Small G-varieties

An affine varieties with an action of a semisimple group $G$ is called"small"if every non-trivial $G$-orbit in $X$ is isomorphic to the orbit of a highest weight vector. Such a variety $X$ carries a canonical action of the multiplicative group $\mathbb{K}^*$ commuting with the $G$-action. We show that $X$ is determined by the $\mathbb{K}^*$-variety $X^U$ of fixed points under a maximal unipotent subgroups $U$ of $G$. Moreover, if $X$ is smooth, then $X$ is a $G$-vector bundle over the quotient $X// G$. If $G$ is of type $A_n$ ($n>1$), $C_n$, $E_6$, $E_7$ or $E_8$, we show that all affine $G$-varieties up to a certain dimension are small. As a consequence we have the following result. If $n>4$, every smooth affine $SL_n$-variety of dimension $<2n$ is an $\mathrm{SL}_n$-vector bundle over the smooth quotient $X//\mathrm{SL}_n$, with fiber isomorphic to the natural representation or its dual.


Introduction
Our base field K is algebraically closed of characteristic zero.If a semisimple algebraic group G acts on an affine variety X, then the closure of an orbit Gx is a union of G-orbits and contains a unique closed orbit.A very interesting special case is when the closure is the union of the orbit Gx and a fixed point x 0 ∈ X: Gx = Gx ∪ {x 0 }.Such an orbit is called a minimal orbit.It turns out that this condition does not depend on the embedding of the orbit Gx into an affine G-variety.In fact, the minimal orbits are isomorphic to highest weight orbits O λ in irreducible representations V λ of G.
If a G-variety X is affine and all orbits in X are either minimal or fixed points, then the variety X is called small.
The following result shows that smooth small G-varieties have a very special structure.
Theorem 1.1.Let G be a simple group and X a smooth irreducible small G-variety.Then G ≃ SL n or G ≃ Sp 2n , and the algebraic quotient X → X/ /G is a G-vector bundle with fiber • the standard representations K n or its dual • the standard representation K 2n if G = Sp 2n .In particular, every fiber is the closure of a minimal orbit.
For G = SL n or G = Sp 2n it turns out that an affine G-variety is small if its dimension is small enough.More precisely, we have the following result.
(1) For n ≥ 5 an irreducible affine SL n -variety X of dimension < 2n − 2 is small.In particular, if X is also smooth, then X is an SL n -vector bundle over X/ / SL n with fiber K n or (K n ) ∨ .(2) For n ≥ 3, an irreducible affine Sp 2n -variety X of dimension < 4n − 4 is small.In particular, if X is also smooth, then it is an Sp 2n -vector bundle over X/ / Sp 2n with fiber K 2n .
In general, we have the following theorem about the structure of a small G-variety where G is a semisimple algebraic group.As usual, we fix a Borel subgroup B ⊂ G and a maximal unipotent subgroup U ⊂ B. For a simple G-module V λ of highest weight λ we denote by O λ ⊂ V λ the orbit of highest weight vectors, and by P λ the corresponding parabolic subgroup, i.e. the normalizer of V U λ .For any minimal orbit O there is a well-defined cyclic covering O λ → O where λ is an indivisible dominant weight, i.e. λ is not an integral multiple of another dominant weight.This λ is called the type of the minimal orbit O.
An action of a reductive group G on an affine variety X is called fix-pointed if the closed orbits are fixed points.
Theorem 1.3.Let X be an irreducible small G-variety.Then the following holds.
(1) The G-action is fix-pointed and in particular X G ∼ − → X/ /G.(2) All minimal orbits in X have the same type λ.
(3) The quotient X → X/ /U − restricts to an isomorphism X U ∼ − → X/ /U − .In particular, X is normal if and only if X U is normal.
(4) There is a unique K * -action on X which induces the canonical K * -action on each minimal orbit of X and commutes with the G-action.Its action on X U is fix-pointed, and (5) The morphism G × X U → X, (g, x) → gx, induces a G-equivariant isomorphism where K * acts on O λ by (t, x) → λ(t −1 ) • x. (6) We have Stab G (X U ) = P λ , and the G-equivariant morphism is proper, surjective and birational, and induces an isomorphism between the algebras of regular functions.
The proofs are given in Proposition 4.3 for the statements (1)-(3) and in Proposition 4.4 for the statements (4)-(6).We define the canonical K * -action on a minimal orbit in Section 2.4.
As a consequence, we obtain the following one-to-one correspondence between irreducible small G-varieties of a given type and certain irreducible fix-pointed affine K * -varieties.The K * -action on a variety Y is called positively fix-pointed if for every y ∈ Y the limit lim t→0 ty exists and is therefore a fixed point.
Corollary 1.4.For any indivisible highest weight λ ∈ Λ G , the functor F : X → X U defines an equivalence of categories irreducible small G-varieties X of type λ F −→ irreducible positively fix-pointed affine K * -varieties Y .
The inverse of F is given by Y → O λ × K * Y where the K * -action on O λ × Y is defined as t(v, y) → (λ(t −1 )v, ty).
Our Theorem 1.1 above is a special case of the following description of smooth small Gvarieties.
Theorem 1.5 (see Theorem 4.12).Let X be an irreducible small G-variety of type λ, and consider the following statements.
(i) The quotient π : X → X/ /G is a G-vector bundle with fiber V λ .
(ii) K * acts faithfully on X U , the quotient X U → X U / /K * is a line bundle, and V λ = O λ .
(iii) The quotient X U \ X G → X U / /K * is a principal K * -bundle, and V λ = O λ .
(iv) The closures of the minimal orbits of X are smooth and pairwise disjoint.
(v) The quotient morphism π : X → X/ /G is smooth.Then the assertions (i) and (ii) are equivalent and imply (iii)-(v).If X (or X U ) is normal, all assertions are equivalent.
Furthermore, X is smooth if and only if X/ /G is smooth and π : X → X/ /G is a G-vector bundle.
In order to see that small-dimensional G-varieties are small (see Theorem 1.2) we have to compute the minimal dimension d G of a non-minimal quasi-affine G-orbit.In fact, if the dimension of the affine G-variety X is less than d G , then every orbit in X is either minimal or a fixed point, hence X is small.
We define the following invariants for a semisimple group G.The following theorem lists m G , d G and r G for the simply connected simple groups, and also gives the closure O of the minimal orbits realizing m G and the reductive subgroup H of G realizing r G .In the last column the null cone N V appears only if N V V .
Theorem 1.6.Let G be a simply connected simple group.Then the invariants m G , r G , d G are given by the following table.In particular, d G = r G except for E 7 and E 8 .The third and last columns of Table 1 will be provided by Lemma 5.3, the fourth column by Proposition 5.7 and the fifth and sixth columns by Lemma 5.5.
Note also that Theorem 1.2 is a consequence of Theorem 1.1 and Theorem 1.6, because X is a small G-variety in case dim X < d G .

