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Automorphisms of moduli spaces of symplectic bundle

Let X be an irreducible smooth complex projective curve of genus at least 3. Fix a line bundle L on X. Let M_{Sp}(L) be the moduli space of symplectic bundles (E, ExE --->L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_{Sp}(L) is generated by automorphisms sending E to ExM, where M is a 2-torsion line bundle, and automorphisms induced by automorphisms of X.


Introduction
Let X be a smooth complex projective curve of genus g, with g ≥ 3. A set of generators of the automorphism group of the moduli space of semistable vector bundles over X of rank r with fixed determinant L was obtained by Kouvidakis and Pantev in [KP]. More precisely, they proved that the automorphism group is generated by the automorphisms of X, automorphisms of the form E −→ E ⊗ M , where M is a line bundle with M ⊗r ∼ = O X , and, if r divides 2 deg L, automorphisms of the form E −→ E ∨ ⊗ N , where N is a line bundle with N ⊗r ∼ = L ⊗2 . In the same paper they prove a Torelli theorem for these moduli spaces. The proofs of their results crucially use the Hitchin map defined on the moduli of Higgs bundles. In [HR], Hwang and Ramanan gave a different proof of the above results using Hecke curves, which are minimal rational curves constructed using Hecke transformations.
Fix a holomorphic line bundle L on X, and consider the moduli space M Sp (L) of stable symplectic bundles (E, ϕ : E ⊗ E −→ L) of rank 2n and with values in L. Take a line bundle M on X with M ⊗2 ∼ = O X . Fix an isomorphism β : M ⊗2 −→ O X . Then we have an automorphism of M Sp (L) defined by (E, ϕ) −→ (E ⊗ M, ϕ ⊗ β).
More generally, let σ : X −→ X be an automorphism, and let M be a line bundle on X such that M ⊗2 ∼ = L ⊗ (σ * L) ∨ . Fix an isomorphism β as above. Then (E, ϕ) −→ (M ⊗ σ * E, β ⊗ σ * ϕ) is an automorphism of M Sp (L). We remark that, in both cases, the automorphism does not depend on the choice of β.
In Theorem 6.5 we show that these are all the automorphisms of M Sp (L). More precisely, the automorphism group Aut(M Sp (L)) fits in a short exact sequence of groups e −→ J(X) 2 −→ Aut(M Sp (L)) −→ Aut(X) −→ e , where J(X) 2 is the group of line bundles on X of order two (see Proposition 6.7). Since the above mentioned automorphisms of M Sp (L) extend to the moduli space of semistable symplectic bundles, it follows that Aut(M Sp (L)) coincides with the automorphism group of the moduli space of semistable symplectic bundles (see Lemma 6.8).
We also prove a Torelli type theorem for this moduli space (Theorem 4.3). This was proved earlier in [BH] by a different method.
Let us comment on our method of proof. The computation in [KP] of the automorphism group of the moduli space of vector bundles uses a delicate argument in which the fibers of the Hitchin map are studied over singular curves with non-generic singularities. Such argument is not easy to generalize to other groups (like the symplectic group). The proof of [HR] is simpler in spirit: it determines geometrically the Hitchin discriminant (the locus of singular spectral curves), and then uses the theory of minimal rational curves to prove that the dual variety of this Hitchin discriminant is a locus of Hecke transforms. Neither of the theory of minimal rational curves nor the constructions of Hecke transforms can be generalized to other groups to an extent that cover the arguments.
We were lead to take the proof of [HR] for the automorphism group of the moduli space of vector bundles and simplify it by removing the use of the dual varieties. Actually, we found that the Hitchin discriminant is enough to recover the nilpotent cones, and from this to get the automorphism group. Therefore, the proof given in this paper is the extension to the case of the moduli space of symplectic vector bundles of a proof for the moduli space of vector bundles which is not in the literature, and which simplifies both [KP] and [HR]. It is an extension of C * by the symplectic group Sp(2n, C) e −→ Sp(2n, C) −→ Gp(2n, C)

Moduli space of symplectic bundles
where q(A) = c for any A and c as in (2.1). From the definition of the homomorphism q it follows immediately that for all A ∈ Gp(2n, C), Let X be an irreducible smooth complex projective curve of genus g, with g ≥ 3. A symplectic bundle on X of rank 2n with values in a holomorphic line bundle L is a pair (E, ϕ), where E is a holomorphic vector bundle of rank 2n and ϕ : E E −→ L is a homomorphism of coherent sheaves which is fiberwise nondegenerate. The line bundle det(E) is canonically a direct summand of (E E) ⊗n , and the composition is an isomorphism. Giving a symplectic bundle is equivalent to giving a principal Gp(2n, C)-bundle.
See [BG] for more on symplectic bundles. We denote by M Sp (L) the moduli space of stable symplectic bundles with values in a fixed line bundle L.
Proof. Using Riemann-Roch, dim M Sp (L) = n(2n + 1)(g − 1). By semicontinuity, is a Zariski closed subset. The lemma will be proved by showing that the codimension of this subset is positive.
