On a smoothness characterization for good moduli spaces

Let $\mathcal{X}$ be a smooth Artin stack with properly stable good moduli space $\pi\colon\mathcal{X} \to X$. The purpose of this paper is to prove that a simple geometric criterion can often characterize when the moduli space $X$ is smooth and the morphism $\pi$ is flat.


Introduction
Let K be an algebraically closed field of characteristic 0 and X be a K-smooth Deligne-Mumford stack with coarse space p : X → X. Applying the Purity of the Branch Locus Theorem to the proper quasi-finite morphism p yields a necessary condition for X to be smooth: the branch locus of p must be pure of codimension-one; here the branch locus is the complement of the largest open set U ⊂ X over which p is étale.On the other hand, the beautiful theorem of Chevalley-Shephard-Todd gives a simple group-theoretic criterion which is sufficient for determining when X is smooth.Specifically, if x ∈ X (K) with stabilizer group G x , then X is smooth at p(x) if and only if the G x -action on the tangent space T X,x is generated by pseudo-reflections, i.e.G is generated by elements g ∈ G x whose fixed locus is a hyperplane.Whenever X is smooth, p is automatically flat.
For smooth Artin stacks we consider the following situation analogous to the Deligne-Mumford setting.Let X be a smooth Artin stack with properly stable good moduli space p : X → X; this means that there is a dense set of points x in X which have 0-dimensional stabilizer and are also p-saturated, i.e., p −1 p(x) = x [ER,Definition 2.5].In this case the good moduli space morphism is not separated but shares some properties of a proper quasi-finite morphism: it is universally closed and if x ∈ X is a closed point, then there is a unique closed point in the fiber of p over x.Such X arise naturally in the context of GIT, e.g. if G is a reductive group with properly stable action on a variety U and if U ss denotes the semistable locus, then p : [U ss /G] → U/G is a properly stable good moduli space.
Despite the fact that smooth Artin stacks with properly stable good moduli spaces are analogous to Deligne-Mumford stacks, there are no general necessary and sufficient criteria to determine when X is smooth and p is flat.Indeed, one cannot invoke the Purity of the Branch Locus Theorem as p is not proper quasi-finite, and there is no known analogue of the Chevalley-Shephard-Todd Theorem since smooth Artin stacks do not have tangent bundles.
The starting point for this paper is to instead take a GIT point of view.By [AHR,Theorem 4.12] and [Lun], at a closed point x of X , the map p is étale locally isomorphic to [V /G x ] → V /G for some representation V of the stabilizer group G x .Thus, the problem of determining when X is smooth and p is flat reduces to the case where X = [V /G] and V is a representation of a linearly reductive group G.A natural analogue of the branch locus is then the image in X of points in V which have a positive dimensional stabilizer group.This is exactly the image of the GIT strictly The first author was supported by Simons Collaboration Grant 315460.The second author was partially supported by a Discovery Grant from the National Science and Engineering Board of Canada as well as a Mathematics Faculty Research Chair from the University of Waterloo.The third author was partially supported by an Undergraduate Student Research Award from the National Science and Engineering Board of Canada.
semi-stable points of X. Inspired by the Purity of the Branch Locus Theorem, a naïve guess is that the following condition is necessary for X to be smooth: (⋆) The image of the strictly semi-stable points must be of pure codimension-one.
The main results of this paper imply that condition (⋆) goes a long way toward determining when V /G is smooth and p is flat.Specifically, we prove for irreducible representations of simple groups, condition (⋆) is both necessary and sufficient for V /G to be smooth and [V /G] → V /G to be flat.In addition we show that when G is a torus, a slight strengthening of condition (⋆) is necessary and sufficient to characterize when V /G is smooth and [V /G] → V /G is flat.
To state our results precisely we introduce Definition 1.2 after recalling some basic notions.
Definition 1.1.Let V be a representation of a reductive group G.A vector v ∈ V is G-stable if Gv is closed and v is not contained in the closure of any other orbit.A vector v ∈ V is G-properly stable if v is stable and dim Gv = dim G.
A representation V is stable (resp.properly stable) if it contains a stable (resp.properly stable vector).In this case, the set V s = V s (G) of G-stable (resp.properly stable) vectors is Zariski open.We denote by V sss = V sss (G) the closed subset V V s ; vectors v ∈ V V s are said to be G-strictly semi-stable.
Definition 1.2.Let V be a stable representation of a connected reductive group G and let π : V → V /G be the quotient map.Then V is pure if π(V sss ) is pure of codimension-one in V /G.
an arbitrary representation of a torus with non-trivial invariant ring, then V is cofree if and only if the stable submodule V ′ is coprincipal.
Our main results relating coregularity, cofreeness, purity, and coprincipality for stable representations are summarized in the diagram below.Note that when V is irreducible and G is simple, all four notions coincide.coregular [KPV] 1.1.Questions and examples.The fact that coprincipality characterize cofreeness for irreducible representations of simple Lie groups as well as reducible representations of tori suggestions that it may be a useful class of representations in greater generality.We pose the following questions.
Question 1.9.Let V be a stable representation of a connected reductive group G.
(2) Let G be semisimple and V be irreducible.If V is pure (equivalently coprincipal), then is it cofree?
Remark 1.10 (Relationship to a result of Brion).Michel Brion pointed us to a result of his [Bri,4.3Corollaire 1] which gives some evidence for an affirmative answer to Question 1.9(1).Precisely, Brion proves that if V is a properly stable representation of a reductive group (not necessarily connected), and if codim(V V pr ) ≥ 2, then K[V ] cannot be a free K[V ] G module.Here V pr is the locus of orbits of principal type.Since V pr ⊂ V s , it follows that such representations are not pure.
