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(G,P)-opers and global Slodowy slices

In this paper we introduce a generalization of G-opers for arbitrary parabolic subgroups P < G. For parabolic subgroups associated to “even nilpotents” we parameterize (G,P)-opers by an object generalizing the base of the Hitchin fibration. In particular, we describe families of opers associated to higher Teichmüller spaces.


Introduction
A complex projective structure on a Riemann surface X is a maximal atlas of charts with values in CP 1 and transition functions which are restrictions of Möbius transformations. Historically, these structures arose from the study of monodromies of second order ordinary differential equations on CP 1 minus a finite number of points. These structures were related to the uniformization problem, since a hyperbolic metric on X induces a complex projective structure via the natural inclusion of the upper half plane H ⊂ CP 1 . Interest later turned to higher order ordinary differential equations on H ⊂ CP 1 and their relationship to uniformization. Imposing certain invariance properties on these equations led to what are now known as developing maps of PSL n C-opers on X.
For G a connected, complex semisimple Lie group, the notion of a G-oper structure on X was codified by Beilinson-Drinfeld [2]. In their language, a Goper is a triple (E G , E B , ω), where E G is a holomorphic principal G-bundle on X, E B is a holomorphic reduction to a Borel subgroup B < G, and ω is a holomorphic connection on E G which satisfies certain transversality and nowhere vanishing properties with respect to the reduction E B . When G = PSL 2 C, a G-oper on X is equivalent to a CP 1 -structure on X.
Around the same time, Hitchin [24] introduced a special component of representations π 1 (X) → G R , where G R < G is a split real form and X is a compact Riemann surface of genus g ≥ 2. For example, PSL n R is the split real form of PSL n C. This component is now called the Hitchin component. When G = PSL 2 C, the Hitchin component corresponds to holonomy representations of hyperbolic structures on the underlying topological surface, and so, is identified with Teichmüller space.
Although representations in Hitchin components and G-opers are very different geometrically, these spaces share many common features. Most notably, both spaces are parameterized by a vector space of holomorphic differentials on X called the base of the Hitchin fibration [2,24]. More recently, in [15], the authors showed that an operation called the conformal limit identifies these parameterizations, proving a conjecture of Gaiotto [17].
In [30], Labourie proved that the representations in Hitchin components generalize many geometric features of Teichmüller space; in particular, they are discrete and faithful. Moreover, he showed that Hitchin representations satisfy a very strong dynamical property called the Anosov property. Since Labourie's work, there has been intense activity aimed at organizing and classifying Anosov representations. Notable among these spaces is the class of so called higher Teichmüller spaces (see [10,22]). In the theory of Anosov representations, a parabolic subgroup P < G plays a fundamental role; for Hitchin components the appropriate parabolic is the Borel subgroup.
Motivated by these spaces of representations and the conformal limit correspondence, in this paper, we introduce the notion of a (G, P)-oper for P < G a parabolic subgroup. When P = B is a Borel subgroup, we recover Beilinson-Drinfeld's notion of G-opers. Our main results (Theorems 5.8, 5.11 and 5.13) generalize the parameterization of Hitchin components and G-opers by the base of the Hitchin fibration.
Although many of the notions introduced involve a significant amount of Lie theory, in most cases these objects can be made relatively explicit with matrices and vector bundles. Throughout the paper, we have included many such examples to guide the reader. In a subsequent paper, we will develop the theory of (G, P)-opers using developing maps of geometric structures.
(G,P)-opers. Let X be a compact Riemann surface of genus g ≥ 2 and let P < G be a parabolic subgroup of a connected complex semisimple Lie group G. Roughly, a (G, P)-oper on X is a triple (E G , E P , ω), where E G is a holomorphic principal G-bundle on X, E P is a holomorphic reduction to the parabolic P < G, and ω is a holomorphic connection on E G which satisfies certain transversality and nowhere vanishing properties with respect to the reduction E P . The subtle point is defining the correct notion of nowhere vanishing (see Definition 4.1).
A simple example is given by G = SL 2n C and P is the stabilizer of a half dimensional subspace C n ⊂ C 2n . In this case, a (G, P)-oper can be defined as a triple (V 2n , V n , ∇), where V 2n is a rank 2n holomorphic vector bundle with trivial determinant, V n ⊂ V 2n is a rank n holomorphic subbundle and ∇ is a holomorphic connection on V 2n such that the induced map ∇ : V n → K ⊗ V 2n /V n is an isomorphism.
We briefly describe the picture we wish to generalize. In [23], Hitchin introduced the moduli space M 0 X (G) of polystable G-Higgs bundles on X. Some relevant aspects of Higgs bundles are: (1) There is a homeomorphism T : M 0 X (G) → M 1 X (G) from Higgs bundles to the moduli space of holomorphic G-connections on X [13,14,23,34]. This is known as the nonabelian Hodge correspondence. (2) There is a fibration M 0 X (G) → rkG j=1 H 0 (K m j +1 ) [25], where K is the canonical bundle of X and {m j } are the exponents of g.
(3) The fibration has a section s h : rkG j=1 H 0 (K m j +1 ) → M 0 X (G) and the Hitchin components are the holonomy representations of T • s h [24]. (4) Each connected component of isomorphism classes of G-opers is parameterized by rkG j=1 H 0 (K m j +1 ) [2]. We summarize this in the following diagram: In fact, for λ ∈ C, the above moduli spaces fit into a family of moduli spaces M λ X (G) of λ-connections. Moreover, there is a family of maps op λ : Our parameterization theorems do not apply to all parabolic subgroups but to a class of parabolics called even Jacobson-Morozov parabolics (see Definition 2.8). Such parabolics P are associated to embeddings of PSL 2 C in the adjoint Lie group, or equivalently, even sl 2 C subalgebras of the Lie algebra g. There are many examples of such parabolics, for instance, the Borel subgroup B < G and the stabilizer of a subspace C n ⊂ C 2n . The object which parameterizes such (G, P)-opers (i.e., the generalization of the Hitchin base) is a groupoid which we call the Slodowy category. The root of this additional structure stems from the centralizer of the associated sl 2 C subalgebra. Briefly, the centralizer of an sl 2 C subalgebra is discrete if and only if it is a so called principal subalgebra, in which case, the associated parabolic is the Borel B < G.
Given an even sl 2 C subalgebra s ⊂ g, let S < G be the connected Lie group with Lie algebra s and C < G be the centralizer of S. The s-Slodowy category B s (G) has objects (E C , ψ 0 , ψ m 1 , · · · , ψ m N ), where E C is a holomorphic Cbundle, ψ 0 is a holomorphic connection on E C and, for 1 ≤ j ≤ N, ψ m j is a holomorphic section of an associated vector bundle E C [Z 2m j ] ⊗ K m j +1 ; the vector spaces Z 2m j are zero weight spaces of certain sl 2 C representations (see Definition 5.1). When B is the Borel, C is the center of G and B s is the product of the Hitchin base with the finite set of C-bundles on X. When P < SL 2n C is the stabilizer of a subspace C n ⊂ C 2n , objects of the Slodowy category correspond to triples (W, ψ 0 , ψ 1 ), where W is a rank n holomorphic vector bundle on X with det(W) ⊗2 trivial, ψ 0 is a holomorphic connection on W and ψ 1 ∈ H 0 (K 2 ⊗ End(W)).