Minimal G-orbits
In this paragraph we introduce and study minimal orbits of a semisimple group G.We will use the standard notation below and refer to the literature for details (see for instance [Bor91,FH91,Hum78,Hum75,Jan03,Kra84,Pro07]).
Let G be a semisimple group.We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ B, and denote by U := B u the unipotent radical of B.
2.1.Highest weight orbits.Let Λ G ⊂ X(T ) := Hom(T, K * ) be the monoid of dominant weights of G.A simple G-module V is determined by its highest weight λ ∈ Λ G which is the weight of the one-dimensional subspace V U , and we write V = V λ .The dual module of a G-module W will be denoted by W ∨ , and for the highest weight of the dual module V ∨ λ we write λ ∨ .Remark 2.1.Define Λ := r i=1 Nω i ⊆ Λ ⊗ Z Q where ω 1 , . . ., ω r are the fundamental weights.We have Λ G ⊆ Λ with equality if and only if G is simply connected.In general, let γ : G → G be the universal covering, i.e.G is simply connected and γ is a homomorphism with a finite kernel F , then F is contained in the center Cent G of G which acts canonically on Λ, and we have Λ G = X(T ) ∩ Λ = Λ F .
For an affine G-variety X, we denote by π : X → X/ /G the algebraic quotient, i.e. the morphism defined by the inclusion O(X) it is closed and contains with any v the line Kv ⊂ V .
Let V = V λ be a simple G-module of highest weight λ ∈ Λ G .Then dim V U = 1, and we define the highest weight orbit to be O λ := Gv ⊂ V where v ∈ V U \ {0} is an arbitrary highest weight vector of V .It is a cone, i.e. stable under scalar multiplication.These orbits and their closure have first been studied in [VP72].
For a subset S of a G-variety X, the normalizer of S is defined in the usual way: Norm G (S) := {g ∈ G | gS = S}.The group of G-equivariant automorphisms of X will be denoted by Aut G (X).
Lemma 2.2.Let V = V λ be a simple G-module of highest weight λ, and let v ∈ V U be a highest weight vector.Then the following holds. ( In particular, O λ is not affine. (3) We have O U λ = K * v, and so Proof.(1)-(2) These two statements can be found in [VP72, Theorem 1 and 2]. ( ) G acts on the projective space P(V ), and the projection p : V \ {0} → P(V ) is Gequivariant and sends closed cones to closed subsets.In particular, p(O λ ) = G p(v) is closed, and so The covariant ϕ k : V λ → V kλ is a finite morphism of degree k and induces a bijective morphism φk : , where µ k acts by scalar multiplication on V λ .
In particular, the induced map ) is closed, and the fibers of ϕ k are the µ k -orbits.This yields the first statement.The last statement follows from the fact that O kλ is normal, by Lemma 2.2(1).
Remark 2.4.The following remarks are direct consequences of the lemma above.
(1) For k > 1 we have ϕ k (V λ ) V kλ , because the quotient V λ /µ k is always singular in the origin.In particular, dim the origin, by Lemma 2.2(7).
The following lemma states that orbits of the form O λ are minimal among G-orbits.
Lemma 2.5.Let W be a G-module and w ∈ W a nonzero element.If p : W ։ V is the projection onto a simple factor V ≃ V λ of W such that p(w) = 0, then dim Gw ≥ dim O λ .
Proof.If v := p(w) = 0, then dim Gw ≥ dim Gv > 0. Hence we can assume that W = V is a simple G-module and p the identity map.
Given a closed subset Y ⊂ V of a vector space, one defines the associated cone CY ⊂ V to be the zero set of the functions gr f , f ∈ I(Y ) ⊂ O(V ), where gr f denotes the homogeneous term of f of maximal degree.If Y is irreducible, G-stable and belongs to a fiber π −1 (z) of the quotient morphism π : V → V / /G, then CY ⊆ N V , and CY is G-stable and equidimensional of dimension dim Y , see [BK79,§3].Lemma 2.2(6) now implies that the highest weight orbit O ⊂ V belongs to CY , and the claim follows.
Example 2.6.The simple SL 2 -modules are given by the binary forms The form y m ∈ V m is a highest weight vector whose stabilizer is fixed by the diagonal torus T ⊂ SL 2 , and the orbit O = SL 2 x k y k is closed and isomorphic to SL 2 /T for odd k and to SL 2 /N for even k where N ⊂ SL 2 is the normalizer of T .It is easy to see that in both cases the associated cone CO is equal to O m .

2.2.
Stabilizer of a highest weight vector and coverings.Let O λ = Gv ⊂ V λ be a highest weight orbit where v ∈ V U λ .We have seen in Lemma 2.2(4) that It follows that the weight λ extends to a character of P λ defining the action of P λ on Kv: Note that G v = ker λ, and so For an affine algebraic group H, we denote by H • its connected component.
Example 2.11.Let V ε 1 = K n be the standard representation of SL n .For any k ≥ 1 the minimal orbit 1 where e 1 = (1, 0, . . ., 0), and O ε 1 = K n \ {0} → O kε 1 is the universal covering which is cyclic of degree k and extends to a finite morphism where k 1 , . . ., k m are coprime and all k i > 1.For w = (e k 1 1 , . . ., e km 1 ) ∈ W we have an SL n -equivariant isomorphism O ε 1 ∼ − → O := SL n w which extends to a bijective morphism ϕ : V ε 1 → O.But ϕ is not an isomorphism, because T 0 O is a submodule of W , hence cannot be isomorphic to V ε 1 .In particular, O is not normal.The fixed point set O U is the cuspidal curve given by the image of the bijective morphism K → K m , c → (c k 1 , . . ., c km ) which shows again that O is not normal.
The following result collects some important properties of minimal orbits.
Proposition 2.12.Let X be an affine G-variety and O ⊂ X a G-orbit. ( In that case, λ is indivisible.
For the proof we will use the following lemma.
Lemma 2.13.Let X, Z be affine G-varieties and O ⊂ Z a G-orbit.Assume that O \ O is a fixed point in Z G , and denote by η : Y → O the normalization.
(1) The morphism η induces an isomorphism η −1 (O) (3) The orbit O is a minimal orbit, as well as its image ϕ(O) ⊂ X, and both have the same type.
Proof. ( 1) and X is affine, the G-equivariant morphism ϕ : O → X induces a G-equivariant morphism φ : Y → X.There is a closed G-equivariant embedding of X into a G-module W , X ֒→ W , and a linear projection pr V λ : W → V λ onto a simple G-module V λ such that ϕ(O) is not in the kernel of pr V λ .
Set ψ := pr V λ • φ : Y → V λ .Since a unipotent group U does not have isolated fixed points on an irreducible affine U -variety (see e.g.[Kra16, Theorem 5.8.8]), we get O U = ∅, and so ψ(O) U = ∅.This implies that ψ(O) = O λ and ψ(Y ) = O λ .We have ψ −1 (0) = {y}, and so ψ is finite and surjective.In particular, O is a minimal orbit of the same type as O λ , by Lemma 2.7(3).From the factorization we see that both maps are finite, and so ϕ(O) is a minimal orbit as well, of the same type as O λ , again by Lemma 2.7(3).
(3) We can assume that Thus, O 1 and O 2 have the same type by Remark 2.10(1).Moreover, n and hence O λ are smooth, and so O λ = V λ by Lemma 2.2(7).In particular, λ is indivisible by Lemma 2.7(2d).The other implication is obvious.
2.4.The canonical K * -action on minimal orbits.In this section we show that there exists a unique K * -action on every minimal orbit O with the following properties.(2) The G-equivariant morphisms η and η are unique, up to scalar multiplication. ( (5) The action by scalar multiplication on O λ corresponds to the representation of Proof.
(1) This follows from Proposition 2.12(2) and the fact that (2) This is clear since Aut which is generated by the scalar multiplication, see Lemma 2.2(5).
(4) The formula obviously holds for the morphism minimal orbits commutes with the scalar multiplication.This follows from (1) and the fact that the scalar multiplication is the unique faithful K * -action on O commuting with the G-action such that the limits lim t→0 t • y exist in O n .
(5) This is clear from (3)&(4): the scalar multiplication on V λ induces the multiplication by t −n on the homogeneous component of O(V λ ) of degree n.
Using this result we can now define the canonical K * -action on minimal orbits.(2) (3) The canonical K * -action on O extends to O.
(4) For any x ∈ O the limit lim t→0 t ℓ • x exists in O and is equal to x 0 .In particular, the canonical K * -action on O extends to an action of the multiplicative semigroup (K, •).Proof.
(1) The first claim follows from Lemma 2.14(5) and obviously implies the second.
(2) This is an immediate consequence of Lemma 2.14, statements (3) and (4). ( Since the homogeneous components V kλ ∨ are simple and pairwise non-isomorphic G-modules we see that O(O) is a graded subalgebra, hence stable under the canonical K * -action.
(4) This obviously holds for the scalar multiplication on O λ ⊂ V λ , hence in the case where O is normal.By (3) it is true in general. ( . This shows that the action of P λ on O U is given by the canonical K * -action.It follows from