Remark 1.11 (Reducible representations).We note that the irreducibility assumption in Question 1.9 (2) cannot be dropped, even when G is simple.The smallest example we know to illustrate this, which we learned from Gerald Schwarz, is the SL 3 -representation V = Sym 2 (C 3 ) ⊕ (C 3 ) ⊕2 .The fact that the irreducibility assumption cannot be dropped is completely analogous to the picture for finite groups.Indeed, for a faithful representation V of a finite group G, if V f denotes the open set on which G acts freely, then the condition that V V f is a divisor is necessary but not sufficient for cofreeness of V .For a simple example, consider the µ 4 -action on C 2 with weights (1, 2).Although reducible, pure, non-cofree representations do exist, the conditions of purity and cofreeness are both quite rare for reducible representations.Indeed, for semisimple groups, any representation with no trivial summands and at least two properly stable summands cannot be cofree; similarly, for a reductive group, any representation with at least two properly stable summands cannot be pure.To see the former statement, note that any properly stable G-representation has dimension at least dim G + 1, and by [PV,Theorem 8.9] if G is semisimple then any coregular (and thus cofree) representation with no trivial summands has dimension at most 2 dim G.To see the latter statement, note that if V and W are properly stable representations then the Hilbert-Mumford criterion implies that ( 2. Outline of the proofs of the main theorems 2.1.Theorem 1.3: "only if " direction.This is the easier direction of Theorem 1.3.Recall that a representation of V is polar if there is a subspace c ⊂ V and a finite group W such that The basic example of a polar representation is the adjoint representation g; here c is a Cartan subalgebra and W is the Weyl group.Using results of Dadok and Kac [DK] we prove that any stable polar representation (not necessarily irreducible) is pure, see Proposition II.2.1.On the other hand, Dadok and Kac proved that any irreducible cofree representation of a simple group is polar.Thus, we conclude that any stable irreducible cofree representation of a simple group is pure.
2.2.Theorem 1.3: "if " direction.This is the most involved result of the paper.In Section II.3, we show that if G is reductive and V is an pure G-representation, then there is a hyperplane H in the character lattice of V tensored with R satisfying the following special condition: H contains at least dim V − dim G + 1 weights when counted with multiplicity.In particular, this implies that when G is semisimple, every irreducible pure representation V has dimension bounded by a cubic in rk(G).In Section II.4,we further show that if V is pure, then its highest weight lies on a ray (or possibly a 2-dimensional face, if G = SL n ) of the Weyl chamber.Comparing with the known list of cofree representations of simple groups, we are reduced to checking that 11 infinite families and 94 more sporadic cases, are not pure.These calculations, performed in Section II.5 are mostly done by computer, but a number must be done by hand, and will show the nature of the computer analysis done.
2.3.Theorem 1.6.We prove Theorem 1.6 for tori T by inducting on dim V .The key to the proof is showing in Proposition III.1.6that if V is a coprincipal representation of a torus, then V splits as This argument makes essential use of the fact that the images of the irreducible components of V sss are Cartier divisors.
Part II.Representations of simple groups: proof of Theorem I. 1.3This part is organized as follows.In §1 we prove some basic facts about pure representations that we will use throughout.In §2, we prove that every stable cofree irreducible representation of a connected simple group G is pure, that is, we prove the "only if" direction of Theorem I.1.3.In §3, we prove the key result that if V is any pure representation, then there is a hyperplane in the weight lattice containing most of the weights.This implies that up to isomorphism there are a finite number of pure representations not containing a trivial summand.
In §4, we apply the criteria in §3 to create a finite list on which all irreducible pure representations of a simple group may be found.Then, in §5, we demonstrate that representations on this list which are not cofree are also not pure, thus proving Theorem I.1.3.

Basic facts about pure representations
Lemma 1.1.If V is a representation of a reductive group H, and f : G → H is a surjection, then V is pure when considered as an H-representation if and only if V is pure when considered as a G-representation.
Proof.Since f is surjective, Hv = f (G)v, which is Gv by definition of the G-action.So all the orbits of the actions are the same, and hence their purity is the same.
In the event that G is a simple Lie group which is not simply connected, it has a universal covering G ′ → G with finite kernel K.There is a one-to-one correspondence between G-representations and G ′ -representations which are trivial when viewed as a K-representations.By Lemma 1.1, we have Corollary 1.2.To prove Theorem 1.3 for all simple Lie groups, it suffices to prove it for the exceptional groups and the groups SL n , Sp 2n , Spin n for all n.
Lemma 1.3.Let be G a reductive group with no non-trivial characters.If V is stable representation for which V sss is pure of codimension-one, then V is coprincipal.In particular, every pure representation of G is coprincipal.
Proof.Since G is connected, every component of V sss is G-invariant.Thus, the equation f ∈ K[V ] of the component must be an eigenfunction for the action of G on K[V ]; that is, g • f = λ(g)f for all g ∈ G. Since G has no non-trivial characters, f must in fact be invariant.Hence the image of Lemma 1.4.If ρ, ρ ′ are representations of G on a vector space V , and there exists an automorphism f of G exchanging ρ and ρ ′ , then ρ is pure (resp.cofree) if and only if ρ ′ is pure (resp.cofree).
Proof.Purity and cofreeness are both determined at the level of the image ρ Remark 1.5.Lemma 1.4 will be used for the spin groups in the following fashion: for each spin group Spin 2n of even order, there exists an outer automorphism exchanging the half-spinor representation Γ ω 2n and the half-spinor representation Γ ω 2n−1 , and it follows that the positive half-spinor representation is cofree and pure if and only if the negative half-spinor representation is cofree and pure.
For Spin 8 there is a triality which gives automorphisms exchanging the half-spinor representations and the standard representation.Since the standard representation is cofree and pure, both halfspinor representations of Spin 8 are cofree and pure.Moreover, these outer automorphisms send the symmetric square Sym 2 C 8 with highest weight 2ω 1 to the representations Γ 2ω 7 , Γ 2ω 8 , which are the irreducible components of the wedge product 4 C 8 .Since Sym 2 C 8 is cofree and pure as a Spin 8 representation, so too are both irreducible components of 4 C 8 .
For low dimensional special orthogonal Lie algebras, there exist exceptional isomorphisms sp(4) = so(5) and sl(4) = so(6).These obviously preserve purity and cofreeness of representations, and so it suffices for the classical groups to prove Theorem 1.3 for SL n when n ≥ 2, Spin 2n+1 when n ≥ 2, Sp 2n when n ≥ 3, and Spin 2n when n ≥ 4. 2. Proof of the "only if" direction of Theorem I.1.3 The proof of the "only if" direction of Theorem I.1.3is relatively straightforward thanks to the work of Dadok and Kac on polar representations [DK].Recall that a representation V of a reductive group G is polar if there exists a subspace c, called a Cartan subspace, such that the map c → Spec K[V G ] is finite and surjective.