To avoid complications with moduli spaces we work with groupoids. Denote the category of (G, P)-opers on X by Op X (G, P). For an even sl 2 C subalgebra s ⊂ g, let S < G be the associated connected Lie group and P < G be the associated even Jacobson-Morozov parabolic. In Theorem 5.5 we establish the following: For each S-oper Θ on X, there is a natural functor (the (Θ, s)-Slodowy functor) which is a bijection on isomorphism classes (see also Theorem 5.11). The Slodowy functor is defined in Definition 5.3. Roughly, the functor uses the fixed S-oper to interpret the sections ψ m j as coefficients in certain highest weight spaces of sl 2 C representations. When P = B is a Borel subgroup, this is analogous to the Hitchin section.
To remove the choice of S-oper, we define a subcategory B s (G) of B s (G) which is analogous to setting the quadratic differential in the Hitchin base to be zero. We now have a functor F : Op X (S, B S ) × B s → Op X (G, P) defined by F (Θ, Ξ) = F Θ (Ξ) which we call the s-Slodowy functor. Theorem 1.2. (Theorem 5.8) Let X be a compact Riemann surface of genus g ≥ 2 and G be a complex connected semisimple Lie group. Let s ⊂ g be an even sl 2 C subalgebra, S < G be the associated connected subgroup and P < G be the associated even Jacobson-Morozov parabolic. Then the s-Slodowy functor is an equivalence of categories when S ∼ = PSL 2 C and essentially surjective and full when S ∼ = SL 2 C. In particular, the Slodowy functor induces a bijection on isomorphism classes. Remark 1.3. For G = SL kr C and the parabolic P stabilizing a partial flag C r ⊂ C 2r ⊂ · · · ⊂ C kr , the above theorem recovers results of Biswas in [4]. Partially motivated by the work presented in the current article, the Biswas-Schaposnik-Wang recently considered certain symplectic and orthogonal analogues of such partial flags, see [5].
The category of (G, P)-opers and the Slodowy category have natural λconnection generalizations, denoted by Op λ X (G, P) and B λ s (G) respectively. The Slodowy functor also naturally generalizes to this context, and in Theorem 5.13 we establish the analogue of the above theorem in this setting.
For λ = 0, a (λ, G, P)-oper is a special type of Higgs bundle. For G = SL 2n C and P the stabilizer of a subspace C n ⊂ C 2n , the above objects can be made very explicit. The objects of B 0 s (G) consist of triples (W, ψ 0 , ψ 1 ), where W is a holomorphic vector bundle of rank n such that det(W) ⊗2 is trivial, ψ 0 ∈ H 0 (K ⊗ End(W)) is a K-twisted traceless endomorphism and ψ 1 ∈ H 0 (K 2 ⊗ End(W)) is a K 2 -twisted endomorphism. A triple (W, ψ 0 , ψ 1 ) is an object of B 0 s (G) if and only if Tr(ψ 1 ) = 0. For the (0, SL 2 C, B)-oper Θ = (K 1 2 ⊕ K − 1 2 , 0 q 2 1 0 ), the Slodowy functor is defined by When ψ 0 = 0, this is an SU n,n -Higgs bundle, and the above description recovers the so called Cayley correspondence for maximal SU n,n -Higgs bundles of [7]. Moreover, when the associated Higgs bundles are polystable, the nonabelian Hodge correspondence identifies them with the higher Teichmüller spaces known as maximal SU n,n -representations.
Remark 1.4. The relation of our construction with other higher Teichmüller spaces is described in §6. While there are (G, P)-opers associated to each expected higher Teichmüller space, there are many examples of (G, P)-opers which are not related to higher Teichmüller spaces. For P an arbitrary even Jacobson-Morozov parabolic, it would be interesting to understand whether the representations associated to applying nonabelian Hodge to the set of (0, G, P)-opers have any special geometric significance.
Finally, we discuss the relationship between (G, P)-opers, Simpson's partial oper stratification and the conformal limit. In [15], the authors showed that the space of (G, B)-opers are related to the Hitchin section by an operation called the conformal limit, proving a conjecture of Gaiotto [17]. For G = SL n C, the space of B-opers and the Hitchin section are each a closed stratum of Bialynicki-Birula stratifications introduced by Simpson [33]. Moreover, under a smoothness assumption, the conformal limit correspondence for arbitrary strata was established in [11]. In the generality of this paper, it is natural to expect that the conformal limit also identifies (0, G, P)-opers with (1, G, P)-opers. Indeed, for the case G = SL 2n C and P the stabilizer of a subspace C n ⊂ C 2n , when the bundle W in (1.1) is a stable bundle, this follows from the work in [11].

Lie theory preliminaries
In this section, we recall most of the Lie theory which will be used throughout the paper. For this section, G will be a connected, complex semi-simple Lie group with Lie algebra g and Killing form B g .

Parabolics. A Borel subgroup B < G is a maximal solvable subgroup.
There is a unique Borel subgroup up to conjugation. A subgroup P < G is called a parabolic subgroup if it contains a Borel subgroup. This is equivalent to the homogeneous space G/P being a projective variety.
For a parabolic P < G, let U < P be the unipotent radical. There is a canonical filtration of U = U 1 , The quotient L = P/U is a reductive group called the Levi factor of P. Choosing a splitting of the sequence 1 → U → P → L → 1 defines a Levi subgroup of P.
Let u ⊂ p be the Lie subalgebra associated to the subgroup U < P. There is a canonical P-invariant Lie algebra filtration of g defined by where g j = {x ∈ g | ad x (u) ⊂ g j+1 }. In particular, g 1 = u and g 0 = p.
A choice of a Levi subgroup L < P defines a splitting p = l ⊕ u, where l is the Lie algebra of L. Moreover, g acquires an L-invariant grading which is L-equivariantly isomorphic to the associated graded of the canonical filtration defined above. Thus, g j ∼ = g j /g j+1 as L-modules and g k ∼ = j≥k g j as P-modules. In particular, By a theorem of Vinberg (see [27,Theorem 10.19]), each graded piece g j has a unique dense open L-orbit O j ⊂ g j . Since the action of P and L on g j /g j+1 are the same, there is a unique of open dense P-orbit in g j /g j+1 . (1) The group P = B which stabilizes a complete flag C ⊂ C 2 ⊂ C 3 ⊂ C 4 consists of unit determinant upper triangular matrices. The unipotent radical U < B consists upper triangular matrices with 1's on the diagonal. The diagonal matrices define a Levi subgroup L < B, and, with this choice, the graded pieces g j are given by the j th -super diagonal. In particular, g −1 is given by matrices of the form 0 The unique open L-orbit O −1 ⊂ g −1 is defined by x j = 0 for all j. (2) The group P < SL 4 C which stabilizes a partial flag C 2 ⊂ C 4 consists of unit determinant (2 × 2)-block matrices of the form The unipotent radical U < P is the subgroup where A = Id = D, and subgroup where B = 0 is defines a Levi. This choice gives The unique open L-orbit O −1 ⊂ g −1 is defined by det(C) = 0. (3) The group P < SL 4 C which stabilizes a partial flag C 1 ⊂ C 3 ⊂ C 4 is given by unit determinant matrices of the form Block diagonal matrices define a Levi subgroup, and, with this choice, The unique open L-orbit O −1 ⊂ g −1 is given by (4) For the group P < SL 4 C which stabilizes a partial flag C ⊂ C 4 , block diagonal matrices define a Levi. With this choice, we have The unique open L-orbit O −1 ⊂ g −1 consists of nonzero vectors.
The next example involves some basic root theory.
Example 2.2. For P = B, a Levi subgroup L < B is a choice of Cartan subgroup. This data defines a set of positive simple roots {α 1 , · · · , α rk(g) }.
