Tamely ramified geometric Langlands correspondence in positive characteristic

We prove a version of the tamely ramified geometric Langlands correspondence in positive characteristic for $GL_n(k)$. Let $k$ be an algebraically closed field of characteristic $p>n$. Let $X$ be a smooth projective curve over $k$ with marked points, and fix a parabolic subgroup of $GL_n(k)$ at each marked point. We denote by $\text{Bun}_{n,P}$ the moduli stack of (quasi-)parabolic vector bundles on $X$, and by $\mathcal{L}oc_{n,P}$ the moduli stack of parabolic flat connections such that the residue is nilpotent with respect to the parabolic reduction at each marked point. We construct an equivalence between the bounded derived category $D^{b}(\text{Qcoh}({\mathcal{L}oc_{n,P}^{0}}))$ of quasi-coherent sheaves on an open substack $\mathcal{L}oc_{n,P}^{0}\subset\mathcal{L}oc_{n,P}$, and the bounded derived category $D^{b}(\mathcal{D}^{0}_{{\text{Bun}}_{n,P}}\text{-mod})$ of $\mathcal{D}^{0}_{{\text{Bun}}_{n,P}}$-modules, where $\mathcal{D}^0_{\text{Bun}_{n,P}}$ is a localization of $\mathcal{D}_{\text{Bun}_{n,P}}$ the sheaf of crystalline differential operators on $\text{Bun}_{n,P}$. Thus we extend the work of Bezrukavnikov-Braverman to the tamely ramified case. We also prove a correspondence between flat connections on $X$ with regular singularities and meromorphic Higgs bundles on the Frobenius twist $X^{(1)}$ of $X$ with first order poles .

