Generalized B-Opers

Opers were introduced by Beilinson-Drinfeld [arXiv:math.AG/0501398]. In [J. Math. Pures Appl. 82 (2003), 1-42] a higher rank analog was considered, where the successive quotients of the oper filtration are allowed to have higher rank. We dedicate this paper to introducing and studying generalized $B$-opers (where"$B$"stands for"bilinear"), obtained by endowing the underlying vector bundle with a bilinear form which is compatible with both the filtration and the connection. In particular, we study the structure of these $B$-opers, by considering the relationship of these structures with jet bundles and with geometric structures on a Riemann surface.


Introduction
The study of opers within geometry and mathematical physics has received much attention in the last years, and in particular in connection with Higgs bundles in the recent years. In fact, certain opers arise naturally as limits of Higgs bundles in the Hitchin components. Recall that the Higgs bundles, corresponding to a complex Lie group G C , on a compact Riemann surface Σ are given by the solutions of Hitchin's equations: In [23] Gaiotto conjectured that the space of opers should be obtained as the -conformal limit of the Hitchin section: taking the limits R → 0 and ξ → 0 simultaneously while holding the ratio = ξ/R fixed. The conjecture was recently established for general simple Lie groups by Dumitrescu, Fredrickson, Kydonakis, Mazzeo, Mulase and Neitzke in [17], who also conjectured that this oper is the quantum curve in the sense of Dumitrescu and Mulase [18], a quantization of the spectral curve S of the corresponding Higgs bundle by topological recursion [19] -see also references and details in [16]. Moreover, very recently Collier and Wentworth showed in [12] that the above conformal limit exists for spaces other than the Hitchin components.

Opers
With views towards understanding other conformal limits of Higgs bundles and their appearance through quantum curves, in this note we introduce generalized B-opers and begin a program to study their geometry and topology, leaving for future work the extension of the above results to this new setting. Opers were introduced by Beilinson-Drinfeld [3,4]; they were motivated by Drinfeld-Sokolov [14,15]. Given a semisimple complex Lie group G, a G-oper on a compact Riemann surface Σ is • a holomorphic principal G-bundle P on Σ with a holomorphic connection ∇, and • a holomorphic reduction of structure group of P to a Borel subgroup of G, such that reduction satisfies the Griffiths transversality with respect to the connection with the corresponding second fundamental forms being isomorphisms. They admit reformulation in terms of holomorphic differential operators on a Riemann surface. For example, SL(n, C)opers on Σ are precisely holomorphic differential operators on Σ of order n and symbol 1 with vanishing sub-principal term.
Opers also provide a coordinate-free description of the connections in the base space of a generalized KdV hierarchy 1 . These connections can be translated to the usual n'th order differential operators according to Drinfeld and Sokolov [14,15]. The study of opers has been carried out from many different perspectives: as an example, Ben-Zvi and Frenkel studied how opers arise from homogeneous spaces for loop groups and found a natural morphism between the moduli spaces of spectral data and opers [5]. Moreover, Frenkel and Gaitsgory studied opers on the formal punctured disc in [22] and showed that there is a natural forgetful map from such opers to local systems on the punctured disc, which appears to be of much importance for the geometric Langlands correspondence. Opers are also related to the representations of affine Kac-Moody algebras at the critical level. In particular, from [21], the algebra of functions on the space of L G-opers, where L G is the Langlands dual of G, on the formal disc is isomorphic to the classical W-algebra associated to the Langlands dual Lie algebra L g.
In recent years, new types of opers were introduced, such as g-opers 2 and Miura opers [21]. We dedicate this paper to introducing and studying generalized B-opers (Definition 2.11; where "B" stands for "bilinear" and not "Borel") which we show are closely related to geometric structures on the base Riemann surface Σ (Theorem 4.6), and which encode certain classical G-opers. These are a generalization of [8], a higher rank analog of the work in Beilinson-Drinfeld [3,4], where the successive quotients of the oper filtration are allowed to have higher rank. In our new setting, when the bilinear form is skew-symmetric and the rank of the bundle is 4, it allows us to recover some of the Sp(4, C)-structure studied in [11].
The relationship between opers and geometric structures has been of much interests in the last half century, with the first results on this going back to work of R.C. Gunning in [25], where it was shown that there is a one-to-one correspondence between SL(n, C)-opers on a compact Riemann surface Σ and complex projective structures on Σ. One should note that the term oper did not exist at the time, and hence Gunning calls his family of flat bundles associated to the complex projective structures indigenous bundles on the Riemann surface. Much work has been done sine then on geometric structures arising through opers, and the reader may want to refer to [30] and references therein for further details.
In [8], a more general class of opers was studied, where rank(E) = nr and the rank of each successive quotient E i /E i−1 is r (the above two conditions remain unchanged). In the present paper, we incorporate a non-degenerate bilinear form B and require the (not necessarily full) filtration and the connection appearing in a G-oper to be compatible with it, thus leading to the natural objects which we call generalized B-opers, and for which we consider geometric structures arising through them in our main Theorem 4.6.
The paper is organized as follows. We first review some of the basic definitions and properties of classical opers, following [4] and [5] (done in Section 2.1), and then introduce in Section 2.2 what we call generalized B-opers and their main properties. These are triples (E, F, D), where F is a B-filtration of a holomorphic vector bundle E associated to a non-degenerate bilinear form B (as in Definition 2.6) and D is a B-connection on E (as in Definition 2.7), such that for every 1 ≤ i ≤ n − 1 the condition D(E i ) ⊂ E i+1 ⊗ K Σ holds, and the resulting homomorphism (2.7)) is an isomorphism. Generalized B-opers are closely related to jets, and we investigate this correspondence in Section 3, where in particular we show in Theorem 3.17 and Proposition 4.5 that there is a correspondence between the generalized B-opers (E, F, D) on Σ and triples consisting of the following: (1) a fiberwise non-degenerate symmetric bilinear form B n on a holomorphic connection on Q * that preserves this bilinear form B n , and (3) a classical Sp(n, C)-oper for n even and a classical SO(n, C)-oper for n odd.
The above correspondence of spaces can be taken further, which is done in Section 4. More precisely, the known relation between classical opers and projective structures can be extended to this generalized setting, leading to one of our main results: Theorem 4.6. For integers n ≥ 2, n = 3 and r ≥ 1, the space of all generalized B-opers of filtration length n and rank(E i /E i−1 ) = r is in correspondence with where C Σ denotes the space of all flat orthogonal bundles of rank r on Σ, which is independent of i, and P(Σ) is the space of all projective structures on Σ.
We conclude the paper describing certain naturally defined Higgs bundles appearing through generalized B-opers which carry certain Lagrangian structure. In Section 5 we initiate the study of those Higgs bundles, leaving to future work their further study.