In [DK,Theorem 2.9], Dadok and Kac proved that if V is polar with Cartan subspace c, then the group [DK,Theorem 2.10], every polar representation is cofree.Furthermore, using the classification of irreducible cofree representations of simple groups, they showed that every irreducible cofree representation of a simple group is polar.
As a result, to prove the "only if" direction of Theorem I.1.3,it is enough to show that polar representations are pure.We are grateful to Ronan Terpereau for suggesting this proof.
Proposition 2.1.If V is a stable polar representation (not necessarily irreducible), then it is pure.
Proof.Let c be a Cartan subspace, and following [DK,p. 506], let c reg be the set of regular points.By definition, v ∈ c reg if and only if Gv is closed and of maximal dimension among closed orbits.If V is a stable representation then this condition is equivalent to the stability of v. Since the Cartan subspace contains a point of each closed G orbit, it follows that Gc reg = V s .
By [DK,Lemma 2.11], c sing = c c reg is a finite union of hyperplanes, and V sss = Gc sing by definition.If W is as above, then the image of c sing under the quotient map [DK,Theorem 2.10].By By Lemma 1.3, we see V is pure.
Remark 2.2.As noted by Victor Kac, there are polar representations with non-trivial rings of invariants where our proposition does not apply.However, an analogous statement holds with V s replaced by the G-saturation of the locus of closed orbits, which are of maximal dimension among closed orbits.

Bounding the dimension of a pure representation
We begin by obtaining results that show pure representations are relatively rare; specifically, any simple group has a finite number of pure representations that do not contain a trivial summand.
The following result holds for any reductive group, not just simple or semi-simple ones.
Proposition 3.1.Let V be a stable representation of a reductive group G. Suppose V sss contains a divisorial component that maps to a divisor in V/G, e.g.V is pure.Then there exists one-parameter subgroup λ such that λ ⊂ V denotes the 0-weight space of λ; said differently, there are at most dim G + 1 weights that do not lie on the hyperplane of the weight space defined by λ.
Proof.First assume that V is properly stable.In this case, every stable vector has finite dimensional stabilizer.Hence, a vector v is not stable if and only if it contains a point with positive dimensional stabilizer in its orbit closure.Any closed orbit in a representation is affine so its stabilizer is reductive by [MM], so if it is positive dimensional then it contains a one-parameter subgroup.Thus v ∈ V sss if and only if there is a 1-parameter subgroup λ such that v ∈ V ≥0 λ , where V ≥0 λ is the subspace of V whose vectors have non-negative weight with respect to λ.
Since all one-parameter subgroups are G-conjugate, it follows that V sss = ∪ λ∈N (T ) GV ≥0 λ where N (T ) is the group of one-parameter subgroups of a fixed maximal torus T .Since V is finite dimensional, it contains a finite number of weights, so there are only finitely many distinct subspaces V ≥0 λ as λ runs through the elements of N (T ).Hence, there exists a one-parameter subgroup λ such that When V is stable but not properly stable, it is still the case that any strictly semi-stable point contains a point with positive dimensional stabilizer in its orbit closure.The same argument used above implies that V sss ⊂ λ∈N (T ) GV ≥0 λ .It follows that any divisorial component of V sss is contained in GV ≥0 λ for some 1-parameter subgroup λ.If this divisorial component maps to a divisor, then image of V 0 λ contains a divisor, so we conclude that dim Remark 3.2.It is clear that if a representation can be shown to not be pure by Proposition 3.1, then its dual representation will also not be pure, since it will have the same number of weights in a hyperplane.
Example 3.3.In this example, we illustrate that if V is not properly stable, then V sss may be a proper subset of λ∈N (T ) GV ≥0 λ .Let V be the adjoint representation of SL 2 .Then the strictly semi-stable locus V sss is the divisor defined by the vanishing of the determinant.However, since the torus of SL 2 is rank one, the fixed locus V 0 λ is the same for all λ and is one dimensional.In this case, We are grateful to the referee for pointing out the following consequence of Proposition 3.1, which has been used to significantly simplify the computations in the paper: Lemma 3.4.Suppose {t 1 , . . ., t k } are a complete set of Weyl group orbit representatives of order ℓ toral elements1 in the maximal torus of a simple Lie group where N t i denotes the normalizer of t i .
Proof.Any 1-parameter subgroup λ contains some toral element t of order ℓ.Thus an upper bound for dim π(V λ ) is max{dim π(V t ) | t of order ℓ}.In particular, if each π(V t ) has codimension at least 2, then π(V λ ) has codimension at least 2, and by Proposition 3.1, dim V sss ≥ 2.Moreover, the morphism In practice, Lemma 3.4 is much easier to use than Proposition 3.1.Software packages like LiE [vLCL] are capable of computing dim V t given a representation of a semisimple Lie group and a toral element t of finite order, and so it is possible to show many representations are not pure by checking that dim V t ≤ dim V − dim G − 2 for all t up to the action of the Weyl group.For representations where this bound is not a priori clear and for representations occurring in infinite families, a more careful analysis can be done by hand.In some exceptional cases, it is still easier to count the number of weights occurring in a hyperplane.
An important use of Proposition 3.1 is to show that a simple Lie group has finitely many pure representations.This can be shown by establishing a bound on the dimension of a pure representation which is polynomial in the rank of the group.To show this bound, we separately bound with multiplicity the number of zero and non-zero weights that can occur in the representation.Proposition 3.5.Let V be a (stable) representation of a simple Lie group G, and suppose that V sss contains a divisorial component.Then there are at most rk(G)(dim(G) + 1) non-zero weights counted with multiplicity.
