Essentially finite $G$-torsors

Let $X$ be a smooth projective curve of genus $g$, defined over an algebraically closed field $k$, and let $G$ be a connected reductive group over $k$. We say that a $G$-torsor is essentially finite if it admits a reduction to a finite group, generalising the notion of essentially finite vector bundles to arbitrary groups $G$. We give a Tannakian interpretation of such torsors, and we prove that all essentially finite $G$-torsors have torsion degree, and that the degree is 0 if $X$ is an elliptic curve. We then study the density of the set of $k$-points of essentially finite $G$-torsors of degree $0$, denoted $M_{G}^{\text{ef},0}$, inside $M_{G}^{\text{ss},0}$, the $k$-points of all semistable degree 0 $G$-torsors. We show that when $g=1$, $M_{G}^{\text{ef}}\subset M_{G}^{\text{ss},0}$ is dense. When $g>1$ and when $\text{char}(k)=0$, we show that for any reductive group of semisimple rank 1, $M_{G}^{\text{ef},0}\subset M_{G}^{\text{ss},0}$ is not dense.


Introduction
Let X be a smooth projective connected curve over an algebraically closed field k. Let g = g(X) be the genus of X. In 1938 Weil introduced the notion of a finite vector bundle; a vector bundle E is called finite if there are two distinct polynomials, f, g ∈ N[x], such that the vector bundle f (E) is isomorphic to g(E) (see [Wei38]). For k = C, he proved that a vector bundle is finite if and only if it arises from a representation of π 1 (X) which factors through a finite group. Almost 40 years later, in [Nor76], Nori introduced the notion of an essentially finite vector bundle as a subquotient of a finite one. The category of essentially finite vector bundles forms a Tannakian category, and the corresponding group is known as the Nori fundamental group, a pro-group scheme over k whose k points are isomorphic to the étale fundamental group, π et 1 (X), when k is of characteristic 0 (see [Sza09,Corollary 6.7.20] and also e.g., [EHS08]).
Viewing a vector bundle as a GL n -torsor, we are led to the question: can we generalise the notion of an essentially finite vector bundle, to a notion of an essentially finite G-torsor, for G an affine algebraic group? Nori proved that a vector bundle E is essentially finite if and only if there exists a finite group scheme Γ, a Γ-torsor F Γ and a representation V of Γ such that E ∼ = F Γ × Γ V . Hence, we are led to the following definition Definition 1.1. An essentially finite G-torsor is a G-torsor over X which admits a reduction to a finite group.
Under the correspondence between vector bundles and GL n -torsors, this agrees with the known definition of essentially finite vector bundles. We prove the following.
Theorem 1.2. Let G be a connected, reductive group. Then for any G-bundle F G , the following are equivalent.
1. The G-bundle F G is essentially finite.
2. There exists a faithful representation ρ : G → GL V such that ρ * F G is an essentally finite vector bundle.
3. For every representation ρ : G → GL V , ρ * F G is an essentally finite vector bundle.
4. There exists a proper surjective morphism f : Y → X such that f * F G is trivial.
Note also that since semistability can be checked on the adjoint bundle, every essentially finite G-torsor is semistable. We give a self-contained proof of this fact, not using the adjoint representation.
Let now M ss G denote the moduli space of semistable G-bundles over X, for G a connected reductive group. Recall that the connected components of M ss G are indexed by the algebraic fundamental group of G, π 1 (G). If a G-bundle, F G , lies in a component corresponding to d ∈ π 1 (G), then it is said to have degree d. Essentially finite vector bundles always have degree 0. We prove the following. Theorem 1.3. For any connected reductive group G, every essentially finite G-torsor over X is of torsion degree.
Again this generalises the case for G = GL n , since in this case π 1 (G) = Z, which is torsionfree. We also show that if X is an elliptic curve then all essentially finite G-bundles have degree 0.