Generalized B-opers
In what follows we shall first give a brief overview of classical G-opers for G a connected reductive complex Lie group (this is done in Section 2.1), and then introduce our generalization, what we call "Generalized B-opers", in Section 2.2.
Let Σ be a compact connected Riemann surface of genus g; its holomorphic cotangent bundle will be denoted by K Σ .

G-Opers
We shall recall here the following definitions from Beilinson and Drinfeld's paper [4]. Fix a Borel subgroup B ⊂ G together with a maximal torus H ⊂ B. Let h ⊂ b ⊂ g be the Lie algebras of H ⊂ B ⊂ G respectively. Fix a set of positive simple roots Γ ⊂ h * with respect to b. For each α in the dual space h * , set Then, there is a unique Lie algebra grading g = k g k such that the following holds Let P be a holomorphic principal B-bundle on Σ, and let E P be the Lie algebroid over Σ of infinitesimal symmetries of P . This E P is same as the Atiyah bundle for P [1]. Let E g P be the Lie algebroid over Σ of infinitesimal symmetries of the holomorphic principal G-bundle Q = P × B G obtained by extending the structure group of P to G. Let b P := P (b) (respectively, g P := P (g)) be the holomorphic vector bundle over Σ associated to P for the B-module b (respectively, g); so b P is the adjoint bundle ad(P ). Then, there is a filtration g k P ⊂ g P , and we shall let E k P be the preimage of g k P /b P ⊂ g P /b P = E g P /E P . One can then define another filtration E k P ⊂ E g P , k ≤ 0. Considering g −α P the P -twist of the B-module g −α , one has that E −1 Definition 2.1 (G-oper). A G-oper on Σ is a holomorphic principal B-bundle P on Σ with a holomorphic connection ω : T Σ −→ E g P such that ω(T Σ) ⊂ E −1 P , and for every α ∈ Γ, the following composition of homomorphisms is an isomorphism In order to draw a parallel between G-opers and our generalization, we shall focus now on SL(n, C)-opers, which can be described as follows from [4]. Indeed, opers are some vector bundles satisfying certain conditions that give them an oper structure. Definition 2.2. A GL(n, C)-oper on Σ is a holomorphic vector bundle E of rank n with a complete filtration of holomorphic subbundles 3) (non-degeneracy) and D induces an isomorphism We denote this GL(n, C)-oper by (E, {E i } , D).
3. An SL(n, C)-oper is a GL(n, C)-oper (E, {E i } , D) on Σ such that the determinant line bundle n E is holomorphically trivial and the connection on n E induced by D coincides with the trivial connection on O Σ . An Sp(2n, C)-oper is an SL(2n, C)-oper (E, {E i } , D) with a horizontal symplectic form on E such that E ⊥ i = E n−i . An SO(n, C)-oper is an SL(n)oper (E, {E i } , D) with a horizontal non-degenerate symmetric bilinear form B on E such that

Second fundamental form and generalized B-opers
In order to generalize the above definitions to account for the structure appearing through bilinear forms and their preserved filtrations, consider a pair (E, B), where E is a holomorphic vector bundle over Σ, and is a holomorphic homomorphism such that for every point x ∈ Σ, the bilinear form is non-degenerate, meaning for every point x ∈ Σ and each nonzero vector v ∈ E x there is some The bilinear forms we shall consider in the present paper will be either symplectic or orthogonal. Let be the natural projection. For any holomorphic subbundle F ⊂ E, define the subbundle be the Dolbeault operator defining the holomorphic structure of the holomorphic vector bundle E. The given condition that the bilinear form B is holomorphic implies that for all locally defined C ∞ sections s 1 and s 2 of E. Since F is a holomorphic subbundle of E, it follows that ∂(s) is a C ∞ (local) section of F ⊗ Ω 0,1 Σ for every (local) C ∞ section s of F . Therefore if s 1 is a C ∞ locally defined section of F ⊥ and s 2 is a C ∞ locally defined section of F , then from (2.2) it follows immediately that The bilinear form B identifies E with its dual E * . Consider the following composition of homomorphisms which is injective. The image of F ⊥ in E * evidently coincides with the image of the natural injective homomorphism Finally, from the definition of F ⊥ it follows immediately that (F ⊥ ) ⊥ = F .
In the case of Hermitian holomorphic vector bundles, Kobayashi showed in [29] that F ⊥ is not a holomorphic subbundle if F is not preserved by the Chern connection on E (see Definition 2.9). Definition 2.6. A B-filtration of a holomorphic vector bundle E is an increasing filtration F of holomorphic subbundles for which the following two conditions hold: 1) the length n ∈ N of the filtration is even when B is symplectic, and it is odd when B is orthogonal; Holomorphic connections where introduced by Atiyah in [1] (see also [2]). We recall that a holomorphic connection on a holomorphic vector bundle W on Σ is a holomorphic differential operator for all locally defined holomorphic sections s of W and all locally defined holomorphic functions f on Σ.
Note that the Leibniz condition in (2.4) implies that the order of the differential operator D is one. Moreover, a holomorphic connection on Σ is automatically flat because the sheaf of holomorphic two-forms on Σ is the zero sheaf. Through this, we can impose the following compatibility condition with respect to the bilinear form B.
Definition 2.8. The holomorphic section given by the composition of homomorphism in (2.6) is called the second fundamental form of F for the connection D.
The above construction of the second fundamental form in Definition 2.8 can be generalized from subbundles of E to filtration of subbundles of E as follows.
Definition 2.9. Let D be a holomorphic connection on E, and let be holomorphic subbundles such that Then, the second fundamental form of (F 1 , F 2 , F 3 , F 4 ) for the connection D is the map that sends any locally defined holomorphic section s of where s is any locally defined holomorphic section of the subbundle F 2 that projects to s.
In view of (2.4), note that the above conditions in Definition 2.9 imply that F 1 ⊂ F 3 and Proof . This follows from the condition D(F 1 ) ⊂ F 3 ⊗K Σ . Indeed, for s a local section of F 2 /F 1 , let s 1 , s 2 be two lifts of s to F 2 . This means s 1 − s 2 is a local section of F 1 . Therefore, from the given condition that From (2.4) it follows immediately that for f a locally defined holomorphic function on Σ and s a locally defined holomorphic section of F 2 /F 1 to F 2 , then the equality holds. In other words, S(D; F 1 , F 2 , F 3 , F 4 ) is an O Σ -linear homomorphism of coherent analytic sheaves.
Recall from Section 2.1 that a GL(n, C)-oper on Σ is a GL(n, C)-local system (that is a holomorphic vector bundle E with a holomorphic connection) equipped with a complete flag satisfying some extra conditions (2) and (3) in Definition 2.2. The following definition generalizes these two conditions.
) is an isomorphism. For simplicity of notation, we shall sometimes write S i (D) for the above map S(D; Remark 2.12. Note that properties (1) and (2) from Definition 2.11 are natural counterparts to (2) and (3) in Definition 2.2. Moreover, the above definition can be seen as a generalization of the opers studied in [8]. Here we have the additional structure of a bilinear form B, and it fits within the broader picture of (G, P )-opers which is introduced in the work of Collier and Sanders in [11].