Graded G-algebras
Let G be a semisimple group.An affine G-varieties whose nontrivial G-orbits are minimal orbits is called a small G-variety.We will show that the coordinate ring of a small G-variety is a graded G-algebra, a structure that we introduce and discuss in this paragraph.
As in the previous section, we fix a Borel subgroup B ⊂ G, a maximal torus T ⊂ B and denote by U := B u the unipotent radical of B which is a maximal unipotent subgroup of G.
Definition 3.1.A finitely generated commutative K-algebra R with a unit 1 = 1 R , equipped with a locally finite and rational action of G by K-algebra automorphisms is called a G-algebra.
For any G-algebra R we have the isotypic decomposition , and so it is a graded G-algebra of type λ ∨ 0 .Note that, by Definition 2.15, this grading is induced by the canonical Definition 3.3.Let H be a group, and let W be an H-module.Define and denote by π H : W → W H the projection.Then π H has the universal property that every H-equivariant linear map ϕ : W → V where V carries the trivial action of H factors uniquely through π H .We call π H : W → W H the universal H-projection or simply the H-projection.If another group N acts linearly on W commuting with H, then N acts linearly on W H and π H is N -equivariant.Note that if W is finite dimensional, then π H is the dual map to the inclusion (W ∨ ) H ֒→ W ∨ .
Example 3.4.Let V be a simple G-module of highest weight λ and consider the universal U -projection π U : V → V U with respect to the action of the maximal unipotent group U ⊂ G. Since T normalizes U we see that π U is T -equivariant and that the kernel is the direct sum of all weight spaces of weight different from the lowest weight −λ ∨ .If U − ⊂ G denotes the maximal unipotent subgroup opposite to U , then V U − is the lowest weight space and thus the composition Lemma 3.5.Let R be a graded G-algebra.Then the kernel of the universal U -projection π U : R → R U is a graded ideal, and the composition Example 3.6.Let X be an affine G-variety and assume that O(X) is a graded G-algebra.
Then O(X U ) = O(X) U and quotient map X → X/ /U induces an isomorphism X U ∼ − → X/ /U − .In fact, we have O(X U ) = O(X)/ √ I where I is the ideal generated by the linear span Lemma 3.7.Let ϕ : R → S be a G-equivariant linear map between G-modules.If the induced linear map ϕ U : R U → S U or ϕ U : R U → S U is injective, then ϕ is injective.In particular, Proof.Let V ⊂ ker ϕ be a nontrivial simple G-submodule.Then V U and V U are both nontrivial.The claims follow if we show that V U ⊂ ker ϕ U and that Now consider the action of G × G on G by left-and right-multiplication, i.e.
(g, h) With respect to this action one has the following well-known isotypic decomposition: This means that the only simple G×G-modules occurring in O(G) are of the form V ⊗V ∨ , and they occur with multiplicity 1.The embedding The action of U ⊂ G on G by right-multiplication induces a G-equivariant isomorphism O(G/U ) ≃ O(G) U with respect to the left-multiplication of G on G/U and on G, and we obtain the following isomorphisms of G-modules  Let ε : O(G/U ) → K denote the evaluation map f → f (eU ).This is the comorphism of the inclusion ι : {eU } ֒→ G/U .Lemma 3.9.The induced linear map Proof.We first consider the evaluation map ε : O(G) → K, f → f (e), which is the comorphism of the inclusion ι : {e} ֒→ G.We claim that on the isotypic components One can use the isomorphisms ελ to define elements f λ := ε −1 λ (1) ∈ O(G/U ) U − with the following properties: f λ • f µ = f λ+µ and f 0 = 1.This means that they form a multiplicative submonoid of O(G/U ) U − isomorphic to Λ G .In fact, there is a canonical isomorphism 3.2.The structure of a graded G-algebra.It is a basic fact from highest weight theory that the structure of a G-module M is completely determined by the T -module structure of M U .In this section we show that the structure of a graded G-algebra R is completely determined by the structure of R U or of R U − as a T -algebra.
Theorem 3.10.Let R be a G-module.Then there are two canonical G-equivariant isomorphisms where the T -action on O(G/U ) is by right-multiplication and on R U , R U − induced by the G-action on R. If R is a graded G-algebra, then Ψ and Ψ ′ are isomorphisms of K-algebras.
For the proof we introduce an intermediate In particular, we have isomorphisms Lemma 3.11.There is a canonical G-equivariant isomorphism where T acts by right-multiplication on O(G/U ).Then there are canonical T -equivariant isomorphisms We first show that for every dominant weight λ there is a canonical isomorphism It is easy to see that the map ϕ λ : Hom G (O(G/U ) λ , R) → (R λ ) U defined by α → ᾱ(1) has the required properties.
(2) Next we show that for every dominant weight λ there is a canonical isomorphism Here we us the elements f λ := ε −1 λ (1) defined after Lemma 3.9, and set ψ λ (α) := α(f λ ).Now the claim follows from (1), because ε λ (f λ ) = 1 and so π λ (α(f λ )) = α(1), i.e. πλ • ψ λ = ϕ λ where πλ : R U Proof of Theorem 3.10.From Lemma 3.11 above we obtain a canonical isomorphism (O(G/U )⊗ A R ) T ∼ − → R of G-modules.Now the first part of the theorem follows from Proposition 3.12.For the last claim, we have to work out the multiplication * on A = A R given by the isomorphism ψ By construction, γ is G-equivariant and has the property that γ(p by uniqueness, and so ( †) follows.
Remark 3.13.We will later need an explicit description of the isomorphism Ψ from Theorem 3.10.Let f ∈ O(G/U ) λ , h ∈ (R λ ) U .Proposition 3.12 shows that there is a unique Gequivariant homomorphism α : O(G/U ) λ → R λ such that π λ (α(f )) = h, and then Ψ(f ⊗ h) = α(f ) by Lemma 3.11: 3.3.Deformation of G-algebras.In this subsection we give an application of the methods developed above.The results are interesting in their own, but they will not be used in the remaining part of the paper. 1et R be a G-algebra with isotypic decomposition R = λ∈Λ G R λ .We define a graded G-algebra gr R in the following way.As a G-module, we set gr R := λ∈Λ G R λ , and the multiplication is defined by the symmetric bilinear map It is not difficult to see that this multiplication is associative, hence defines a K-algebra structure on gr R such that gr R becomes a graded G-algebra.We now generalize Theorem 3.10 to non-graded G-algebras.
Proposition 3.14.For any G-algebra R there is a canonical G-equivariant isomorphism of K-algebras The definition of the multiplication on gr R implies that the subalgebra (gr R) Applying Theorem 3.10 to the graded G-algebra gr R, we get hence the claim.
Lemma 3.15.Let R be a G-algebra.There exists a K[t]-algebra R with the following properties: (1) R is a free K[t]-module and, in particular, flat over K[t]; (2) R/m 0 R ≃ gr R; (3 On Λ G we have a partial ordering: -module with basis {t λ r λ,j } λ,j where {r λ,j } j is a basis of R λ , proving (1).
We have .
Remark 3.16.Let X be variety.For simplicity we assume that X is affine.Then a flat family (A x ) x∈X of finitely generated K-algebras is a finitely generated and flat O(X)-algebra A such that A x := A/m x A where m x is the maximal ideal of x ∈ X.
The above lemma tells us that for a given G-algebra R there is a flat family (R x ) x∈K r of finitely generated G-algebras such that R 0 ≃ gr R and R x ≃ R for all x from the dense open set K r \ i Ke i , where e 1 , . . ., e r is the standard basis of K r .
We say that a property P for finitely generated K-algebras is open if for any flat family Proposition 3.14 together with the Deformation Lemma 3.15 allows to show that certain properties of the U -invariants R U also hold for R.
Example 3.17.The following result is due to Since gr R ≃ R 0 is normal, the Deformation Lemma implies that R x is normal for all x in an open neighborhood W of 0 ∈ K r .Since W meets K r \ i Ke i it follows that R is normal.The argument from this example can be formalized in the following way.
Proposition 3.18.Let P be a property for finitely generated K-algebras which satisfies the following conditions.
(i) P is open; (ii) O(G/U ) has property P; (iii) If R and S have property P, then so does R ⊗ S; (iv) If R is a T -algebra with property P, then R T has property P. Then a finitely generated G-algebra R has property P if R U has property P.
Proof.If R U has property P, then so does R U − .Hence, assumptions (ii)-(vi) imply that (O(G/U ) ⊗ R U − ) T has property P. In particular, gr R has property P by Proposition 3.14.Now (i) implies that R has property P as well.
Another very interesting property satisfying the assumption of the proposition above is that of rational singularities, see [Bou87].