Proof.Since G is simple, the image of a divisorial component of V sss in V /G is also a divisor by Lemma I.1.3.Hence by Proposition 3.1 there is a 1-parameter subgroup λ such that V 0 λ contains at least dim V − dim(G) − 1 weights.Let H be the hyperplane in the character lattice determined by this one-parameter subgroup, so Proposition 3.1 says that H contains at least dim V − (dim(G) + 1) such weights.The Weyl group conjugates of H also contain at least dim V − (dim(G) + 1) weights.We claim that H has at least rk(G) linearly independent conjugates under the Weyl group.Suppose v is a normal vector to H.If W is the Weyl group, then it acts linearly on h * , and the representation so obtained is irreducible by [FH,Lemma 14.31].Since the linear span of the orbit W v is a nonzero W -invariant subspace of h * , it must be all of h * , and so the number of linearly independent conjugates of v under the Weyl group is dim h * = dim h = rk(G).
Let H = H 1 , . . ., H rk(G) be rk(G) such conjugate hyperplanes whose normal vectors are linearly independent.By inclusion-exclusion, H 1 ∩ H 2 contains at least dim V − 2(dim(G) + 1) weights counted with multiplicity, since weights counted with multiplicity.In other words, the multiplicity of the 0 weight in V is at least dim V − rk(G)(dim(G) + 1).
Lemma 3.6.Let V be an irreducible representation of a semi-simple Lie group G, and let α i be the positive simple roots.If a is any non-highest weight for V , then Proof.Consider the linear map V a → i V a+α i given by v → (e i (v)) where e i is the root vector in the Lie algebra for α i .Since a is not a highest weight, V a does not contain a highest weight vector, that is, no vector v ∈ V a is killed by all positive simple roots.Hence, the above map is injective.
Lemma 3.7.Let V be a representation of a simple Lie group G which contains no trivial summands.Then .
Proof.It clearly suffices to prove the lemma for every non-trivial irreducible subrepresentation of V , and so we may assume V is irreducible.Let d = dim V , let d 0 = dim V 0 be the dimension of the 0-weight space, and let d α = dim V α be the dimension of the weight space of any root α.There are dim(G) − rk(G) total roots, so we obtain the inequality (dim . Proposition 3.8.Let V be a (stable) representation of a simple Lie group G which contains no trivial summands and such that V sss contains a divisor.If .
Proof.Proposition 3.5 proves the claim when 0 is not a weight.Assume 0 is a weight.Since V sss is a divisor, it follows from Proposition 3.5 that by Lemma 3.7, which implies that For the classical groups, we make use of slightly weaker bounds than those in Proposition 3.8.
Definition 3.9.Let G be a classical group and V an irreducible G-representation.
In this case, we often say the highest weight of V is small enough.
Remark 3.10.By Proposition 3.8, a pure irreducible representation of a simple Lie group is small enough.The problem of enumerating the pure representations may thus be reduced to first enumerating the small enough representations, and then determining which of them are pure.
Remark 3.11.The bound of Proposition 3.8 holds for representations of semisimple Lie groups also.Unlike the case of simple Lie groups, for semisimple Lie groups difficulties arise in trying to create lists of representations meeting these bounds primarily because if G 1 is a simple Lie group with large rank and G 2 is a simple Lie group with much smaller rank, then V 1 ⊗ V 2 frequently gives a representation which is small enough but is not pure or cofree.The sizes of the lists in this case are infeasible to handle with the methods presented here.

Small enough representations
Let g be a simple Lie algebra with Cartan algebra h and fundamental weights ω 1 , . . ., ω n ∈ h * .Denote by Γ a 1 ω 1 +•••anωn the representation of g with highest weight vector a 1 ω 1 + . . .+ a n ω n .Define the width of the vector a 1 ω 1 + . . .+ a n ω n to be a 1 + • • • + a n , and define its support to be the number of i such that a i = 0.If ω = a 1 ω 1 + . . .+ a n ω n is a weight, and i, j are such that a i , a j = 0, define the shift ω i→j to be ω + a i (ω j − ω i ).Note that ω and ω i→j both have the same width, and the support of ω i→j is one less than the support of ω.We use the following lemma, which is proven along the way to [GGS,Lemma 2 In particular, if ω is small enough, then one of ω i→j , ω j→i must be small enough.This suggests the following algorithm to classify the small enough representations: (1) Find a closed form for the dimension of the irreducible representation with highest weight ω s for each s, and classify the 1-supported small enough weights using these closed forms.(2) If ω is a small enough weight with support {i, j}, use Lemma 4.1 to restrict the possibilities for ω and classify the 2-supported small enough weights.
(3) If the list of 2-supported small enough weights is not empty, repeat the above proceduce for 3-supported small enough weights, and so forth, until the list of K-supported small enough weights is empty for some K. (4) If ω is a weight with support greater than K, by repeatedly applying Lemma 4.1, its dimension can be bounded below by that of a K-supported weight.Since no K-supported weight is small enough, it follows that ω is not small enough for any support greater than K, and the list is complete.
Reading the tables.Each of the families of Lie groups in this section are accompanied by a list of all their small enough representations, see Tables 1-4.The representations are described by their highest weights, in the notation of [FH].
The final column in each table describes whether the indicated representation is cofree and/or pure for the indicated values of n.By the "only if" direction of Theorem I.1.3,cofree implies pure for a stable irreducible representation of a simple group, so "cofree" is written to mean "cofree, pure, and stable".The representations meeting this criterion may be looked up in a list such as [KPV].Note that these lists contain representations up to outer automorphism; as a result, entries on the tables in this section occasionally do not appear on them.Such representations are be indicated as cofree with a reference to the relevant tool needed to find it in the list of [KPV].
An entry of "impure" similarly means "stable, impure, and not cofree."When it is not otherwise specified, a representation marked "Impure" was proven to not be pure by a computer analysis using the software LiE [vLCL] and Lemma 3.4.For this paper, toral elements of orders 2 and 3 were used in this fashion, and LiE was used to compute the dimensions dim V t for all of these t.In the event that dim V t is too large, or for infinite families of representations, a reference is be given to a statement in Section 5 in which the impurity of the representation is shown.
Representations that are not stable further have "unstable" in the final column.The fact that each small enough representation is either unstable, cofree, or impure, proves Theorem 1.3 for simple simply connected Lie groups, and then for all simple Lie groups by Corollary 1.2.