Let now M ef,0 G denote the k-points of the essentially finite G-torsors of degree 0, inside M ss,0 G , and let G = GL n . If n = 1, then essentially finite G-bundles correspond to essentially finite line bundles, which correspond to torsion line bundles (see Lemma 3.1 [Nor76]). Hence, M ef GL 1 is dense inside M ss,0 GL 1 = Jac 0 (X) since torsion points are dense in any abelian variety. In positive characteristic Ducrohet and Mehta have shown that M ef,0 GLn ⊂ M ss,0 GLn is dense for all n when g ≥ 2, and similarly for vector bundles with trivial determinant (they show in fact that a smaller set of objects, called Frobenius periodic vector bundles, are dense; see [DM10]). However, in characteristic zero much less seems to be known about the density of essentially finite bundles when the rank is greater than 1. Hence, we may ask whether M ef,0 GLn is dense in M ss,0 GLn for n > 1, when char(k) = 0. More generally, we are interested in the question of whether M ef,0 G is dense in M ss,0 G for arbitrary connected reductive groups G over an arbitrary, algebraically closed field k.
Hence it is clear that every essentially finite G-torsor over P 1 is trivial. We give a self-contained proof of this result using a Tannakian interpretation of both the classification of G-torsors over P 1 (see [Ans18]) and the definition of essentially finite torsors. If g = 1, that is if X is an elliptic curve, then we prove that M ef,0 G is dense in M ss,0 G for all connected, reductive groups. This follows from work of Frăţilă [Fră21] and Laszlo [Las98]. On the contrary, if g ≥ 2 and char(k) = 0, then we show the following.
Theorem 1.4. Let char(k) = 0. For all connected, reductive groups of semisimple rank 1, The main work lies in proving the theorem for PGL 2 -torsors. Note also that this shows that M ef GL 2 is not dense in M ss,0 GL 2 . In characteristic 0, Weissman [Wei22] has independently obtained this non-density result for M ef GLn for all n ≥ 1. By the theorem of Narasimhan and Seshadri, the points of M ss,0 GLn (C) are also the isomorphism classes of representations π 1 (X) −→ U n (C), i.e., there is an analytic homeomorphism between M ss,0 GLn (C) and the character variety Hom(π 1 (X), U n (C))/ ∼. In particular finite vector bundles correspond to unitary representations of π 1 (X) which factor through finite groups. As the Zariski topology is coarser than the analytic topology we see as a corollary to non-density for rank n vector bundles that the set of rank n unitary representations of π 1 (X) which factor through finite groups is not dense inside Hom(π 1 (X), U n (C))/ ∼.
The outline of the text is as follows. In Section 2 we introduce the necessary notations and background. In Section 3 we define essentially finite G-torsors, generalising the notion of essentially finite vector bundles. We prove that such torsors are (strongly) semistable of torsion degree. Finally, in Section 4 we prove the above mentioned statements about density of M ef,0 G in M ss,0 G .

Acknowledgements
We would like to thank our respective advisors, Carlo Gasbarri and Wushi Goldring, for their support during this project. We would also like to thank Dragoş Frăţilă, João Pedro dos Santos, Emiliano Ambrosi, Florent Schaffhauser, and Georgios Kydonakis for their remarks and questions that have greatly improved the quality of this work. Finally, we thank an anonymous referee for explaining to us that we could relax the assumptions in Proposition 3.5 and how this gives a much simpler proof of the implication "3. implies 1." in Theorem 3.10 (which also allowed us to remove a restriction on the characteristic in an earlier version), as well as for several other useful comments that greatly improved the quality of the paper.