The underlying bundle of a generalized B-oper
Let (E, F, D) be a generalized B-oper as in Definition 2.11 above. In what follows we shall construct a natural K ⊗(n−1) Σ -valued symmetric form S on the subbundle E 1 in Definition 2.11. For this, first note that for any 1 ≤ i ≤ n − 1, the isomorphism By composing the above maps from (2.9) we obtain an isomorphism (2.10) Lemma 2.13. The bilinear form B produces the following isomorphism on a B-filtration as in The lemma follows from this.
Using the identification between E * 1 and E n /E n−1 in Lemma 2.13, the isomorphism S in (2.10) can be seen as a holomorphic section Lemma 2.14. Take a holomorphic vector field v ∈ H 0 (U, T Σ) on an open subset U ⊂ Σ, and define the map where as usual, H 0 (U, E) denotes the space of all holomorphic sections of E over U . The map V from (2.12) satisfies the identity Proof . This can be proved using (2.5). Indeed, since D is a B-connection, the following equality of 1-forms on Σ holds (see (2.5)), and thus contracting both sides of (2.13) by the vector field v the identity in the lemma is obtained.
Lemma 2.15. Let (E, F, D) be a generalized B-oper as in Definition 2.11. Take s, t ∈ H 0 (U, E 1 ). Then, for the map V in (2.12).
Proposition 2.16. The section S in (2.11) lies in the subspace Proof . We shall first assume that B is symmetric, hence an orthogonal form. In this case, from Definition 2.6(1) we know that the integer n in Definition 2.11 is odd. Since B is symmetric, and n is odd, from (2.14) in Lemma 2.15 it follows immediately that S in (2.11) is a section of Moreover, the K ⊗(n−1) Σ -valued bilinear form S on E 1 is fiberwise non-degenerate, because the homomorphism S in (2.10) is an isomorphism. Now assume that B is a symplectic form. In this case, from Definition 2.6(1) we know that the integer n in Definition 2.11 is even. Since B is anti-symmetric, and n is even, from (2.14) it follows that S in (2.11) is a section of Moreover, as in the previous case, the K ⊗(n−1) Σ -valued bilinear form S on E 1 is fiberwise nondegenerate, because the homomorphism S in (2.10) is an isomorphism.

Jet bundles
We briefly recall the definition of a jet bundle of a holomorphic vector bundle on the Riemann surface Σ. Consider the product Σ × Σ, and let be the projection to the i-th factor. Let be the (reduced) diagonal divisor.
Definition 3.1. Given a holomorphic vector bundle W of rank r 0 on Σ and any nonnegative integer m ≥ 0, the mth order jet bundle J m (W ) of W is the holomorphic vector bundle of rank (m + 1)r 0 given by the direct image Lemma 3.2. For j ≥ 1, the kernel of the above natural projection Proof . Whilst this is something well known, we shall include here a short proof in order to be somewhat self-contained. We will show this algebraically, following the proof given by Eisenbud and Harris for bundles of principal parts [20, p. 247]. Since Σ can be viewed as a smooth projective curve, it suffices to prove the claim on Spec(R), where R = C[z]. The vector bundle W on R = Σ corresponds to a R-module, which we also denote by W . The pullback bundle p * 2 W = S ⊗ R W = R ⊗ C W is just an S-module from changing base ring. Let I be the ideal sheaf of the diagonal ∆. Then I is generated by z ⊗ 1 − 1 ⊗ z, where z is the generator of R. In this set-up, we see the R-module associated to the jet bundle is viewed as an R-module with multiplication given by r −→ r ⊗ C 1. Then the kernel of the natural projection J j (W ) −→ J j−1 (W ) is given by I m /I m+1 ⊗ C W . It is easy to see that I j /I j+1 is generated by z m ⊗ 1 − 1 ⊗ z m and is isomorphic to Sym m (Ω R/C ) = Ω ⊗m R/C as R-modules, where the latter is generated by dz ⊗m . Finally, the canonical bundle coincides with the sheaf of differentials for a curve. Hence the short exact sequence The following two definitions will be of much use here, and the reader should refer for instance to [8] and references therein for further details on them. Definition 3.3. Given holomorphic vector bundles W , W on Σ, we let Diff m Σ (W, W ) be the holomorphic vector bundle on Σ given by the sheaf of holomorphic differential operators of order m from W to W , that is, In what follows we shall construct an O Σ -linear homomorphism from the holomorphic vector bundle E of a generalized B-oper (E, F, D) to the jet bundle J k (Q), where k is any nonnegative integer and Q = E/E n−1 ; this will be done using the connection D.
Take a generalized B-oper (E, F, D). Consider a point x ∈ Σ, and take a point in the corresponding fiber v ∈ E x . We shall denote by v the unique locally defined flat section of E for the flat connection D, for which v(x) = v (note that v is defined on any simply connected open neighborhood of x). Consider the holomorphic section the O Σ -linear homomorphism constructed this way. From the construction of the homomorphisms f k in (3.3) it follows immediately that the following diagram of homomorphisms is commutative: The holomorphic vector bundle J k (Q) has an increasing filtration of holomorphic subbundles of length k + 2 where Q i , for every 0 ≤ i ≤ k, is the kernel of the natural projection J k (Q) −→ J k−i (Q) obtained by iterating the projection in Lemma 3.2.
The following result from [8] will be of much use.  Proposition 3.7. Let V and W be holomorphic vector bundles on Σ, and let D W be a holomorphic connection on W . Then for any integer n ≥ 1, there is a natural holomorphic isomorphism Proof . Take any point x ∈ Σ and a vector w ∈ W x in the fiber of W over x. Let s w denote the unique holomorphic section of W , defined on a simply connected open neighborhood of x ∈ Σ, satisfying the following two conditions: • s w (x) = w, and • s w is flat with respect to the integrable connection D W on W .
Let s w ∈ J n (W ) x be the element obtained by restricting s w to the n-th order infinitesimal neighborhood of x. Consequently, we get a holomorphic homomorphism that for any x ∈ Σ, sends any w ∈ W x to s w ∈ J n (W ) x constructed as above. Given any two holomorphic vector bundles V 1 and V 2 , there is a natural holomorphic homomorphism