The case of SL
It therefore suffices to classify the representations with at least half their support in a 1 , . . ., a ⌊n/2⌋ , and the others are the dual representations.By Remark 3.2, duality does not change purity or cofreeness as long as our proofs use Proposition 3.1 to show impurity, and hence these representations shall be be ignored.For example, in the case of 1-supported weights, we only consider highest weights tω s where n ≥ 2s − 1.
For SL 2 , every irreducible representation is of the form Sym k V for some k, where V is the defining representation.The maximum number of weights contained in a hyperplane is then 0 if k is odd, or 1 if k is even.So if k > 4, then Sym k V is not pure by Proposition 3.1; one can check that when k ≤ 4, the representation is both cofree and pure.In the sequel we consider representations of SL n for n > 2.
Lemma 4.2.The small enough irreducible representations of SL n with 1-supported highest weights are the entries (1)-( 20) of Table 1.
Proof.Recall that Γ ωs = s SL n and so has dimension n s .When 4 ≤ s ≤ n 2 , it is clear that this dimension is asymptotically larger than n 3 , and so there can be at most finitely many small enough such ω s .We verify that the following are the only possible representations: We may also verify that no non-trivial scalar multiples of these representations are small enough.

Highest weight Restrictions on n Purity/Cofreeness kω
, and we see again that for t ≥ 4 there can be at most finitely many solutions.We can verify that these solutions are exactly By [GGS,Lemma 2.2], dim Γ tωs ≥ dim Γ tω 1 for all t, s; using this we check that tω 2 and tω 3 are never small enough for t ≥ 4.
For t = 2, 3, we have dim Γ tω 1 ≤ n 3 for all n, and so these representations are both small enough.It remains only to determine the n for which 2ω 2 , 2ω 3 , 3ω 2 and 3ω 3 are small enough.The dimension of 2ω 2 is n 2 (n 2 −1)

12
, which is at most n 3 when 4 ≤ n ≤ 12, and we check for n in this range that 3ω 2 is only small enough when n = 4.The weight 2ω 3 is only small enough when n = 6, and 3ω 3 is not small enough for this n.This completes the list of 1-supported small enough weights.
Proof.By Lemma 4.1, if ω = ω i + ω j is small enough, then one of ω i→j = 2ω j and ω j→i = 2ω i must be small enough.Therefore, assuming without loss of generality that i ≤ ⌊n/2⌋, we have that i is one of 1, 2, 3, with the latter two cases only possible if n ≤ 12 and n = 6 respectively.If i < j, then we check that i = 3 is never small enough, and for i = 2 we have the possible cases ω 2 + ω 3 for n = 5, and ω 2 + ω 3 , ω 2 + ω 4 for n = 6.
Proof.A small enough 3-supported highest weight ω must have width exactly three, since there is no small enough 2-supported weight with width larger than 3. Thus, ω = ω i + ω j + ω k for some i, j, k, and since one of ω j→i , ω j→k must be small enough, we have either ω j→i = 2ω 1 + ω n−1 or ω j→k = 2ω 1 + ω n−1 .Suppose it is the former without loss of generality.Then i = 1 and k = n − 1, so ω = ω 1 + ω j + ω n−1 .Note that Γ ω is contained in Γ ω 1 +ω j ⊗ V * .The exact decomposition is given by [FH,Proposition 15.25(ii)]: we conclude that for j = 2, and so Γ ω is small enough in this case only when n = 4 and ω = ω 1 + ω 2 + ω 3 .When 2 < j < n − 1, there are only finitely many (j, n) for which ω 1 + ω j is small enough, and we may check that none of them yield small enough ω 1 + ω j + ω n−1 .
Proposition 4.5.No k-supported highest weight of SL n is small enough for k > 3, and hence the lists of the previous three lemmas are complete.
Proof.If ω is k-supported for k > 3, then ω has width at least k.By applying k − 3 shift operations in every possible way, we get a list of 3-supported weights α 1 , . . ., α m , all of width at least k, such that dim Γ ω ≥ min i dim Γ α i by Lemma 4.1.But the only small enough weights of support 3 have width 3, so none of the α i are small enough, and hence ω is not small enough.
The cases of Sp 2n , Spin 2n+1 , and Spin 2n proceed in the same fashion as SL n and are, in fact, easier to handle so their proofs are omitted.for n ≥ 2. Let L 1 , . . ., L n be the standard dual basis in h * , so the fundamental weights are

Highest weight Restrictions on n
Proposition 4.8.Table 4 is a complete list of the highest weights of the small enough irreducible rational representations of the odd spin groups Spin 2n for n ≥ 4. 4.5.The exceptional groups.Proposition 4.9.Theorem 1.3 holds for all exceptional groups.
Proof.A computer check reveals that any highest weight for an exceptional group either has dimension greater than the bound of Proposition 3.8, is the adjoint representation, the smallest nontrivial representation, or is the representation 2ω 1 of G 2 .The adjoints and smallest non-trivial representations are cofree for all the exceptional groups.If V is the G 2 -representation 2ω 1 , one checks that every order 3 toral element t has dim V t = 9 ≤ dim V − dim G 2 − 2 = 27 − 14 − 2 = 11, and so 2ω 1 is also not pure by Lemma 3.4.Thus pure implies cofree for irreducible representations of exceptional groups.

Remaining cases
In this section we complete the proof of Theorem 1.3 by demonstrating that the representations in Tables 1-4 that were not cofree are not pure.By Lemma 3.4, it suffices to check that dim V t /N t is at most dim V − dim G − 2 for all toral elements of any fixed order, up to the action of the Weyl group, which we will do for every representation but the spinor representation of Spin 15 .
5.1.The case of SL n .