The spaces g j consist of direct sums of root spaces g α associated to roots In particular, the g −1 space consists of a direct sum of negative simple root spaces. In this case, an element is in the open dense L-orbit O −1 ⊂ g −1 if and only if x j = 0 for all j.

Nilpotents and the Jacobson-Morozov
Theorem. An element e ∈ g is a nilpotent element if ad e : g → g is a nilpotent endomorphism of g. Denote the centralizer of a nilpotent e ∈ g by V (e) = V , V = ker(ad e : g → g) .
The Jacobson-Morozov theorem defines a bijective correspondence between conjugacy classes of sl 2 C subalgebras of g and conjugacy classes of nonzero nilpotents in g (see for example [12, §3] Throughout the paper, we will use the letter s to denote an sl 2 -subalgebra. Fix an sl 2 -triple f, h, e = s ⊂ g. This data decomposes g into as sl 2 Cmodules and ad h -weight spaces. Namely, where W j is isomorphic to a direct sum of n j -copies (with n j ≥ 0) of the unique (j + 1)-dimensional sl 2 C representation, and Note that e ∈ g 2 and f ∈ g −2 . Note also that the trivial representation W 0 is a reductive subalgebra since it is the centralizer of subalgebra s. We will denote this subalgebra by W 0 = c(s) = c. Let b s = h, e ⊂ g to be the Borel subalgebra of s. The centralizer V of e decomposes (c ⊕ b s )-invariantly as The affine space is called the Slodowy slice through e. It is transverse to the G-orbit of e in g [37, §7.4].
The following proposition is immediate and will be useful later.
The decomposition (2.4) associates a parabolic subalgebra p to an sl 2triple f, h, e , Moreover, this parabolic comes with a choice of Levi subalgebra l = g 0 ⊂ p. Note that p also decomposes as In particular, for all j ≥ 1 and u ∈ U j we have where a ∈ ad f (u j ) is nonzero and b ∈ u j The following theorem relates the open dense L-orbit O −2 ⊂ g −2 with the sl 2 -triple. It is a combination of results of Kostant and Malcev, see [27, §10] or [12, §3.4] for details.
Theorem 2.5. Let f, h, e ⊂ g be an sl 2 -triple, and consider the Z-grading g = j∈Z g j from (2.4). Let p and l be the associated parabolic and Levi subalgebras with associated Lie subgroups L < P < G. Then the nilpotent elements e and f are contained in the unique open dense L-orbits O 2 ⊂ g 2 and O −2 ⊂ g −2 , respectively. Moreover, the L-stabilizer of e is the Lie group C < G which centralizes the sl 2 -subalgebra.

2.3.
Even Jacobson-Morozov parabolics. We will call a parabolic which arises from an sl 2 -subalgebra a Jacobson-Morozov parabolic (abbreviated JM-parabolic). Not all parabolics are JM-parabolics. For example, the stabilizer of a line in C n is not a JM-parabolic when n > 2. Moreover, different conjugacy classes of sl 2 -triples can define the same JM-parabolic.
Remark 2.7. An sl 2 -triple f, h, e ⊂ g is an even if and only if g 2j+1 = 0 for all j in decomposition (2.4). Equivalently, the sl 2 C-module decomposition from (2.3) consists only of odd dimensional irreducible sl 2 -representations. Another equivalent definition of s = f, h, e being even is that the associated subgroup S < G Ad of the adjoint group is isomorphic to PSL 2 C. Definition 2.8. A parabolic subgroup P < G will be called an even JMparabolic if it arises from an even sl 2 -triple.
Remark 2.9. Since the JM-parabolic p associated to an sl 2 -triple f, h, e comes with a choice of Levi subalgebra, we have two Z-gradings. Namely, the ad h -weight space decomposition (2.4) and the associated graded of the canonical filtration (2.2). For even JM-parabolics, the j th -graded piece of (2.2) is the 2j th -graded piece of (2.4).
The following proposition is an easy consequence of [12, §3.8].
Proposition 2.10. There is a one-to-one correspondence between conjugacy classes of even sl 2 -triples and conjugacy classes of even JM-parabolics.
Fix an even sl 2 -triple f, h, e ⊂ g. Let 1 = m 1 < m 2 < · · · < m N be such that the sl 2 -representations W 2m j from (2.3) are nonzero. We have The Slodowy data from (2.5) is Note that ad the defines an c ⊕ h -invariant decomposition of the Levi subalgebra of the even JM-parabolic. This decomposition will be important later.
Proposition 2.11. The Slodowy data of an even JM-parabolic defines an c ⊕ h -invariant decomposition of the Levi subalgebra g 0 Example 2.12. The first three parabolic subgroups of SL 4 C described in Example 2.1 are even JM-parabolics. In fact these are the only even JMparabolic subgroups of SL 4 C. Below, we use the same numbering as the items in Example 2.1.
The next example sketches the construction and states some properties of principal sl 2 -triples, for more information see [28] or [12, §4].
Example 2.13. The Borel subgroup B < G is always an even JM-parabolic. An associated even sl 2 -triple is called a principal sl 2 -subalgebra. Fix a Cartan subalgebra g 0 ⊂ g and a set of simple roots {α 1 , · · · , α rk(g) }. For each α j choose e j ∈ g α j and Then The centralizer C of s is the center of the group G, thus c = W 0 = {0}. The associated s-module decomposition is g = j W 2m j where {m j } are the exponents of g. The multiplicities satisfy n m j = 1 for all j, with the exception that n 4n−2 = 2 for g = so 2n C.
Finally, we recall a generalization of a theorem of Kostant proven by Lynch in [31, Theorem 1.2]. It will play a crucial role later in the paper.
Theorem 2.14. Let P < G be an even JM-parabolic subgroup determined by an sl 2 -triple f, h, e . Let U m < U m−1 < · · · < U 1 < P be the filtration of P by unipotent subgroups and g m ⊂ · · · ⊂ g 0 = p be the corresponding canonical filtration of p. Then, for V = ker(ad e : g → g), the map is an isomorphism of affine algebraic varieties for every 0 < i ≤ m.
Remark 2.15. In [18, Lemma 2.1], a more general version of the above theorem is proven for any (not necessarily even) sl 2 -triples with a choice of isotropic subspace of g 1 from (2.4). It would be interesting to consider how such data could be used to generalize the results of this paper to arbitrary parabolics.

Geometric preliminaries
In this section we recall the necessary definitions concerning principal bundles, connections and flag varieties.
3.1. Bundles and connections. Throughout this paper, all principal bundles are taken to be right principal bundles. If G is a Lie group and E G is a principal G-bundle, then we denote the G-invariant vertical vector field on E G induced by Y ∈ g by Y ♯ . If V is a complex vector space equipped with a holomorphic G-action, we will suppress the action and denote the associated holomorphic vector bundle by E G (V ). For example, E G (g) is the holomorphic vector bundle determined by the adjoint action of G on g.
For the rest of this section, E G will be a holomorphic principal G-bundle over a compact connected Riemann surface X and K will denote the holomorphic cotangent bundle of X.
On a Riemann surface, the curvature F (ω) = dω+ 1 2 [ω, ω] of a holomorphic connection ω always vanishes 1 . Thus, all holomorphic connections on E G are flat, and we refer to the pair (E G , ω) as holomorphic flat G-bundle.
Let {U α } be a trivializing open cover of for a holomorphic bundle E G defined by local sections s α : U α → E G with transition functions g αβ : U αβ → G. For a holomorphic connection ω on E G , define the local connection forms ω α := s * α ω ∈ Ω 1 (U α , g). These local connection forms satisfy the relation Finally, let E G , F G be holomorphic principal G-bundles and η be a holomorphic connection on F G . Let s : U → E G and t : U → F G be local sections over an open set U ⊂ M. Consider a morphism Φ : E G → F G and the local connection forms η U := t * η and η Φ U := t * (Φ * η). Then, with respect to the trivializations induced by s and t, Φ is determined by a holomorphic map φ : U → G and

3.2.