Now consider the composition of homomorphisms
where η is the homomorphism in (3.5). It is straightforward to check that this composition of homomorphisms is fiberwise injective. This implies that it is an isomorphism, because rank(J n (V ) ⊗ W ) = rank(J n (V ⊗ W )).
The following is a special case of Proposition 3.7.
Corollary 3.8. Let V be a holomorphic vector bundle on Σ, and L be a holomorphic line bundle on Σ of order r. Then there is a canonical holomorphic isomorphism for every j ≥ 1.
Proof . Take a holomorphic isomorphism There is a unique holomorphic connection D L on L such that the isomorphism ξ takes the trivial connection on O Σ , given by the de Rham differential d, to the connection on L ⊗r induced by D L . This connection D L does not depend on the choice of the isomorphism ξ, because any two choices of ξ differ by an automorphism of O Σ given by a nonzero scalar multiplication. Note that a nonzero scalar multiplication preserves the trivial connection on O Σ .
Since L is equipped with a canonical connection D L , the required isomorphism is given by Proposition 3.7.

A natural symmetric form
Let g be the genus of Σ. Recall that a theta characteristic on Σ (or spin structure) is a holomorphic line bundle ξ on Σ of degree g − 1 equipped with a holomorphic isomorphism of ξ ⊗2 with K Σ . Fix a theta characteristic on Σ, and denote it by K 1/2 Σ . For any integer m, the holomorphic line bundle K 1/2 Σ ⊗m will be denoted by K m/2 Σ . Take a generalized B-oper (E, F, D) on Σ. For ease of notation we define the holomorphic vector bundle (3.6) Having studied the natural K ⊗(n−1) Σ -valued symmetric form S on E 1 in Section 2.3, in what follows we shall construct an equivalent symmetric form on Q.
Using the isomorphism E * 1 = Q from Lemma 2.13, the section S in (2.11) produces a section In view of Proposition 2.16, the section S satisfies the following: Proposition 3.9. The section S in (3.7) lies in the subspace Moreover, the symmetric bilinear form on Q * defined by S is fiberwise non-degenerate.
Proof . This follows immediately from Proposition 2.16 and Lemma 2.13.
Since the symmetric bilinear form S in (3.7) is fiberwise non-degenerate, it produces a symmetric bilinear form on Q. In view of (3.6), this S ∨ defines a homomorphism (see [8, p. 18, equation (4.6)]). As before, ∆ ⊂ Σ × Σ is the divisor in (3.1). Using D one can construct a holomorphic section of p * 1 Q ⊗ p * 2 Q * over the non-reduced divisor 2∆ as in [8, p. 27] (note that in [8] this section is called κ| 2∆ ⊗ s). Moreover, as shown in [8, p. 27], the restriction of this section to ∆ ⊂ 2∆ coincides with the identity map of Q. Therefore, this section of p * 1 Q ⊗ p * 2 Q * over 2∆ defines a holomorphic connection on the holomorphic vector bundle Q [8, p. 7], [13]; this connection on Q will be denoted by ∇.
Let ∇ denote the holomorphic connection on Sym 2 (Q) induced by the connection ∇ on Q. To complete the proof of the proposition, we need to show that the section S of Sym 2 (Q) in Proposition 3.9 is flat (covariant constant) with respect to this induced connection ∇. For that we need to recall the construction, as well as some properties, of the differential operator D in (3.9).
On the other hand, Q is identified with its dual Q * using the pairing S in Proposition 3.9. Consequently, we conclude that Now from the construction of κ it follows that the section κ is symmetric, meaning where η : Σ × Σ −→ Σ × Σ is the involution defined by (x, y) −→ (y, x); in (3.12), the section κ is considered as the section in (3.11).
In view of η, the following lemma implies that the section S of Sym 2 (Q) is covariantly constant with respect to the connection ∇.
Lemma 3.11. Let F be a holomorphic vector bundle over Σ equipped with a holomorphic connection ∇ F . Let θ F be the section of (p * 1 F )⊗(p * 2 F * ) over 2∆ corresponding to the connection ∇ F . Let be a nondegenerate symmetric bilinear form on F * . Then the following statements hold: • Using B, the section θ F produces a section θ F of (p * 1 F ) ⊗ (p * 2 F ) over 2∆. • For the connection ∇ F on F ⊗ F induced by ∇ F , where η is the involution of Σ × Σ in (3.12).
In particular, B is covariant constant with respect to the connection ∇ F if and only if η * θ F = θ F .
Proof . Since B identifies F with F * , we have (p * 1 F ) ⊗ (p * 2 F * ) = (p * 1 F ) ⊗ (p * 2 F ). So θ F produces a section θ F of (p * 1 F ) ⊗ (p * 2 F ) over 2∆. The restriction of θ F to ∆ ⊂ 2∆ is Id F . Hence we have This, and the fact that O Σ×Σ (−∆)| ∆ = K ∆ with K ∆ being the holomorphic cotangent bundle of ∆, together imply that where S ∨ 0 is the homomorphism in (3.8). Consider H as a paring −, − between Q and Q ⊗ K n Σ with values in K Σ . Then it can be shown that the differential operator D in (3.9) satisfies the equation for any locally defined holomorphic sections s and t of Q. Indeed, (3.13) is an exact reformulation of (3.12). In other words, the differential operator D is "self-adjoint".
Remark 3.12. Given the above defined pairing −, − between Q and Q ⊗ K n Σ with values in K Σ , for any differential operator D n ∈ H 0 Σ, Diff n Σ Q, Q ⊗ K n Σ , the adjoint differential operator D * n defined by the equation where s and t are any locally defined holomorphic sections of Q, is similar to the classical Lagrange adjoint [28]. More precisely, when the rank of Q is one, the above defined adjoint map coincides with the Lagrange adjoint.