Notations, conventions and background
Throughout the text, let k be an algebraically closed field and let X be a smooth, projective, connected curve over k. Recall that if G denotes an algebraic group over k, then a G-torsor over X is a scheme F G over X with an action of G such that there exists an fppf cover, (U i → X) i∈I such that for each i ∈ I there is a G| U i -equivariant isomorphism F G | U i ∼ = G| U i . We will also use the term G-bundle as synonym for G-torsor. If ϕ : H → G is a group morphism and F H is an H-torsor, then we denote by ϕ * F H the G-torsor ϕ * F H := F H × H G. In the special case when ϕ : G → GL V is a representation of G, we denote ϕ * F G by V F G (following [Sch15]). If F G is a G-torsor such that F G ∼ = ϕ * F H for some triple (H, ϕ, F H ) as above, then we say that F G admits a reduction of structure group to H. We denote by Rep k (G) the category of finite-dimensional representations of G over k. Recall that to give a G-torsor over X is equivalent to give an exact, k-linear, tensor functor Rep k (G) → Vec X , where Vec X denotes the category of vector bundles over X. We will use the same notation for the bundle seen as a functor. Now suppose that G is a connected, reductive group. Given a maximal torus T ⊂ G let X * (T ) denote the characters of T and let X * (T ) denote the cocharacters. Let further Φ ⊂ X * (T ) denote the corresponding roots and let Φ ∨ ⊂ X * (T ) denote the corresponding coroots. We let π 1 (G) denote the algebraic fundamental group of G, namely, π 1 (G) = X * (T )/ span{Φ ∨ }.
Let M G denote the stack of G-torsors over X, let M ss G denote the substack of semistable G-torsors and let M ss G denote the moduli space of semistable G-torsors (see [Ram96a], [Ram96b] and [GLS + 08]). If we consider another curve, Y , then for clarity we may also write M G,Y to denote the stack of G-torsors over Y . We define M ss G,Y and M ss G,Y analogously. Recall that the connected components of M G are labeled by π 1 (G), that is, (2.2) Definition 2.1. If F G is an object of Mλ G , then F G is said to be of degreeλ.
We also have that π 0 (M P ) ∼ = π 0 (M L ) = π 1 (P ) and we similarly say that a P -torsor is of degreeλ P if it lies in the component corresponding toλ P .
Lemma 2.2. Suppose that ϕ : G → H is a morphism of smooth connected algebraic groups and let F G be a G-torsor of degree 0. Then ϕ * F G has degree 0.
Proof. By [Hof10] we have a commutative diagram of pointed sets where all morphisms are the natural ones induced by ϕ and where the left vertical map is a group morphism. The statement follows.

Semistable torsors
Let T be a maximal torus of G and let B ⊃ T be a Borel containing T . Then the center of G can be described as By composition via the inclusion Z(G) → T we have a natural map Upon tensoring with Q this induces an isomorphism X * (Z(G)) Q ∼ = π 1 (G) Q . Following [Sch15] the definition of the slope map and subsequently the definition of a semistable G-torsor is as follows.
Definition 2.4. For a parabolic subgroup, P , such that B ⊂ P ⊂ G, with corresponding Levi L, the slope map φ P : π 1 (P ) → X * (T ) Q is the map given by Example 2.5. For G = GL n , we will describe the slope map φ G . We have that L = G so Z(L) = scalar n , the scalar matrices of rank n. We also have the standard identifications X * (scalar n ) ∼ = Z and X * (T ) ∼ = Z n . Further, we may write π 1 (G) = Z · e 1 , where e i : t → diag(1, ..., 1, t, 1, ..., 1) with t in the i th position, and (−) represents the image in π 1 (G).
(2.5) Now let P be an arbitrary parabolic of G = GL n , with Levi factor L = m i=1 GL n i . Then (a 1 , ..., a m ) → (a 1 , ..., a 1 , a 2 , ..., a 2 , ..., a m , ..., a m ) → (n 1 a 1 , n 2 a 2 , ..., n m a 2 ), where a i occurs n i times in the tuple in the middle. Thus, the slope map φ P is given by Definition 2.6. Let F G be a G-torsor of degreeλ. We say that F G is semi-stable if for each parabolic P ⊂ G and each reduction F P of F G to P , of degreeλ P , we have that Example 2.8. Again let G = GL n , we show why this definition gives back the usual slope semi-stability for vector bundles. Recall first that the slope µ(E) of a vector bundle E is defined as µ(E) = deg(E) rk(E) and that E is called slope semi-stable if for any subbundle F we have that µ(F ) ≤ µ(E).