Extensions and reductions of structure group. Given a holomorphic principal G 1 -bundle E G 1 and a homomorphism b : G 1 → G 2 between complex Lie groups, there is an associated holomorphic principal G 2 -bundle for g 1 ∈ G 1 and the action of G 2 is given by right multiplication on the second factor. Transition functions of E G 1 (G 2 ) are defined by post-composing transition functions of E G 1 with the map b : G 1 → G 2 . This process is usually called extending the structure group.
Let ω be a holomorphic connection on E G 1 and θ G 2 be the left invariant Maurer-Cartan form of G 2 . Then the G 2 -equivariant map descends to define a connection on the extended bundle E G 1 (G 2 ). We will denote the connection on E G 1 (G 2 ) induced by ω by the same symbol ω.
If E G 1 , E G 2 are holomorphic G 1 , G 2 bundles respectively, then their fiber product defines a holomorphic principal ( subgroups which commute with each other, then the multiplication map m : G 1 × G 2 → G is a group homomorphism. In this case, two holomorphic bundles with holomorphic connection (E G 1 , ω), (E G 2 , η) induce a holomorphic G-bundle with connection which we denote by Note that the transition functions of E G 1 ⋆ E G 2 (G) are just the product of the transition functions of E G 1 and E G 2 in G and that locally ω ⋆ η is just a sum of the connection forms. Let V be a complex vector space with a G-action.
There is a canonical isomorphism between equivariant, horizontal V -valued holomorphic 1-forms and holomorphic sections the vector bundle K ⊗ E G (V ) over X.
If H < G is a complex Lie subgroup, then a structure group reduction of a holomorphic principal G-bundle E G to H is a holomorphic subbundle E H ⊂ E G . Given a holomorphic connection ω on E G , the second fundamental form of a structure group reduction E H ⊂ E G measures the failure of ω to be induced by a connection on E H .
Finally, we define a notion of relative position for a holomorphic principal with holomorphic connection with respect to a reduction of structure. 3.3. Flag varieties. Let G be a connected, complex semisimple Lie group and P < G a parabolic subgroup. The associated flag variety G/P is a smooth complex projective algebraic variety. In particular, G/P a compact complex manifold, and G → G/P is a holomorphic principal P-bundle. There is a canonical G-equivariant isomorphism between the holomorphic tangent bundle T (G/P) and the associated vector bundle G(g/p). Namely, where the P-action is defined by (g, v) · p = (g · p, Ad(p −1 )(v)).
Every P × C * -invariant subset O ⊂ g/p determines a holomorphic fiber bundleÔ := G × P O → G/P which is equipped with a canonical C * -equivariant bundle morphism  3.4. λ-connections and Higgs bundles. In this subsection we generalize the notion of a holomorphic connection to depend on a parameter λ ∈ C. These objects have been studied by Simpson [32], and many others. Let G be a complex semisimple Lie group and E G be a holomorphic principal G-bundle on a closed Riemann surface X.
Remark 3.8. A holomorphic 1-connection is just an ordinary holomorphic connection, and if ω is a λ-connection for λ = 0, then 1 λ ω is an ordinary connection. In contrast, a 0-connection ω vanishes on vertical vectors, thus ω is equivalent to a holomorphic section ω ∈ H 0 (X, K ⊗ E G (g)), such a section is usually called a Higgs field. Remark 3.9. Recall that local connection forms transform as in (3.1) and (3.2). Local λ-connections transform the same way with the Maurer-Cartan form θ G in (3.1) and (3.2) replaced by λθ G .
Note that there is a natural projection map π : QF(G) → C defined by π(λ, E G , ω) = λ. The fibers π −1 (λ) = F λ (G) are the categories of λconnections for a fixed λ. There is also a C * -action on QF(G): The map π is equivariant with respect to the standard action of C * on C.
When λ = 0, the category of 0-connections F 0 (G) is also called the category of G-Higgs bundles on X. In this case a holomorphic zero connection (0, E G , ω) is equivalent to the data (E G ,ω) whereω ∈ H 0 (K ⊗ E G (g)).
Let {p 1 , · · · , p rk(g) ) be a basis of the G-invariant polynomials on g with deg(p j ) = m j + 1. With this choice we can define the Hitchin fibration .
The vector space rk(g) j=1 H 0 (K m j +1 ) is usually called the Hitchin base. In [24], Hitchin used the special features of principal sl 2 -triples to constructed a section of this map.
Since the new objects introduced in this paper generalize the Hitchin section, we sketch the construction here. Let f, h, e ⊂ g be a principal sl 2 -triple from Example 2.13. Recall that the Lie algebra decomposes into ad h -eigenspaces g = j∈Z g 2j . Consider the holomorphic Lie algebra bundle the holomorphic frame bundle E G Ad has structure group G Ad . The Lie algebra bundle E G Ad [g] is isomorphic to E g .
Recall also that V = ker(ad e ) = rk(g) . Choose a vector e 2m j in each V 2m j . Note that f defines a holomorphic section of K ⊗ (K −1 ⊗ g −2 ) and q m j +1 ⊗ e 2m j defines a holomorphic section Consider the map s H : In [29], Kostant proved that there is a basis (p 1 , · · · , p rk(g) ) of the G-invariant polynomials on g with deg(p j ) = m j + 1 such that Defining the Hitchin fibration with respect to this choice of basis for the invariant polynomials makes s H a section.

The category of (G,P)-opers
In this section we define the main new object of the paper, (G, P)-opers. Let X be a compact, connected Riemann surface of genus at least two, G a connected complex semisimple Lie group, and P < G a parabolic subgroup. ω) is a holomorphic flat G-bundle over X, (2) E P ⊂ E G is a holomorphic reduction of structure to the parabolic subgroup P < G, (3) the position of ω relative to E P satisfies pos E P (ω) = O.
Remark 4.2. When P = B, using Example 2.13, the third condition if Definition 4.1 is that the projection of ω to every negative simple root space is nowhere vanishing. This recovers Beilinson-Drinfeld's definition of an oper in [2].
A morphism between two (G, P)-opers (E G , E P , ω), (F G , F P , η) is an isomorphism Φ : E G → F G of holomorphic principal bundles such that Φ| E P : E P → F P is an isomorphism and Φ * η = ω. The category of (G, P)-opers over X is denoted Op(G, P).
Remark 4.3. The definition of a (G, P)-oper extends immediately to the context of λ-connections. We denote the category of (G, P, λ)-opers over X as λ ranges over C by QOp(G, P). Namely, the objects of QOp(G, P) consists of tuples (λ, E G , E P , ω), where λ ∈ C and (E G , E P , ω) is a λ-oper.
Using Example 2.1, we can rephrase the above definition for G = SL 4 C in terms of rank 4-vector bundles. (1) Since B is the stabilizer of full flag in C 4 , a holomorphic B-reduction E B ⊂ E defines a holomorphic filtration where rk(V i ) = i. The third condition of Definition 4.1 is that, for each i, ∇(V i ) ⊂ V i+1 ⊗ K and ∇ induces a bundle isomorphism (2) For P the stabilizer of a subspace C 2 ⊂ C 4 , a holomorphic Preduction E P ⊂ E defines a holomorphic filtration where rk(V i ) = i. The third condition of Definition 4.1 is that ∇ induces a bundle isomorphism (3) For P the stabilizer of a partial flag C 1 ⊂ C 3 ⊂ C 4 , a holomorphic P-reduction E P ⊂ E defines a holomorphic filtration where rk(V i ) = i. The third condition of Definition 4.1 is that ∇(V 1 ) ⊂ V 3 and ∇ 2 induces a bundle isomorphism (4) For P the stabilizer of a subspace C 1 ⊂ C 4 , a holomorphic Preduction E P ⊂ E defines a holomorphic line subbundle V 1 ⊂ V.