1. Introduction 1.1.Geometric Langlands in positive characteristic.Let X be a smooth projective curve over C. Let G be a reductive group over C and let G be its Langlands dual group.The geometric Langlands correspondence (GLC), as proposed by Beilinson and Drinfeld in [7], is a conjectural equivalence between the (appropriately defined) category of D-modules on the moduli stack Bun G of G-bundles on X, and the (appropriately defined) category of quasi-coherent sheaves on the moduli stack Loc G of G-local systems on X.A precise statement of this conjecture can be found in [2].
In [8], a generic version of the GLC in positive characteristic is established for G = GL n (k).The D-modules are interpreted in terms of crystalline differential operators.Using the Azumaya property of crystalline differential operators and a twisted version of the Fourier-Mukai transform, the authors prove a generic version of the GLC over the open subset of the Hitchin base where the spectral curves are smooth.In the case of G = GL n (k), the results of [8] are generalized in various directions.In [22], the mirabolic version of this correspondence is established.In [27], the author proved the quantum version of this correspondence.In [17], the equivalence in [8]  1 Defined in the sense of [8], Section 3.13.See Section 3.3 and Section 5.1.
The equivalence in Theorem 1.1 satisfies the following Hecke eigenvalue property: There is an isomorphism of functors: Now let (E, ∇) be a k-point of Loc 0 n,PD .We denote by M E,∇ the image of (E, ∇) under Φ P .By Theorem 1.2, M E,∇ satisfies 1.3.Summary of the proof.We fix a k-point q ∈ X and a parabolic subgroup P of GL n (k).For the purpose of simplifying notations, our exposition will be restricted to the case of D = q and P D = P from now on.The only proof that will be different in the more general setting is the proof of Proposition 2.7 in the case of X = P 1 , m ≥ 4 and all P i are Borel subgroups.We discuss this case in Remark 2.11.
Our proof of Theorem 1.1 is based on the same strategy as used in [8], but some new ingredients come into play.Note that in [8], the geometric Langlands correspondence is established over the open subset of the Hitchin base where the spectral curves are smooth.Compared to the unramified case in [8], one of the main difficulties in the tamely ramified case is that unless P is a Borel subgroup of GL n (k), there are no smooth spectral curves.We resolve this situation by considering the normalization of the spectral curves.It is observed in [24] that under generic restrictions on the spectral curves, a fiber of the Hitchin map h P : Higgs n,P −→ B P is isomorphic to the Picard stack of the normalization of the corresponding spectral curve.In Section 2.3, we extend this observation to a family version.More precisely, we prove: Theorem 1.3.There exists a Zariski open dense subset B 0 P ⊂ B P and a flat family of smooth projective curves Σ −→ B 0 P such that Higgs n,P × BP B 0 P ∼ = Pic( Σ/B 0 P ).
For each b ∈ B 0 P (k), Σ b is the normalization of the spectral curve Σ b .In Section 4, we establish a correspondence between flat connections on X with regular singularity at q and Ω X (1) (q)-twisted Higgs bundles on the Frobenius twist X (1) of X, which can be thought of as a characteristic p version of the non-abelian Hodge correspondence in [25].Let a be an unordered n-tuple of elements in k.We denote by Higgs n,a (X (1) ) the moduli stack of Ω X (1) (q)-twisted Higgs bundles (E, φ) on X (1) such that the tuple of eigenvalues of the residue res q (φ) of the Higgs field at q is a.Let B (1) a be the image of Higgs n,a (X (1) ) under the Hitchin map h (1) .We fix a set-theoretic section σ of the Artin-Schreier map k −→ k that maps t to t p − t.We denote by Loc n,σ(a) the moduli stack of flat connections (E, ∇) with regular singularity at q such that the tuple of eigenvalues of res q (∇) is σ(a).The pcurvature of (E, ∇) (see Section 4.1) defines the Hitchin map h ′ for flat connections with regular singularity at q: h ′ : Loc n,σ(a) −→ B (1)  a .
We will define an open substack (see Section 4.2) Loc r n,σ(a) ⊂ Loc n,σ(a) and prove the following theorem: Theorem 1.4.
Note that for an arbitrary reductive group G, a similar construction is used in [10] to establish the characteristic p version of the non-abelian Hodge correspondence for flat connections without singularities.
One of the key steps in our proof of Theorem 1.4 is to show that the map h ′ : Loc r n,σ(a) −→ B (1)   a is surjective.Since we consider flat connections with singularity at q, we cannot apply the Azumaya property of differential operators on X directly.Instead, we construct a flat connection on X\q using the Azumaya property, construct a flat connection on the formal disk around q by explicitly solving a differential equation for the connection form, and glue them together using the Beauville-Laszlo theorem [5].Note that Loc n,(0) is the moduli stack of flat connections with regular singularity and nilpotent residue at q. Restricting the isomorphism in Theorem 1. 4(2) to (B 0 P ) (1) and combining Theorem 1.3, we deduce that Loc 0 n,P := Loc n,P × B (1) is a Pic( Σ (1) /(B 0 P ) (1) )-torsor.It is proved in [8] that for a smooth algebraic stack Z that is good in the sense of [7](i.e.Z satisfies dim T * Z = 2 dim Z), there is a natural sheaf of algebras D Z on T * Z (1) that satisfies π (1) * D Z ∼ = Fr * D Z , and the restriction of D Z to the maximal smooth open substack (T * Z 0 ) (1) ⊆ (T * Z) (1) is an Azumaya algebra of rank p 2 dim Z (see Section 3.3 for a review of this construcion).Here π (1) : T * Z (1) −→ Z (1) is the projection and Fr : Z −→ Z (1) is the relative Frobenius.The stack Bun n,P "almost" satisfies those two properties, and we can still construct a sheaf of algebras D Bunn,P that satisfies π n,P × B (1) ) is an Azumaya algebra.We associate with D 0 Bunn,P its stack of splittings , which is a G m -gerbe over the Picard stack Pic( Σ (1) /(B 0 P ) (1) ).In Section 5.2, we show that D 0 Bunn,P has a tensor structure, therefore Y D 0 Bun n,P has the structure of a commutative group stack, and there is a short exact sequence Bun n,P −→ Pic( Σ (1) /(B 0 P ) (1) ) −→ 0.
By taking dual, we get another short exact sequence: In Section 5.5, we prove that (Y ∨ D 0 Bun n,P ) 1 := π −1 (1) is isomorphic to Loc 0 n,P as Pic( Σ (1) /(B 0 P ) (1) )-torsors, therefore we can apply a twisted version of the Fourier-Mukai transform (reviewed in Section 5.4) to prove the equivalence in Theorem 1.1.For the proof of this isomorphism, we show that the tautological 1-form θ (1) on T * (X\q) (1) extends to a 1-form θ (1) on Σ (1) , and both Pic( Σ (1) /(B 0 P ) (1) )-torsors are isomorphic to the moduli stack of rank one flat connections on Σ with p-curvature θ (1) .1.4.Structure of the article.In Section 2, we first review some basic constructions related to the Hitchin fibration.Then we define the Zariski open dense subset B 0 P ⊂ B P and establish the correspondence between parabolic Higgs bundles and the Picard stack of the normalization of spectral curves over B 0 P .In Section 3, we first review some properties of crystalline differential operators in positive characteristic, including the Azumaya property and the Cartier descent.Then we describe the correspondence between modules over an Azumaya algebra and twisted sheaves associated to its G m -gerbe of splittings.Finally we review the definition of tensor structures on Azumaya algebras over group stacks.In Section 4, we first construct the Hitchin map for flat connections with regular singularities.Then we prove the non-abelian Hodge correspondence between Loc n,σ(a) and Higgs n,a (X (1) ).In Section 5, we first define the sheaf of algebras D Bun n,P and construct a tensor structure on D 0 Bunn,P .Then we review the Fourier-Mukai transforms on commutative group stacks and use this framework to prove the main theorem.Finally we discuss the Hecke eigenvalue property of this equivalence.1.5.Notations and definitions.Unless otherwise mentioned, k is an algebraically closed field of characteristic p > 0. We consider the general linear group GL n (k) and assume p > n.Let gl n (k) be the Lie algebra of GL n (k).We denote by N the nilpotent cone in gl n (k).Let P be a parabolic subgroup of GL n (k).The Lie algebra of P decomposes as Lie(P ) ∼ = l ⊕ n + P .We denote by O P the Richardson orbit corresponding to P , which is the unique nilpotent orbit in gl n (k) such that the intersection with n + P is open dense in n + P .Let X be a smooth projective algebraic curve over k.Let g X be the genus of X.We fix a k-point q ∈ X.For any k-scheme S, we denote by ι q : S −→ S × X the base change of q : Spec(k) −→ X.We denote by p X the projection from S × X to X, and by p S the projection to S. Definition 1.5.An S-family of (quasi-)parabolic vector bundles on X is a vector bundle E of rank n on S × X with a P -reduction along S × q.We denote the moduli stack of such objects by Bun n,P .To be more precise, Bun n,P classifies triples (E, E P , τ ), where E P is a P -bundle on S × q and τ is an isomorphism . Let Bun n be the moduli stack of rank n vector bundles on X.There is a canonical map from Bun n,P to Bun n , which is defined by forgetting the P -reduction.
Remark 1.6.Let B be the Borel subgroup of GL n (k) that consists of upper triangular matrices.There is a one-to-one correspondence between the set of parabolic subgroups of GL n (k) containing B and the set of ordered n-tuples of positive integers µ = (µ 1 , µ 2 , . . ., µ s ) such that s i=1 µ i = n.This correspondence can be described as follows.We consider the standard representation of GL n (k) acting on k n .Let e 1 , e 2 , . . ., e n be the standard basis of k n .For i = 1, 2, . . ., s, let V i = mi j=1 ke j where be the conjugate partition to µ.The Richardson orbit corresponding to P µ consists of M λ the nilpotent matrix with Jordan blocks of sizes Let E be a rank n vector bundle on S × X.A P µ -reduction of the structure group along S × q corresponds to a partial flag structure: , where E i is a vector bundle of rank m i on S.