Let now E be a vector bundle, let P ⊂ G be a parabolic with Levi factor L = m i=1 GL n i and let F P be a reduction of E to P . This amounts to giving a filtration 0 where π i : P → L → GL n i is the composition of the projections P → L and L → GL n i . Then we see that (2.9) Since φ G (deg(E)) = (µ(E), ..., µ(E)) we see that Definition 2.6 agrees with the usual slope semi-stability definition.
Now we recall some results of [Sch15] regarding the slope map which we will need to prove that essentially finite torsors are semi-stable. To this end, let λ ∈ X * (T ) be a dominant character and let V be a finite-dimensional G-representation of highest weight λ. If P is a parabolic with Levi factor L, where Φ L are the roots of the Levi L. Then we have the following result.
Proposition 2.9 ([Sch15] Proposition 3.2.5(b),(c)). Keep the notation as above. Let F G be a G-torsor of degreeλ G . Then the slope of the vector bundle V F G is given by (2.12)

Essentially finite torsors
We begin with the main object of study in this article.
Definition 3.1. An essentially finite G-torsor is a G-torsor over X which admits a reduction to a finite group.
Remark 3.2. Although we have fixed a smooth, projective, connected curve X over k for simplicity of the exposition, this definition makes sense over an arbitrary scheme. Similarly, we may use the same definition for arbitrary affine groups, not necessarily connected reductive.
Example 3.4. 1. The trivial G-torsor G × X is essentially finite since it admits a reduction to the trivial group.
3. Note that if α : G → G ′ is a morphism of algebraic groups and F G is an essentially finite G-torsor, then α * F G is an essentially finite G ′ -torsor.
Let us phrase two equivalent conditions for a G-bundle to be essentially finite; one in terms of the Nori fundamental group, and one Tannakian interpretation. Since k is algebraically closed, there is a rational point x of X. Let π N 1 (X, x) denote the Nori fundamental group of X and let X denote the universal Proof. Let F G be an essentially finite G-torsor, let ι : Γ ֒→ G be a finite subgroup of G and let j : F Γ → X be a Γ-torsor such that ι * F Γ ∼ = F G . Let y be a rational point of F Γ such that j(y) = x. Then j defines a pointed finite torsor (F Γ , y) → (X, x). By [Nor76, Proposition 3.11], there is a morphism π N 1 (X, x) → Γ, which we compose with ι to get a morphism ρ : π N 1 (X, x) → G such that F G ∼ = ρ * X. Conversely, suppose that we have a morphism ρ : π N 1 (X, x) → G such that ρ * X ∼ = F G . Since π N 1 (X, x) = lim ← −i A i is the inverse limit of its finite quotients A i (see [Nor82]), there is some i and a morphism ρ i : A i → G such that ρ factors where π i is the projection. Since ρ * X ∼ = ρ i, * (π i, * X) we see that F G is essentially finite.
Proof. If F G is essentially finite, coming from a finite group Γ, a group morphism ϕ : Γ → G and a Γ-torsor F Γ , then we take α to be the induced functor from ϕ. Conversely, every such α, by [DM82, Corollary 2.9], comes from a group morphism ϕ : Γ → G.
Remark 3.7. If a G-torsor F G is essentially finite then there exists a finite group Γ and a Γ-torsor j Γ : F Γ → X such that j * Γ F G is trivial. Proposition 3.8. Under the correspondence between vector bundles of rank n and GL ntorsors, a GL n -torsor is essentially finite if and only if the corresponding vector bundle is essentially finite.
Proof. Let F GL n be a GL n -torsor, and let Γ be a finite subgroup of GL n , α : Γ → GL n and let j : F Γ → X be a Γ-torsor such that F GLn = α * F Γ . Then F GLn is trivialised by j : F Γ → X so the corresponding vector bundle E is also trivialised by j : F Γ → X. Thus, E is essentially finite.