Proof.Let t be a toral element of order 2 and let k = dim SL t n ; note that n − k is always even.As a representation of G ′ := SL k × SL n−k , we have, Let H be the stabilizer of a generic point for the G ′ -action on ( 3 SL n ) t .Since G ′ normalizes t, from Lemma 3.4 it follows that if , where H ′ is the stabilizer of a generic point for the action of SL k on 3 SL k .For k ≥ 10, this representation is properly stable, so dim H ′ = 0; for k ≤ 9, H ′ is listed in the generic stabilizer table of [PV,p. 261 Comparing with the previous equation, it is enough to show The table of [PV,p. 261] reveals that dim H ′ ≤ 16 always, and as a result we can compute that if n ≥ 15, then (5.3) holds for any k.When 10 ≤ n ≤ 14, one checks that (5.3) holds for all (n, k) except for (10, 2), (10, 4), ( for example, if k = 7, then [PV,p. 261] shows that dim H ′ = 14 and (5.3) holds for n ∈ {11, 13}.It remains to handle the (n, k) pairs listed above.We begin with the case where k = 1.Then 6], we see the dimension of the generic stabilizer is 3 2 (n − 2), so again showing (5.2) holds.
With the exception of (n, k) = (10, 6), for all remaining (n, k), the G ′ -representation SL k ⊗ 2 SL n−k falls within Case 6 of [ É2], but does not appear on Table 6 of (loc.cit.), so dim H = 0.It follows that dim and so (5.2) holds.
We now turn to the last case: (n, k) = (10, 6).We must show that the quotient of First, note that N contains the image of the map GL 6 × SL 4 → SL 10 given by (g, h) → diag(g, det g −1 h).The kernel of this map is finite, so dim V/N ≤ dim V/(GL 6 × SL 4 ).We will show that the latter quotient has dimension at most 19.
To simplify notation, let G = GL 6 and H = SL 4 .Note that (V ⊕W )/(G×H) = ((V ⊕W )/G)/H.As a G = GL 6 -module, V = SL ⊕6 6 , and G acts generically freely on V ; in particular, the quotient is 0-dimensional.Thus dim(V + W )/G = dim V /G + dim W = 20.To prove the assertion, we need to show that H acts non-trivially on this quotient and that the generic orbit is closed.The action of h ∈ H on a G-orbit (v, w) is given by h(v, w) = (hv, hw) = (hv, w), where the last equality is because H = SL 4 acts trivially on W = 3 SL 6 .
If v ∈ V is a generic vector in V then Stab G v = 1.In addition H acts trivially on V /G (as it is 0-dimensional), so if h ∈ H, there is a unique g ∈ V such that hv = gv.This defines a morphism ϕ v : H → G, and we can rewrite the action of H on (V +W )/G as h(v, w) = (hv, w) = (v, ϕ(h) −1 w).
The map ϕ v must necessarily have finite kernel, because H = SL 4 is a simple group and the image is non-trivial-H acts non-trivially on V , so we can pick v such that Stab v H = H.Thus h stabilizes (v, w) if and only ϕ v (h) ⊂ Stab G w. Since w is independent of v, we claim that there must exist w ∈ W such that the image of ϕ v (H) is not contained in Stab G w.This would mean that it is a strictly smaller dimensional subgroup of ϕ v (H), since a proper subgroup of a connected group always has strictly smaller dimension.
To prove the claim, suppose to the contrary that ϕ v (H) ⊂ Stab G w for all w in a dense open set.Then ϕ v (H) is a subgroup of the kernel of the G action on 3 SL 6 .But the kernel of this action is trivial.
Proof.We first consider the case of V := Γ ω 1 +ω 2 .From the decomposition we see that if t = diag(1, . . ., 1, −1, . . ., −1) is an order 2 toral element with k ones, then With these constraints, one checks that the above cubic is maximized at k = n − 2, so for n ≥ 7, we have showing that V is not pure by Lemma 3.4.The cases 4 ≤ n ≤ 6 are handled as follows.When n = 6, one may use LiE to directly verify that dim V t ≤ dim V − dim SL 6 −2 for all t in a set of toral representatives of order 3, and so V is not pure by Lemma 3.4.When n = 5, one check by computer that the maximum number of weights lying on a hyperplane in the weight space for V is 14 < 15 = dim V − dim SL 5 −1, so by Proposition 3.1 it is not pure.
For n = 4, we check using toral elements of order four that we always have dim V t / N t ≤ 3, branching the representation as an SL 2 × SL 2 module as necessary.We are grateful for the referee for suggesting a simpler proof of this case.There are five toral elements of order four up to Weyl group conjugacy, of which two have no fixed points, and the remaining three are the elements diag(i, −i, −1, −1), diag(i, i, −1, 1), diag(i, −i, 1, 1) where i is a primitive fourth root of unity.Take for example t = diag(i, −i, 1, 1) which is normalized by {1} × SL 2 .As a representation of SL 2 × SL 2 , we have where (Sym 2 SL 2 ) 0 denotes the 0 weight space of Sym 2 SL 2 .From here we can see that dim V t /N t ≤ dim V t /{1} × SL 2 ≤ 3. A similar analysis can be performed for the other two toral elements, showing that V is not pure for n = 4.
The cases Γ 2ω 1 +ω n−1 and Γ ω 1 +ω n−2 follow similarly using the decompositions Γ 2ω In the former case, this cubic is always non-negative for n > 6, meaning that the representation is not pure for n ≥ 6 by Lemma 3.4.In the latter case, this cubic is always non-negative for n > 6 and 1 ≤ dim SL t n ≤ n, and in the last case where dim SL t n = 0, then t = −I.
The remaining for both representations where n = 4, 5, 6 are handled with LiE using toral elements of order 3 and using Lemma 3.4.Lastly, for V := Γ 3ω 1 ≃ Sym 3 SL n , we have When n ≥ 7, then one can check dim V t ≤ dim V − dim SL n −2 for all k except k = n when n is even; but then t = −I.For 4 ≤ n ≤ 6 we adopt a different approach.It is proved in [MFK] that every smooth cubic hypersurface of degree 3 is GIT stable.This implies that if V = Sym 3 SL n then V sss is contained in the discriminant divisor.For n = 4, 5, 6 (cubic surfaces, threefolds and fourfolds) work in GIT [ACT, All, Laz] implies that the generic singular hypersurface is GIT stable.Since the discriminant divisor is irreducible this implies that V sss cannot have codimension one since it is a proper algebraic subset of the discriminant.5.2.The case of Sp 2n .