Take a pair ((E, ϕ) , s) such that (E, ϕ) ∈ M Sp (L) and s ∈ H 0 (E) \ {0}. It defines a short exact sequence where D is the effective divisor defined by s. Let K be the kernel of the composition This pairing is perfect because ϕ is pointwise nondegenerate. In particular, Note that under the homomorphism the class ξ 1 ∈ Ext 1 (Q, M ) for the bottom exact sequence in (2.4) maps to the class ξ 2 ∈ Ext 1 (M ∨ ⊗ L, F ) for the vertical exact sequence in the right of (2.4). For a general point (E ′ , ϕ ′ ) ∈ M Sp (L), the underlying vector bundle E ′ is stable. If E is a stable vector bundle, then Hom(Q, M ) = 0, because any nonzero homomorphism from Q to M produces a nilpotent endomorphism of E. Let deg M = ℓ and deg E = n · deg L = d. If Hom(Q, M ) = 0, then . Now let us see that the symplectic form ω F : F ⊗ F −→ L determines the symplectic form on E. First, ω F extends uniquely to a homomorphism F ⊗K −→ L, which extends naturally to a homomorphism Q ⊗ K −→ L; both these extensions are consequences of the fact that ϕ(M ⊗ K) = 0. Any two extensions of the pairing F ⊗ K −→ L to a pairing Q ⊗ K −→ L differ by a section contained in Hom((Q/F ) ⊗ K, L) = Hom((M ∨ ⊗ L) ⊗ K, L) = Hom(K, M ).
First, if a homomorphism K −→ M composed with M −→ K is non-zero, then it produces a splitting of the short exact sequence which is a contradiction. Therefore, (2.6) is proved. Now the homomorphism Q ⊗ K −→ L extends uniquely to a map E ⊗ K −→ L. This again extends to the map ω E : E ⊗ E −→ L, up to an indeterminacy contained in Hom(E⊗(E/K), L) = Hom(E⊗(M ∨ ⊗L), L), which actually lives in the subspace Hom((E/M ) ⊗ (M ∨ ⊗ L), L) = Hom(Q, M ) = 0.
Then the dimension of the family of bundles parametrizing (2.3) is . This completes the proof of the lemma.
For a symplectic bundle (E, ϕ), let be the set consisting of symmetric symplectic endomorphisms of E. For any divisor Lemma 2.2. Let D be an effective divisor of degree ℓ, with g ≥ max{2ℓ, ℓ + 2}.
Proof. By tensoring with a suitable line bundle, we may assume that L has degree ǫ ∈ {0, 1}. This makes the slope of any symplectic bundle to be ǫ 2 < g − 1. Moreover, we may assume that L is generic in the sense that H 0 (L(D)) = 0 and H 0 (L * (D)) = 0 , for any D effective divisor of degree ℓ ≤ g − 1 − ǫ. Let X (ℓ) = Sym ℓ (X) be the set of effective divisors of degree ℓ.
For n = 1, consider any stable vector bundle F of rank two with determinant L. It has a symplectic structure: The symplectic bundle (F, ω) is automatically stable. However, we are going to construct an specific bundle F 0 for later use. Consider an extension These extensions are parametrized by elements in H 1 (L * ), and dim H 1 (L * ) = g − 1 + ǫ. Consider an effective divisor D ∈ X (ℓ) . Then the exact sequence gives an exact sequence we see that if g − 1 + ǫ > 2ℓ, then there is an extension (2.7) whose class ξ ∈ H 1 (L * ) goes to a non-zero element under the homomorphism H 1 (L * ) −→ H 1 (L * (D)) for any D ∈ X (ℓ) . Now the connecting homomorphism H 0 (O X (D)) −→ H 1 (L * (D)) for the dual sequence −→ 0 of (2.7), which is multiplication by ξ, is injective; indeed, if a section s ∈ H 0 (O X (D)), defining a divisor D ′ , maps to zero, then the extension class ξ goes to zero under the homomorphism H 1 (L * ) −→ H 1 (L * (D ′ )), but this is not the case by construction. This implies that H 0 (L * (D)) = H 0 (L * ⊗ F 0 (D)) , which is zero by assumption. Also the exact sequence implies that H 0 (F 0 (D)) = H 0 (O X (D)). And finally, the exact sequence gives that H 0 (End F 0 (D)) = H 0 (O X (D)). Hence H 0 (End 0 F 0 (D)) = 0, where End 0 denotes the space of trace-free endomorphisms. Note that End Sp F 0 = End 0 F 0 , so H 0 (End Sp F 0 (D)) = 0. Now for n > 1, consider a general symplectic bundle (F 1 , ω 1 ) of rank 2n − 2. By induction hypothesis, H 0 (End Sp F 1 (D)) = 0, for any effective divisor D of degree ℓ. Consider the symplectic bundle E = F 0 ⊕ F 1 . This is a symplectic semistable bundle of rank 2n. Let us see that This would imply that also for a general stable bundle E, we have that for all D ∈ X (ℓ) (note that X (ℓ) is a complete variety). The vector space H 0 (End Sp E(D)) has four components: • H 0 (End Sp F 0 (D)) = 0, by construction.
