Higgs bundles, abelian gerbes and cameral data
aa r X i v : . [ m a t h . AG ] F e b HIGGS BUNDLES, ABELIAN GERBESAND CAMERAL DATA
OSCAR GARC´IA-PRADA AND ANA PE ´ON-NIETO
Abstract.
We study the Hitchin map for G R -Higgs bundles on a smooth curve, where G R is a quasi-split real form of a complex reductive algebraic group G . By looking at themoduli stack of regular G R -Higgs bundles, we prove it induces a banded gerbe structureon a slightly larger stack, whose band is given by sheaves of tori. This characterizationyields a cocyclic description of the fibres of the corresponding Hitchin map by means ofcameral data. According to this, fibres of the Hitchin map are categories of principaltorus bundles on the cameral cover. The corresponding points inside the stack of G -Higgs bundles are contained in the substack of points fixed by an involution induced bythe Cartan involution of G R . We determine this substack of fixed points and prove thatstable points are in correspondence with stable G R -Higgs bundles. Contents
1. Introduction 2Acknowledments 52. Higgs bundles and the Hitchin map 53. The complex group case 74. The local situation: untwisted G R -Higgs bundles 114.1. Quasi-split real groups and the local Hitchin map 114.2. The case of real forms 144.3. The associated gerbe of N G ( G R )-Higgs bundles 154.4. Involutions on the local stack 184.5. An alternative description of the band: the semisimple locus 194.6. An alternative description of the band: back to arbitrary Higgs fields 205. The global situation: twisted Higgs bundles 245.1. Cameral data 255.2. Cameral data for fixed points under involutions 26Appendix A. Lie theory 29Appendix B. The geometry of regular centralisers 30References 33 Date : 16/02/2019.The work of the first author was partially supported by the Spanish MINECO under ICMAT SeveroOchoa project No. SEV-2015-0554, and under grant No. MTM2013-43963-P. The second author wassupported by the FPU grant from Ministerio de Educaci´on. Introduction
Higgs bundles over a compact Riemann surface for complex reductive Lie groups wereintroduced by Hitchin in [H1], and have since been object of intensive study, due to the richgeometry of their moduli spaces. A particularly interesting aspect is the Hitchin integrablesystem. This is a fibration by complex Lagrangian tori over the so called Hitchin base[H2, Si, Sco, S, DG, N]. This process, called abelianization, has proven useful in the studyof the geometry of moduli spaces [H2]. It constituted moreover one of the hints that ledto the link with mirror symmetry [HaT, DP].Higgs bundles for real Lie groups arise naturally via the non abelian Hodge correspon-dence, which establishes a homeomorphism of the moduli space of Higgs bundles withthe moduli space of representations of the fundamental group. Higgs bundles provide,as already ilustrated by Hitchin in [H1] a very powerful tool to study the geometry andtopology of the moduli spaces of representations. In particular, this approach has beenvery useful in identifying and studying higher Teichm¨uller spaces (see [G] for a survey).It is interesting to note that Higgs bundles for real Lie groups generalize the complexgroup case (as a complex group is a real group with extra structure), but they also ariseas fixed points of Higgs bundles for complex groups via involutions [GR]. We will considerboth aspects in different sections of this article.A Hitchin map can be defined for Higgs bundles for real groups. The goal of this paperis to study this map for quasi-split real groups. Simple quasi-split real groups include splitforms and groups whose Lie algebra is su ( n, n ), su ( n, n + 1), so ( n, n + 2), and e . It iswell known that any real form is isomorphic via inner equivalence to a quasi-split group.From Cartan’s point of view, the class of any outer automorphism of order two containsa quasi-split real form. It turns out that in this quasi-split situation the Hitchin fibrationcan be abelianized [P1].The Hitchin map fibres the non compact moduli spaces of Higgs bundles onto an affinespace, the Hitchin base. As a result, it allows to split the moduli space into non com-pact subspaces (sections