Proof.The representation Γ ω 3 is the kernel of the contraction 3 Sp 2n → Sp 2n using the bilinear form, of dimension 2n 3 − 2n.Let t be a toral element of order 2, and let k = dim Sp t 2n .Note that since with respect to the standard maximal torus, t = diag(x 1 , . . ., x n , x −1 1 , . . ., x −1 n ) for some x i = ±1, it follows that k is even.By decomposing Sp 2n into its positive and negative eigenspaces under t and expanding the exterior product as a sum of tensors, we find by the exactness of taking invariants that For 0 ≤ k ≤ 2n − 2, this cubic is maximized at k = 2n − 2 for n ≥ 6, and if k > 2n − 2, then since k must be even, in fact k = 2n and t = 1 is not of order 2. Thus we have an upper bound dim Γ t ω 3 ≤ 2n−2 3 for all n ≥ 6.We then check that 2n−2 3 ≤ dim Γ ω 3 − dim Sp 2n −2 for n ≥ 6, showing that Γ ω 3 is not pure for such n by Lemma 3.4.
For n = 5, we find that dim V t ≤ dim V − dim Sp 2n −2 for toral elements of order 3 except Otherwise, note that an order two toral element acting on Spin 2n+1 always has odd-dimensional positive eigenspace; as a matrix, it can be taken to be of the form diag(x 1 , . . ., x n , x −1 1 , . . ., x −1 n , 1) in the standard maximal torus, where each x i = ±1, and so the number of entries with value 1 is always odd.
For Γ ω 3 , we note that it is equal to the third exterior power 3 Spin 2n+1 for n ≥ 4. As a result, if t is a toral element of order 2, and k and it can be checked that this is less than dim Γ ω 3 − dim Spin 2n+1 −2 for all (n, k) when n ≥ 4 and 0 ≤ k ≤ 2n + 1 except when k ≥ 2n.As we remarked above, since k must be odd, k ≥ 2n implies k = 2n + 1 and then t = 1 is not of order 2, and thus by Lemma 3.4, Γ ω 3 is not pure for any n ≥ 4.
Lemma 5.7.The spinor representation of Spin 15 is not pure.
Proof.Let V = Spin + 16 be the positive half-spinor representation, which is a cofree representation of Spin 16 , hence pure by Proposition 2.1.Then the spinor representation Spin 15 is the restriction of V under the natural inclusion Spin 15 ⊂ Spin 16 of Lie groups.To show Spin 15 is not pure, we employ the following strategy.Since Spin 15 and Spin + 16 are properly stable representations, by the Hilbert-Mumford Criterion, we have an inclusion V sss (Spin 15 ) ⊆ V sss (Spin 16 ).
We know that V sss (Spin 16 ) is pure of codimension 1, so to prove Spin 15 is not pure, it suffices to show the above inclusion is strict.
For each S ∈ S, let a S,i = 1 if i ∈ S and a S,i = −1 if i / ∈ S. We then obtain a set of 8 weights for Spin + 16 .For each µ ∈ W, choose a non-zero weight vector v µ and let v = µ v µ .Since every µ has a positive L 8 -coefficient, v ∈ V sss (Spin 16 ).
We claim that v / ∈ V sss (Spin 15 ).To see this, let µ ′ be the weight of Spin 15 induced by µ.Then when V is viewed as a Spin 15 -representation, our set of weights W map to the set and moreover, µ ′ ∈W ′ µ ′ = 0. Therefore, the origin is the interior of the convex hull of W ′ .We further see that the weights of W ′ satisfy a strong orthogonality relation: any two µ ′ 1 , µ ′ 2 ∈ W ′ differ by 4 sign flips, so µ ′ 1 − µ ′ 2 is not a root.It follows from [DK,Proposition 1.2] that v ∈ V s (Spin 15 ), the complement of V sss (Spin 15 ).
Remark 5.8.Lemma 5.7 did not rely on Proposition 3.1 to show that the spinor representation of Spin 15 was not pure, and so the content of Remark 3.2 does not apply.However, since this representation is self-dual, the dual representation does not need to be handled differently.
Lemma 5.9.The spinor representation of Spin 17 is not pure.
Proof.Let V = Spin 17 be the spinor representation of the group Spin 17 and let t = diag(ζ m 1 , . . ., ζ m 8 ) be a toral element with ζ a primitive 3rd root of unity.Using the fact that t acts trivially on the weight space of 1 2 8 i=1 a i L i if and only if n i=1 a i m i = 0 mod 3, one checks that dim V t ≤ 118 = 2 8 − 17 2 − 2 unless t = diag(1, . . ., 1, ζ, ζ m ) with m = ±1.To handle this remaining case, we begin by constructing a copy of Spin 13 in the centralizer of t.In the notation of [FH,p. 370 (23.8)], we have Let S = {7, 8, 15, 16}.Since e i is orthogonal to e j for i / ∈ S and j ∈ S, the even Clifford algebra generated by the e i for i / ∈ S yields a copy of Spin 13 ⊂ Spin 17 that commutes with t.Next, since V t is the direct sum of the weight spaces with weights 1 2 8 i=1 a i L i and a 7 + ma 8 = 0 mod 3, as a Spin 13 -representation, we have V t = (Spin 13 ) ⊕2 , i.e., it is two copies of the spinor representation.From [PV,p. 262], we see the generic stabilizer of Spin 13 is 16-dimensional, so dim(V t /N t ) ≤ dim(V t /Spin 13 ) ≤ 2 7 − 13 2 + 16 = 66 ≤ 118.
Thus, V is not pure.
Proof.This proof is very similar to the proof of Lemma 5.6, so we carry it out with somewhat less detail.Note that Γ ω 3 = 3 Spin 2n for n ≥ 5, and so if t is an order 2 toral element and k = dim Spin t 2n , then dim As we remarked previously for the odd spin groups, the only possibilities for k will be even, and if k = 2n then t = 1 is not of order 2; so only 0 ≤ k ≤ 2n − 2 must be considered.Except for n = 5, the maximum is always reached at the endpoint 2n − 2, where we can check that dim Γ t ω 3 ≤ dim Γ ω 3 − dim Spin 2n −2, and so none of these representations are pure by Lemma 3.4.When n = 5 we can check again that no choice of 0 ≤ k ≤ 8 gives dim Γ t ω 3 ≤ dim Γ ω 3 −dim Spin 10 −2, so once more by Lemma 3.4, Γ ω 3 is not pure for n = 5.
