On the topology of geometric and rational orbits for algebraic group actions over valued fields

Lemma 1.1 (see [12, Lemma C.4.1]).

Let X be an affine k-scheme. Then there is a unique geometrically reduced closed subscheme X+ of X such that

for all separable extensions F/k. The formation of X+ is functorial in X and commutes with the formation of products over k and separable extensions of the base field.

Definition 1.2 (Gabber’s condition, see [14], [15, Definition 2.4.3]).

We say that a k-group G satisfies the (*)-condition (or the Gabber condition) if all k¯-tori of Gk¯ are tori of (G+)k¯.

Definition 1.3 (see [10, p. 7]).

Assume that G is a k-group scheme, X is a G-scheme, and let π:X→Y be a G-invariant morphism, where Y is a scheme. We say that X is an fppf G-torsor over Y if π is faithfully flat and the morphism

is an isomorphism. Here p2 is the projection onto the second coordinate.

Remark 1.4.

Let G be a smooth k-group of finite type acting on a variety V defined over k, and let v be a k-point. Then it is well known that G.v is open in its closure G.v¯. Particularly, G.v is a locally closed subvariety of V. Furthermore, by [13, Chapter 3, Proposition 2.2], the orbit map G→G.v, g↦g.v is also a Gv-torsor. Here Gv is considered as a k-group scheme which is not necessarily smooth, and the action of Gv on G is given by h⋅g:=gh-1. In fact, the orbit map G→G.v is surjective by construction, and flat over a non-empty open subset of G.v by generic flatness. On the other hand, since the map is G-equivariant, it is also flat everywhere. So the orbit map G→G.v is faithfully flat. Furthermore, the natural morphism Gv×G→G×G.vG, (g,x)↦(g⋅x,x), here g⋅x:=xg-1, is an isomorphism by checking directly that the morphism induces a bijection on R-points for any k-algebra R. (We thank M. Brion for this argument.)

Theorem 1.5 (see [15, Sections 1.2, 1.4] and [16, Theorem 7.2.1]).

Suppose that k=(k,v) is an admissible valued field, and let G be an affine algebraic k-group scheme. Let f:X→Y be an (fppf) G-torsor where X,Y are algebraic k-varieties.

1. The image I:=f(X(k)) is locally closed in Y⁢(k)Top, and the induced map X⁢(k)Top→I is a principal topological G⁢(k)Top-fibration. In particular, the mapping fTop is strict.

2. If G satisfies the condition (*), I is clopen (closed and open) in Y⁢(k)Top.

Theorem 1.6 (see [6, Theorem 1.2] for the case of very good characteristics and [17, Theorem 3.3]).

Let G be a smooth connected reductive algebraic group over k. Assume the characteristic of k is zero or pretty good. Then all centralizers of closed subgroup schemes in G are smooth, or equivalently, all closed subgroup schemes are separable.

Corollary 1.7 (see [6, Corollary 3.4] for the case of very good characteristics).

Let G be a smooth connected reductive algebraic group, and suppose that p is pretty good for G. Let g1,…,gn∈G, and let x1,…,xm∈g:=Lie(G). Then the orbit maps G→G.(g1,…,gn) and G→G.(x1,…,xm) are separable.

Lemma 1.8 (see [17, Lemma 3.5]).

Let H be a closed subgroup scheme of GLn. Then the centralizer CGLn⁢(H) is smooth.

Corollary 1.9.

With the assumption as in Corollary 1.7, let G=GLn be the general linear group. Then the orbit maps G→G.(g1,…,gn), G→G.(x1,…,xm) are separable.

Definition 1.10.

We say that the orbit G⁢(k).v is cocharacter-closed over k if whenever limα→0⁡μ⁢(α).v=v′ (not necessarily in G.v) exists for some k-defined cocharacter μ, then v′∈G⁢(k).v. Here a k-defined cocharacter μ of G is a homomorphism μ:𝔾m→G defined over k.