Small G-varieties
Recall that a affine G-variety is small if every nontrivial orbit is a minimal orbit.We will show that the coordinate ring of a small G-variety is a graded G-algebra and then use the results of the previous section to obtain important properties of small G-varieties and a classification.
Remark 4.1.The G-action on a small G-variety X is fix-pointed which means that the closed orbits are fixed points.This has some interesting consequences.For example, it is not difficult to see that for a fix-pointed action the algebraic quotient π : X → X/ /G induces an isomorphism X G ∼ − → X/ /G, cf.For an affine G-variety X we have a canonical G-equivariant morphism and a T -action on G/U × X U given by (t, (gU, x)) → (gt −1 U, tx).They respectively extend to a morphism ϕ : G/ /U × X U → X and a T -action on G/ /U × X U .It follows that ϕ is constant on the T -orbits, and thus induces a G-equivariant morphism Proposition 4.2.Let X be an affine G-variety and assume that O(X) is a graded G-algebra.
Then the canonical morphism Φ : G/ /U × T X U → X is a G-equivariant isomorphism.Its comorphism is the inverse of the isomorphism Ψ from Theorem 3.10.Moreover, the quotient X → X/ /U − induces an isomorphism X U ∼ − → X/ /U − .

Proof. By definition, the comorphism Φ
U , then Φ * is an isomorphism by Lemma 3.7.We have where O(X U ) µ is the T -weight space of O(X U ) of weight µ.Since the evaluation map , is the universal U -projection (Lemma 3.9), we see that the is the U -projection as well, and the claim follows.
It remains to see that Φ * is equal to the inverse of Ψ from Theorem 3.10.Using again Lemma 3.7 it suffices to show that the diagram commutes.This is stated in Remark 3.13.The last claim is proved in Example 3.6.proved 4.2.The structure of small G-varieties.
Proposition 4.3.Let X be an irreducible small G-variety.Then the following holds.
(1) The G-action is fix-pointed, and all minimal orbits in X have the same type λ.
(3) The quotient X → X/ /U − restricts to an isomorphism X U ∼ − → X/ /U − .In particular, X is normal if and only if X U is normal.
We call such a variety X a small G-variety of type λ.
We can assume that X is a closed G-stable subvariety of a G-module W .Let O ⊂ X be a non-trivial orbit.There is a linear projection p : W → V onto a simple G-module V of highest weight λ such that O ker p. Proposition 2.12(5) implies that p(O) = O λ and that O is of the same type as O λ .The same is true for all orbits O ′ from the open subset X ′ := X \ ker p of X.Since X is irreducible all minimal orbits are of type λ.
(2) Let λ be the type of the minimal orbits of X, and let O ≃ O µ be an orbit in X, µ = ℓλ.As explained in Example 3.2, we have with distinct ℓ j and at least one ℓ j = k 1 + k 2 .There exists a non-trivial orbit O such that all functions f 1 , f 2 , h j do not vanish on O.It follows that the restrictions f1 , f2 , hj ∈ O(O) are nonzero, that f1 which contradicts our assumptions on the ℓ j .
Proposition 4.4.Let X be an irreducible small G-variety of type λ.
(1) There is a unique K * -action on X which induces the canonical K * -action on each minimal orbit and commutes with the G-action.The action on X U is fix-pointed, and where K * acts on O λ by the inverse of the scalar multiplication: (t, x) → t −1 • x.
(3) We have Stab G (X U ) = P λ , and the G-equivariant morphism is proper, surjective and birational and induces an isomorphism between the algebras of regular functions.
Proof.(1) By Proposition 4.3(2) O(X) is a graded G-algebra of type λ ∨ .If we define the K *action on O(X) such that the isotypic component of type nλ ∨ has weight n, then this action is fix-pointed and restricts to the the canonical K * -action on the closure of each minimal orbit (Proposition 2.16(3)).Since X is the union of the closures of the minimal orbits, this K * -action is unique.By Proposition 2.16(1) the K * -action and the G-action commute on the closure of every minimal orbit, hence they commute on X.We have X G = (X U ) K * and the K * -action on X U is fix-pointed, since this holds for the closure of a minimal orbit.This implies that − → X U / /K * , and X G ∼ − → X/ /G since the G-action is fix-pointed which yields the remaining claims.
(2) Choose x 0 ∈ O U λ and consider the G-equivariant morphism η : G/ /U → O λ induced by gU → gx 0 .Define D := ker λ ⊂ T .We claim that η is the algebraic quotient under the action of D. In fact, the action of t ∈ T on O(G/U ) µ is by scalar multiplication with µ ∨ (t) (Remark 3.8(1)).Hence, the action of D is trivial if and only if µ is a multiple of λ ∨ .This implies that see Lemma 2.2(2).In particular, the T -action on O λ factors through λ : T → K * and the induced K * -action is the canonical K * -action.Since D acts trivially on By construction, the T -action on O λ × X U is given by t(v, x) = (λ(t) −1 • v, tx), i.e. by the inverse of the canonical K * -action on O λ and the given action on X U .Hence (O λ × X U )/ /T = O λ × K * X U , and the claim follows from Proposition 4.2.
(3) Consider the action of P λ on G × X U given by p(g, x) = (gp −1 , px).Then the action map G × X U → X, (g, x) → gx, factors through the quotient For Θ we have the following factorization: where the first map is a closed immersion and the second an isomorphism.Since G/P λ is complete it follows that Θ is proper.Moreover, Θ is surjective, because every G-orbit meets X U .We claim that Θ induces a bijection G It remains to see that the comorphism of Θ is an isomorphism on the global functions.Let K λ be the kernel of the character λ : P λ → K * .Then G/K λ ≃ O λ , and the action of P λ on G by right-multiplication induces an action of K * = P λ /K λ on G/K λ by right-multiplication corresponding to the canonical action on O λ .This gives the G-equivariant isomorphisms and the claim follows from (2).
Example 4.5.Let X := O µ ⊂ V µ be the closure of the minimal orbit in V µ , and let µ = ℓλ where λ is indivisible.Then X U = K, and from Proposition 4.4(2) we get an isomorphism where K * acts on O λ by the inverse of the canonical action, (t, x) → λ(t −1 ) • x, and by the canonical action on K = O µ which is the scalar multiplication with µ(t).
The second statement of Proposition 4.4 says that a small G-variety X can be reconstructed from the K * -variety X U .In order to give a more precise statement we introduce the following notion.A K * -action on an affine variety Y is called positively fix-pointed if for every y ∈ Y the limit lim t→0 ty exists and is therefore a fixed point.
For a fix-pointed K * -action on an irreducible affine variety Y either the action is positively fix-pointed or the inverse action (t, y) → t −1 y is positively fix-pointed.In fact, for any y ∈ Y either lim t→0 ty or lim t→∞ ty exists.Embedding Y equivariantly into a K * -module one sees that the subsets Y + := {y ∈ Y | lim t→0 ty exists} and Y − := {y ∈ Y | lim t→∞ ty exists} are closed.As Y is irreducible, this yields the claim.(The claim does not hold for connected K * -varieties, as the example of the union of the coordinate lines in the two-dimensional representation t(x, y) := (tx, t −1 y) shows.)Remark 4.6.A positively fix-pointed K * -action on Y extends to an action of the multiplicative semigroup (K, •), and the morphism K × Y → Y , (s, y) → sy, induces an isomorphism where the compositions of the horizontal maps are the identity.
Lemma 4.7.Let Y be a positively fix-pointed affine K * -variety and let λ ∈ Λ G be indivisible.Consider the K * -action on O λ × Y given by t(v, y) := (λ(t) −1 • v, ty).Then is a small G-variety of type λ where the action of G is induced by the action on O λ .Moreover, there is canonical K Proof.By definition, X is an affine G-variety.
hence a minimal orbit of type λ (Proposition 2.12( 5)).As a consequence, X is a small G-variety of type λ.Furthermore, since the canonical K * -action on O λ n commutes with the G-action, we have where the last morphism is given by [t, y] → ty which is an isomorphism, as explained in Remark 4.6 above.