Remark 1.7.In the work of Mehta-Seshadri [21], a parabolic vector bundle is defined as a quasi-parabolic vector bundle together with a set of real numbers (α 0 , α 1 , α 2 , . . ., α s ) satisfying called parabolic weights.The parabolic weights can be used to define a stability condition on such objects, which is necessary for the construction of a moduli space.Since we focus on studying the moduli stack of such objects, we do not introduce the parabolic weights in this paper.Definition 1.8.An S-family of (quasi-)parabolic Higgs bundles on X is a parabolic vector bundle (E, E P , τ ) together with a Higgs field φ ∈ Γ(End(E) ⊗ p * X (Ω X (q))), such that the residue of φ at q, which we denote by res q (φ) ∈ End(ι * q E), lies in Γ(S, E P × P n + P ).In other words, if the parabolic reduction gives the following partial flag structure: We denote the moduli stack of such objects by Higgs n,P .
We denote by Higgs n,q the moduli stack of Ω X (q)-twisted Higgs bundles (E, φ), ).There is a canonical map from Higgs n,P to Higgs n,q , which is defined by forgetting the P -reduction.Remark 1.9.Higgs n,P ∼ = T * Bun n,P .
Remark 1.10.A parabolic version of the Hitchin moduli stack is previously considered in the work of Yun [28].Definition 2.1.1 in [28] is different from our Definition 1.8 in two aspects: the marked point q on X is allowed to move in [28], and the Higgs field φ is only required to preserve the flag structure instead of being nilpotent with respect to the flag structure.Definition 1.11.An S-family of parabolic flat connections on X is a parabolic vector bundle (E, E P , τ ) together with a flat connection with regular singularity at q ∇ : E −→ E ⊗ p * X (Ω X (q)), (i.e.∇ is a O S -linear map of sheaves that satisfies the Leibniz rule), such that the residue res q ∇ of ∇ at q lies in Γ(S, E P × P n + P ).We denote the moduli stack of such objects by Loc n,P .
We denote by Loc n,q the moduli stack of flat connections of rank n on X with regular singularity at q.There is a canonical map from Loc n,P to Loc n,q , which is defined by forgetting the P -reduction.
1.6.Acknowledgements.I would like to thank my advisor Tom Nevins for many helpful discussions on this subject, and for his comments on this paper.I would like to thank Christopher Dodd, Michael Groechenig and Tamas Hausel for helpful conversations.I would like to thank Tsao-Hsien Chen and Siqing Zhang for useful comments on an earlier version of this paper.
2. Spectral data of parabolic Higgs bundles 2.1.Basic constructions.In this Subsection, we discuss the construction of the Hitchin map, spectral curves and spectral sheaves in [100] and [6] in the parabolic setting.By taking the coefficients of the characteristic polynomial of the Higgs field, we get the Hitchin map: where B = n i=1 Γ(X, Ω X (q) i )2 .If we require the residue of the Higgs field to be nilpotent, the image of this map lies in Let T * X(q) = Spec X (Sym OX T X (−q)), where T X (−q) is the sheaf of vector fields on X that vanish at q. Let π be the projection π : T * X(q) −→ X.We denote by y the tautological section of π * (Ω X (q)).For b = (b 1 , b 2 , . . ., b n ), b i ∈ Γ(X, Ω X (q) i ), we define the spectral curve Σ b to be the zero-subscheme of the section Let (E, φ) be a k-point of Higgs n,q such that h(E, φ) = b.We can think of φ as a morphism φ : T X (−q) −→ End(E).By Cayley-Hamilton, there is a coherent sheaf F on Σ b such that π * (F ) = E.We call F the spectral sheaf corresponding to (E, φ).Conversely, let G be a coherent sheaf on Σ b , there is a canonical section φ can ∈ Γ(X, End(π * (G)) ⊗ Ω X (q)) obtained by adjunction.It is proved in [6] that if Σ b is reduced, the Hitchin fiber h −1 (b) is isomorphic to the stack of torsion free sheaves on Σ b , and if Σ b is smooth, If we require the residue of φ to be nilpotent, then π −1 (q) is a single point q ′ that lies in the zero-section of T * X(q).Let V = spec(A) be an affine open neighborhood of q in X.Let x be an element of A that is mapped to a local parameter of X at q. Shrinking V if necessary, we assume dx x is a nowhere vanishing section of Ω V (q).Let U = π −1 (V ).The section dx x gives a trivialization of T * X(q)| V and π * (T * X(q))| U .Under this trivialization, the tautological section y is equal to x∂ x considered as an element in 2.2.The parabolic Hitchin base B P .Now let P be a parabolic subgroup of GL n (k), and we assume the Richardson orbit O P of P contains the nilpotent matrix with Jordan blocks of sizes Composing h with the forgetful map from Higgs n,P to Higgs n,q , we get In order to describe the image of h P , we define the following sets of formal power series.We denote by P η the set of formal power series of the form and by P 0 η the subset of elements in P η that satisfy a i (x, y) ∈ k[[x, y]] × .In particular, if η = (m), P 0 m is the set of formal power series of the form y m + a(x, y)x, where a(x, y This lemma follows from a direct computation, see Proposition 22 in [3].It follows from Lemma 2.2 that h P factors through the affine space To be more explicit, we have m = 1 λ1 2 λ2 • • • r λr , meaning that the first λ 1 terms are 1, the next λ 2 terms are 2,..., and the last λ r terms are r.
(1) Let f be a formal power series in where each f i is a formal power series in P 0 η l i i , (2) Let ĝ = y ηl + a 1 (x, y)xy η(l−1) + a 2 (x, y)x 2 y η(l−2) + • • • + a l (x, y)x l be a power series in P 0 η l .We write a i (0, 0) for the constant term of a i ∈ k [[x, y]].Assume the polynomial y l + a 1 (0, 0)y l−1 + a 2 (0, 0)y l−2 + • • • + a l (0, 0) has distinct roots.Then ĝ factorizes uniquely as ĝ = g 1 g 2 In order to show that f factorizes as required, it is enough to show that f factorizes as f = gh, where Comparing coefficients, we have where y stands for raising y to some positive integer power.Since a s is invertible b s−lt is also invertible by the last equation.For Part (2), since we assume y l + a 1 (0, 0)y l−1 + a 2 (0, 0)y l−2 + • • • + a l (0, 0) has distinct roots, by Hensel's lemma, they lift to distinct roots of the polynomial gives the desired factorization.Since In order to obtain a spectral description of parabolic Hitchin fibers, we define the following open subset of the Hitchin base B P .Definition 2.4.We define B 0 P to be the subset of B P such that b ∈ B 0 P is characterized by the following properties: for some s = t, then the constant terms of a s and a t are not equal to each other.
In particular, if P is a Borel subgroup of GL n (k), B 0 P is characterized by the spectral curve being smooth.Lemma 2.5.For every b ∈ B 0 P , there exists a k-point of Higgs n,P that is mapped to b under the Hitchin map h P .
Proof.Let Σ b −→ Σ b be the normalization of the spectral curve Σ b and let π : Σ b −→ X be the projection to X. Let D = Spec ÔX,q be the formal disk around q.By (2.2), Σ b × X D is the disjoint union of Σ i , where Let L be an invertible sheaf on Σ b , then π * (L) defines a k-point ( π * (L), φ) of Higgs n,q such that h( π * (L), φ) = b and res q φ ∈ O P the Richardson orbit of P .Therefore we can find a partial flag structure on π * (L) q such that res q φ is nilpotent with respect to this partial flag structure.
Remark 2.6.Let (E, E P , τ, φ) be a k-point of Higgs n,P that is mapped to b ∈ B 0 P .Condition (2.2) on Σ b enforces that res q (φ) lies in the Richardson orbit O P .Note that O P n + P consists of a single P -orbit.Since we are in type A, for any x ∈ O P n + P , the centralizer of x in GL n (k) lies in P .Therefore there is a unique partial flag structure on E q that is compatible with res q (φ).
Proposition 2.7.In the following two cases: (1) P is Zariski open dense in B P .Moreover, B P is the scheme-theoretic image of the Hitchin map h P , i.e. the smallest closed subscheme of B through which h P factors.
Proof.The first statement together with Lemma 2.5 implies the second statement.For the first statement, we only need to show that both (2.1) and (2.2) define a non-empty open subset in B P .
We start by showing that (2.1) defines a non-empty open subset in B P .We denote by B sm P the locus in B P where the spectral curves are smooth away from q ′ .Since B sm P ⊂ B P is open, it is enough to show that it is non-empty.Case 1. g X ≥ 2, except for the case when g X = 2, n = 2, P = GL 2 (k).We use the following version of Bertini's theorem in [12]: Theorem 2.8 (cf.[12], Corollary 1).Let V be a smooth algebraic variety over an algebraically closed field k.Let S be a finite-dimensional linear system on V .Assume that the rational map V P N corresponding to S induces (whenever defined) separably generated residue field extensions.Then a generic element of S defines a subscheme of V that is smooth away from the base locus of S.
Let π be the projection π : T * X(q) −→ X.We denote by y the tautological section of π * (Ω X (q)).Let S be the linear system of sections in π * (Ω X (q) n ) spanned by y n and π The section y n is not contained in the span of π * (b i )y n−i .The set of spectral curves Σ b with b ∈ B P corresponds to the open subset of S defined by the coefficient of y n being non-zero.Let N = dim(S)−1.We denote by f S : T * X(q) P N the map induced by S. In order to apply Theorem 2.8, we show that f S is unramified away from π −1 (q), which will imply that f S induces finite separable extensions on the residue fields when restricted to T * (X\q).By the exact sequence it is enough to show that for any k-point p ′ on T * X(q) such that π(p ′ ) = p = q, the map ν induces a surjection onto the fiber of Ω T * (X\q) at p ′ .Let V = Spec(A) be an affine open neighborhood of p in X.Let x be an element of A that is mapped to a local parameter of X at p. Shrinking V if necessary, we assume q / ∈ V and dx is a nowhere vanishing section of Ω 1 V .Let U = π −1 (V ).The section dx gives a trivialization of T * X(q)| V and π * (T * X(q))| U .Under this trivialization, the tautological section y is equal to ∂ x considered as an element in O U , and Ω U is a free O U -module generated by dx and dy.The fiber of Ω U at p ′ is a k-vector space of dimension two spanned by dx and dy.
Under our assumptions on g X , n and P , we have then d(s 2 /s 1 ) and d(s 3 y/s 1 ) span the fiber of Ω U at p ′ .Now we apply Theorem 2.8 to the restriction of the linear system S to T * (X\q).Since q ′ ∈ π −1 (q) is the only base point of S, a spectral curve Σ b is smooth away from q ′ for a generic b ∈ B P .
Case 2. g X = 2, n = 2, P = GL 2 (k).By the same arguments as in Case 1, the map f S : T * (X\q) −→ P N is unramified away from the union of π −1 (p) for all p ∈ X\q that satisfies O(2p) ∼ = Ω X .There are finite many points of X with this property, therefore the fact that a generic spectral curve is smooth away from q ′ follows from the following lemma: Lemma 2.9.Let p ∈ X\q.For a generic b ∈ B P , the spectral cover Σ b −→ X is étale around p.
Proof.This follows easily from the calculation Case 3. g X = 1.We consider the subspace