The proof for Γ 3ω 1 is similar to the case of the third symmetric power for odd spin groups: we have Γ 3ω 1 = ker(Sym 3 Spin 2n → Spin 2n ) and so for a toral element t of order two, dim Γ t 3ω 1 = dim(ker Sym 3 Spin 2n → Spin 2n ) t = dim(Sym 3 Spin 2n ) t − dim Spin t 2n .If we write k = dim Spin t 2n , then and we can again check that for 0 ≤ k ≤ 2n − 2, this is less than dim Γ 3ω 1 − dim Spin 2n −2, so by Lemma 3.4, these representations are not pure.
Lemma 5.11.The half-spinor representations of Spin 18 are not pure.
Proof.Since there is an outer automorphism of Spin 18 interchanging the two half-spinor representations, it suffices by Remark 1.5 to consider the representation V = Spin + 18 .Then the weights of V are 1 2 9 i=1 a i L i with a i = ±1, and an even number of a i = −1.We follow the same strategy of proof as in Lemma 5.9.One checks that if t is an order 3 toral element, then dim V t ≤ 101 = 2 8 − 18 2 − 2 unless t = diag(1, . . ., 1, ζ, ζ m ) with m = ±1.It remains to handle this latter case.
As in the proof of Lemma 5.9, there is again a copy of Spin 14 in the centralizer of t.Note that V t is the direct sum of the weight spaces where weights 1 2 9 i=1 a i L i with a i = ±1, a 8 + ma 9 = 0 mod 3, and i a i = 1; said another way, it is a direct sum of weight spaces with weights of the form 1 2 ( i=7 a i L i + a 8 (L 8 − mL 9 )), where −m = 7 i=1 a i .Thus, viewing V t as a Spin 14 -representation, we have V t ≃ (Spin − 14 ) ⊕2 , m = 1 (Spin + 14 ) ⊕2 , m = −1.From [PV,p. 262], we see the generic stabilizer of Spin 14 is 28-dimensional, so dim(V t /N t ) ≤ dim(V t /Spin 14 ) ≤ 2 7 − 14 2 + 28 = 65 ≤ 101.
Thus, V is not pure.

Part III. Actions of tori
We now turn our attention to torus representations.We prove Theorem I.1.6 in §1.In §2, we give examples that distinguish the classes of representations pure, coprincipal, and coregular.The most subtle of these is Example 2.3, which shows that coprincipal is not equivalent to pure; this is in contrast to Lemma I.1.3which shows that pure and coprincipal are equivalent for connected G with no non-trivial characters.
1. Proof of Theorem I.1.6 Our initial goal is to prove the following proposition.This is done after several preliminary lemmas.Throughout this section, if V is a G-representation, then we denote by V sss (G) the strictly semi-stable locus for the action of G.
Proposition 1.1.Let V 1 and V 2 be stable representations of a torus T .Let V = V 1 ⊕ V 2 be a decomposition as T -representations and assume that V /T = V 1 /T × V 2 /T .Then V is cofree (resp.coprincipal) if and only V 1 and V 2 are cofree (coprincipal) representations.
Remark 1.2.Note the condition that V /T = V 1 /T × V 2 /T is a very strong since it implies that

Table 2 .
Small enough representations of Sp 2n 4.2.The case of Sp 2n .Listed in Table 2 are the small enough representations of Sp 2n for n

Table 3 .
Table 2 is a complete list of the highest weights of the small enough irreducible rational representations of the symplectic groups Sp 2n for n ≥ 3. Small enough representations of Spin 2n+1 4.3.The case of Spin 2n+1 .Listed in Table 3 are the small enough representations of Spin 2n+1

Table 4 .
Table 3 is a complete list of the highest weights of the small enough irreducible rational representations of the odd spin groups Spin 2n+1 for n ≥ 2. Small enough representations of Spin 2n 4.4.The case of Spin 2n .Listed in Table 4 are the small enough representations of Spin 2n for n ≥ 4. Let L 1 , . . ., L n be the standard basis in h * , and then we have the fundamental weights 1, 1), up to Weyl group conjugacy.For this t, we have However, since Sp 8 ⊕ 3 Sp 8 is a reducible representation of a simple Lie group that does not appear on the table of[ É1], it follows that Sp 8 acts with trivial generic stabilizer on this representation,Sp 8 − dim Sp 8 = 64 − 36 = 28 ≤ 53 = dim V − dim Sp 10 −2,and the representation is not pure by Lemma 3.4.The proof for Γ 3ω 1 = Sym 3 Sp 2n is similar: let t be a toral element of order 2, and let k = ≤ dim Γ 3ω 1 − dim Sp 2n −2 for all n ≥ 5, so these representations are not pure by Lemma 3.4.5.3.The case of Spin 2n+1 .Lemma 5.6.The representations Γ ω 3 for n ≥ 4 and Γ 3ω 1 for n ≥ 2 of Spin 2n+1 are not pure.Proof.The representation Γ 3ω 1 has dimension 2n+1+2 3 −(2n+1), and is the kernel of the contraction Sym 3 Spin 2n+1 → Spin 2n+1 by the symmetric bilinear form.For any toral element t, exactness of taking invariants gives dim ker Sym 3 Spin 2n+1 → Spin 2n+1 t = dim(Sym 3 Spin 2n+1 ) t − dim Spin t 2n+1 , so if t is of order 2 and k = dim Spin t 3Sp 8 by the exactness of taking invariants.Since Sp 8 × Sp 2 ⊆ N t , we finddim Γ t ω 3 /N t ≤ dim 3 Sp 8 /(Sp 8 × Sp 2 ) = dim 3 Sp 8 /Sp 8 .andfind again that on 0 ≤ k ≤ 2n − 2, it is maximized at k = 2n − 2. So dim Γ