4.3.
Smoothness of small G-varieties.Before describing the smoothness properties of small varieties, let us look at some examples.As before, G is always a semisimple algebraic group.
Remark 4.8.Let W be a G-module whose non-trivial orbits are all minimal.We claim that W is a simple G-module and contains a single non-trivial orbit which is minimal.In particular, the highest weight of W is indivisible.Indeed, all minimal orbits in W have the same type by Lemma 4.3(1) and therefore the same dimension d > 1 by Remark 2.10(2), and every minimal orbit meets W U in a punctured line, by Lemma 2.2(3).This implies that dim W = dim W U − 1 + d.Let W = m i=1 V i be the decomposition into simple G-modules.Every factor contains a dense minimal orbit, all nontrivial orbits are minimal and hence of the same type by Lemma 4.3 (1).By Lemma 2.2(6), a simple G-module contains at most one minimal orbit, hence dim W = md.Since dim W U = m we find md = m − 1 + d, and so m = 1.
Remark 4.9.If a small G-variety X is smooth and contains exactly one fixed point, then X is a simple G-module V λ containing a dense minimal orbit, and λ is indivisible.Indeed, smoothness and having exactly one fixed point imply by Luna's Slice theorem [Lun73, §III.1 Corollaire 2] that X is a G-module, and the rest follows from the remark above.
Example 4.10.Let K n be the standard representation of SL n , and set W where e 1 = (1, 0, . . ., 0), and set X := SL n Y ⊆ W . Since Y is B-stable and closed it follows that X is a closed and SL n -stable subvariety of W with the following properties (cf.Example 2.11).
(1) X contains a single closed SL n -orbit, namely the fixed point {0}.
(2) Every nontrivial orbit O ⊂ X is minimal of type ε 1 , and O ≃ K n as an SL n -variety.
In particular, X is a small SL n -variety.(3) Since X U = Y is normal (even smooth), X is also normal, by Lemma 4.3(3).However, by Remark 4.9 and (2), X is not smooth if m > 1.
Example 4.11.Let W := K 3 be the K * -module with weights (2, 1, 0), i.e. t(x, y, z) := (t 2 • x, t • y, z).The homogeneous function f := xz − y 2 defines a normal K * -stable closed subvariety Y = V(f ) ⊂ K 3 with an isolated singularity at 0. The invariant z defines the quotient π = z : Y → K = Y / /K * .The (reduced) fibers of π are isomorphic to K, but Y is not a line bundle, because the zero fiber is not reduced.The action of K * is given by (t, s) → t • s on the fibers over K \ {0} and by (t, s) → t 2 • s on the zero fiber.In fact, the zero fiber contains the point (1, 0, 0) which is fixed by {±1}, but not by K * .
By Lemma 4.7, (1).All fibers of the quotient map π : X → X/ /G = K different from the zero fiber are isomorphic to Concerning the smoothness of small G-varieties we have the following rather strong result, cf.Theorem 1.5.
Theorem 4.12.Let X be an irreducible small G-variety of type λ, and consider the following statements.
(i) The quotient π : X → X/ /G is a G-vector bundle with fiber V λ .
(ii) K * acts faithfully on X U , the quotient X U → X U / /K * is a line bundle, and (iv) The closures of the minimal orbits of X are smooth and pairwise disjoint.
(v) The quotient morphism π : X → X/ /G is smooth.Then the assertions (i) and (ii) are equivalent and imply (iii)-(v).If X (or X U ) is normal, all assertions are equivalent.
Furthermore, X is smooth if and only if X/ /G is smooth and π : − → X where K * acts by the inverse of the scalar multiplication on V λ , see Proposition 4.4(2)).If X U → X U / /K * is a line bundle, then it looks locally like K × W pr W −→ W , and K * acts by scalar multiplication on K. Hence V λ × K * X U looks locally like The (reduced) fibers of π : X → X/ /G are small G-varieties with a unique fixed point.If such a fiber F is smooth, then F ≃ V λ and V λ = O λ by Remark 4.9.
(iv) ⇒ (iii): If the closure of a minimal orbit O is smooth, then O ≃ O λ and O λ = V λ , again by Remark 4.9.It follows that the action of K * on X U \ X G is free and so bundle and L := K × K * P → X U / /K * the associated line bundle, then there is a canonical morphism (see Remark 4.6) It remains to prove the last statement where one implication is clear.Assume that X is smooth.Since the G-action is fix-pointed, it follows from [BH85, (10.3)Theorem] that π : X → X/ /G is a G-vector bundle.