This condition defines a non-empty open subset of B
under our assumptions on g X , n and P .The fact that the second condition in (2.2) defines a non-empty open subset follows easily from the uniqueness part of Lemma 2.3.
Remark 2.10.The second statement in Proposition 2.7 was previously obtained in [3] using different methods.
Remark 2.11.Proposition 2.7 also holds for the case of X = P 1 with ramification at D = q 1 + q 2 + • • • + q m , m ≥ 4 and each parabolic subgroup P i is a Borel subgroup.We need to show that for a generic b Therefore it is enough to show that there exists b ∈ B PD (k) such that Σ b is smooth away from π −1 (q i ).If n ≥ 3, the same arguments as in Case 1 of the proof of Proposition 2.7 would work.If n = 2, we consider the subspace Γ(P 1 , Ω P 1 ) ⊕ Γ(P 1 , Ω ⊗2 P 1 (q 1 + q 2 + q 3 + q 4 )) ⊆ B PD .Since Γ(P 1 , Ω P 1 ) = 0 and Γ(P 1 , Ω ⊗2 P 1 (q 1 + q 2 + q 3 + q 4 )) ∼ = Γ(P 1 , O P 1 ) = k, the spectral curve Σ b is étale away from π −1 (q i ) for any b ∈ k × .Proof.We've already constructed a map Pic( Σ b ) −→ h −1 P (b) in the proof of Lemma 2.5, therefore it is enough to construct the inverse map.Let (E, φ) ∈ h −1 P (b), and we denote by F ∈ Coh(Σ b ) the corresponding spectral sheaf.Our goal is to show that there is a natural sheaf Note that since the action of y on F /xF as a matrix with Jordan blocks of type λ, the element v 1 := y λ1 /x acts on the spectral sheaf F sheafifies over Σ1 b defined by The first l 1 components are formal disks that correspond to Σ 1j , j = 1, 2, . . ., l 1 in the normalization curve.The spectral sheaf over those components must be line bundles, and each contributes a Jordan block of size λ 1 to the residue of the Higgs field at the marked point q.Let F 1 be the spectral sheaf over the last component of Σ1 b in (2.1).Since y acts on F 1 /xF 1 as a matrix with Jordan blocks of type Repeating the same procedure for t times, the spectral sheaf F over Σb that we start with decomposes as where each L ij is a line bundle over Σ ij .Since the normalization curve Σ b locally is the disjoint union of those Σ ij , we get the desired statement that the spectral sheaf F sheafifies over Σ b .
For the purpose of this paper, we need to develop a family version of Theorem 2.12.The first step is to construct a simultaneous normalization of the family of spectral curves above B 0 P .This can be done since the spectral curves above B 0 P are equisingular.To be more precise, let Σ ⊆ B 0 P × T * X(q) be the global spectral curve above B 0 P ; we will construct a new family of curves Σ −→ B 0 P with a proper birational morphism σ : Σ −→ Σ such that for each b ∈ B 0 P (k), the morphism The construction is as follows.Recall that q ′ is the closed point of T * X(q) above q ∈ X that lies in the zero section of T * X(q).We blow up B 0 P × T * X(q) along B 0 P × q ′ , and denote the strict transform of Σ by Σ 1 .Let V be an open neighborhood of q and We denote by q ′ 1 the point defined by y = u = 0.By assumption (2.1) and the second part of assumption (2.2) in the definition of where t is the largest integer so that λ t − 1 > 0. Let (y λi−1 + a i (yu, y)u) and g i = y λi−1 + a i (yu, y)u, so g factorizes as g = g 1 g 2 • • • g t .In each g i , there is a unique monomial of the form y m , and the degree of such monomial is in decreasing order.Compared to f 1 , the degree of such monomial in g 1 is lower by 1.This observation guarantees that the family of curves Σ can be resolved simultaneously by λ 1 steps of blow-ups.Now we blow up Spec(O U [u]/(x − yu)) along B 0 P × q ′ 1 , and denote the strict transform of Σ 1 by Σ 2 .Repeating this procedure, we get a series of families of curves above b is a projective curve of the same genus.The morphism Σ λ1 −→ Σ is proper and birational by properties of strict transforms.We set Σ ∼ = Σ λ1 .Remark 2.13.After our paper appeared on the arXiv, similar results as in Theorem 2.12 were also obtained in [26], see Theorem 1.1.In [26] the authors also considered the generic fiber of so-called weak parabolic fibrations, in which the residue of the Higgs field is not required to be nilpotent.We will prove Theorem 2.14 the family version of Theorem 2.12, which did not appear in [26].Now we are ready to state the following theorem, which is a family version of Theorem 2.12.We denote Higgs n,P × BP B 0 P by Higgs 0 n,P .Theorem 2.14.The correspondence between Higgs bundles and spectral sheaves induces an isomorphism of stacks over B 0 P : Proof.Let S be a k-scheme.Since both Higgs 0 n,P and B 0 P are locally of finite type over k, we can assume S is locally of finite type over k.Let (E, φ) be an S-point of Higgs 0 n,P such that h(E, φ) = b ∈ B 0 P (S).We denote by F the corresponding spectral sheaf on Σ b .The goal is to construct a sheaf The strategy is to construct by induction a series of sheaves . We assume that we already have F 0 , F 1 , . . ., F t−1 with the required property and aim to obtain F t .Note that above V an open neighborhood of q, while obtaining Σ k b , we add a new variable and impose u k−1 = u k y, starting from u 0 = x.Therefore in order to construct F t so that (p t ) * F t = F t−1 , all we need to do is to define an action of u t−1 /y on F t−1 .Note that for any s : Spec , therefore if such an action exists, it is unique.For the existence of such an action, we consider the coherent sheaf There exists an action of u t−1 /y on F t−1 if and only if G = 0.By Theorem 2.12, such an action exists when restricted to s, so s * G = 0 for all closed points s of S. Therefore G = 0.
We set F = F λ1 .Since Σ b is smooth, F is an invertible sheaf.Now let (E 1 , φ 1 ) and (E 2 , φ 2 ) be two S-points of Higgs 0 n,P , both mapped to b under the Hitchin map, and we denote the corresponding spectral sheaves by F 1 and F 2 .The construction of F implies that there is an isomorphism Therefore we have a morphism of stacks Higgs 0 n,P −→ Pic( Σ 0 /B 0 P ).The inverse of this morphism is constructed as follows.Let L be an invertible sheaf on Σ b .Since Σ b ⊆ S × T * X(q), there is a morphism By adjunction, we get a morphism By Remark 2.6, there is a unique parabolic reduction of π * L at q that is compatible with this Higgs field.