Conversely suppose E is an essentially finite vector bundle. Then there is a finite group ι : Γ → GL n and a Γ-torsor F Γ → X such that E = F Γ × Γ A n . Then we have that whence the vector bundle associated to ι * F Γ is E. Hence, the bundle corresponding to E is isomorphic to ι * F Γ , hence essentially finite.
Lemma 3.9. Let Y be a proper and connected scheme over k. A G-bundle F G over Y is trivial if and only if for any faithful representation ρ : G → GL V , ρ * F G is trivial.
Proof. The idea of this can be found in [BD13, Lemma 4.5], but we spell out the details since their assumptions on the base scheme are different from ours. Suppose that ρ : G → GL V is any faithful representation. Consider the long exact sequence of pointed sets (see [DG70, III, §4, 4.6]) where π : is affine and hence, using that Y is proper and connected, y is constant. That is, we have a factorisation y : Y → Spec k → GL V /G. Since k is algebraically closed, (GL V /G)(k) = GL V (k)/G(k), and hence y being constant implies that there is a liftỹ : Y → GL V of y.
By the universal propery of fiber products we thus see that δ(y) admits a section, whence δ(y) is trivial. Hence, by exactness of (3.4) a G-bundle F G is trivial if and only if ρ * F G is trivial.
Theorem 3.10. Let G be a connected, reductive group and let F G be a G-bundle. The following are equivalent.
1. The G-bundle F G is essentially finite.
2. There exists a faithful representation ρ : G → GL V such that ρ * F G is an essentally finite vector bundle.
3. For every representation ρ : G → GL V , ρ * F G is an essentally finite vector bundle.
4. There exists a proper surjective morphism f : Y → X such that f * F G is trivial.
First suppose that 2. holds, let ϕ : G → GL W be a faithful representation such that ϕ * F G is essentially finite and let ρ : G → GL V be an arbitrary representation. Since ϕ * F G is essentially finite there is a proper surjective morphism f : Y → X such that f * ϕ * F G is trivial. Since any restriction of f * ϕ * F G to a connected component of Y is trivial, we may assume that Y is connected. Thus, since f * ϕ * F G ∼ = ϕ * f * F G , we see from Lemma 3.9 that f * F G is trivial. Hence, f * ρ * F G ∼ = ρ * f * F G is trivial, which implies that ρ * F G is essentially finite (again by [BdS11]). This proves that 2. implies 3. Now assume that 3. holds. Then the functor F G : Rep k (G) → Vec X factors through the category of essentially finite vector bundles, hence induces a group morphism ρ : π N 1 (X, x) → G such that ρ * X ∼ = F G . Thus, by Proposition 3.5 F G is essentially finite.
Proof. Let F G be such a torsor. Let further P ⊂ G be a parabolic of G, let λ be a dominant character and let V be a representation of highest weight λ. Since F G is essentially finite, the associated vector bundle V F G is essentially finite, hence semistable. Hence, using Proposition 2.9, we have that (3.5) That is, for every dominant character λ ∈ X * (T ) Q we have that Since the cone of cocharacters with non-negative pairing with all dominant characters is double-dual to the cone of simple coroots, we see that Theorem 3.12. Let F G be an essentially finite G-torsor. Then its degree is torsion as an element of π 1 (G).
Proof. Let F G be such a bundle. Let j : F Γ → X be a finite bundle such that F G ∼ = F Γ × Γ G. Let T be a maximal torus and B ⊃ T a Borel containing T , and choose a reduction F B of F G to a Borel. We know that j * F G is trivial. Since we see that j * F B × B G is trivial. We have that π 0 (M B,F Γ ) = π 0 (M T,F Γ ) = X * (T ) and this maps surjectively onto π 0 (M G,F Γ ). The fact that j * F B maps to the trivial torsor means that it corresponds to 0 in π 1 (G) = X * (T )/Φ ∨ = π 0 (M G,F Γ ). This implies that the degree of j * F B , seen as an element in X * (T ), is a sum of coroots. The equality π 0 (M B ) = π 0 (M T ) is induced by the morphism π T : B → T , so π T, * j * F B also corresponds to a sum of coroots. Since π T, * j * F B = j * π T, * F B , the conclusion follows if we can show that the morphism has the property that, if j * F T has degree in Φ ∨ , then the same holds for a multiple of deg(F T ).