, which is zero as above. This completes the proof of the lemma.

Hitchin discriminant
Let us recall the definition of the Hitchin map (see [Hi,Section 5.10]). The holomorphic cotangent bundle of X will be denoted by K X . A symplectic Higgs bundle is a triple (E, ω, θ), where (E, ω) is a symplectic bundle and θ : E −→ E⊗K X is a symmetric map with respect to ω: Let M Sp (L) be the moduli space of semistable symplectic Higgs bundles of rank 2n.
As before, M Sp (L) is the moduli space of stable symplectic bundles. The cotangent bundle T * M Sp (L) ⊂ M Sp (L) is an open subset. Consider the affine space: . This extends to the Hitchin map on the moduli space M Sp (L) of semistable symplectic Higgs bundles, For an element s = (s 2 , . . . , s 2n ) ∈ W , the spectral curve X s associated to s is the curve in the total space V(K X ) of K X defined by the equation x is a coordinate for X, and y is the corresponding tautological coordinate dx along the fibers of the projection V(K X ) −→ X).

Consider the compactification
Let p : S −→ X be the projection. Giving a Higgs bundle (E, θ : E → E ⊗ K X ) is equivalent to giving a coherent sheaf A of rank one supported on some spectral curve S ⊂ V(K X ). Indeed, E = p * A, and the Higgs field θ corresponds to the homomorphism A −→ A ⊗ p * K X defined by multiplication with the tautological section of ). The support of A is given by the equation (3.1). For more details, see [Hi], [BNR] and [Si].
The symplectic bundle structure ω : E ⊗ E −→ L corresponds to an isomorphism where σ : S −→ S is the involution y −→ −y (note that the spectral curve is invariant under this involution because all the exponents of y in (3.1) are even integers). Indeed, applying p * to this isomorphism we obtain the symplectic structure: The second equality is proved in two steps. There is a spectral sequence . Since A has support of dimension 1, we obtain , and the relative Serre duality for the projective morphism p gives We can think of A as a sheaf on the spectral curve X s . If this is integral, then A is torsionfree as a sheaf on X s , and then If A is locally free on X s , then A ∨ is the usual dual line bundle on X s . Fix once and for all a square root line bundle R = (K Xs ⊗ π * K −1 X ⊗ π * L) 1/2 . If we denote U = A ⊗ R, then σ * U ∼ = U ∨ . In other words, U is an element of the Prym subvariety of the compactified Jacobian J(X s ) and, conversely, an element of this Prym produces a symplectic Higgs bundle whose spectral curve is X s . Therefore, the fiber of H over s ∈ W is isomorphic to Prym(X s , σ), and the isomorphism depends only on the choice of square root R. The dimension of this Prym variety is Let Y be an integral curve whose only singularity is one simple node at a point y. Let π Y : Y −→ Y be the normalization, and let x and z be the pre-images of y in Y . The compactified Jacobian J(Y ), which parametrizes torsionfree sheaves of rank 1 and degree 0 on Y , is birational to a P 1 -fibration P over J( Y ), whose fiber over any L ∈ J( Y ) is P 1 (L x ⊕ L z ). The morphism P −→ J (Y ) is constructed as follows. A point of P corresponds to a line bundle L on Y and a one dimensional quotient q : L x ⊕L z ։ C (up to scalar multiple). This is sent to the torsionfree sheaf L ′ on Y defined as For the proof, see [Bh,Theorem 4].
Assume that Y has an involution σ. It lifts to an involution σ of Y . This induces an involution in P . Indeed, if (L, q) is a point, and q : It is a uniruled variety, because it has a surjective morphism from the P 1 fibration P | Prym defined by the pullback Analogously, if Y is an integral curve with two simple nodes, and Y is its normalization, then J(Y ) is birational to a P 1 × P 1 -bundle P on J( Y ). If the nodes are called y 1 and y 2 , then the two P 1 -factors in the Cartesian product correspond to one dimensional quotients q 1 : Let σ be an involution of Y interchanging the two nodes. It lifts to an involution of Y , and also to an involution of P , sending (L, q 1 , q 2 ) to ( σ * L ∨ , q ∨ 2 , q ∨ 1 ), and this induces an involution of J(Y ). A fixed point (L, q 1 , q 2 ) in P for this involution has L ∼ = σ * L ∨ and q 2 = q ∨ 1 , hence it is a P 1 -fibration on Prym( Y ), and the image of this map is the fixed point locus on J(Y ), which is denoted by Prym(Y, σ). We again obtain that this Prym is a uniruled variety.
Consider D ⊂ W the divisor consisting of characteristic polynomials with singular spectral curves. This has two components D = D 1 ∪ D 2 , where D 1 consists of those curves for which s 2n has a double root (then (3.1) has a node at the horizontal axis), and D 2 consists of curves with two symmetrical nodes (i.e., y n + s 2 (x)y n−1 + . . . + s 2n−2 (x)y + s 2n (x) = 0 has a node). Let D o i ⊂ D i , i = 1, 2, be the locus of all those curves that do not contain extra singularities.