Theorem 1.11 (see [4, Theorem 1.5, Corollary 7.2]).

If (Gv)k¯,red is defined over k, then the descent property of cocharacter-closedness is true.

Definition 1.12.

Let λ:𝔾m→G be a cocharacter. We say that P is the R-parabolic subgroup relative to λ if P is the parabolic subgroup given by

The R-Levi subgroup corresponding to the cocharacter λ is the subgroup of the form

It is worth noting that if G is connected, the notion of R-parabolic subgroup matches the usual notion of parabolic subgroup.

Definition 1.13.

A subgroup H of G is called G-completely reducible (G-cr) if, whenever H is contained in an R-parabolic subgroup P of G, there exists an R-Levi L subgroup of P containing H. Similarly, a subgroup H of G is called G-completely reducible over a field k if, whenever H is contained in a k-defined R-parabolic subgroup P of G, there exists a k-defined R-Levi subgroup L of P containing H.

Definition 1.14 ([4, Definition 9.2]).

Let H be a subgroup of G, and let G↪GLm be an embedding (over k¯) of algebraic groups. Then h¯∈Hn is called a generic tuple of H with respect to the embedding G↪GLm if we have 〈h¯〉=〈H〉 in Matm. We call h¯∈Hn a generic tuple of H if it is a generic tuple of H for some embedding G↪GLm. Here 〈h¯〉 (resp., 〈H〉) is the subalgebra of Matm generated by h¯ (resp., H).

Theorem 1.15 (see [4, Theorem 9.3]).

Let H be a subgroup of G (not necessarily connected), h¯∈Hn a generic tuple of H, and consider the action of G on Gn by simultaneous conjugation. Then H is G-completely reducible over k if and only if G⁢(k).h¯ is cocharacter-closed over k.

Theorem 1.16.

There are examples that H≤G is G-cr over k¯ but not G-cr over k.

1. Let k be an imperfect field of characteristic 2 , and let G be the split simple group of exceptional type E6,E7,E8,G2. Then there exists a k-defined subgroup H of G such that H is G-cr over k¯ but not G-cr over k.

2. (see [23, Theorem 1.2]) Let k be a nonperfect separably closed field of characteristic 2 , and let G be a simple k-group of type D4. Then there exists a k-subgroup H of G that is G-cr over k¯ but not G-cr over k.

Theorem 4.1 (Relative version of Kostant–Rosenlicht’s theorem).

Assume that k=(k,v) is an admissible valued field, and let G=U be a smooth unipotent group acting on an affine variety V defined over k, and v∈V⁢(k) is a rational point. Then G⁢(k).v is always closed with respect to the induced topology from k.

Proof.

First, by Remark 1.4, for any action of smooth k-group G on V, the orbit map f:G→G.v is an (fppf) Gv-torsor. Further, since U is unipotent, the stabilizer Uv is also unipotent (not necessarily smooth). Since the k¯-tori in Uk¯ are trivial, the k¯-tori of (Uv)k¯ are trivial. It follows that Gv satisfies the Gabber condition. So, by Theorem 1.5 (2), I=f⁢(U⁢(k))=U⁢(k).v is clopen in (U.v)(k). Since U.v is always closed by Kostant–Rosenlicht’s theorem, the set (U.v)(k) is closed in V⁢(k). Thus, U⁢(k).v is always closed when U is unipotent and k is an admissible (e.g., local) valued field. ∎

Example 4.2.