Azumaya property of differential operators in positive characteristic
3.1.Frobenius twist of a k-scheme.Let Y be a scheme over an algebraically closed field k of characteristic p. Recall that the absolute Frobenius F Y : Y −→ Y is the map that fixes the underlying topological space and takes f to f p on regular functions.The Frobenius twist Y (1) of Y is the k-scheme that fits into the following pull-back diagram: The relative Frobenius Fr : Y −→ Y (1) is the unique map that makes the following diagram commute.
/ / Y Spec k Since Fr induces a bijection on k-points, we will not distinguish between k-points on Y and k-points on Y (1) .Let F and G be two

Azumaya property of differential operators.
In this section we review the Azumaya property of crystalline differential operators in characteristic p, following [8].Let Y be a smooth variety over k.We denote by D Y the sheaf of crystalline differential operators on Y , i.e. the sheaf of algebras generated by O Y and T Y subject to the relations: Therefore Fr * D Y sheafifies on T * Y (1) , i.e. there exists a sheaf of algebras D Y on T * Y (1) that satisfies π The following theorem is proved in [9].Let A be an Azumaya algebra on Y .A splitting of A is defined to be a pair (E, ρ), where E is a locally free sheaf on Y and ρ : A ≃ − → End(E) is an isomorphism of O Y -algebras.Such a (E, ρ) induces an equivalence between the category QCoh(Y ) of quasi-coherent sheaves on Y and the category A -mod of A-modules, which maps F ∈ QCoh(Y ) to E ⊗ F .We define an equivalence from an Azumaya algebra A to another Azumaya algebra B to be a splitting of A op ⊗ B. Such a splitting induces an equivalence from the category of A-modules to the category of B-modules.Note that if there is a locally free sheaf E that gives a splitting of A op ⊗ B, then Hom OY (E, O Y ) gives a splitting of A ⊗ B op .
Let f : Z −→ Y be a morphism between smooth k-varieties.We denote by df (1)  the Frobenius twist of the map induced by the differential of f : (1) .
We review the Cartier descent for flat connections with zero p-curvature.Let F be a quasi-coherent sheaf on Y (1) .There is a canonical D Y -action on Fr * (F ) ∼ = O Y ⊗ O Y (1) F , which comes from the canonical action of D Y on O Y .Therefore we have a flat connection (Fr * F , ∇ can ).This construction induces a functor from the category of quasi-coherent sheaves on Y (1) to the category of D Y -modules on Y with zero p-curvature.Theorem 3.4 (Cartier descent, cf.[19] Theorem 5.1).Let Y be a smooth variety over k.Then the construction of (Fr * F , ∇ can ) induces an equivalence between the category of quasi-coherent sheaves on Y (1) and the category of D Y -modules on Y with zero p-curvature.

Differential operators on smooth stacks.
Let Y be a smooth irreducible algebraic stack over an algebraically closed field k.When k is the field of complex numbers C, for Y that is good in the sense that it satisfies dim T * Y = 2 dim Y , the sheaf of differential operators on Y is defined in [7] as a sheaf of algebras D Y on the smooth topology Y sm .We review this definition as follows.The objects of Y sm are k-schemes S together with a smooth morphism f S : S −→ Y , and the morphisms between (S, f S ) and (S ′ , f S ′ ) are pairs (φ, α) containing a smooth morphism φ : S −→ S ′ and α : where φ −1 is the sheaf-theoretic inverse image.We call D Y the sheaf of differential operators on Y .It is observed in [8] that the isomorphism (3.2) no longer holds when k is of characteristic p > 0. But meanwhile, Fr * D Y is a quasi-coherent sheaf on Y (1) , and the authors constructed a coherent sheaf of algebras D Y on T * Y (1) that satisfies π  (1) .We have the following proposition: Let A be an Azumaya algebra on Y .We associate with it a G m -gerbe Y A over Y , which is defined as follows.For f : S −→ B a map of schemes, Y A (S) classifies triples (y, E, σ) where y ∈ Y (S), E is a vector bundle on S, and σ : is an isomorphism of algebras over S. We call Y A the stack of splittings of A. We have the following lemma:  (1) , which is defined as follows.For any smooth morphism f S : S −→ Y , we associate with it a G m -gerbe (G Y ) S on (T * Y ) (1) (1) , which is defined to be the pull-back of the G m -gerbe of splittings of the Azumaya algebra D S along df S −→ T * S (1) .For any smooth morphism φ : S −→ S ′ over Y , we have an isomorphism ( φ (1) and the two Azumaya algebras (dφ (1) ) * D S and p * 2 D S ′ are equivalent by Proposition 3.2.It is shown in [27] that the category of D-modules on Y is equivalent to the category of twisted quasi-coherent sheaves associated to G Y , see Theorem 2.3 in [27].Now we assume Y satisfies dim T * Y = 2 dim Y , and denote by T * Y 0 the maximal smooth open substack of T * Y .Recall that in Section 3.3, we defined a coherent sheave of algebras D Y on T * Y (1) , such that its restriction to (T * Y 0 ) (1) is an Azumaya algebra of rank p 2 dim Y .The G m -gerbe of splittings of this Azumaya algebra is isomorphic to the restriction of G Y to (T * Y 0 ) (1) , see Proposition 2.7 in [27].Therefore the category of -modules is a localization of the category of (crystalline) D-modules on Y .

3.5.
Tensor structures on Azumaya algebras.Let G be a commutative group stack over B, and A an Azumaya algebra over G.We denote the multiplication on G by µ : G × G −→ G. Following [23], we define a tensor structure on A to be an equivalence of Azumaya algebras from µ * A to A ⊠ A, which is a bimodule M that induces a Morita equivalence, together with an isomorphism of bimodules that satisfies the pentagon condition [13](1.0.1).
A tensor structure on the Azumaya algebra A induces a group structure on the stack Y A of splittings of A as follows.Let S be a k-scheme.An S-point of Y A is a pair (a, E), where a ∈ G(S) and E is a splitting module for a * A. Let (a, E) and (b, F ) be two such pairs.The locally free sheaf E ⊠ F is a splitting module for a * A⊠b * A. Applying the equivalence between µ * A and A⊠A and then pulling-back along the diagonal map ∆ S : S −→ S × S, we get a splitting module for µ(a, b) * A. The construction of this group structure implies that the projection map Y A −→ G is a group homomorphism, therefore we have a short exact sequence: 4. A non-abelian Hodge correspondence between Loc n,q and Higgs n,q 4.1.Spectral data for flat connections with regular singularities.Let (E, ∇) be a flat connection of rank n on X with regular singularity at q.We associate with it the p-curvature ψ ∇ , which is a O X -linear map It is associated with the p-linear map ) for any ∂ ∈ T X (−q)(U ) and U ⊆ X open.We can think of ψ ∇ as a twisted Higgs field The coefficients of its characteristic polynomial define a point b of Let Fr * : n i=1 Γ(X (1) , Ω X (1) (q) i ) ֒→ n i=1 Γ(X, (Fr * Ω X (1) (q)) i ) be the pull-back map.It follows from a similar argument as in [20] Proposition 3.2 that b actually lies in the image of Fr * , and we also denote by b the corresponding point in B (1) Γ(X (1) , Ω X (1) (q) i ).We call this map h ′ : Loc n,q −→ B (1) the Hitchin map for flat connections with regular singularity at q.The corresponding spectral curve Σ ′ b lies in the total space of Fr * Ω X (1) (q), which is isomorphic to X × X (1) T * X(q) (1) .Since b ∈ B (1) , Σ ′ b fits into the following pull-back square: Fr / / X (1)   where Σ (1) b ⊂ T * X(q) (1) is the spectral curve above b ∈ B (1) as defined in Section 2.1.We denote by E ′ ∈ Coh(Σ ′ b ) the spectral sheaf corresponding to ψ ∇ , so E ′ satisfies π ′ * (E ′ ) ∼ = E. Let x be a local parameter of O X,q .Let (E, ∇) be a flat connection with regular singularity at q. Restricting ψ ∇ (x∂ x ) to q, we get res q (ψ ∇ ) ∈ End(E q ), which we call the residue of ψ ∇ at q. Lemma 4.1.res q (ψ ∇ ) = (res q ∇) p − res q ∇.Proof.This equation follows from the computation (x∂ x ) [p] = x∂ x .Remark 4.2.If we assume res q ∇ is nilpotent, since p > n, (res q ∇) p = 0.So res q (ψ ∇ ) = − res q ∇.In particular, they lie in the same nilpotent orbit.4.2.Statement of the theorem.Let a be an unordered n-tuple of elements in k.We denote by Higgs n,a (X (1) ) the moduli stack of Ω X (1) (q)-twisted Higgs bundles (E, φ) on X (1) such that the unordered n-tuple of eigenvalues of res q (φ) is a.Let B (1) a be the image of Higgs n,a (X (1) ) under the Hitchin map h (1) .Note that when a = (0, 0, . . ., 0), B (1) N .We fix a set-theoretic section σ of the Artin-Schreier map k −→ k that maps t to t p − t.Let Loc n,σ(a) be the substack of Loc n,q that classifies flat connections (E, ∇) such that the unordered n-tuple of eigenvalues of res q (∇) is σ(a).Note that by Lemma 4.1, h ′ (Loc n,σ(a) ) ⊆ B (1) Loc r n,σ(a) has a natural structure of a Pic(Σ (1) /B (1) Higgs n,a (X (1) ).
Before getting into the proof of Theorem 4.3, we state two corollaries.