If F T corresponds to the cocharacter µ F T , then j * F T corresponds to the cocharacter µ j where α i are the simple roots and µ ∈ X * \ Φ ∨ then We now apply this to our situation above, i.e., with F T := π T, * F B , and since π 1 (G) = X * (T )/Φ ∨ we can conclude that deg(F G ) is torsion.
Proposition 3.13. Let G be a connected, reductive group. If X is an elliptic curve, then every essentially finite G-bundle over X has degree 0.
Proof. We argue by induction on the dimension of G. If dim(G) = 1 then G ∼ = G m and the result follows since it is true for all vector bundles. Suppose now that dim(G) = n > 1. Let F G be an essentially finite G-bundle of degree d. By [Fră21] there is a proper Levi L and a degree d ′ ∈ π 1 (L) such that the inclusion ι : L → G induces a surjection M d ′ L,X → M d G,X . Let F L be a reduction of structure group of F G to L. Since F G is essentially finite there is a faithful representation ρ : G → GL V such that ρ * F G ∼ = (ρ • ι) * F L is essentially finite. By Theorem 3.10 this implies that F L is essentially finite. Since L is a proper Levi, by induction d ′ = 0, whence d = 0.
If the characteristic of k is positive, there is a stronger notion of semistability, defined as follows. Let σ X : X → X denote the absolute Frobenius of X.
Definition 3.14. A G-torsor F G is said to be strongly semistable if for all n > 0, (σ n X ) * F G is semistable. Proposition 3.15. Every essentially finite G-torsor is strongly semistable.
Proof. For any algebraic group H, and any H-torsor, if σ H : H → H denotes the absolute Frobenius of H, then we have that Let now F G be an essentially finite G-torsor. Let j : F Γ → X be a finite bundle such that F G ∼ = F Γ × Γ G. Then by (3.10) applied to Γ and since the push-forward along group morphisms commutes with pullbacks, we have that Hence (σ X ) * F G is essentially finite and thus semistable. The statement follows similarly via induction.

The prestack of essentially finite torsors
Let M ef G denote the functor Proof. First suppose that f : U ′ → U is a morphism in Aff op k and suppose F G is an essentially finite G-torsor over U × X. Let (U i → U ) be a cover and (g ij : g ij ∈ G(U ij )) a cocycle for F G . Then (f * U i → U ′ ) is a cover of U ′ and (f * g ij ) ij is a cocycle for f * F G . Indeed, since g ij g jk = g ik we see that (3. 2) The torsor f * F G is also essentially finite since if g ij ∈ Γ(U ij ) ⊂ G(U ij ) for some finite group Γ, then f * g ij = g ij • f also takes values in Γ. Since M ss G is a lax functor we see that M ef G is one as well.
Next it is clear that if F G , F ′ G ∈ M ef G (U ), then Isom(F G , F ′ G ) : Aff /U → Set is a sheaf since homomorphisms of finite G-torsors are simply homomorphisms of G-torsors and M ss G is a stack.
Remark 3.17. Note however that M ef G is not a stack since the descent data is not necessarily effective. Indeed, let G = GL n and let E be a vector bundle which is not essentially finite. Let further (U i → X) be a trivialising cover of E, with trivilising morphisms φ i : . Now, if E| X×X is essentially finite, then so is E. Indeed, by [BdS11] we have a proper surjective morphism f : Y → X × X such that f * E X×X is trivial, and by composing with the projection X × X → X we have a proper surjective morphism g : Y → X such that g * E is trivial. Since E was assumed not to be essentially finite, we conclude that E| X×X is not essentially finite and the descent data constructed is not effective.
