from Lemma 9.8 that for some g 0 ∈ G, λ ∈ Y (g 0 Kg −1 0 ) and (H × g 0 Kg −1 0 ) · L λ is dense in G. By Lemma 9.3, there is no harm in assuming that g 0 Kg −1 0 = K -i.e., that λ ∈ Y (K ) and (H × K ) · L λ is dense in G-and we shall do this for notational convenience.
To prove that generic (H × K )-orbits on G are closed, it is therefore enough to show that (H × K ) · l is closed for generic l ∈ L λ .L e tH 2 = L λ (H ) and let K 2 = L λ (K ); then H 2 × K 2 = L (λ,λ) (H × K ). Consider the double coset action of H 2 × K 2 on L λ . Let l ∈ L λ .Then(λ, λ) fixes l,so(H 2 × K 2 ) l = L (λ,λ) ((H × K ) l ), which is reductive if (H × K ) l is. Hence generic stabilizers of (H 2 × K 2 ) on L λ arereductive.AsG is semisimple, dim(L λ )<dim(G), so generic (H 2 × K 2 )-orbits on L λ are closed by induction. It follows from Remark 4.2(iii) that (H × K ) · l is closed for generic l ∈ L λ , so we are done as (H × K ) · L λ is dense in G. Now consider the case when generic stabilizers of H × K on G are finite. Suppose generic (H ×K )-orbits on G are not closed. Then G cl is a proper closed subset of G, so the union of the non-closed orbits contains a nonempty open subset of G. For each g ∈ G such that (H × K )·g is not closed, choose nontrivial τ g ∈ Y (H × K ) such that τ g destabilizes g. By an argument similar to the one in the positive-dimensional case above, there exist λ ∈ Y (H ) and g 0 ∈ G such that λ ∈ Y (g 0 Kg −1 0 ) and (H ×g 0 Kg −1 0 )· P λ is dense in G. As before, we can assume that g 0 Kg −1 0 = K .NowR u (P −λ )(H )P λ (H ) and P λ (K )R u (P −λ )(K ) are nonempty open subsets of H and K respectively [10, Proposition 14.21(iii)], so R u (P −λ )(H )P λ R u (P −λ )(K ) is dense in G,a sHP λ K is. It follows that dim(R u (P −λ )(H )) + dim(R u (P −λ )(K )) + dim(P λ ) ≥ dim(G),sodim(R u (P −λ )(H ))+dim(R u (P −λ )(K )) ≥ dim(G)−dim(P λ ) = dim(R u (P λ )).
By hypothesis, we can choose g ∈ P λ such that (H × K ) g is finite. Write g = ul, where l = L λ and u ∈ R u (P λ );t h e nl = lim a→0 (λ, λ)(a) · g. We show that l is (H × K )-conjugate to g. Consider the double coset action of R u (P λ (H )) × R u (P λ (K )) on G.
Now (H ×K ) l is finite, since l is (H ×K )-conjugate to g.But(λ, λ) fixes l, a contradiction. We deduce that generic (H × K )-orbits on G are closed after all. This completes the proof. ⊓ ⊔ Remark 9.9 One can prove the following more general statement of Theorem 9.1 for nonconnected reductive G.LetG 1 ,...,G r be the minimal subsets of G having the property that each G i is (H × K )-stable and contains some connected component of G. Each G i is a union of certain connected components of G;ifH and K are connected then the G i are precisely the connected components of G. Here is our result: for each i, G i has good dimension for the (H × K )-action if and only if generic stabilizers of H × K on G i are reductive if and only if H ∩ gKg −1 is reductive for generic g ∈ G i . To see this, note first that we can assume that H and K are connected, by the proof of Theorem 9.1; hence we can assume that each G i is a connected component of G. We can now choose g ∈ G such that G i g = G 0 , and use the map r g to translate the case of G i into the case of the connected group G 0 (cf. the proof of Lemma 9.3). We leave the details to the reader.
We record a useful corollary.

Corollary 9.10 Suppose one of H and K is a torus. Then G has good dimension for the (H × K )-action.
Proof This is immediate from Theorem 9.1, since any subgroup of a torus is reductive. ⊓ ⊔ We now consider a concrete example; our methods allow us to deal with arbitrary characteristic. Note that we use Theorem 1.1 in parts (a) and (b) below.