The third condition of Definition 4.1 is that the map V 1 → V/V 1 ⊗ K induced by ∇ is an injective bundle map.
Remark 4.5. Each of the above examples generalizes naturally to higher rank holomorphic vector bundles. For the first example, we recover the classical vector bundle definition of an oper. Namely a (SL n C, B)-oper consists of a rank n holomorphic vector bundle V equipped with a holomorphic filtration For G = S is isomorphic to PSL 2 C or SL 2 C, the Borel B S < S is the only proper parabolic subgroup. In terms of vector bundles, an SL 2 C-oper is a (V, V 1 , ∇), where rk(V) = 2 with Λ 2 V ∼ = O, rk(V 1 ) = 1 and ∇ is a holomorphic connection such that the induced map ∇ : V 1 → V/V 1 ⊗ K is an isomorphism. Thus, V 1 ∼ = K 1 2 is one of the 2 2g square roots of K and V/V 1 ∼ = K − 1 2 . The associated holomorphic vector bundle V is isomorphic to the unique non-split extension Choose a smooth identification of V with K 1 2 ⊕ K − 1 2 . In this splitting, an explicit example of an oper is given by the following Dolbeault operator and holomorphic connection Here, h is the uniformizing hyperbolic metric on X,∂ 1/2 and∂ −1/2 are Dolbeault operators on K 1 2 and K − 1 2 respectively, ∂ h 1/2 and ∂ h −1/2 are the Chern connections of the induced metric on K 1 2 and K − 1 2 , respectively, and q ∈ H 0 (K 2 ). In fact, every SL 2 C-oper is isomorphic to (4.1). This will be generalized in Theorem 5.11 for (G, P)-opers when P is an even JMparabolic.
Remark 4.6. In the above discussion, we have a choice of a square root of K 1 2 . Each such choice defines the same PSL 2 C-oper.

The global Slodowy slice for even JM-parabolics
In this section we define the s-Slodowy category B s (G), associated to any even sl 2 -triple s = f, h, e ⊂ g. We first prove that the category B s (G) is equivalent to the category Op(G, P) of (G, P)-opers, where P is the even JM-parabolic associated to s. This equivalence can be interpreted as a "global slice theorem". When s is a principal subalgebra and G is the adjoint group, B s (G) is the Hitchin base and we recover the Hitchin section and more generally Beilinson-Drinfeld's parameterization of (G, B)-opers.
Let S < G be the connected subgroup with Lie algebra s. The functor realizing the equivalence depends on the choice of an S-oper. We then show that there is a subcategory B s (G) of B s (G) and a natural functor Op(S, B S )× B s (G) → Op(G, P) which is an equivalence when S ∼ = PSL 2 C and essentially surjective and full when S ∼ = SL 2 C. Finally we extend these results to the category of (λ, G, P)-opers.
5.1. The s-Slodowy category. Let G be a connected, complex semi-simple Lie group. Fix an even sl 2 -triple s = f, h, e ⊂ g, let b s = h, e ⊂ g and let B S < S < G be the associated connected subgroups of G. Recall from (2.8) that we have s-module decomposition of g Recall also that the Slodowy data and Levi data of s given by Moreover, from Proposition 2.4, we have V 2 = e ⊕V 2 and Z 2 = h ⊕Ẑ 2 . The Slodowy data of s is the Lie theoretic mechanism which defines the generalization of the Hitchin base.
Definition 5.1. Let G be a complex semisimple Lie group and s = f, h, e ⊂ g be an even sl 2 -triple with centralizer C < G. The s-Slodowy category B s (G) has objects (E C , ψ 0 , ψ m 1 , · · · , ψ m N ), where and morphisms given by isomorphisms of holomorphic flat C-bundles which identify holomorphic sections of E C (Z 2m j ) ⊗ K m j +1 for every j.
Remark 5.2. For a principal sl 2 -triple s ⊂ g, C < G is the center of G and ψ m j ∈ H 0 (K m j +1 ). In this case, B s (G) is the product of the Hitchin base with finite set of holomorphic C-bundles on X. In particular, for an adjoint group G, B s (G) is the Hitchin base.
Let Θ = (F S , F B S , η) be an (S, B S )-oper. For every holomorphic C-bundle E C , the second fundamental form of η induces a holomorphic isomorphism where we recall that ⋆-notation from (3.3). is defined by where Ξ = (E C , ψ 0 , ψ m 1 , · · · , ψ m N ) is an object in B s (G) and we have identified each ψ m j with a holomorphic section of F B S ⋆ E C (V 2m j ) ⊗ K using the isomorphism (5.1). If f : Remark 5.4. The Slodowy functor can be extended to a functor .
Given morphisms f 1 : Θ → Θ ′ and f 2 : Recall that a functor defines an equivalence of categories if it is essentially surjective and fully faithful, i.e. every isomorphism class is in the image and the functor induces a bijection on morphisms.
Theorem 5.5. Let G be a complex connected semisimple Lie group, s = f, h, e ⊂ g be an even sl 2 -triple and P < G be the associated even JMparabolic. For every (S, B S )-oper Θ, the Θ-Slodowy functor is an equivalence of categories.
Remark 5.6. When s is a principal nilpotent and P = B is the Borel subgroup, Theorem 5.5 is due to Beilinson-Drinfeld [2]. Note that, in this case, the Slodowy functor is defined similarly to the Hitchin section (3.6), see (5.9) for the explicit relation.
To remove the choice of (S, B S )-oper from Theorem 5.5, we introduce a subcategory B s (G) of B s (G). Since, the space Z 2 decomposes C-invariantly as h ⊕Ẑ 2 , for any holomorphic C-bundle E C we have Thus, for any object (E C , ψ 0 , ψ m 1 , · · · , ψ m N ) in B s (G), ψ m 1 decomposes as . Thus, the s-Slodowy category B s (G), factors as the product of the Definition 5.7. Let s = f, h, e ⊂ g be an even sl 2 -triple. Define the traceless quadratic s-Slodowy category to be the subcategory B s (G) of B s (G) whose objects are (E C , ψ 0 , ψ m 1 , · · · , ψ m N ) with ψ m 1 =ψ m 1 .
Theorem 5.8. Let G be a complex connected simple Lie group, s = f, h, e ⊂ g be an even sl 2 -triple, S < G be the connected subgroup with Lie algebra s and P < G be the associated even JM-parabolic. Then the functor is an equivalence of categories when S ∼ = PSL 2 C and essentially surjective and full when S ∼ = SL 2 C.
Remark 5.9. Recall that when G is an adjoint group, the subgroup S < G is always isomorphic to PSL 2 C. Thus, for adjoint groups, the above functor always defines an equivalence of categories.

5.2.
Proofs of Theorems 5.5 and 5.8. In this section we will prove Theorems 5.5 and 5.8. The key technical result is Theorem 5.11. For this section we fix an even sl 2 -triple s = f, h, e ⊂ g and retain the notation from the previous sections. We begin with a local result.
where θ P is the left invariant Mauer-Cartan form on P. Moreover, Ψ is unique up to right multiplication by a holomorphic map D → C.