The following statement is immediate, but will be important for us in the final section.
Proposition 3.18. Let G and G ′ be reductive groups. The isomorphism M ss Proof. The isomorphism on objects is given by where π : G × G ′ → G and π ′ : G × G ′ → G ′ are the projections. If Γ ⊂ G × G ′ is a finite structure group of F G×G ′ , then π(Γ) and π ′ (Γ ′ ) are evidently finite structure groups of F G and F G ′ respectively. Similarly, finite structure groups Γ and Γ ′ of F G , respectively F G ′ , give a finite structure group, Γ × Γ ′ of F G × F G ′ .

Genus 0
Let now X = P 1 k , where k is an arbitrary algebraically closed field. By Proposition 3.5 we immediately have the following statement.
Proposition 4.5. Every essentially finite G-bundle over X is trivial.
Proof. Since π N 1 (X, x) is trival, the statement follows from Proposition 3.5.
It is also well-known that M ss,0 G (k) is a singleton so the density statement is immediate. For the remainder of this section, we give a different proof of Proposition 4.5, which might be interesting in its own right. We do this by using the Tannakian interpretation of essentially finite G-bundles and the classification of G-bundles on X.
The classification of G-bundles on X was initially done by Grothendieck [Gro57] and by Harder [Har68] for characterstic p. In [Ans18] Anschütz gives a Tannakian interpretation of this classification. We thus begin by introducing the relevant notions from [Ans18].
Over X there is a canonical G m -torsor often called the Hopf bundle. Pushforward along this bundle defines an exact, faithful tensor functor Taking the Harder-Narashiman filtration of a vector bundle over X defines a fully faithful tensor functor HN : Vec X → FilVec X (4.3) from Vec X to the category of filtered vector bundles. Finally we can take the graded pieces of a filtered vector bundle and this defines an exact tensor functor where GrVec X is the category of graded vector bundles.
is an equivalence of tensor categories onto its essential image, which consists of graded bundles E = n∈Z E i such that each E i is semistable of slope i.

The main Theorem of Grothendieck, restated in the Tannaka language by Anschütz is now given by
Proposition 4.7 (Anschütz, [Ans18], Theorem 3.3). Let G be a reductive group over k. The composition with E defines a faithful functor which induces a bijection on isomorphism classes.
The inverse of this is given by composition with E −1 Gr • Gr • HN. Using this we can now describe all essentially finite G-bundles on X.
Proposition 4.8. Every essentially finite G-torsor over X is trivial.
Proof. Let F G : Rep k (G) → Vec X be an essentially finite torsor. By Proposition (3.6) there exists a commutative diagram of tensor functors for some finite group Γ. By [Ans18] this sits inside the following larger diagram where f is defined to be the composition Since all functors are tensor functors, so is f . By [DM82] f is induced by a morphism Since G m is connected and Γ is discrete we see thatf and thus f is the trivial map. But this implies that is the trivial torsor.

Genus 1
In the case when X is an elliptic curve, the density result follows almost immediately from known properties of M ss G , studied by Laszlo [Las98] in characteristic 0 and Frăţilă in charactierstic p [Fră21].

Genus g ≥ 2
Let now X be of genus g ≥ 2. Suppose first that char(k) = p > 0 and let σ X denote the absolute Frobenius of X. Then a vector bundle E is called periodic under the action of Frobenius if E ∼ = (σ n X ) * E for some integer n ≥ 1. If E is such a vector bundle, then, we know that E is trivialized by an étale cover [BD07, Theorem 1.1]. Hence, E is essentially finite [BdS11, Theorem 1]. In [DM10, Proposition 4.1 and corollary 5.1] the authors proved that, for any n > 0, the set of k-points in M ss,0 GLn (resp M ss SLn ) periodic under the action of Frobenius is dense. Hence, the set of k-points corresponding to essentially finite vector bundles is also dense. Hence, we may state the following. Proof. This follows from the previous discussion and the fact that the projection GL n → PGL n induces a surjection M ss,0 GLn → M ss,0 PGLn (see [Ser58,Proposition 18]) which takes essentially finite GL n -bundles to essentially finite PGL n -bundles.