Example 9.11
Let G be simple of type B 2 and fix a maximal torus T of G.L e tA be the subgroup of G generated by the long root groups with respect to T .If p = 2thenletB be the subgroup of G generated by the short root groups with respect to T . The groups A and B are normalized by N G (T ).
(a). Let p be arbitrary and let H = K = A.Sincedim(G) = 10 and dim(H ) = dim(K ) = 6, dim(H × K ) g ≥ 2forallg ∈ G, with equality if and only if (H × K ) · g is dense in G. Let λ ∈ Y (T ) be nontrivial. We show first that for generic l ∈ L λ , (H × K ) · l is closed. If L λ = T or L λ is a long-root Levi subgroup (that is, a Levi subgroup L such that [L, L] is the subgroup of type A 1 corresponding to some long root) then L λ ≤ A,so(H × K ) · l = A is closed. Note that in this case, (H × K ) · L λ = A is not dense in G.
So suppose L λ is a short-root Levi subgroup. As in the positive-dimensional case in the proof of Theorem 9.1, it is enough to show that (L λ (H ) × L λ (K )) · l is closed for We give a direct proof of this. The orbit O is of the form (H × K ) · g,whereg ∈ G has the property that u := ττ g is a regular unipotent element of G and u is inverted by τ (see [17,Proposition 3.1]). In fact, we can choose g to be a regular unipotent element of G such that g 2 = u and τ inverts g (take g to be u s if p > 0, where 2s ≡ 1mod|u|). Set U = g ;then τ normalizes U ,asτ inverts g. There exists λ ∈ Y (G) such that lim a→0 λ(a)g ′ λ(a) −1 = 1 for all g ′ ∈ U . We can choose λ to be optimal in the sense of [9, Defn. 4.4 and Theorem 4.5] (cf. Sect. 2.5). Then τ normalises P λ .No wN Aut(G) (P λ ) is an R-parabolic subgroup of the reductive group Aut(G) [35,Proposition 5.4(a)], so N G (P λ ) = P μ for some μ ∈ Y (G).As τ ∈ P μ and τ is linearly reductive, we can choose μ to centralize τ : that is, we can choose μ to belong to Y (H ).

Applications to G-complete reducibility
We finish with some applications of ideas from Sects. 3 and 9 to G-complete reducibility. Our next lemma gives a basic structural result about G and its subgroups which can quickly be proven using the framework we have now set up; the setting is as in Sect. 9 but more general, since we allow one of the subgroups to be non-reductive (cf. [15]). The argument used is taken from the proof of [29, Kap. III.2.5, Satz 2]; note that although the reference [29] works with groups and varieties defined over the complex numbers, many of the arguments are completely general. For convenience, we reproduce the details here.

Lemma 10.1 Suppose K is a subgroup of G and let H be a reductive subgroup of G that contains a maximal torus of K . Then H K is a closed subset of G.
Proof First suppose that K is unipotent. The quotient X = G/H is affine and H is the stabilizer in G of the point x = π G,H (1) ∈ X .S i n c eK is unipotent, and all orbits for unipotent groups on affine varieties are closed [10, Proposition 4.10], K · x is closed, so KH (and hence HK)isclosedinG by Lemma 3.5. Now, in the general case, let T be a maximal torus of K contained in H and let B be a Borel subgroup of K containing T with unipotent radical U .ThenUH = BH is closed in G by the first paragraph, and the following argument from [29, Kap. III.2.5, Satz 2] gives us what we want. We have a sequence of morphisms where φ(g ′ , g) := (g ′ , g ′ g) for g ′ ∈ K , g ∈ G, π K ,B is the quotient morphism K → K /B and pr 2 is the projection of K /B × G onto the second factor. Let Y = K × BH.SinceBH is closed in G, Y is closed in K × G.Sinceφ is an isomorphism of varieties, φ(Y ) is closed in K × G and therefore ρ(φ(Y )) is closed in K /B × G. Finally, since K /B is complete, pr 2 (ρ(φ(Y ))) is closed in G. But it is easy to see that pr 2 (ρ(φ(Y ))) = KH, so we are done.
⊓ ⊔ The lemma above allows a quick proof of the following result.

Proposition 10.2
Suppose G is reductive, X is an affine G-variety and x ∈ X. If H is a reductive subgroup of G containing a maximal torus of G x ,thenH· xi sc l o s e di nG· x. In particular, if G · x is closed in X then H · xi sc l o s e di nX.
Remark 10.7 Note that every pair (G, H ) of reductive groups with H ≤ G is a reductive pair in characteristic 0 and the separability hypothesis is also automatic. In characteristic p > 0, every subgroup of G is separable as long as p is "very good" for G;see [7,Theorem 1.2].
As a final remark, we note that there are Lie algebra analogues of Propositions 10.5 and 10.6, where we replace the subgroup A with a Lie subalgebra of Lie(H ). For details of how to make such translations, see [9,Sect. 5], for example.