Consider the field k defined by F. K. Schmidt, and mentioned in [16, Subsection 6.4.3]. Namely, choose an element s∈𝔽p⁢[[t]] which is transcendental over 𝔽p⁢(t). We consider the subfield 𝔽p⁢(t,sp) in 𝔽p⁢((t)) and let k:=𝔽p(t,sp)h be the separable closure of 𝔽p⁢(t,sp) in 𝔽p⁢((t)). Then k is henselian and separable over 𝔽p⁢(t,sp), but the completion k^=𝔽p⁢(t,sp)^ is not separable over k. This implies that kp is not Hausdorff closed in k (see [16, Lemma 6.4.1]). Let G=𝔾a be the additive group acting on X=𝔸1 as g.v=gp+v. Then, for v=0, the stabilizer is Gv={gp=0}=αp with the reduced part (Gv)k¯,red={g=0} defined over k, but the rational orbit G⁢(k).v={gp∣g∈k}=kp is not Hausdorff closed in k. Likewise, this example also shows that the relative version of Kostant–Rosenlicht’s theorem is not true if we only assume the henselian condition for k.

Theorem 4.3.

Let k=(k,v) be an admissible valued field, and let G be a smooth commutative group acting on an affine variety V, all defined over an admissible valued field k. Assume that v∈V⁢(k). Then the following types of closedness of orbits are equivalent:

1. G.v is closed in V with respect to the Zariski topology.

2. G⁢(k).v is closed in V⁢(k) with respect to the topology induced from k.

Proof.

(a) ⇒ (b): This is similar to the proof of Theorem 4.1. By Remark 1.4, we consider the orbit map f:G→G.v which is also an (fppf) Gv-torsor. Since G is commutative, the stabilizer Gv is also a commutative group scheme (not necessarily smooth). We claim that Gv satisfies the Gabber condition. Indeed, since Gv is commutative, its base change (Gv)k¯ contains a unique maximal torus T. By [12, Lemma C.4.4], the unique maximal torus T of (Gv)k¯ is defined over k, i.e. T=(T1)k¯, where T1 is the maximal torus of Gv. Furthermore, since any torus is geometrically reduced, T1 is smooth, and hence T1⊆Gv+. Therefore, T=(T1)k¯ is a torus of (Gv+)k¯. Thus the unique maximal torus of (Gv)k¯ is contained in (Gv+)k¯. It follows that Gv satisfies the Gabber condition. Taking Theorem 1.5 (2) into account, we get the orbit G⁢(k).v is clopen in (G.v)(k). Hence, from the assumption that G.v is Zariski closed, the relative orbit G⁢(k).v is also (Hausdorff) closed.

(b) ⇒ (a): This fact is implied by [2, Proposition 3.2.2]. In fact, this implication still holds for any nilpotent group. ∎

Remark 4.4.

(a) Theorem 4.3 is an extension of our previous result in [1, Theorem, p. 1062] which says that the assertion holds for groups of multiplicative type. Furthermore, we note that Theorem 4.3 is not true if we only assume that k is henselian. Indeed, we consider k as in Example 4.2. We see from Example 4.2 that kp is not closed in k; thus k×,p is not closed in k×. So we choose the action of 𝔾m on V:=𝔸1∖{0} by g.v=gp⁢v and let v=1. Hence, the (schematic) stabilizer Gv=μp={gp=1} contains the trivial reduced part (Gv)k¯,red={*} which is k-defined. But in this case, the relative orbit G⁢(k).v=k×,p is not closed in V⁢(k)=k×.

(b) By the same argument as in the proof of Theorem 4.3, we see that if Gv is commutative and k is an admissible valued field, we have the (Zariski) closedness of geometric orbit G.v deduces the (Hausdorff) closedness of G⁢(k).v. We note that this conclusion does not hold if we only assume the commutativity of the group of geometric points Gv⁢(k¯), or the commutativity of the reduced part (Gv)k¯,red of the stabilizer. Indeed, in Example 2.4, the geometric points (as well as the reduced part) of stabilizer is commutative,

but the stabilizer

is not a commutative group scheme (see Example 2.4 (4)). In fact, this group is solvable of dimension 1 with trivial largest smooth subgroup (Gv)+={e} and does not satisfy the Gabber condition.

(c) One may ask the equivalence between Zariski closedness of G.v and Hausdorff closedness of G⁢(k).v for nilpotent groups G. It would lead to the problem of studying Gabber’s condition for nilpotent group schemes.