Let now k be of characteristic zero. We restrict ourselves to split reductive groups of semisimple rank 1. By classical results (see e.g., [Mil17,Chapter 21]) these are all given by the following list.
Proposition 4.11. Let G be a split reductive group of semisimple rank 1. Then, up to isomorphism, G is one of the following groups:  Thus, to show non-density for split reductive groups of semisimple rank 1 it suffices to show it for PGL 2 , which we do now.
To do this we need a bound on the dimension of M ss O(2) . For a connected reductive group G it is well-known that dim M G = dim(G)(g − 1) (see e.g. [Sor00]). Since O(2) is not connected, we compute dim M O(2) following the approach for connected reductive groups. , which is equal to −χ(X, E). By Riemann-Roch we thus have that (4.3) By identifying O(2) as the matrices Proposition 4.14. The subset of essentially finite PGL 2 -torsors is not dense inside M ss,0 PGL 2 .
Proof. By [NvdPT08] the finite subgroups of PGL 2 are given by S 4 , A 5 , A 4 and for all n ∈ N, µ n and D n . Furthermore, for each finite subgroup there is only one conjugacy class by Proposition 4.1 in [Bea10]. Hence, for a given finite subgroup Γ, we may choose any embedding ι : Γ ֒→ PGL 2 and unambiguously consider ι * M Γ ⊂ M ss,0 PGL 2 . Now, for any such group Γ, ι * M Γ ⊂ M ss,0 PGL 2 is a finite number of points. Indeed, we have that H 1 et (X, Γ) = Hom(π 1 (X), Γ) (4.5) and since π 1 (X) is (pro)finitely generated, we see that H 1 et (X, Γ) is a finite set. Hence, to prove the proposition it is enough to show that the essentially finite torsors whose finite group is isomorphic to D n or µ n for some n > 0, is not dense. By abuse of notation, we still denote this subset by M ef,0 PGL 2 . Let π : GL 2 → PGL 2 denote the quotient morphism. From [NvdPT08] Section 2 we thus see that we may choose the embedding such that for every such Γ, we have a commutative diagram Now, for any essentially finite PGL 2 -torsor, F PGL 2 , by (4.6) we may assume that F PGL 2 = ι ′ * F O(2) where F O(2) is an essentially finite O(2)-torsor. Hence, we have a finite type morphism f : U → M ss,0 PGL 2 of projective varieties such that M ef,0 PGL 2 ⊂ f (U ). (4.9) Thus, it suffices to show that f is not dominant. Suppose it was. Then we obtain an inclusion of functions fields k(M ss,0 PGL 2 ) ֒→ k(U ). (4.10) This implies that 3g − 3 = dim M ss,0 PGL 2 = tr.deg k k(M ss,0 PGL 2 ) ≤ tr.deg k k(U ) = dim U = dim M ss O(2) ≤ g − 1, (4.11) where the last inequality follows from Lemma 4.12.
From the statement for PGL 2 we obtain the same statement for SL 2 .
Corollary 4.15. The subset of essentially finite SL 2 -torsors is not dense inside M ss,0 SL 2 .
Proof. Since the map M ss SL 2 → M ss,0 PGL 2 is dominant this follows from Proposition (4.14).
From this we obtain the same statement for GL 2 .
Corollary 4.16. The subset of essentially finite GL 2 -torsors is not dense inside M ss,0 GL 2 .
Proof. The same proof as above applies, or we have the following. Consider the map det : M ss,0 GL 2 → Jac 0 (X). (4.12) Since det −1 (O X ) = M ss SL 2 by Corollary (4.15) we obtain the desired result.
Finally, the complete statement is the following.
Corollary 4.17. For any split reductive group G‚ of semi-simple rank 1, the essentially finite G-torsors are not dense in M ss,0 G .