Corollary 4.4. There exists an étale cover
a , such that We denote by Higgs N (X (1) ) the moduli stack of Ω X (1) (q)-twisted Higgs bundles (E, φ) on X (1) such that res q (φ) is nilpotent, and by Loc N the substack of Loc n,q that classifies (E, ∇) with nilpotent res q (∇).Then we have: Corollary 4.5.
Our definition of Loc r n,σ(a) and formulation of Theorem 4.3 is motivated by the work of Chen-Zhu [10] on the characteristic p version of the non-abelian Hodge correspondence for flat connections without singularities.The strategy of proof is similar to [10] besides the proof of the surjectivity result Proposition 4.6.The rest of this section is devoted to the proof of Theorem 4.3.We start by showing: a is surjective.
We need to show that for any b ∈ B a (k), there exists (E, ∇) ∈ Loc r n,σ(a) (k) that is mapped to b under the Hitchin map.The idea of constructing (E, ∇) is as follows: we construct a flat connection (E 0 , ∇ 0 ) on X\q and a flat connection ( Ê, ∇) on the formal disk around q, such that both flat connections have the correct p-curvature.Then we glue (E 0 , ∇ 0 ) and ( Ê, ∇) together using the Beauville-Laszlo theorem [5]  a (k).Let π ′ : Σ ′ −→ X and π (1) : Σ (1) −→ X (1) be the corresponding spectral covers as described in Section 4.1.We will construct (E, ∇) such that h ′ (E, ∇) = b and the spectral sheaf E ′ is invertible.
(5.2') the spectral curve of . Let e be its generator.A meromorphic flat connection with spectral curve Σ′ i is determined by the connection acting on e, which can be written as . By (5.1') and (5.2'), ∇ need to satisfy the following: We look for solutions of the form so (5.1") is automatically satisfied, and (5.2") is equivalent to We look for solutions of the form g , for which the existence of solutions follows from Hensel's lemma.
Step 3. Let D × = Spec(k((x))) be the punctured disk around q, and let Σ× = give splittings of the Azumaya algebra D X | Σ× .Since all invertible sheaves on Σ× are trivial, we have an isomorphism of connections We fix such an isomorphism.By the theorem of Beauville-Laszlo [5], E 0 and Ê can be glued together to get a rank n vector bundle E on X.Since the gluing data is compatible with the connections, ∇ 0 and ∇ are glued together to get a flat connection ∇ on E with regular singularity at q.This connection (E, ∇) satisfies all the properties we need.4.4.D X -modules on spectral covers of X.Let b ∈ B (1) , and let π ′ : Σ ′ −→ X and π (1) : Σ (1) −→ X (1) be the corresponding spectral covers as described in Section 4.1.We have the following pull-back square: There is a canonical , which comes from the canonical action of D X on O X .Similarly, for any quasi-coherent sheaf M on Σ (1) , the pull-back sheaf ρ * M ∼ = O X ⊗ O X (1) M has a canonical D X -action.We denote by ∇ can the corresponding map Definition 4.8.We define a D X -module on Σ ′ to be a quasi-coherent sheaf F on Σ ′ together with a k-linear map Let D X (−q) be the subsheaf of algebras of D X generated by O X and T X (−q).Similarly we define D X (−q)-modules on Σ ′ .The only difference is that now ∇ is a map We have the following lemma concerning this definition.
(1) The structure sheaf O Σ ′ is a D X -module on Σ ′ .For any quasi-coherent sheaf M on Σ (1) , the pull-back ρ * M is a D X -module on Σ ′ .( 2) Let (E, ∇) be a flat connection with regular singularity at q such that h In all of the cases above, we denote by ∇ can the corresponding map induced by the action of T X (−q).Let (F , ∇) be a D X (−q)-module on Σ ′ such that π ′ * (F ) is locally free, then π ′ * (F ) has the structure of a flat connection with regular singularity at q. 4.5.Proof of Theorem 4.3.Now we construct the map Φ that induces the isomorphism in Theorem 4.3.Let b ∈ B (1) a .Let (E, ∇ E ) ∈ Loc r n,σ(a) and (M, φ) ∈ Higgs n,a (X (1) ), both mapped to b under the Hitchin map.We denote the spectral sheaf of (E, ∇ E ) by E ′ ∈ Coh(Σ ′ ) and the spectral sheaf of (M, φ) by M ∈ Coh(Σ (1) ).Lemma 4.10.Let G ∈ Coh(Σ (1) ) and let L be an invertible sheaf on Σ ′ .The push-forward π ′ * (L ⊗ ρ * (G)) is a locally free sheaf of rank n on X if and only if π (1) * (G) is a locally free sheaf of rank n on X (1) .By Lemma 4.9 and Lemma 4.10, we get a flat connection (π ′ * (E ′ ⊗ρ * (M)), ∇ can ) on X with regular singularity at q. Lemma 4.11.
Let (E, ∇ E ) ∈ Loc r n,σ(a) such that h ′ (E, ∇ E ) = b, and denote the corresponding spectral sheaf by E ′ ∈ Coh(Σ ′ ).Let L be an invertible sheaf on Σ (1) .Then by Lemma 4.11, This construction defines an action of Pic(Σ (1) /B Proof.Let S be a k-scheme.Let b be an S-point of B a .We need to show that the action of Pic(Σ that is mapped to b under the Hitchin map.We denote the corresponding spectral sheaves by E ′ and F ′ By the discussion after Lemma 4.12, there exists a quasicoherent sheaf M on Σ which is an isomorphism since Ψ produces its inverse. We denote by Loc n,q is the moduli stack of flat connections on X with regular singularity at q, without constraints on the eigenvalues of the residue.Let Loc r n,q ⊂ Loc n,q be the substack characterized by the spectral sheaf being invertible.We have the following proposition: Proposition 4.14.The map h ′ : Loc r n,q −→ B (1) is smooth.Before getting into the proof of Proposition 4.14, we state a corollary that is going to be used in the proof of Theorem 4.3.
a is smooth.
Proof.The map Loc r n,q × B (1) B (1) a