Computations
In this paragraph we calculate the invariants m G , d G and r G which are defined for any simple algebraic group G in the following way: For any orbit O in an affine G-variety X we have dim O ≥ dim O λ (Lemma 2.5).An orbit O ≃ G/H with H reductive is affine and thus cannot be minimal (Lemma 2.2(2)).
If O ⊂ X is an orbit of dimension m G , then it is either minimal or closed.In fact, if O is not closed, then O \ O must be a fixed point since it cannot contain an orbit of positive dimension.This implies, by Proposition 2.12(1), that O is minimal.This shows that if If d G > m G , then an irreducible G-variety X of dimension < d G is small and we can apply our results about small G-varieties.
For simplicity, we assume from now on that G is simply connected.The nodes of the Dynkin diagram of G are the simple roots ∆ G .We will use the Bourbakilabeling of the nodes: We also have a canonical bijection between the simple roots ∆ G = {α 1 , . . ., α r } and the fundamental weights {ω 1 , . . ., ω r } induced by the Weyl group invariant scalar product (•, For any root α we denote by σ α the corresponding reflection of X(T ) R : 2. Parabolic subgroups.We now recall some classical facts about parabolic subgroups of G, cf.[Hum75, §29-30].
If R ⊂ ∆ is a set of simple roots and I := ∆ \ R the complement we define P (R) := BW I B ⊆ G where W I ⊆ W is the subgroup generated by the reflections σ i corresponding to the elements of I. Any parabolic subgroup of G containing B is of the form P (R), and we have R ⊆ S if and only if P (S) ⊆ P (R), with R = S being equivalent to P (R) = P (S).In particular, P (∅) = G and P (∆) = B, and the P (α i ) := P ({α i }) are the maximal parabolic subgroups of G containing B.
Consider the Levi decomposition P (R) = L(R)⋉U (R), where U (R) is the unipotent radical of P (R) and L(R) the Levi part of P (R) containing T , i.e.L(G) = Cent G (Z) where Z := α∈I ker α ⊆ T .In particular, L(R) is reductive, and so its derived subgroup (L(R), On the level of Lie algebras, we have If Φ I ⊆ Φ is the subsystem generated by I we get Furthermore, (2) and (3) yield Remark 5.1.The following facts will be important in our calculations of the invariants m G and d G .From the Dynkin diagram of G we can read off the semisimple type of L(R) by simply removing the nodes corresponding to the roots in R.Moreover, we have g α ⊂ u(R) for any α ∈ R, and one can determine the irreducible representation It is easy to see that the Lie subalgebra generated by V (α) consists of all root spaces g β where β is a positive root containing α.In the special case R = {α i } this implies that u(α i ) is equal to the Lie subalgebra generated by V (α i ).
5.3.The parabolic P λ .Recall that for a simple G-module with highest weight λ the subgroup ) is a parabolic subgroup of G, and λ induces a character λ : As above there is a well-defined Levi decomposition P λ = L λ ⋉ U λ where T ⊆ L λ , which carries over to the Lie algebra: Since P λ contains B it is of the form P (R) where the subset R ⊆ ∆ G has the following description.

The invariant m
So, it suffices to calculate (5) p For this it is clearly sufficient to consider the maximal parabolic subgroups P ω i = P (α i ).
Lemma 5.3.The following table lists the invariants m G and p G for the simple groups G, the corresponding maximal parabolic subgroups P ω as well as the dimensions of the fundamental representations V ω .The last column gives some indication about O ω where the null cone N V appears only in case see Lemma 2.2(1) and (4) and Section 5.2 above.We now apply the above strategy to each simple group G.In each case, dim d i turns out to be quadratic in i and achieves its minimum on the interval [1, n].Hence, if d i is of maximal dimension, then i is either 1 or n.
(row A n ) For i = 1, . . ., n, we obtain d i = sl i ⊕ sl n−i+1 .It is of maximal dimension for i = 1, n.Furthermore, V ω 1 = K n+1 and V ωn = (K n+1 ) ∨ are the the standard representation of SL n+1 and its dual which yields codim is the standard representation of SO 2n+1 , and the quotient V ω 1 / / SO 2n+1 ≃ K is given by the invariant quadratic form.In particular, dim N Vω 1 = 2n, and SO 2n+1 acts transitively on the isotropic vectors N Vω 1 \ {0}, hence O ω 1 = N Vω 1 .This gives the row B n , n ≥ 3, and half of the row B 2 .
If n = 2, then V ω 2 is the standard representation K 4 of Sp 4 , hence O ω 2 = K 4 \ {0}, giving the other part of the row B 2 .
(row C n ) Here we get d i = sl i ⊕ sp 2(n−i) which is of maximal dimension for i = 1.Furthermore, V ω 1 = K 2n is the standard representation of Sp 2n , and O ω 1 = V ω 1 , hence m Sp 2n = 2n.
(rows D 4 and D n ) For i = 1, . . ., n − 3, we get d i = sl i ⊕ so 2(n−i) .Moreover, d n−2 = sl n−2 ⊕ sl 2 ⊕ sl 2 and d n−1 = d n = sl n .They are maximal dimensional for i = 1 if n ≥ 5 and for i = 1, 3 and 4 if n = 4. Furthermore, V ω 1 = K 2n is the standard representation of SO 2n , and we get the claim for D n , n ≥ 5 and for V ω 1 in case n = 4.In this case, V ω 3 and V ω 4 are conjugate to the standard representation V ω 1 = K 8 by an outer automorphism of D 4 .For the standard representation V we have V / /G = K, given by the invariant quadratic form, and the nullcone consists of two orbits, {0} and the minimal orbit of nonzero isotropic vectors.
(row E 6 ) Here we find d 1 = d 6 = so 10 , d 2 = sl 6 , d 3 = d 5 = sl 2 ⊕ sl 5 , d 4 = sl 3 ⊕ sl 2 ⊕ sl 3 .The maximal dimension is reached for i = 1, 6, and we get p E 6 = 16.The representations V ω 1 and V ω 6 of dimension 27 are dual to each other.The quotient V ω 1 / /E 6 = K is given by the cubic invariant of V ω 1 (see [Sch78,   (row G 2 ) We have d 1 = d 2 = sl 2 and hence p G 2 = 5 and dim O ω i = 6.Furthermore dim V ω 1 = 7, dim V ω 2 = 14 and G 2 preserves a quadratic form on Remark 5.4.The lemma above has the following consequence.Let G be a simple group.If O λ is smooth, then we are in one of the following cases: (1) G = SL n and λ = ω 1 or λ = ω n , i.e.O λ is the standard representation or its dual.
In fact, if O λ is smooth, then O λ = V λ by Lemma 2.2(7), and the claim follows from the last column of Table 2 in Lemma 5.3.Now we can prove the first theorem from the introduction.
Proof of Theorem 1.1.Theorem 1.5 implies that X → X/ /G is a G-vector bundle with fiber V λ , where λ is the type of X, and the minimal orbits are smooth.This means that O λ = V λ , by Lemma 2.2(7), and the claim follows from Remark 5.4 above.5.5.The invariant r G .In this section, we compute the invariant which is the minimal dimension of a nontrivial affine G-orbit.These orbits are never minimal orbits, by Lemma 2.2(2).
Lemma 5.5.The following table lists the types of the proper reductive subgroups H of the simple groups G of maximal dimension, their codimension r G = codim G H and the invariant m G from Lemma 5.3.(In the table T 1 denotes the 1-dimensional torus.)  5 and 6, pp.234-235).From these tables one gets the following candidates for reductive subalgebras of minimal codimension.
Our claim is that c G = r G , i.e. that we have found the minimal codimensions of reductive subgroups of the classical groups.In order to prove this we have to show that [GOV94, Theorems 3.3] does not give any reductive subgroup of smaller codimension: 2 One has to be careful since the tables contain several errors.