Proof. Let L < P be the Levi subgroup with Lie algebra g 0 . Since L acts transitively on O −1 , L preserves p, and f ∈ O s , there is a holomorphic map Ψ 0 : D → L such that Since C < L is the stabilizer of f , Ψ 0 is unique up to right multiplication by any holomorphic map D → C. Define By Theorem 2.14, there exists a unique holomorphic Ψ 1 : D → U 1 such that Then, define Using Theorem 2.14 inductively for each i ≥ 2, we obtain a unique Ψ i : To complete the process, we claim there exists a unique Ψ m N +1 : D → U S such that Then Ψ = Ψ 0 · · · Ψ m N +1 : D → P is the desired map.
A direct calculation shows For the uniqueness of Ψ, any other map satisfying (5.5) can be written as Ψ · Φ for Φ : D → P. By (2.7), the projection of Φ onto U is the identity. Moreover, Ad(Φ −1 ) acts trivially on O s . Thus, the projection of Φ to L is contained in C. Hence, Ψ is unique up to right multiplication by a holomorphic map D → C.
The proofs of Theorems 5.5 and 5.8 follow almost immediately from the following parameterization theorem.
Theorem 5.11. Let G be a connected complex semisimple Lie group, s = f, h, e ⊂ g be an even sl 2 -triple, S < G be the connected subgroup with Lie algebra s, C < G be the centralizer of s and P < G be the associated even JM-parabolic. Let Θ = (F S , F B S , η) be an (S, B S )-oper and (E G , E P , ω) a (G, P)-oper. Then, there is a unique (up to isomorphism) flat C-bundle (E C , ψ 0 ) and a holomorphic map Ψ : By Lemma 5.10, we have {Ψ α : U α → P} so that Let E Q be the principal P-bundle defined by the transition functions Q := {q αβ = Ψ −1 β p αβ Ψ α : U αβ → P}. By construction, the locally defined Ψ α patch to a globally defined isomorphism Ψ : E Q → E P .
Next, let ω Ψ α = Ad(Ψ −1 α ) • ω α + Ψ * α θ P denote the local expression of Ψ * (ω) is the coordinate chart U α . Then In particular, Equation (5.6) is a direct sum of graded pieces with respect to the canonical grading (2.2). The −1-graded piece is the first non-zero term. Since the (−1)-graded pieces of ψ α , Ad(q −1 αβ ) • ψ β and q * αβ θ P vanish, we have This implies that the image of q αβ is contained in the subgroup of P generated by C, B S and U. In particular, the Maurer-Cartan form q * αβ θ P is valued in the subalgebra c ⊕ b s ⊕ u.
For the zeroth graded piece we have , this implies that the image of q αβ is contained in the subgroup of P generated by C, B S and U 2 . The Maurer-Cartan form q * αβ θ P is thus valued in c ⊕ b s ⊕ u 2 . Moreover, the b s part of q * αβ θ P is b * αβ θ B S . For the first graded piece is valued in e and given by So, using (2.7), it follows that q αβ takes values in the group generated by B S , C and U 3 . Moreover, we have Thus, using (2.7), (Ad(q −1 αβ ) • η β ) 2 = 0 and the is image of q αβ is contained in the subgroup of P generated by C, B S and U 4 . Now, for j ≥ 2 assume 0 = (q * αβ θ P ) j . The same argument as in the previous case implies that the image of q αβ is contained in the subgroup of P generated by C, B S and U j+2 and Since U m N +1 = 0, we conclude that q αβ is valued in the subgroup generated by B S and C. Since B S and C commute we have where c αβ : U αβ → C are the transition functions of a holomorphic C-bundle E C . Thus, E Q = E C ⋆ F B S (P) and, by (5.7), {(ψ α ) 0 } defines a holomorphic connection on E C . Moreover, by (5.8) The uniqueness of (E C , ψ 0 ) up to isomorphism follows from Lemma 5.5. Namely, the maps Ψ α are unique up to right multiplication by a map C α : U α → C. The different choices define isomorphic C-bundles.
We now prove Theorem 5.5 and Theorem 5.8.
Proof of Theorem 5.5. By Theorem 5.11, for each (S, B S )-oper Θ, the Slodowy functor F Θ is essentially surjective. Given a morphism f : . This is clearly faithful. To see that it is full, consider such that g * (η ⋆ψ ′ 0 ) = η ⋆ψ 0 . Locally, g = {g α : U α → P}, where each g α acts trivially on s. Thus g α : U α → C, and g : Proof of Theorem 5.8. Fix an (S, B S )-oper Θ = (F S , F B S , η) and let (E G , E P , ω) be a (G, P)-oper. By Theorem 5.11, there is Ξ where, using the isomorphism (5.1), ψ m j ∈ H 0 (F B S ⋆ E C (V 2m j ) ⊗ K). By (5.4), ψ m 1 decomposes as ψ m 1 = q +ψ m 1 , where LetΞ = (E C , ψ 0 ,ψ m 1 , ψ m 2 , · · · , ψ m N ) be the associated object in B s (G). Note that Θ q = (F S , F B S , η + q) is another (S, B S )-oper and Thus, the functor F : For fullness, consider such that g * (η ′ ⋆ ψ ′ 0 +ψ ′ 1 + · · · + ψ ′ m N ) = η ⋆ ψ 0 +ψ 1 + · · · + ψ m N . Locally, g = {g α : U α → P}, where each g α preserves s + V . By (2.7), we have g α : U α → C ⋆ B S , and g : Such a g can be written as a product g C · g B S where g C : E C → E ′ C and g B S : F B S → F ′ B S . For faithfulness, note that the multiplication map S × C → S ⋆ C < G is injective if and only if S ∼ = PSL 2 C. Thus, the functor F is faithful if and only if S ∼ = PSL 2 C. (1) For s = f, h, e ⊂ sl 4 C a principal sl 2 , we have C = √ −1Id . The objects of B s (SL 4 C) consists of tuples where L is a holomorphic line bundle on X such that L 4 = O and ψ j ∈ H 0 (K j+1 ) for j = 1, 2, 3.
We now describe the Slodowy functor for each of these examples. To make the descriptions more concrete we will choose a smooth isomorphism between filtered objects and their associated graded. The holomorphic bundles with connections will then be expressed as Dolbeault operators and connections on the smooth bundle.
Recall Examples 4.4 and Example 2.12. Let Θ the SL 2 C-oper from (4.1). The ordering of the below cases is the same as the previous examples.
where (∂ V , ∇) are Dolbeault operators and holomorphic connections on the smooth bundle LK where ∇ L is the holomorphic connection which induces the trivial connection on L 4 = O.
where (∂ V , ∇) are Dolbeault operators and holomorphic connections on the smooth bundle W ⊗ K 1 2 ⊕ W ⊗ K − 1 2 given bȳ where (∂ V , ∇) are Dolbeault operators and holomorphic connections on the smooth bundle LK ⊕ L ⊕ L −3 ⊕ LK −1 given bȳ 5.4. The (s, λ)-Slodowy functor. We now describe appropriate generalizations of Theorems 5.5, 5.8 and 5.11 to the category QOp(G, P) of (λ, G, P)-opers. Since the proofs and statements of the results are almost identical to the λ = 1 case, we mostly omit the details. First note that the definition of the s-Slodowy category B s (G) generalizes immediately to a (λ, s)-Slodowy category QB s (G) whose objects consist of tuples (λ, E C , ψ 0 , ψ m 1 , · · · , ψ m N ), where (λ, E C , ψ 0 ) is an object in QF(C) and ψ m j ∈ H 0 (E C (Z 2m j ) ⊗ K m j +1 ). Similarly, we define the traceless quadratic part Q B s (G) to be the locus where ψ m 1 =ψ m 1 .