Symplectic and orthogonal opers
In what follows, using the connection ∇ on Q from Proposition 3.10, we shall construct a holomorphic isomorphism between the bundles J n−1 (Q) and Q ⊗ J n−1 K (1−n)/2 Σ , which in turn would enable us to express generalized B-opers in terms of classical opers together with a fiberwise non-degenerate symmetric bilinear form on Q * , and a holomorphic connection on Q * that preserves this form.
As before, (E, F, D) is a generalized B-oper on Σ.
Given any point x ∈ Σ and v ∈ Q x , let v be the unique flat section of Q, defined around x, such that v(x) = v. Then, for each k, just as the map f k in (3.3) is constructed, we have a homomorphism γ k ∇ : Q −→ J k (Q) that sends any v ∈ Q x , x ∈ Σ, to the element of J k (Q) x obtained by restricting the flat section v to the k-th order infinitesimal neighborhood of x. Moreover, we shall denote by the homomorphism given by the composition of homomorphisms where the homomorphism J k (Q)⊗J k K for any vector bundles A, B on Σ. For ease of notation, in what follows we denote by J k the jet bundle Then, the following lemma is established.
in (3.14) is an isomorphism for each k. Hence the isomorphism f n−1 in Theorem 3.5 produces a holomorphic isomorphism of E = J n−1 (Q) with Q ⊗ J n , where J n is defined in (3.15).
Proof . We will show that the homomorphism Γ k ∇ is fiberwise injective. For that, take any point x ∈ Σ, and consider the restriction of Q to a simply connected open neighborhood U x of x. Recall from Proposition 3.10 that Q is equipped with a holomorphic connection ∇. As mentioned earlier, any holomorphic connection on a Riemann surface is automatically flat. Using this flat connection ∇ on Q, we trivialize the restriction Q| Ux of Q to U x (this is possible because U x is simply connected). Since Q ⊗ K (1−n)/2 Σ = Q (see (3.6)), using this trivialization of Q| Ux , the restriction Q| Ux of Q to U x gets identified with K (1−n)/2 Σ ⊕r | Ux , where r = rank(Q). Using this identification of Q| Ux with K (1−n)/2 Σ ⊕r | Ux , the restriction of the homomorphism Γ k ∇ in (3.14) to U x gets identified with the natural homomorphism The homomorphism in (3.16) is evidently fiberwise injective. Consequently, Γ k ∇ is fiberwise injective. The homomorphism in (3.16) is also fiberwise surjective, using which it can be deduced that Γ k ∇ is an isomorphism. Alternatively, since Γ k ∇ is fiberwise surjective, because it is fiberwise injective.
Corollary 3.14. For a generalized B-oper (E, F, D) on Σ, the holomorphic vector bundle E is canonically identified with Q ⊗ J n−1 K Proof . This follows from the combination of Theorem 3.5 and Lemma 3.13.
Lemma 3.15. Associated to (E, F, D), there is a natural homomorphism Proof . Combining the isomorphism in Lemma 3.13 for k = n with the definition of Diff m Σ (W, W ) in (3.2) for m = n, W = Q and W = Q ⊗ K n Σ , we have Using the trace homomorphism the isomorphism in (3.17) gives the homomorphism Although the holomorphic connection ∇ on Q does not explicitly arise in the statement or proof of Lemma 3.15, the homomorphism τ in Lemma 3.15 crucially uses ∇. Indeed, the isomorphism Γ k ∇ in Lemma 3.13, whose construction uses ∇, is the key ingredient in the construction of τ in Lemma 3.15.
Through the map τ from Lemma 3.15 we shall see how generalized B-opers relate to the classical Sp(n, C) and SO(n, C)-opers as introduced in Section 2.1.
Proposition 3.16. Consider the differential operator D in (3.9). For the map τ in Lemma 3.15, The differential operator is a classical Sp(n, C)-oper for n even, and a classical SO(n, C)-oper for n odd.