Now the table above implies the following. Assume
and so H can be omitted.
The following table contains the minimal dimensions of irreducible representations of the simply connected exceptional groups.They have been calculated using [FH91, Exercise 24.9] which says that one has only to consider the fundamental representations.

H
E Comparing the values of r G and m G in Table 3 of Lemma 5.5 we get the following result.(1) G is of type A 1 and d G = 2; (2) G is of type B n and d G = 2n; (3) G is of type D n , n ≥ 4, and d G = 2n − 1; (4) G is of type F 4 and d G = 16; (5) G is of type G 2 and d G = 6.In all other cases we have r G ≥ d G > m G .
Proposition 5.7.The following table lists the invariants r G , d G and m G for the simply connected simple algebraic groups G. G Thus the only cases to be considered are A n for n ≥ 4, C n for n ≥ 3 and E 6 , E 7 , E 8 .
We have seen in Section 5.3 that for a dominant weight λ ∈ Λ G the corresponding parabolic subgroup P λ ⊂ G and its Lie algebra p λ have well-defined Levi decompositions P λ = L λ ⋉ U λ where T ⊆ L λ and p λ := Lie P λ = l λ ⊕ u λ .In addition, we define the closed subgroup P (λ) := ker(λ : P λ → K * ) which has the Levi decomposition P (λ) = L (λ) ⋉ U λ , L (λ) := ker(λ : L λ → K * ), and its Lie algebra By construction, the semisimple Lie algebra [l λ , l λ ] is contained in l (λ) , and they are equal in case λ is a fundamental weight ω i .Note also that P For an affine G-variety X and x ∈ X we set g x := Lie G x and denote by n x ⊆ g x the nilradical of g x .
The method for proving Proposition 5.7 was communicated to us by Oksana Yakimova who also worked out the result for the symplectic groups and for E 6 .It is based on the following lemma which is a translation of a fundamental result of Sukhanov, see [Suk90, Theorem 1].
Lemma 5.8.Let O be a quasi-affine G-orbit.Then there exist λ ∈ Λ G and x ∈ O such that g x ⊆ p (λ) and u x ⊆ u λ .In particular, we get an embedding l Theorem 1] implies that such an L is subparabolic which means that there is an embedding L ֒→ Q such that L u ֒→ Q u where Q is the isotropy group of a highest weight vector.Translating this into the language of Lie algebras we get the first part of the lemma.
For the second part, we note that g x p (λ) , so that and the claim follows.
The strategy of the proof of Proposition 5.7 is the following.Let O = Gx ⊂ X be a nonminimal nontrivial orbit, and consider an embedding g x ֒→ p (λ) given by the lemma above.
(1) Since O is not minimal, we have dim O ≥ dim u λ + 2. Thus one has only to consider those p λ with dim u λ + 2 < d G .For this one first calculates dim u ω i , i = 1, . . ., n, and then uses that dim u λ ≥ dim u ω i for all i such that ω i appears in λ, see Lemma 5.2.

The type C
Suppose that G = Sp 2n and g = sp 2n where n ≥ 3, and let O be a non-minimal and non-trivial quasi-affine orbit.We have to show that dim O ≥ 4n − 4. We have seen above that it suffices to consider those embeddings g x ⊂ g λ where dim n λ < 4n − 6.
(3) The case p (ω 6 ) is similar to p (ω 1 ) from (1).5.6.5.The type E 8 .Let G be simply connected of type E 8 and g = Lie G, and let O be a non-minimal and nontrivial quasi-affine orbit.We have to show that dim O ≥ 86.We have seen above that it suffices to consider those embeddings g x ⊂ p (λ) where dim u λ < 84.
If λ is a fundamental weight, then we find p (ω 1 ) = so 14 ⊕ u ω 1 , dim u (4) Denote by n ± l i ⊂ l i the sum of the positive resp.negative root spaces.Then g α i is annihilated by n − l i , because α i − β is never a root for a positive root β of l i , by (3).(5) The irreducible s i -submodule V (α i ) ⊂ g generated by g α i has g α i as lowest weight space, by (4).Denoting by λ i the weight α i | s i of s i , then V (α i ) ≃ V ⊥ −λ i .In fact, the dual module V (α i ) ⊥ has highest weight −λ i .(6) The submodule V (α i ) consists of those root spaces g β where β is a positive root containing α i with multiplicity 1. (7) If the center z i has weight κ on g α i , then V (α i ) is the κ-eigenspace of z i .This follows from (6), because z i has weight 0 on all of l i .These remarks offer several possibilities to calculate the dimension of V (α i ) using the computer algebra program LiE [vLCL92].5.7.1.Computation of the κ-eigenspace.The rows of the Cartan matrix Cartan(G) are the coordinates of the simple roots α j in the basis of the fundamental weights ω k .It follows that the ith column of the inverse of the Cartan matrix i_Cartan(G) defines the homomorphism K → h = K r with image z i .(In LiE i_Cartan(G) is the inverse of Cartan(G) multiplied by the determinant in order to have integral entries.)Now one uses the function spectrum(adjoint(G),t,G) which calculates the dimensions of the eigenspaces of the toral element t.
and the claim follows.(6) Let Y := Gv ⊂ N V which implies that 0 ∈ Y .Since Y is irreducible, the fixed point set Y U does not contain isolated points (see e.g.[Kra16, III.5, Theorem 5.8.8]), and so Y U = {0}.Hence Y contains a highest weight vector, and so Y ⊃ O λ .(7) The tangent space T (a) Every G-equivariant morphism η : O → O ′ between minimal orbits is also K * -equivariant.(b) If O ⊂ X is a minimal orbit in an affine G-variety X, then the K * -action on O extends to the closure O. (c) If O ⊂ X is as in (b), then the limit lim t→0 ty exists for all y ∈ O and is equal to the unique fixed point x 0 ∈ O.(d) If O = O λ where λ is indivisible, then the canonical action is the scalar multiplication.Let O ≃ O λ be a minimal orbit of type λ 0 , i.e. λ 0 is indivisible and λ = ℓλ 0 for some ℓ ∈ N, see Definition 2.8.Since Aut G (O) ≃ K * by Lemma 2.2(5), there are two faithful K * -actions on O commuting with the G-action.Both extend to the normal closure O n , and for one of them we have that lim t→0 ty exists for all y ∈ O and is equal to the unique fixed point in O n .This action corresponds to the scalar multiplication in case O = O λ ⊂ V λ .We call it the action by scalar multiplication and denote it by (t, y) → t • y.Lemma 2.14.Let O, O ′ be minimal orbits, and let η : O → O ′ be a G-equivariant morphism.(1) O and O ′ are of the same type, and η extends to a finite G-equivariant morphism η : O n → O ′ n .

( 1 )
Definition 2.15.Let O ≃ O λ be a minimal orbit of type λ 0 , where λ = ℓλ 0 .The canonical K * -action on O is defined by (t, y) → t ℓ • y for t ∈ K * and y ∈ O.It follows that this K * -action extends to O n such that the limits lim t→0 t ℓ • y exist in O n .If λ is indivisible, then the canonical action on O λ coincides with the scalar multiplication, but it is not faithful if λ is not indivisible.Proposition 2.16.Let O ≃ O λ be a minimal orbit of type λ 0 where λ = ℓλ 0 .The canonical K * -action on O corresponds to the representation on O(O) which has weight −n on the isotypic component O(O) nλ ∨ 0 .In particular, it commutes with the G-action.