. Proposition 3.1. As before, h : T * M Sp (L) −→ W is the Hitchin map. The following statements hold: (1) For w ∈ W − D, the fiber h −1 (w) is an open subset of an abelian variety (actually a Prym variety).
is an open subset of the uniruled variety Prym(X w , σ).
is an open subset of the uniruled variety Prym(X w , σ). The complement of the open subsets in each of the cases is of codimension at least 2 (at least for generic w in the corresponding set).
Proof. The map H : M Sp (L) −→ W is proper. By [Hi], H −1 (w) is an abelian variety for w ∈ W − D. The complement is of codimension ≥ 3 (the assumption that g ≥ 3 is used here). In [Fa,Theorem II.6 (iii)] it is proved that the complement has codimension ≥ 2 under a weaker assumption, but if we assume g ≥ 3, then the same proof gives that the codimension is ≥ 3. Therefore Proof. We need to see that h −1 (D * ) is of codimension at least two in T * M Sp (L). This follows easily from Theorem II.5 of [Fa], which says that the fibers of the Hitchin map H : M Sp (L) −→ W are Lagrangian (hence of half-dimension). So the fibers of H are equidimensional, and in particular the codimension of h −1 (D * ) is that of D * ⊂ W , which is at least two. Proof. Let l ∼ = P 1 ⊂ h −1 (D). Then h(l) ⊂ W . As it is a complete curve, it should be a point. So l is included in a fiber. By Proposition 3.1, it cannot be contained in a fiber over w ∈ W − D.
Now let w ∈ D o . Then Proposition 3.1 again shows that there is a family of P 1 covering these fibers. Now using Proposition 3.2, we get that the closure is the entire h −1 (D).

Torelli theorem
This section is devoted to a Torelli type theorem for the moduli space M Sp (L), i.e., to prove that the moduli space determines the curve X up to isomorphism. Moreover, consider the standard dilation action of C * on the fibers of T * M Sp (L). Then there is a unique C * -action "·" on W such that h is C * -equivariant, meaning h(E, λθ) = λ · h(E, θ).
Proof. This holds for the Hitchin map H on the moduli of semistable symplectic Higgs bundles M Sp (L) (cf. [Hi]). On the other hand, the generic fiber of H is smooth and the codimension of T * M Sp (L) ⊂ M Sp (L) on these fibers is at least two (cf. [Fa,Theorem II.6 (i)], and note that T * M Sp (L) is a subset of the moduli M 0 Sp (L) of stable Higgs bundles). Therefore, it follows that the lemma also holds for the restriction of the Hitchin map to the cotangent bundle T * M Sp (L).
Lemma 4.1 allows us to recover the base W of the Hitchin fibration as an algebraic manifold. Although there is an isomorphism W ∼ = n k=1 H 0 (K 2k X ), we do not know at this point how to recover the spaces W 2k ⊂ W corresponding to H 0 (K 2k X ). Lemma 4.1 also gives us the natural C * -action on W . This gives us the origin of W (as the only fixed point of the action). Also the subspaces W ≥2k = n t=k W 2t , for each t = 2, . . . , n, are uniquely determined (these are the spaces where the ratio of convergence is bigger than λ 2k , for λ −→ 0). In particular, W 2n ⊂ W is welldetermined. Moreover, the C * -action on W 2n determines the usual C * -action of weight one, which is multiplication by scalars. This determines the vector space structure of W 2n . So we recover W 2n = H 0 (K 2n X ).
is the dual variety of X ⊂ P(W * 2n ) for the embedding given by the linear series |K 2n X |. Proof. A spectral curve corresponding to a point of s 2n ∈ H 0 (K 2n X ) has equation y 2n + s 2n (x) = 0, and this curve is singular at the points with coordinates (x, 0) such that x is a zero of s 2n of order at least two. Clearly C = D 1 ∩ W 2n . On the other hand, D 2 ∩ W 2n ⊂ C, since it consists of singular curves. Therefore, the first statement follows.
The elements b ∈ W 2n correspond to spectral curves of the form . From this the second statement follows, taking into account that the linear system |K 2n X | is very ample, so X is embedded.
and taking the bundle of tangent hyperplanes to X ⊂ P(W * 2n ), we have We shall also need to consider the bundle of hyperplanes through a given point of . This is intrinsically defined once we have obtained X.
The following theorem is proved in [BH] by a different method.
Theorem 4.3 (Torelli). Let X and X ′ be two smooth projective curves of genus g ≥ 3, and let M Sp (L) and M ′ Sp (L ′ ) be moduli spaces of stable symplectic bundles over X and X ′ respectively. If the variety The C * -actions by dilations on the fibers of T * M Sp (L) and T * M ′ Sp (L ′ ) induce C * -actions on W and W ′ , and f should be C * -equivariant (as dΦ is C * -equivariant). Therefore f : W 2n −→ W ′ 2n , and it is linear.