For the group SL 4 C, explicit models of the (λ, s)-Slodowy category are given by replacing the C-connection in §5.3 with a (λ, C)-connection. Explicit descriptions of the λ-Slodowy functor are defined by replacing the SL 2 C-oper from (4.1) with the (λ, SL 2 C)-oper τ 1 (λ) or τ 2 (λ) defined above. In particular, the associated Higgs bundles are obtained by setting λ = 0.
Finally, analogous to Theorem 5.8, the above equivalence can be upgraded to remove choice of section τ. There are natural C * -actions on QB s (G) and QOp(G, P) defined by (5.10) ξ · (λ, E G , E P , ω) = (ξλ, E G , E P , ξω) Denote the fiber product of the categories QOp(S, B S ) and Q B s (G) with respect to the natural projections to C by The diagonal C * -action on QOp(S, B S )×Q B s (G) induces a natural C * -action on the fiber product. The proof of the following theorem is almost identical to the proof of Theorem 5.8.
Theorem 5.13. Let G be a connected complex semisimple Lie group, s = f, h, e ⊂ g be an even sl 2 -triple, S < G the connected subgroup with Lie algebra s and P < G be the associated even JM-parabolic. Let F Θ λ be the Θ λ -Slodowy functor associated to an (λ, S, B S )-oper Θ λ . Then, the functor is an equivalence of categories when S ∼ = PSL 2 C and essentially surjective and full when S ∼ = SL 2 C. Moreover, QF is equivariant with respect to the natural C * -actions from (5.10).

More explicit examples and related objects
So far all of our explicit examples have been for G = SL 4 C or the parabolic being the Borel subgroup. In this section we discuss a few more explicit examples which generalize the SL 4 C-examples discussed throughout the paper. We also discuss the relation between some of the examples below and so called higher Teichmüller spaces.
The parabolics of the groups SO n C and Sp 2m C will be described in terms of the subspaces they stabilize in C n (n = 2m for Sp 2m C) equipped with a nondegenerate bilinear form which is symmetric for SO n C and skewsymmetric for Sp 2m C. Also a holomorphic vector bundle V will sometimes be equipped with a holomorphic symplectic structure Ω V or a holomorphic orthogonal structure Q V . By this we mean, 6.1. B-opers for classical groups. An (SL n C, B, λ)-oper is a tuple where λ ∈ C, V is a rank n holomorphic vector bundle with det V ∼ = O, V i is a rank i holomorphic subbundle and ∇ V is a λ-connection on V such that For SO 2n+1 C (resp. Sp 2n C), a (B, λ)-oper is an (SL 2n+1 C, B, λ)-oper (resp. (SL 2n C, B, λ)-oper), where V is equipped with a holomorphic orthogonal (resp. symplectic) structure which is preserved by ∇ V and with respect to which V i is isotropic and V n−i = V ⊥ i for 1 ≤ i ≤ n. For G = SO 2n C, (B, λ)-opers are described in §6.3 when 2n = 2k + 2.
For the section τ 2 of QOp PSL 2 C (X) from (4.2), the Slodowy functor F τ 2 : QB s (SL n C) → QOp(SL n C, B) is given by where, ∇ L is the holomorphic λ-connection inducing the trivial connection on L n = O, and (∂ L ⊗∂ V , ∇ L ⊗ ∇ V ) are Dolbeault operators and λ-connections on the smooth bundle K n−1 2 ⊕ K n−3 2 ⊕ · · · ⊕ K 1−n 2 given by Here µ i = i(n−i) and the coefficient on each ψ i are 1 for i > 1. The SO 2n+1 C (resp. Sp 2n C) Slodowy functor is given by the restriction of the SL 2n+1 C (resp. SL 2n C) Slodowy functor. Note that V has natural orthogonal (resp. symplectic) structure when rk(V) = 2n + 1 (resp. rk(V) = 2n) which is preserved by ∇ V when ψ 2j = 0 for all j.

6.2.
Compact dual of Tube-type Hermitian symmetric space. A family of even JM-parabolics P < G arise when G/P is the compact dual of a non-compact Hermitian symmetric space of tube type. There are five such pairs (G, P): (1) G = SL 2n C and P the stabilizer of a C n ⊂ C 2n ; (2) G = Sp 2n C and P the stabilizer of an isotropic subspace C n ⊂ C 2n ; (3) G = SO 4n C and P the stabilizer of an isotropic subspace C 2n ⊂ C 4n ; (4) G = SO n C and P the stabilizer of an isotropic line C ⊂ C n ; (5) G = E 7 and, if 7 i=1 n i α i is an expression of the longest root with respect to a choice of simple roots, P is the maximal parabolic subgroup associated to the unique simple root α k with n k = 1.
Each such parabolic is a maximal parabolic which corresponds to a simple root α k with n k = 1, where the longest root is given by Thus, the Z-gradings (2.2) are g = g −1 ⊕g 0 ⊕g 1 and the sl 2 -module decompositions (2.3) are g = W 0 ⊕ W 2 . The centralizers C and multiplicities (n 0 , n 2 ) are We now describe the vector bundle definition of an (G, P, λ)-oper in cases (1)-(3), case (4) will be described in the next subsection. An (SL 2n C, P, λ)oper consists of a tuple (λ, V, V n , ∇ V ), where λ ∈ C, V is a rank 2n holomorphic vector bundle with Λ 2n V ∼ = O, ∇ V is a λ-connection on V and V n ⊂ V is a holomorphic rank n subbundle such that the induced map ∇ V : V n → V/V n ⊗ K is an isomorphism. For case (2), an (Sp 2n C, P, λ)-oper is an (SL 2n , C, P, λ)-oper (λ, V, V n , ∇ V ) such that V is equipped holomorphic symplectic structure Ω V which is preserved by ∇ V and with respect to which V n is isotropic. For case (3), an (SO 4n C, P, λ)-oper is an (SL 4n , C, P, λ)-oper (λ, V, V 2n , ∇ V ) such that V is equipped holomorphic orthogonal structure Q V which is preserved by ∇ V and with respect to which V 2n is isotropic.