( 5 )
We have Norm G (O U ) = Norm G (O U ) = P λ , and the action of P λ on O U is given by px = λ(p) • x = λ 0 (p) ℓ • x, i.e. it factors through the canonical K * -action.
and since O(X)/I ≃ O(X) U − ⊆ O(X) we finally get I = √ I.It follows that the restriction map ρ : O(X) → O(X U ) can be identified with the universal Uprojection π : O(X) → O(X) U , and thus, by Lemma 3.5 above, the composition O giving the isotypic decomposition of O(G/U ) = O(G) U .Thus O(G/U ) contains every simple G-module with multiplicity 1.Since the torus T normalizes U there is also an action of T on O(G) U induced by the action of G by right-multiplication, and this T -action commutes with the G-action.Thus we have a G × T -action on O(G/U ) = O(G) U .Remark 3.8.

( 1 )
The isomorphism ( * ) above is G × T -equivariant where T acts on O(G/U ) λ ≃ V λ by scalar multiplication with the character λ ∨ .Thus the T -action on O(G/U ) corresponds to the grading given by the isotypic decomposition.In particular, O(G/U ) is a graded G-algebra.(2)The universal U -projection π U : O(G/U ) → O(G/U ) U is equivariant with respect to the T × T -action.On the one-dimensional subspace (O(G/U ) λ ) U ⊂ O(G/U ) U the action of (s, t) ∈ T × T is given by multiplication with λ ∨ (s) −1 λ ∨ (t).
any dominant weight λ.Recall that we have a T -action on O(G/U ) by scalar-multiplication with the character λ ∨ on O(G/U ) λ , see Remark 3.8(1).
R is normal, by Proposition 3.14.Normality is an open property, i.e. in a flat family (A x ) x∈X of finitely generated K-algebras the set {x ∈ X | A x is normal} is open, see [Gro66, Corollaire 12.1.7(v)].

5. 1 .
Notation.Let G be a simple group.As before, we fix a Borel subgroup B ⊂ G, a maximal torus T ⊂ B, and denote by W := Norm G (T )/T the Weyl group.The monoid of dominant weights Λ G ⊂ X(T ) := Hom(T, K * ) is freely generated by the fundamental weights ω 1 , . . ., ω r , i.e.Λ G = r i=1 Nω i (see Section 2.1).We denote by Φ = Φ G ⊂ X(T ) the root system of G, by Φ + = Φ + G ⊂ Φ the set of positive roots corresponding to B and by ∆ = ∆ G ⊂ Φ + the set of simple roots.Furthermore, g := Lie G, b := Lie B and h := Lie T are the Lie algebras of G, B and T , respectively, g α ⊂ g is the root subspace of α ∈ Φ and G α ⊂ G the corresponding root subgroup of G, isomorphic to K + .
If h ⊂ g is a maximal subalgebra, then it is either semisimple or parabolic, [GOV94, Theorem 1.8].Since the Levi parts of the parabolic subalgebras have maximal rank the second case does not produce any new candidate.It is therefore sufficient to look at the maximal semisimple subalgebras.For the exceptional groups G the classification is given in [GOV94, Theorem 3.4], and one finds one new case, namely F 4 ⊂ E 6 which has codimension 26.Thus the claim is proved for the exceptional groups.(c) From now on G is a classical group and we can use [GOV94, Theorems 3.1-3.3].From the first two theorems one finds the new candidates B n−1 ⊂ D n of codimension 2n − 1, including B 2 ⊂ A 3 of codimension 5.This gives the following table.

Lemma 5 . 6 .
Let G be simple and simply connected.If r G = d G = m G , then we are in one of the following cases.

Table 1 .
Vω 1 , N Lie G 2 The invariants m G , r G , d G for the simple groups, the orbit closures realizing m G and the reductive subgroups H G realizing r G .
and O → O λ are both cyclic, of degree m and ℓ respectively, and k = ℓm.Hence O ≃ O mλ 0 and ℓ(mλ 0 ) = λ.2.3.Minimal orbits.In this section we define the central notion of minimal orbits and prove some remarkable properties.Definition 2.8.An orbit O in an affine G-variety X isomorphic to a highest weight orbit O λ will be called a minimal orbit.This name is motivated by Lemma 2.5.The type of a minimal orbit O ≃ O λ is defined to be the indivisible element λ 0 ∈ Qλ ∩ Λ G ≃ Nλ 0 from Lemma 2.7.Two minimal orbits O 1 ≃ O λ 1 and O 2 ≃ O λ 2 are of the same type if and only if Qλ 1 = Qλ 2 (Lemma 2.7).This is the case if and only if for v i ∈ O i the groups G • If O 1 , O 2 are minimal orbits and η : O 1 → O 2 a finite G-equivariant covering, then O 1 and O 2 have the same type, and the covering is cyclic.

Table 3 .
Maximal reductive subgroups of simple groups Proof.The classification of maximal subalgebras h of a simple Lie algebra g is due to Dynkin, see [Dyn52a, Dyn52b].His results are reformulated in [GOV94, chap.6,§1 and §3].(a) If h is maximal reductive of maximal rank ℓ := rank g, then the classification is given in [GOV94, Corollary to Theorem 1.2, p.186] (the results are listed in Tables It remains to consider the simple subgroups H G of classical type where G = SL n , SO n , Sp n .(d 1 ) The irreducible representations H → SL n of minimal dimension of a group H of classical type are given by the following table.It is obtained by using again the fact that one has only to consider the fundamental representations, see [FH91, Exercise 24.9].
They correspond to the standard representations SL n ⊂ GL n , SO n ⊂ GL n and Sp n ⊂ GL n , except for B 2 = C 2 where it is Sp 4 ⊂ GL 4 .If H is not of type A we have codim SLn H > c SLn = 2n − 2 except for type B 2 where codim SL 4 Sp 4 = 5 = c SL 4 .Moreover, if SL k → SL n is not an isomorphism, then k < n and codim SLn SL k > c SLn .(d 2 ) Next we consider irreducible orthogonal representations ρ : H → SO n for H of classical type where n ≥ 5.If H is a candidate not already in (a), then rank H < rank SO n , and one calculates straight forwardly that codim SOn H > c SOn = n − 1. (d 3 ) Finally, we consider irreducible symplectic representations ρ : H → Sp 2m for H of classical type where m ≥ 2. As above, if H is a candidate not already in (a), then rank H < rank Sp 2m = m.Again, an easy calculation shows that codim Sp 2m H > c Sp 2m = 4m − 4. 5.6.The invariant d G .In this section, we compute the invariant

Table 4 .
The invariants r G , d G and m G for the simple groupsThe first and last row of Table4are the rows from Table2 and Table 3.We have seen above that for r G Thus we have s ω 7 = E 6 ⊕sl 2 , and the last output shows that dim l ω 7 = 82 and dim V (α 7 ) = 54.In addition, u ω 7 contains two other irreducible representations, of dimensions 27 and 2. 5.7.2.Computation of the lowest weight λ i := α i | s i .The lowest weight λ i is obtained from the ith row of the Cartan matrix Cartan(G) by removing the ith entry (which is a 2) and and rearranging the other entries according to the numeration of the nodes of the Dynkin diagram of s i obtained by removing the ith node from the Dynkin diagram of G. Example 5.14.Lowest weight of V (α 1 ) for G = E 8 .