We have seen in Proposition 3.1 that the Hitchin discriminant D = D 1 ∪ D 2 ⊂ W is an intrinsically defined subset, and therefore it is preserved by f . So f preserves C = D ∩ W 2n . This induces an isomorphism of the corresponding dual varieties, hence by Proposition 4.2, an isomorphism σ : X −→ X ′ is obtained.
Remark 4.4. Take X and X ′ as in Theorem 4.3. Let M Sp (L) and M ′ Sp (L ′ ) be moduli spaces of semistable symplectic bundles over X and X ′ respectively. The proof of Theorem 4.3 gives that X is isomorphic to Remark 4.5. Let us see that we can extend the previous arguments to recover the (linear) projection π 2n : W −→ W 2n (even though we do not know the linear structure of W ). This means that the map f in the proof of Theorem 4.3 commutes with π 2n .
First of all, the two irreducible components D 1 , D 2 of D are well characterized. Suppose this for a moment. The divisor D 1 consists of (s 2 , . . . , s 2n ) such that there exists a point x ∈ X with s 2n (x) = s ′ 2n (x) = 0. So, writing W ′ = n−1 k=1 W 2k ⊂ W , we have As such, there is a unique rational map D 1 X (up to automorphisms of the base) whose fibers are connected and rational (actually, isomorphic to open subsets of linear spaces). This determines each fiber W ′ × C x , x ∈ X, and a fortiori, the intersection W ′ ⊂ W of all of them. It is not difficult to see that the information of W 2n , W ′ and the C * -action together give the decomposition W = W ′ × W 2n . The projection π 2n follows.
It remains to identify the subvariety D 1 . Linearize the C * -action at the origin of W . According to the weights, the tangent space T 0 W decomposes as n k=1 W 2k (here the decomposition is clearly well-defined), and also W ′ = n−1 k=1 W 2k ⊂ T 0 W . The tangent cone to D 1 is x∈X (W ′ × C x ) ⊂ T 0 W . This is not the tangent cone to D 2 . So this property (for instance) can be used to characterize D 1 .

Nilpotent cone and flag variety
We need some results on linear algebra about the space of symplectic endomorphisms. Let (V, ω) be a symplectic vector space of dimension 2n. In this section, we want to study the symplectic nilpotent cone Lemma 5.1. The following statements hold: (1) N is a 2n 2 -dimensional algebraic variety.
(3) A ∈ N sm (the smooth locus of N ) if and only if rk A = 2n − 1.
(4) Let F l(V, ω) be the set of full isotropic flags on C 2n . Then there is a fibration π : N sm −→ F l(V, ω) with fibers isomorphic to (C * ) n × C n 2 −n .
Finally we prove statement (4). Note that if A ∈ N sm , then rk A = 2n − 1. This determines a well-defined full flag Let us see that ker A i is dual (with respect ω) to ker A 2n−i . For this, note that ker A 2n−i = im A i . If u = A i u 0 , and v ∈ ker A i , then This means that the flag in (5.1) is isotropic (in particular, ker A n is Lagrangian). The fiber over a point of the flag variety is given as follows. Fix a symplectic basis e 1 , . . . , e n , e n+1 , . . . , e 2n , such that the flag is (5.2) e n ⊂ e n−1 , e n ⊂ · · · ⊂ e 1 , . . . , e n ⊂ e 1 , . . . , e n , e n+1 ⊂ · · · ⊂ e 1 , . . . , e 2n .
Then the matrices in the fiber are of the form a n,1 a n,2 . . . a n,n−1 0 with a i+1,i = 0, and B = (b ij ) is symmetric with b 11 = 0. So the fiber of is defined in the statement (4) of the lemma).
We will now prove that N sm determines F l(V, ω). Consider the fibration π in Lemma 5.1 (4). We shall show that the fibers of π are intrinsically defined. Take any F ∈ F l(V, ω). Let U F ⊂ N ⊂ End Sp (V ) be the space of symmetric endomorphisms respecting the flag F . It is a linear subspace of End Sp (V ) of dimension n 2 contained in the nilpotent cone. By Lemma 5.2 below, all subspaces of End Sp (V ) of dimension n 2 contained in the nilpotent cone are of the form U F for some F . Moreover, U F ∩ N sm is the fiber of π over F . Therefore π is uniquely defined, up to automorphism of the base.
Lemma 5.2. Let L ⊂ N ⊂ End Sp (V ) be a linear subspace of dimension n 2 such that L ∩ N sm = ∅. Then there exists a unique flag F such that L = U F . Proof. We shall divide the proof in several steps.
Step 1. tr(A i B) = 0, for any A, B ∈ L, i ≥ 0. If i is even, then A i B is an antisymmetric endomorphism, hence of zero trace. For i odd, note that tr(C 2j ) = 0 for any C ∈ N , j ≥ 0. Then considering that C λ = A + λB ∈ L ⊂ N , we have that tr(C i+1 λ ) = tr((A + λB) i+1 ) = 0. Take the coefficient of λ to get tr(A i B) = 0.
Step 2. Let A ∈ N . Then A ∈ L if and only if tr(AB) = 0, for all B ∈ L.