−→ B
(1) a is smooth by base change, and the fiber product Loc r n,q × B (1) B a is the disjoint union of Loc r n,c , where c ranges from all unordered n-tuples of elements in k that maps to a under the Artin-Schreier map.
We denote by Loc n,q the stack that classifies triples (E, ∇, θ), where E is a vector bundle of rank n on X, ∇ : E −→ E ⊗ Ω X (q) is a flat connection with regular singularity at q and θ : E q ∼ = − → k n is a frame of E at q.The natural action of GL n on the frame θ gives Loc n,q the structure of a GL n -torsor over Loc n,q .Lemma 4.16.Loc n,q and Loc n,q are algebraic stacks locally of finite type over k.
Proof.The 1-morphism Loc n,q −→ Bun n is representable and locally of finite presentation.Since Bun n is an algebraic stack locally of finite type over k and Loc n,q is a GL n -torsor over Loc n,q , both Loc n,q and Loc n,q are algebraic stacks locally of finite type over k.Lemma 4.17.Loc r n,q and Loc r n,q are smooth.
Proof.In order to show that Loc r n,q is smooth, all we need to show is that for any small extension of finite-generated Artinian local k-algebras A ′ −→ A, an A-point of Loc r n,q can be lifted to an A ′ -point of Loc We denote by (E, ∇, θ) the k-point r n,q .The obstruction to the existence of such liftings lies in the second hypercohomology where ∇ End(E) is the canonical connection on End(E) induced by ∇.By Serre duality, ) is isomorphic to the Lie algebra of Aut(E, ∇, θ).Since (E, ∇) ∈ Loc r n,q , we have Aut(E, ∇) = k × by Proposition 4.13.But multiplication by scalars does not preserve the framing θ, therefore Aut(E, ∇, θ) is the trivial group.This implies H A by-product of the proof of Lemma 4.17 is the following computation of the dimension of Loc r n,q .Since H 0 (F Proof of Proposition 4.14.Let b ∈ B (1) (k).By Proposition 4.6 and 4.13, the fiber (Loc r n,q b )-torsor.We compute that dim B (1) = n(n + 1)(2g − 1)/2 + n(1 − g) b ) = dim Loc r n,q − dim B (1) .Since both Loc r n,q and B (1) are smooth, the map h ′ is flat by miracle flatness.Furthermore, Since Pic(Σ Proof of Theorem 4.3.The first part follows from Proposition 4.13 and Corollary 4.15.For the second part, it is easy to see that the morphism Φ defined above induces a morphism Φ : Loc r n,σ(a) × Pic(Σ (1) /B (1) a ) Higgs n,a (X (1) ) −→ Loc n,σ(a) , and Ψ induces the inverse.Now we discuss how the residues of Higgs bundles and flat connections match under Φ.Proposition 4.18.Let (E, ∇ E ) ∈ Loc r n,σ(a) and (M, φ M ) ∈ Higgs n,a (X (1) ) such that h a (k).Denote the image of (E, ∇ E ) and (M, φ M ) under Φ by (F, ∇ F ). Then res q (ψ ∇F ) and res q (φ M ) lie in the same adjoint orbit.
Proof.Let E ′ ∈ Coh(Σ ′ ) be the spectral sheaf of (E, ∇ E ), and let M ∈ Coh(Σ (1) ) be the spectral sheaf of (M, φ M ).Let x be a local parameter of X at q.Note that res q (φ M ) is the action of x p ∂ p x on the fiber π (1) * (M)| q , and res q (ψ ∇F ) is the action of x p ∂ p x on the fiber π In particular, if σ(a) = (0), Proposition 4.18 together with Remark 4.2 implies that res q (∇ F ) and res q (φ M ) lie in the same nilpotent orbit.Therefore we have the following Since the stack Bun n,P does not satisfy the property required in Proposition 3.5, we cannot apply this proposition directly.In order to solve this problem, we introduce a new stack Bun n,P similar to the stack Bun n introduced in [8].The stack Bun n,P classifies the same objects as Bun n,P , but the morphisms are different.Let S be a k-scheme, and let (E, E • q ) and (F, F • q ) be two rank n vector bundles on S × X with partial flag structures of type P (see Remark 1.6) along S × q, then the set of morphisms between (E, E • q ) and (F, F • q ) are defined to be the set of isomorphic classes of pairs (ι, L), where L is a line bundle on S and ι is an isomorphism ι . By taking L = O S , we get a natural map Bun n,P −→ Bun n,P , and Bun n,P is a G m -gerbe over Bun n,P .
Proof.We apply the same strategy as in [16].The main goal is to show that the nilpotent cone N ilp := h −1 P (0) ⊂ T * Bun n,P is isotropic.Then the argument used in the proof of Proposition 7, 8 in [16] applies here to deduce the desired equality.Let B be a Borel subgroup of GL n (k) that is contained in P .We denote by Bun B the moduli stack of B-bundles on X.By Lemma 23 in [18], the natural map f : Bun B −→ Bun n,P is surjective.In order to apply Lemma 5 in [16] to show that N ilp is isotropic, all we need to show is that for any (E, E e. there exists a complete flag structure of E over X such that its restriction to q is compatible with the partial flag structure E • q , and the Higgs field φ is nilpotent with respect to this complete flag structure.We choose a basis (e 1 , e 2 , . . ., e n ) of E q such that the complete flag structure 0 ⊂ e 1 ⊂ e 1 , e 2 ⊂ • • • e 1 , e 2 , . . ., e n = E q is compatible with E • q .Let U = Spec A be an open neighborhood of q over which E and Ω 1 X (q) trivializes.Fixing such trivializations, the Higgs field φ corresponds to an A-linear map A n −→ A n .Since res q (φ) is nilpotent with respect to E • q , φ q (e i ) lies in the k-vector space spanned by e 1 , e 2 , . . ., e i−1 .Shrinking U if necessary, the basis (e 1 , e 2 , . . ., e n ) of E q can be lifted to a basis (ẽ 1 , ẽ2 , . . ., ẽn ) of E over U that still satisfies φ(ẽ i ) ∈ ẽ1 , ẽ2 , . . ., ẽi−1 .The B-reduction of E over U given by 0 ⊂ ẽ1 ⊂ ẽ1 , ẽ2 ⊂ • • • ẽ1 , ẽ2 , . . ., ẽn = E| U can be extended to a B-reduction over X since GL n (k)/B is projective.Such a B-reduction satisfies all the properties we need.Remark 5.2.Over C the field of complex numbers, the analogue of Proposition 5.1 was proved in [4] (see Theorem 6, 7) for a general reductive group G and parahoric P .It is not clear to the author if their arguments can be adapted to the characteristic p setting.Now we apply Proposition 3.5 to Bun n,P and get D Bun n,P .The sheaf of algebras D Bun n,P is defined to be the pull-back of D Bun n,P to Bun n,P .map.Therefore having a triple ((F, F • q , φ F ), x, E ⊂ F ) as above is equivalent to having (b, L, x ′ ), where b ∈ B 0 P , L ∈ Pic( Σ b ) and x ′ ∈ Σ 0 b .
Let S be a k-scheme.Let b be an S-point of (B 0 P ) (1) .Consider the following commutative diagram: σ (1)   !
But since τ * E ′ is an invertible sheaf on Σ b , the residue must be zero, so the flat connection ∇ can has no singularity at q i .By the definition of G ∨ , there is a universal G m -torsor on G × G ∨ , which gives rise to the Poincaré line bundle P G .
In [8], a commutative group stack G is called very nice, if locally in smooth topology, G is a finite product of stacks in the examples above.Under this assumption, the natural map G −→ G ∨∨ is an isomorphism.Therefore there is another Poincaré line bundle P G ∨ on G ∨ × G. ) 1 := π −1 (1) is isomorphic to Loc 0 n,P as P ic( Σ (1) /(B 0 P ) (1) )torsors.
is extended to the Hitchin base of reduced and irreducible spectral Shiyu Shen has received funding from NSF grant DMS-1502125 and the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No. 101034413. of D Bunn,P D and an open substack Loc 0 n,PD of Loc n,PD (see Section 5.1 for precise definitions).We will construct an O Loc 0 n,P D ⊠ D 0 Bunn,P D module P (see Section 5.5) and consider the Fourier-Mukai functor with kernel P Φ P : D b (QCoh(Loc 0 n,PD )) −→ D b (D 0 Bunn,P D -mod) from the bounded derived category of quasi-coherent sheaves on Loc 0 n,PD to the bounded derived category of D 0 Bunn,P D -modules.The main theorem of the paper is the following: Theorem 1.1.Φ P is an equivalence of derived categories.There are natural functors from both sides of the equivalence: the Hecke functor H 0 PD (see Section 5.6) H 0 PD : D b (D 0 Bunn,P D -mod) −→ D b (D 0 Bunn,P D ⊠ D X\D -mod) and the functor W 0 PD W 0 PD : D b (O Loc 0 n,P D -mod) −→ D b (O Loc 0 n,P D ⊠ D X\D -mod) defined by tensoring with the universal flat connection.Let Φ P,X\D be the Fourier-Mukai equivalence induced by the pull-back of P: Φ P,X\D : D b (O Loc 0 n,P D ⊠ D X\D -mod) ≃ − → D b (D 0 Bunn,P D ⊠ D X\D -mod).
Solving this system of equations is equivalent to solving a single equation with variable b s−lt .Indeed, we can solve c lt , c lt−1 , . . ., c 1 in turn as functions of b s−lt from the last l t equations; then we can solve b 1 , b 2 . . ., b s−lt−1 in turn as functions of b s−lt from the first s − l t − 1 equations; then we get the desired equation with variable b s−lt by substituting the other variables as functions of b s−lt in the (s − l t )-th equation.This equation has a solution by Hensel's lemma.Indeed, after reduction to k, this equation has a unique solution b s−lt (0, 0) = a s−lt (0, 0).
and all components in the factorization of f in Lemma 2.3 Part (1) satisfy the assumption in Lemma 2.3 Part (2).It follows that f factorizes as