Proof . Holomorphic differential operators on Σ are given by holomorphic section of suitable bundles on the neighborhood of the diagonal (see [8, p. 25, equation (5.1)] for the precise statement). First we shall describe the homomorphism τ in Lemma 3.15 in terms of such sections. Recall from (3.11) that the differential operator D in (3.9) is given by the section It can be shown that using the flat connection ∇ on Q, the two vector bundles p * 1 Q and p * 2 Q are identified on some open neighborhood of the diagonal ∆ ⊂ Σ × Σ. Indeed, take an open subset U ∆ ⊂ Σ × Σ containing ∆ such that ∆ is a deformation retraction of U ∆ . For i = 1, 2, let q i : U ∆ −→ Σ be the projection to the i-th factor. Consider the two flat bundles (q * 1 Q, q * 1 ∇) and (q * 2 Q, q * 2 ∇) on U ∆ . On the diagonal ∆ ⊂ U ∆ , both are evidently identified with the flat bundle (Q, ∇) (once we identify ∆ with Σ using the map x −→ (x, x)). Since ∆ is a deformation retraction of U ∆ , this isomorphism between (q * 1 Q, q * 1 ∇)| ∆ and (q * 2 Q, q * 2 ∇)| ∆ extends to an isomorphism between (q * 1 Q, q * 1 ∇) and (q * 2 Q, q * 2 ∇) over the entire U ∆ . Using this isomorphism between q * 1 Q and q * 2 Q, the above section κ becomes a section Now using the symmetric bilinear form S * on Q (see (3.7)), the above section κ produces a section Using the isomorphism in [8, p. 25, equation (5.1)], this section κ produces a holomorphic differential operator (3.18) The differential operator D in (3.18) coincides with the differential operator τ (D) in the statement of Proposition 3.16. Let for locally defined holomorphic sections s and t of K The symbol of the differential operator D is Id Q [8, p. 18] (see the paragraph following (4.6)). From this it follows that the symbol of D is the constant function 1. Such a differential operator produces a holomorphic connection on the jet bundle J n−1 (K (1−n)/2 Σ ) (see [7, p. 15, equation (4.3)]; the spaces A and B in [7, equation (4.3)] are defined in [7, p. 13]). The holomorphic connection on J n−1 (K (1−n)/2 Σ ) given by the differential operator D will be denoted by D. Consequently, all the three vector bundles in Corollary 3.14, namely E, Q and J n−1 K (1−n)/2 Σ , are equipped with holomorphic connections. The isomorphism in Corollary 3.14 between E and Q ⊗ J n−1 K (1−n)/2 Σ is connection preserving. In particular, the above connection D is an SL(n, C)-connection; recall that D on E is an SL(nr, C)-connection and ∇ on Q is an SL(r, C)connection, where r = rank(Q).
Therefore, D defines an SL(n, C)-oper on Σ. We recall that the Sp(n, C) opers (n is an even integer) and SO(n, C) opers (n ≥ 5 is an odd integer) have the following property: Let F, ∇ F be a holomorphic principal Sp(n, C)-bundle, or a holomorphic principal SO(n, C)-bundle, F equipped with a holomorphic connection ∇ F , giving such an oper. Consider the holomorphic principal SL(n, C)-bundle equipped with a holomorphic connection obtained by extending the structure group of F, ∇ F using the natural inclusions of Sp(n, C) and SO(n, C) in SL(n, C). Then this holomorphic principal SL(n, C)-bundle equipped with a holomorphic connection is an SL(n, C) oper. It should be mentioned that this is not true for SO(2n, C)-opers. This is one of the reasons why our method does not apply to the even orthogonal cases, which shall be treated separatedly in future work [10].
Since D satisfies (3.19), the SL(n, C)-oper given by it is actually an Sp(n, C)-oper when n is even, and it is an SO(n, C)-oper when n is odd. In other words, the holomorphic connection D on J n−1 K (1−n)/2 Σ defines a Sp(n, C)-oper when n is even, and it defines a SO(n, C)-oper when n is odd. To construct the bilinear form on J n−1 K (1−n)/2 Σ for this Sp(n, C) or SO(n, C) oper structure, consider the isomorphism in Corollary 3.14. This produces an injective homomorphism The bilinear form B on E and the bilinear form S on Q * (see (3.7) and Proposition 3.9) together produce a bilinear form on E ⊗ Q * . The bilinear form on J n−1 K Let ∇ denote the holomorphic connection on Q * induced by the holomorphic connection ∇ on Q. Since the holomorphic connection D (respectively, ∇ * ) on E (respectively, Q * ) preserves the bilinear form B (respectively, S) on E (respectively, Q * ), the holomorphic connection D on J n−1 K (1−n)/2 Σ preserves the bilinear form on J n−1 K (1−n)/2 Σ constructed above.
From the above results, one has the following: Theorem 3.17. Let (E, F, D) be a generalized B-oper over Σ of filtration length n > 0. Then, the following three objects are canonically associated to (E, F, D):