The "only if" part follows from Step 1. To prove the "if" part, suppose that tr(AB) = 0, for all B ∈ L. As A ∈ N , we can choose a flag so that A ∈ U (this is unique if A ∈ N sm ). Taking an appropriate symplectic basis, the flag is the one in (5.2), and there is a well-defined space U of symplectic symmetric matrices (see (5.3)) preserving the flag. Denote . Consider also the space D ⊂ End Sp (V ) consisting of symmetric symplectic diagonal matrices. Therefore The bilinear map q(B 1 , B 2 ) = tr(B 1 B 2 ) is symmetric and non-degenerate. The q-dual of U is U + D. As End Sp (V )/(U + D) = U T , there is an induced perfect pairing q : U × U T −→ C. Let p = p U T : End Sp (V ) −→ U T be the projection. Now L ∩ (U + D) = L ∩ U , since all elements of L are nilpotent. Let us see that L ∩ U is q-dual to p(L). Clearly, q(B, C 1 ) = tr(BC 1 ) = tr(BC) = 0, for B ∈ L ∩ U , C 1 ∈ p(L) and C = C 1 + C 2 ∈ L, with C 2 ∈ U + D. On the other hand, The conclusion is that given B ∈ U , we have B ∈ L∩U if and only if q(B, C 1 ) = 0 for all C 1 ∈ p(L). In short, B ∈ L ∩ U if and only if tr(BC) = 0, for all C ∈ L.
Step 3. If A ∈ L then A 2i−1 ∈ L for any i ≥ 1. Without loss of generality, we can suppose A ∈ N sm . Therefore, A determines a flag F , and a space U = U F of endomorphisms preserving the flag. By Step 1, tr(A 2i−1 B) = 0, for any B ∈ L. By Step 2, we have A 2i−1 ∈ L.
Step 5. Let A ∈ L ∩ N sm . Let F be the (isotropic) flag determined by A, and We claim that either r(B) < 0 or r(B) is odd. Let r = r(B). If r = 0, then B ∈ U +D. As B is nilpotent, it follows that B ∈ U , meaning r < 0. Now we work by induction on r. Suppose that r > 0 and it is even. Write B = (b ij ), in some basis adapted to the flag, and note that b ij = 0 for i−j > r.
It is easy to see that r(C) ≤ r − 2, and that it has coefficients By induction hypothesis, r(C) = r − 2, so r(C) < r − 2 and all c i = 0. From here, b i = 0 for all i, and so r(B) < r.
Step 4 implies As this vector is a multiple of the previous one, it must be This implies the vanishing of all b i unless k = 1. And moreover, if k = 1, then taking the 3 × 3-matrix with rows and columns i 0 + 1, i 0 + 2, i 0 + 3, we get that b i 0 +1 = α, b i 0 +2 = −α, for some α ∈ C, by using that B ′ + λA ′ should be nilpotent.
Step 5, r(B) ≤ 1. By the previous discussion, if r(B) = 1, then the entries (b i ) ∈ W have the form given above.
So p H (U ∩ L) contains matrices with any values in the second diagonal, and with values in the first diagonal of the form (. . . , * , β 1 , β 1 , * , . . . , * , β 2 , β 2 , * , . . .). Now just consider a matrix C ∈ U L with all zeroes on the first diagonal, and just one 1 in the second diagonal, in a position (i 0 , i 0 + 2), so that the 3 × 3-block coming from B + λA + C ∈ L has a 1 in the right-top corner. This matrix is not nilpotent. This is a contradiction.
Step 7. L = U . Fix some A ∈ L ∩ N sm , and the corresponding subspace U . Let B ∈ L, and let r = r(B). We only have to see that r < 0. If r ≥ 0, then r is odd from Step 5. By Step 6, it cannot be r = 1. The same argument as in Step 5, proves that it cannot be r > 1 and odd. So B ∈ U . Hence L ⊂ U , so they are equal by dimensionality.
Remark 5.3. Lemma 5.2 also follows from the main theorem in [DKK]. The above proof, which uses only elementary methods, is included in order to be self-contained.

Automorphisms of the moduli space
In this section we will compute the automorphism group of a moduli spaces of symplectic bundles on X. As before, we assume that g ≥ 3.
Proposition 6.1. Fix a generic stable bundle E ∈ M Sp (L), and consider the map given as composition of the Hitchin map on H 0 (End Sp E ⊗ K X ) = T * E M Sp (L) ⊂ T * M Sp (L), followed by projection π 2n : W → W 2n (see Remark 4.5). Then Proof. First, note that the sequence The result follows from this easy linear algebra fact: if (V, ω) is a symplectic vector space, and A ∈ End Sp (V ) satisfies that det(A + C) = 0 for any C ∈ End Sp (V ) with det(C) = 0, then A = 0.