For case (1), the objects QB s (SL 2n C) consist of tuples (λ, W, ∇ W , ψ 1 ), where λ ∈ C, W is a rank n holomorphic vector bundle with det(W) 2 ∼ = O, ∇ W is a holomorphic λ-connection on W and ψ 1 ∈ H 0 (End(W) ⊗ K 2 ). For the section τ 2 of QOp PSL 2 C (X) from (4.2), the Slodowy functor is where (∂ V , ∇ V ) are Dolbeault operators and λ-connections on the smooth bundle W ⊗ K 1 2 ⊕ W ⊗ K − 1 2 given by (6.2) For case (2), the objects of QB s (Sp 2n C) are objects (λ, W, ∇ W , ψ 1 ) of QB s (SL 2n C) where W is equipped with a holomorphic orthogonal structure Q W which is preserved by ∇ W and with respect to which ψ 1 is symmetric. The Slodowy functor F τ 2 is the restriction of functor on QB s (SL 2n C). The symplectic structure on W ⊗ K 1 2 ⊕ W ⊗ K − 1 2 is defined by For case (3), the objects of QB s (SO 4n C) are objects (λ, W, ∇ W , ψ 1 ) of QB s (SL 4n C) where W is equipped with a holomorphic symplectic structure Ω W which is preserved by ∇ W and with respect to which ψ 1 is symmetric. The Slodowy functor F τ 2 is the restriction of functor on QB s (SL 4n C). The orthogonal structure on W ⊗ K 1 2 ⊕ W ⊗ K − 1 2 is defined by 6.3. Some more opers for SL n C and SO n C. Another family of even JM-parabolics of SL n C for are given by the stabilizers of partial flags where 2k < n. When C n is equipped with an orthogonal structure and the subspaces C i are isotropic for 1 ≤ i ≤ k, the associated parabolic P < SO n C is also an even JM-parabolic. In particular, for the orthogonal group, the parabolic P is the Borel subgroup when n = 2k + 1 and n = 2k + 2, and P is the parabolic from case (4) of the previous section when k = 1. For the above parabolic P, an (SL n C, P, λ)-oper consists of a tuple where λ ∈ C, V is a rank n holomorphic vector bundle with Λ n V ∼ = O, ∇ V is a λ-connection on V and V i ⊂ V is a holomorphic rank i subbundle such that: When G = SO n C, an (SO n C, P, λ)-oper is an (SL n C, P, λ)-oper where the holomorphic bundle V is equipped with an orthogonal structure Q V which is preserved by ∇ V and with respect to which V i is isotropic and We will describe the s-Slodowy category and functor for G = SO n C, the G = SL n C case is left to the reader. The sl 2 -module decomposition of the even JM-parabolic is The centralizer C of the even sl 2 is isomorphic to O n−2k+1 C, and the multiplicities are n 0 = (n−2k−1)(n−2k−2) 2 , n 4j−2 = 1 for 4j − 2 = 2k, and When k is odd, the C-representation space Z 2k decomposes as a direct sum of the one dimensional trivial representation and the standard (n − 2k − 1)dimensional representation twisted by the determinant representation. The objects of the category QB s (SO n C) consists of tuples (λ, W, ∇ W , ψ 1 , · · · , ψ 2k−1 ), where λ ∈ C, W is a rank (n − 2k − 1) holomorphic vector bundle equipped with an orthogonal structure Q W , ∇ W is a λ-connection on W which preserves Q W , ψ 2j−1 ∈ H 0 (K 2j ) for 2j − 1 = k and The Slodowy functor F τ 2 : QB s (SO n C) → QOp(SO n C, P) is given by F τ 2 (λ, W, ∇ W , ψ 1 , · · · , ψ 2k−1 ) = (λ,∂ V , ∇ V ), where (∂ V , ∇ V ) are Dolbeault operators and λ-connections on the bundle V = W ⊕ W 2k+1 ⊗ det W given by Here, • (λ, W 2k+1 , ∇ W 2k+1 ) denotes the (SO 2k+1 C, B, λ)-oper (6.1) associated to the Slodowy functor evaluated on (ψ 1 , ψ 3 , ψ k−2 , q k , ψ k+2 , · · · , ψ 2k ) if k is odd (ψ 1 , ψ 3 , · · · , ψ 2k ) if k is even; •Ψ k : W 2k+1 ⊗ det W → W ⊗ K is a holomorphic bundle map which, in the smooth splitting of W 2k+1 from (6.1), is given bŷ Ψ k = 0 · · · 0ψ k : (K k ⊕ · · · ⊕ K −k ) ⊗ det W → W ⊗ K.
The orthogonal structure Q V on W ⊕ W 2k+1 ⊗ det W is given by

Remarks on Nonabelian Hodge and Higher Teichmüller spaces.
So far, we have avoided the discussion of moduli spaces and stability. However, for every λ, both the B λ s (G) and Op λ (G, P) ⊂ F λ (G) have natural stability conditions and corresponding coarse moduli spaces. One would expect the natural stability conditions on B λ s (G) and F λ (G) to be related in a way such that the Slodowy functor F Θ 0 induces a well defined map on moduli spaces. This is especially desirable when λ = 0 because one can associate a holomorphic connection to a point in the moduli space of Higgs bundles via the Nonabelian Hodge correspondence.
Denote the moduli spaces of polystable G-Higgs bundles and holomorphic connections on a Riemann surface X by M 0 (G) and M 1 (G) respectively. We refer the reader to [35,36] for the construction of this moduli space. Furthermore, the moduli space M 1 (G) is analytically isomorphic to the character variety X (G) of conjugacy classes of reductive representations of the fundamental group of X in G: X (G) = Hom(π 1 (X), G) G.
Note that X (G) only depends on the topological surface underlying the Riemann surface X.
The nonabelian Hodge correspondence (see [13,14,23,34]) defines a real analytic isomorphism between these spaces: For (0, PSL 2 C, B)-opers, the Higgs bundles are always stable and the image under the map T can be identified with the Teichmüller space of the underlying topological surface [23]. More precisely, the image of (0, PSL 2 C, B)opers under the nonabelian Hodge correspondence T consists of all conjugacy classes of holonomies of hyperbolic structures on the underlying topological surface.
For (0, G, B)-opers (i.e., the Hitchin section) it is not hard to show that the Higgs bundles are always stable and so define points in M 0 (G). Moreover, applying nonabelian Hodge to this locus defines a (union of) connected component of the character variety of representations into the split real form of G called the Hitchin component [24]. Moreover, the surface group representations in this component generalize many features of Teichmüller space. In particular, they are all discrete and faithful [16,30] and are holonomies of certain geometric structures on closed manifolds [21,26]. The Hitchin component was the first example of what is now referred to as a higher Teichmüller component.
The family of (0, G, P)-opers described in §6.2 are related to a family of higher Teichmüller spaces known as maximal representations into a real Hermitian Lie group of tube type. In these cases, when the Higgs field ψ 0 is identically zero and q = 0, the Slodowy functor F τ 1 (0) recovers the Cayley correspondence of [3, §5] (see also [7,8,19]). This correspondence relates Higgs bundles with so called maximal Toledo invariant for a real 2 Hermitian Lie group of tube type G R with K 2 -twisted Higgs bundles for another group. For the complex groups SL 2n C, Sp 2n C, SO 4n C, SO n C and E 7 , the Hermitian Lie group G R is SU n,n , Sp 2n R, SO * 4n , SO 2,n−2 and E −25 7 , respectively.
Under the nonabelian Hodge correspondence, the isomorphism classes such Higgs bundles which are polystable are in bijective correspondence with the set of so called maximal representations. Such representations have been studied by many authors, and correspond to unions of connected components of the character variety which consist entirely of discrete and faithful representations [9] and carries a properly discontinuous action of the mapping class group [38].
For the family of (0, SL n C, P)-opers from §6.3 are not related to related to unions of connected components of the character variety of a real form of SL n C. However, the (0, SO n C, P)-opers from §6.3 are related to connected components of character varieties. In this case, when the Higgs field ψ 0 is identically zero and q = 0, the Slodowy functor F τ 1 (0) recovers the generalized Cayley correspondence of [1]. Similar to the case of maximal representations, under the nonabelian Hodge correspondence, to the isomorphism classes such Higgs bundles which are polystable are identified with unions of connected components of the character variety for the real form SO(k−1, n−k+1). From the results of [1] and Guichard-Wienhard's work on Θ-positivity [22] one expects such components to define higher Teichmüller spaces.
Remark 6.1. For a general even JM-parabolic, when λ = 0 and ψ 0 = 0 the Slodowy functor will not be related to Higgs bundles for a real group. However, using Guichard-Wienhard's work on Θ-positivity [22], one expects one more such family for certain real forms of F 4 , E 6 , E 7 and E 8 . This is indeed the case and the relationship between the Slodowy functor and higher Teichmüller spaces will be described in [6].