2. 3 .
Spectral data of parabolic Higgs bundles.The next theorem describes the spectral data of parabolic Higgs bundles.Theorem 2.12 (cf.[24], Theorem 5.16).For b ∈ B 0 P (k), the fiber of the Hitchin map h −1 P (b) is isomorphic to the Picard stack Pic( Σ b ).Here σ : Σ b −→ Σ b is the normalization of the spectral curve Σ b .
and U ⊆ Y open.Since we are in characteristic p, for any ∂ ∈ T Y (U ), ∂ p ∈ D Y acts as a derivation on O U , and we denote this derivation by ∂ [p] ∈ T Y (U ).There is a p-linear map T Y −→ D Y defined by ι(∂) = ∂ p − ∂ [p] .By the discussion above, ι induces an O Y -linear map Fr * T Y (1) ∼ = F * Y T Y −→ D Y .By adjunction, we have an O Y (1) -linear map ι : T Y (1) −→ Fr * D Y .

Theorem 3 . 1
(cf. [8] Theorem 3.3 and [9] Theorem 2.2.3).(1) The map ι induces an isomorphism of sheaves from O T * Y (1) to the center of D Y .(2) The sheaf of algebras D Y is an Azumaya algebra over T * Y (1) of rank p 2d , where d is the dimension of Y .

Proposition 3 . 2
(cf. [8] Proposition 3.7).The Azumaya algebras (df (1) ) * D Z and p * 2 D Y are canonically equivalent.Following [8], we define f !: D Y -mod −→ D Z -mod to be the composition of the pull-back functor D Y -mod −→ p * 2 D Y -mod, the equivalence in Proposition 3.2, and the push-forward functor df * D Z -mod −→ D Z -mod.Similarly, we define f * : D Z -mod −→ D Y -mod to be the composition of the pull-back functor D Z -mod −→ df * D z -mod, the equivalence in Proposition 3.2, and the push-forward functor p * 2 D Y -mod −→ D Y -mod.Let θ (1)
be an object of Y sm .We denote by I the left ideal D S T S/Y ⊂ D S generated by the relative tangent sheaf T S/Y .We define (D Y ) ♯ S := D S /I.It has a D S -action by left multiplication.Let N DS (I) be the normalizer of I in D S .We define (D Y ) S := N DS (I)/I.In other words, we set (D Y ) S = End DS ((D Y ) ♯ S ) op .For any smooth morphism φ : S −→ S ′ over Y , we have a canonical isomorphism (3.1)

( 1 )
* D Y ∼ = Fr * D Y .The construction of D Y is as follows.For any k-scheme S with a smooth morphism f S : S −→ Y , we need to define a coherent sheaf of algebras (D Y ) S on (T * Y ) (1) S := S (1) × Y (1) T * Y (1) .We consider the D S -module (D ♯ Y ) S , and denote by (D ♯ Y ) S the corresponding coherent sheaf on T * S (1) .Since we mod out the left ideal generated by T S/Y when defining (D ♯ Y ) S , the support of (D ♯ Y ) S lies in the closed substack (T * Y ) − → T * S (1) .We set (D Y ) S := End DS ((D ♯ Y ) S ) op .For any smooth morphism φ : S −→ S ′ over Y , isomorphism (3.1) induces an isomorphism ( φ (1) ) * (D Y ) S ′ ∼ = (D Y ) S , where φ is the map (T * Y ) S −→ (T * Y ) S ′ .Therefore (D Y ) S sheafifies to be a coherent sheaf of algebras D Y on T * Y

Proposition 3 . 5
(cf. [8] Lemma 3.14 and [27] Proposition 2.7).The coherent sheaf of algebras D Y satisfies π (1) * D Y ∼ = Fr * D Y .If the stack Y is good in the sense that dim T * Y = 2 dim Y , and we denote by T * Y 0 the maximal smooth open substack of T * Y , then the restriction of D Y to (T * Y 0 ) (1) is an Azumaya algebra of rank p 2 dim Y .3.4.D-modules, Azumaya algebras and G m -gerbes.Let k be an algebraically closed field.Let B be a k-scheme locally of finite type.Let Y be a stack locally of finite type over B. Let Y −→ Y be a G m -gerbe over Y .We denote by QCoh( Y ) the category of quasi-coherent sheaves on Y .We say Y splits if there is an isomorphism Y ∼ = Y × BG m of G m -gerbes.In this case, there is a decomposition QCoh( Y ) ∼ = n∈Z QCoh( Y ) n given by the weight of the G m -action.If Y does not split, we still have such a decomposition by pulling back along the action map a : BG m × Y −→ Y .We call QCoh( Y ) 1 the category of twisted quasi-coherent sheaves associated to Y .

Lemma 3 . 6
(cf. [8] Lemma 2.3 and [14] Example 2.6).There is a canonical equivalence between the category A -mod of A-modules on Y and QCoh( Y A ) 1 .Now let Y be a smooth irreducible algebraic stack over k.A (crystalline) Dmodule M on Y is the datum of a D-module M S on S for each object (S, f S ) in Y sm , and an isomorphism φ !M S ′ ≃ − → M S of D-modules for each morphism (φ, α), φ : S −→ S ′ in Y sm .Here φ denotes the O-module pull-back with the natural D-module structure.Those isomorphisms need to satisfy the cocycle condition for compositions.When k is of characteristic p > 0, D-modules on Y correspond to twisted quasi-coherent sheaves associated to a certain G m -gerbe G Y on T * Y Loc r n,σ(a) the substack of Loc n,σ(a) that classifies flat connections (E, ∇) such that the corresponding spectral sheaf E ′ ∈ Coh(Σ ′ b ) is invertible.We have the following theorem: Theorem 4.3. .

Corollary 4 . 15 .
The map h ′ : Loc r n,σ(a) −→ B r n,q , i.e. we want to produce the dashed arrow for the following commutative diagram:

Corollary 4 . 19 . 5 . 5 . 1 .
The scheme-theoretic image of Loc n,P under the Hitchin map h ′ is B (1) P .Tamely ramified geometric Langlands correspondence in positive characteristic The algebra D Bun n,P .In this subsection we clarify what we mean by D Bun n,P .