1) a fiberwise non-degenerate symmetric bilinear form on
2) a holomorphic connection on Q that preserves this bilinear form, and 3) a classical Sp(n, C)-oper for n even and a classical SO(n, C)-oper for n odd.
Proof . Consider the symmetric bilinear form S on Q * in (3.7). Since S is nondegenerate (Proposition 3.9), it produces a nondegenerate symmetric bilinear form on the dual vector bundle Q.
Then, from Proposition 3.10, there is a holomorphic connection ∇ on Q which preserves the symmetric bilinear form on Q given by S.
Finally, the classical Sp(n, C)-oper for n even, and classical SO(n, C)-oper for n odd, are given by the differential operator τ (D) in the statement of Proposition 3.16.
The correspondence in Theorem 3.17 can be shown to be a one-to-one correspondence, and therefore in the next section we shall prove a converse of Theorem 3.17.

Generalized B-opers and projective structures 4.1 A projective structure from a B-oper
Given a projective structure P on Σ, from [7, p. 13, equation (3.6)] we have a (P dependent) decomposition of the space of differential operators The component in H 0 (Σ, O Σ ) corresponding to a differential operator is the symbol σ(D ) of D . Moreover, given a differential operator , whose symbol is a nonzero (constant) function, there is a unique projective structure P D on Σ such that the component of D in H 0 Σ, K 2 Σ vanishes identically for the decomposition of H 0 Σ, Diff n Σ K (as in (4.1)) associated to P D (see [7, p. 14, equation (3.7)]).
Definition 4.1. Consider the differential operator D ∈ H 0 Σ, Diff n Σ Q, Q ⊗ K n Σ in (3.9), and the map τ in Lemma 3.15. We shall denote by P the unique projective structure corresponding to the differential operator  .15)) which is orthogonal when n is odd and symplectic when n is even.
Proof . Any two choices of theta characteristic on Σ differ by tensoring with a holomorphic line bundle of order two on Σ. If K 1/2 and K 1/2 = K 1/2 ⊗L are two theta characteristics on Σ, where L is a holomorphic line bundle of order two, then K . Therefore, from Corollary 3.8 we conclude that where J k is defined in (3.15).
In view of (4.2), using [6, p. 10, Theorem 3.7], the projective structure P from Definition 4.1 produces a holomorphic isomorphism β : Sym n−1 (J 2 ) −→ J n . (4.3) We note that and hence the fibers of the vector bundle J 2 are equipped with a symplectic structure. For any x ∈ Σ, and for any v, w in the fiber ( be this symplectic pairing. Note that we have v, w 1 = − w, v 1 (4.5) as the pairing −, − 1 is symplectic. We will show that the above symplectic structure −, − 1 in (4.4) on the fibers of J 2 produces a bilinear form on the fibers of the symmetric product Sym d (J 2 ) for every d ≥ 1. For this, take a point x ∈ Σ, and take

Then we have
where −, − 1 is the pairing in (4.4). Note that the pairing −, − d in (4.6) coincides with −, − 1 in (4.4) when d = 1. It is straightforward to check that −, − d in (4.6) produces a nodegenerate bilinear form on Sym d (J 2 ) x . Next note that from (4.5) and (4.6) we have Therefore, the nodegenerate bilinear form −, − d on Sym d (J 2 ) x is symmetric if d is even and −, − d is anti-symmetric if d is odd.
For n odd (respectively, even), using the isomorphism β in (4.3), the orthogonal (respectively, symplectic) form −, − n−1 on Sym n−1 (J 2 ) produces an orthogonal (respectively, symplectic) structure on the fibers of J n . The filtration (see Lemma 3.2) of J 2 produces a filtration of Sym n−1 (J 2 ) such that all the successive quotients are of the form K j/2 Σ , for 1 − n ≤ j ≤ n − 1. Proof . This is straightforward. When genus(Σ) ≥ 2, both these filtrations coincide with the Harder-Narasimhan filtration of Sym n−1 (J 2 ) = J n .

Construction of B-opers
In order to build a converse statement to that of Theorem 3.17, let W be a holomorphic vector bundle on Σ equipped with a symmetric fiberwise non-degenerate bilinear form (4.7) Moreover, consider a holomorphic connection ∇ W on W that preserves the bilinear form S W . For an integer n ≥ 2, we shall let ω be an Sp(n, C)-oper (respectively, SO(n, C)-oper) on Σ if n is even (respectively, odd), which defines a holomorphic connection ∇ ω on the holomorphic vector bundle J n as defined in (3.15). Then, one can show the following. Proof . Consider the filtration of holomorphic subbundles on J n given by where A i is the kernel of the natural projection J n −→ J n−i−1 K (1−n)/2 Σ (similar to the filtration in (3.4)). By tensoring with W , the above filtration produces a filtration of holomorphic subbundles of J n ⊗ W given by (4.8) From [7, p. 13, equation (3.4)], the oper ω produces a holomorphic differential operator of order n from K , and from [7, p. 14, equation (3.7)] this differential operator produces a projective structure on Σ. Then, as in (4.3), from [6, p. 10, Theorem 3.7] this projective structure produces a holomorphic isomorphic between Sym n−1 (J 2 ) and J n . Moreover, as observed in the proof of Lemma 4.2, the vector bundle Sym n−1 (J 2 ) is equipped with an orthogonal (respectively, symplectic) form if the integer n is odd (respectively, even) and thus we get a non-degenerate bilinear form S ω on J n via the above isomorphism.
The form S ω and the form S W in (4.7) together define a bilinear form on J n ⊗ W , which we shall denote by S 0 . Moreover, one can see that the holomorphic connection (∇ ω ⊗ Id W ) ⊕ Id Jn ⊗ ∇ W on J n ⊗ W preserves S 0 , because ∇ W preserves S W and ∇ ω preserves S ω . Then, it is straightforward to check that the triple J n ⊗ W, S 0 , (∇ ω ⊗ Id W ) ⊕ Id Jn ⊗ ∇ W and the filtration in (4.8) together define a generalized B-oper on Σ, completing the proof.
The constructions in Theorem 3.17 and Proposition 4.5 are evidently inverses of each other, and can be further understood in terms of projective structures on the Riemann surface Σ. For this, let P(Σ) denote the space of all projective structures on the Riemann surface Σ. We recall that P(Σ) is an affine space for H 0 Σ, K ⊗2 Σ (see [24]). In the case of classical opers, from [7, p. 17, Theorem 4.9] and [7, p. 19, equation (5.4)], the space of SL(n, C)-opers on Σ is in bijection with Note that the above description of SL(n, C)-opers allows one to relate them with the Hitchin base of the SL(n, C)-Hitchin fibration introduced in [27].
When the integer n is even (respectively, odd) the subclass of Sp(n, C)-opers (respectively, SO(n, C)-opers) corresponds to the subspace In the case of generalized B-opers, combining Theorem 3.17 and Proposition 4.5, we have the following equivalent result. Theorem 4.6. For integers n ≥ 2, n = 3 and r ≥ 1, the space of all generalized B-opers of filtration length n and rank(E i /E i−1 ) = r is in correspondence with where C Σ denotes the space all flat orthogonal bundles of rank r on Σ, which is independent of i, and P(Σ) is the space of all projective structures on Σ.
In Theorem 4.6, the case of n = 3 is excluded because SO(3, C) = Sp(2, C)/(Z/2Z). The summation in Theorem 4.6 should be clarified -the summation is from 2 to [n/2] in increasing order. So