Proposition 6.1 allows to construct the bundle E −→ X whose fiber over x ∈ X is E x = H 0 (End Sp E ⊗ K X (−x)). This is a subbundle of the trivial bundle and there is an exact sequence This is exact on the right by Lemma 2.2. So we recover the bundle Remark 6.2. A natural continuation of argument is as follows. Take an automorphism Φ : M Sp (L) −→ M Sp (L) of the moduli space, let E ∈ M Sp (L) be a generic bundle, and let E ′ = Φ(E). Then, by the previous argument, there is a bundle isomorphism End Sp E ∼ = End Sp E ′ . From here we would like to deduce that E ′ is the twist of E by a line bundle. However, this is not known (at least, to the authors). This would amount to proving the generic injectivity of the map E → End Sp E between the corresponding moduli spaces.
Because of this, we shall take an alternative route, determining first the nilpotent symplectic cones bundle, then the isotropic flag varieties bundle, and from this the Lie algebra structure of End Sp E. This actually determines E up to twist by a line bundle.
We start with the following lemma: Lemma 6.3. The linear structure of the base of the Hitchin map W is uniquely determined. So is the decomposition W Proof. Let M Sp (L) be the moduli space of symplectic bundles, and let Φ : M Sp (L) −→ M Sp (L) be an automorphism. As in the proof of Theorem 4.3, the automorphism Φ yields an isomorphism σ : X −→ X and commutative diagrams (6.2) Let M be a line bundle such that M ⊗2 ∼ = L⊗(σ * L) ∨ . Composing with the automorphism given by σ −1 and M −1 (see the introduction), we may assume that σ = Id. Take a generic bundle E, and let E ′ be its image by Φ. Then we have where f is an automorphism which commutes with the C * -action. To prove the lemma, we have to check that f is linear with respect to a chosen decomposition W = ⊕ n k=1 W 2k . There is a bundle E −→ X whose fiber over any x ∈ X is the subspace . Analogously, there is a bundle E ′ −→ X whose fiber over x is the subspace . By Proposition 6.1, the map dΦ : Going to the quotient bundle (6.1), we have a bundle isomorphism is induced by ϕ.
The Hitchin map h : H 0 (End Sp E ⊗ K X ) −→ W factors as the composition of the map and the trace map By the discussion above, the map dΦ descends to a map and is diagonal with respect to the product decomposition. Therefore, the map f : n k=1 W 2k → n k=1 W 2k is also diagonal. Finally, this means that we have maps f : W 2k → W 2k which are C * -equivariant. As there is just one weight for W 2k , this means that f is linear on W 2k . Therefore f is linear. Our next step is to recover the symplectic nilpotent cone bundle N E −→ X , whose fibers are the symplectic nilpotent cone spaces Note that this nilpotent cone bundle sits as N E ⊂ End Sp E ⊗ K X .
Lemma 6.4. Consider the map given as composition of the Hitchin map on H 0 (End Sp E ⊗ K X ) = T * E M Sp (L) ⊂ T * M Sp (L), followed by the projection W → W 2k (defined thanks to Lemma 6.3). Then the vector subspace generated by the image h 2k (H 0 (End Sp E ⊗ K X (−x))) is )) . Now the vector space generated by h 2k (H 0 (End Sp E ⊗ K X (−x))) equals to the image of X (−2k x)) , followed by the trace map H 0 (End ant E ⊗ K 2k X (−2k x)) −→ H 0 (K 2k X (−2k x)), where End ant E consists of the anti-symmetric symplectic endomorphisms of E (i.e., those ϕ such that ω(ϕ u, v) = ω(u, ϕ v)). Note that the multiples of the identity are in End ant E.
There is a bundle map (first map is composition of endomorphisms, second map is the trace). This is split, therefore the map Consider the vector subspace generated by the image h 2k (H 0 (End Sp E⊗K X (−x))) which is H 0 (K 2k X (−2k x)) ⊂ W 2k = H 0 (K 2k X ), for moving the point x ∈ X. This gives a fiber bundle over X, which has co-rank 2k. The curve X is embedded into P(W * 2k ) via the linear system |K 2k X |, and the osculating 2k-space at x is given as ) . The embedding of the curve X in P(W * 2k ) is recovered from the osculating 2k-spaces. More specifically, if g : X → P N andg : X → Gr(2k +1, N +1) is the map giving the osculating 2k-spaces, theng determines g. This is proved as follows: the pull-back of the universal bundle throughg is the bundle and g is determined by a section of this bundle. As T X is of negative degree, this has only one section.
So we recover the embeddings X ֒→ P(W * 2k ) = P(H 0 (K 2k X ) * ), and hence the hyperplanes This is the pre-image of the nilpotent cone under the surjective map Take its image to get the bundle Now we are ready to prove the main result of the paper. First, as explained in the introduction, line bundles of order two and automorphisms of X produce automorphisms of moduli spaces of symplectic bundles. be an automorphism. Then Φ is induced by an automorphism of X and a line bundle of order two.
Proof. We work as in Lemma 6.3. There is an automorphism σ : X −→ X and a line bundle M , with M ⊗2 ∼ = L ⊗ (σ * L) ∨ , such that, after composing with σ −1 and M −1 (as in the introduction), we have the diagram (6.2) with σ = Id.