Generalized B-opers on jet bundles
Consider now a holomorphic vector bundle W on Σ of rank n equipped with a fiberwise nondegenerate symmetric pairing As before, using ν we have a non-degenerate pairing −, − between W and W ⊗ K n Σ with values in K Σ . The jet bundle J n−1 (W ) has a filtration 0 = S 0 ⊂ S 1 ⊂ · · · ⊂ S n−1 ⊂ S n := J n−1 (W ), which is constructed as done in (3.4) using the natural projections J n−1 (W ) −→ J n−1−i (W ). More precisely, S i is the kernel of the projection J n−1 (W ) −→ J n−1−i (W ). Given a holomorphic differential operator its symbol σ(D) as in Definition 3.4 is a holomorphic section of (T Σ) ⊗n ⊗ Hom W, W ⊗ K n Σ = End(W ).
Definition 4.7. We define the differential operator , where s and t are any locally defined holomorphic sections of the vector bundle W , and , −− is the K Σ valued pairing between W and W ⊗ K n Σ mentioned earlier.
From this definition one can obtain a correspondence between differential operators D as above and generalized B-oper structures on the jet bundle J n−1 (W ). Proof . In (3.13) we saw that the holomorphic differential operator associated to a generalized B-oper satisfies the above condition D * = D. Also, it was noted in the proof of Proposition 3.16 that symbol of the operator is Id (see [8, p. 18] Furthermore, a holomorphic differential D lying in this subset satisfies the equation vanishes for all 1 ≤ j ≤ (n − 1)/2 , from which the proposition follows.

Concluding remarks: generalized B-opers and Higgs bundles
For those having worked with Higgs bundles, introduced in [26] as solutions of the so-called Hitchin equations on Σ, there is proximity between some aspects of B-opers and Higgs bundles on Σ, which are pairs (E, Φ) where • E is a holomorphic vector bundle on Σ, • the Higgs field Φ : E → E ⊗ K, is a holomorphic K-valued endomorphism.
To illustrate this, we shall focus on B-opers of even rank where B is anti-symmetric, which shall naturally lead to symplectic Higgs bundles. For further details on Higgs bundles, the reader may refer to standard references such as Hitchin [26,27] and Simpson [32,33,34]. Finally, we should mention that generalized B-opers carry particularly interesting properties when the length of the filtration is 2, a case which leads to an intermediate generalization of opers with short filtrations, whose study is carried on in more detail in the third author's PhD thesis [36].
In what follows, we shall describe some aspects initiating the program. For this, some properties of determinant bundles shall be of much use, and thus we will give a brief description of them first.
We want to find a formula for the determinant bundle for each subbundle in the filtration of a generalized B-oper. The proof to the next proposition follows the idea given by Wentworth in [35,Lemma 4.9]. where Q = E/E n−1 . In particular, when the B-oper is a complete flag, we have Proof . The composition of second fundamental forms S j (D)•· · ·•S n−1 (D) gives an isomorphism for each associated graded piece: E j /E j−1 E n /E n−1 ⊗ K n−j Σ = Q ⊗ K n−j Σ , ∀ j = 1, 2, . . . , n.
Combining these isomorphisms with the fact that there is a smooth decomposition of E into associated graded pieces one gets the following isomorphism: Since E i has rank ri, the determinant bundle of E i is given by where we have used the exterior product isomorphisms ∧ k (V ⊕ W ) ∧ k V ⊗ ∧ k W and ∧ k (V ⊗ W 1 ) ∧ k V ⊗ W ⊗k 1 , for W 1 a one dimensional vector space.

Generalized B-opers and filtrations of length n = 2
From their construction, given an anti-symmetric form B and a rank 2r holomorphic bundle E, from Definition 2.11 a generalized B-oper (E, F, D) naturally defines a Lagrangian subspace E 1 | x ⊂ E| x for each x ∈ Σ and thus a Lagrangian subbundle E 1 ⊂ E. Then there is a rank 2r Higgs bundle (E, Φ), where the Higgs field is induced by In particular, when r = 1, the Higgs bundle is K 1/2 ⊕ K −1/2 , Φ .
Proof . Note that S 1 (D) induces an isomorphism E 1 → (E/E 1 ) ⊗ K Σ = E * 1 ⊗ K Σ , because the definition of a B-filtration gives E/E 1 E * 1 . Then, E 1 ⊕ (E/E 1 ) E 1 ⊕ E * 1 has the induced Higgs field Φ = 0 0 S 1 (D) 0 : When r = 1 we have E 1 is just a line bundle over Σ. Then the isomorphism given by S 1 (D) implies E 2 1 K Σ . Hence E 1 K 1/2 Σ is a choice of theta characteristic, and E K 1/2

Other induced Higgs bundles
We shall finally consider the appearance of Higgs bundles through generalized B-opers with flags of length bigger than 2. In this case, one has the following Higgs bundles constructed in a natural way from generalized B-opers. Proof . We need to check that (E, Φ) is a stable Higgs bundle. From the decomposition of E, we see that E/E n−1 is the only Φ-invariant subbundle, because Φ(E/E n−1 ) = 0 ⊂ E/E n−1 ⊗ K.
Thus for n > 1 and g > 1, the Higgs bundle (E, Φ) is always stable.
Remark 5.4. One should note that all of the Higgs bundles induced by B-opers we constructed in this section lie in the nilpotent cone in the Hitchin fibration, because the characteristic polynomial of Φ is det(xI − Φ) = x n .

The rank 4 case
We shall conclude here with some further comments on the lowest possible rank of generalized B-opers which have a symmetric bilinear form B. For this, consider now a B-filtration of a holomorphic vector bundle E of rank 4 on Σ, which is given by an increasing filtration of holomorphic subbundles for which E ⊥ i = E 4−i for all 1 ≤ i ≤ 3. Then, from Definition 2.11, we have that a generalized B-oper is a triple (E, F, D), where F is a B-filtration as in (5.2) and D is a B-connection on E, such that D(E i ) ⊂ E i+1 ⊗ K Σ for all 1 ≤ i ≤ 3, and the homomorphisms −→ E/E 3 , and in this case one has that E 2 = E ⊥ 2 and thus we obtain again a Lagrangian sub-fibration. Hence, a rank 4 generalized B-oper (E, E 1 , D) induces the following naturally defined Higgs bundle (P, Φ), where P = E 3 and Φ : E 3 → E 3 ⊗ K is induced by