Take a generic bundle E, and let E ′ be its image by Φ. By the arguments in Lemma 6.3, we have a bundle isomorphism and the diagram (6.3) becomes where h is the Hitchin map and f is a linear isomorphism. By Lemma 6.4 and the discussion following it, f preserves H (2k) x = H 0 (K 2k X (−x)) ⊂ W 2k , for each k = 1, . . . , n, and x ∈ X.
Therefore dΦ preserves k≥1 h −1 2k (H (2k) x ) ⊂ H 0 (End Sp E ⊗K X ). Taking the image under the surjective map H 0 (End Sp E ⊗ K X ) −→ End Sp E ⊗ K X | x , we see that dΦ preserves the symplectic nilpotent cone N E,x , for all x ∈ X. So we have an isomorphism where N E , N E ′ are the corresponding symplectic nilpotent cone bundles. By Lemma 5.2, we get an isomorphism of the corresponding isotropic flag varieties bundles. Going to global vertical fields, we have a (Lie algebra) bundle isomorphism Using Lemma 6.6 below, it follows that E ′ ∼ = E ⊗ M , for some line bundle M with M 2 ∼ = O X . As this holds for a generic E, it holds for all E.
Lemma 6.6. Let (E, E ⊗ E → L) and (E ′ , E ′ ⊗ E ′ → L ′ ) be two symplectic vector bundles such that ad Sp E and ad Sp E ′ are isomorphic as Lie algebra bundles. Then there is a line bundle M such that E ′ ∼ = E ⊗ M . Furthermore, if we assume L ∼ = L ′ , then M ⊗2 ∼ = O X .
Proof. Giving a vector bundle ad Sp E with its Lie algebra structure is equivalent to giving a principal Aut(sp 2n )-bundle P Aut(sp 2n ) which admits a reduction to a principal Gp 2n -bundle P Gp 2n , corresponding to (E, E ⊗ E → L).
Since sp 2n does not have outer automorphisms, all automorphisms are inner, and Aut(sp 2n ) is connected. Therefore, we have Gp 2n ։ PGp 2n = Inn(sl 2n ) = Aut(sl 2n ) Consider the short exact sequence of groups e −→ C * −→ Gp 2n −→ PGp 2n −→ e Hence, the set of reductions of a PGp 2n -bundle to Gp 2n is a torsor for the group H 1 (X, O * X ). Therefore, if (E, E ⊗ E → L) is a symplectic bundle corresponding to a reduction, the other reductions are of the form for any line bundle M .
Finally, it follows from this expression that, if L ∼ = L ′ , then M ⊗2 ∼ = O X .
Let Aut(X) and Aut(M Sp (L)) be the automorphisms of X and M Sp (L) respectively. Let J(X) 2 be the group of line bundles on X of order two.
Proof. From the proof of Theorem 6.5, if follows that we have a surjective homomorphism ρ : Aut(M Sp (L)) −→ Aut(X) .
The homomorphism sends Φ to σ (see the proof of Theorem 6.5). The kernel of ρ is a quotient of J(X) 2 . Therefore, to prove the proposition it suffices to show that the action of J(X) 2 on M Sp (L) is effective. Let M Sp (2, L) (respectively, M Sp (2n − 2, L)) be the moduli space of symplectic bundles of rank 2 (respectively, 2n − 2) such that the symplectic form takes values in L. There is a natural embedding (where M Sp (L) is the moduli space of semistable symplectic bundles), defined by ((E 1 , ϕ 1 ) , (E 2 , ϕ 2 )) −→ (E 1 ⊕ E 2 , ϕ 1 ⊕ ϕ 2 ). To prove that the action of J(X) 2 on M Sp (L) is effective it is enough to show that the action of J(X) 2 on M Sp (2, L) is effective.
First assume that deg L = 2δ, where δ is an integer. Then for a general line bundle M ∈ J δ (X), the symplectic bundle M ⊕ (L ⊗ M * ) ∈ M Sp (2, L) is moved by the action of every nontrivial element of J(X) 2 . Therefore, the action of J(X) 2 on M Sp (2, L) is effective. Now assume that deg L = 2δ + 1. Fix a nontrivial line bundle ξ ∈ J(X) 2 . Take a pair (E, θ), where E is a stable vector bundle of rank two with 2 E = L, and θ : E −→ E ⊗ ξ is an isomorphism. Therefore, E is a fixed point for the action of ξ on M Sp (2, L).
The line bundle ξ defines a nontrivialétale covering f : Y −→ X of degree two, and E produces a line bundle η −→ Y such that f * η = E (see [BNR], [Hi]). Therefore, η lies in the Prym subvariety of J 2δ+1 (Y ) associated to the covering f . The dimension of the Prym variety is g − 1. On the other hand, the dimension of M Sp (2, L) is 3g − 3. Since 3g − 3 > g − 1, we conclude that the action of ξ on M Sp (2, L) is effective. This completes the proof of the proposition.
Lemma 6.8. Let M Sp (L) be the moduli space of semistable symplectic bundles. The automorphism group of M Sp (L) is identified with Aut(M Sp (L)).