isomorphic to the base [H3, GPR]) and compact dimensions (thefibres). Abelianization is certainly important from the gauge-theoretic point of view, asit transforms the objects of study into simpler ones. Here, abelianization should be un-derstood in a broader sense. Even for quasi-split groups, in general, the fibres will notbe abelian varieties (exceptions to this are split groups and SU( p, p )). This is due to thefact that they are contained inside the compactified Jacobians of non-smooth curves (theso called spectral and cameral covers) [Si, S, DG]. However, generic points of the fibresdo define abelian groups. In terms of the stack, the Hitchin fibration of the regular locusdefines an abelian gerbe; in particular, the fibres are categories of tori over cameral covers.In order to study them, we use the cameral techniques introduced by Donagi–Gaitsgory[DG], and follow Ngˆo’s formulation [N]. The spectral data for some quasi-split real formhas been formerly studied by different authors (see [FGN, P2, Sc1, Sc2]).Let G R be a real reductive Lie group. Following Knapp [Kn, § VII.2], by this we meana tuple ( G R , H R , θ, h · , · i ), where H R ⊂ G R is a maximal compact subgroup, θ : g → g isa Cartan involution and h · , · i is a non-degenerate bilinear form on g R , which is Ad( G R )-and θ -invariant, satisfying natural compatibility conditions. We will also need the notionof a real strongly reductive Lie group (see Definition A.1). Let θ be the Cartan involution, IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 3 with associated Cartan decomposition g R = h R ⊕ m R where h R is the Lie algebra of H R . The group H := H CR acts on m := m CR through theisotropy representation.Let X be a compact Riemann surface and L be a holomorphic line bundle over X . An L -twisted G R -Higgs bundle on X is a pair ( E, ϕ ), where E is a holomorphic principal H -bundle over X and ϕ is a holomorphic section of E ( m ) ⊗ L , where E ( m ) = E × H m isthe m -bundle associated to E via the isotropy representation. The section ϕ is called theHiggs field. When L is the canonical line bundle K of X we obtain the familiar theory of G R -Higgs bundles. When G R is compact the Higgs field is identically zero and a L -twisted G R -Higgs bundle is simply a principal G -bundle, where G := G CR . When G R is complex G R = H and the isotropy representation coincides with the adjoint representation of G R .This is the situation originally considered by Hitchin in [H1, H2], for L = K .Let M ( G R ) the moduli space of isomorphism classes of polystable L -twisted G R -Higgsbundles. By considering a basis of homogeneous polynomials p , . . . , p a ∈ C [ m ] H (where a is the rank of G R and deg( p i ) = d i ), one obtains the Hitchin map h : M ( G R ) → A L ( G R )defined by evaluating p , . . . , p a on the Higgs field (see Section 2 for a more intrinsicdefinition and Remark 2.7 for the relation between both). In the above A L ( G R ) ∼ = L ai =1 H ( X, L d i ) is the Hitchin base. This construction also yields a stacky version ofthe Hitchin map [ h ] L on the stack of G R -Higgs bundles Higgs ( G R ).In this paper we study the morphism [ h ] L after imposing a regularity condition on theHiggs field. Namely, we assume that φ ( x ) has a maximal dimensional isotropy orbit forevery x ∈ X . We find that the Hitchin map defines a gerbe structure once the automor-phisms have been extended (yielding a stack Higgs reg,θ ( G R )). This means that locally onthe Hitchin base, the Hitchin fibration can be identified with the classifying stack for somegroup. Moreover, the existence of a section [GPR] proves that this is globally the case.We describe the fibres in terms of cameral techniques for quasi-split real groups, that is,we identify them with categories of tori over cameral covers. We also study the action ofthe natural algebraic involution induced by θ on the stack Higgs reg ( G ), where G = G CR .We describe the fixed points, finding in Corollary 5.10 that for split groups these are ordertwo points (see also [Sc1]). Moreover, we prove that stable points are precisely points of Higgs reg ( G R ).In Section 2 we recall the preliminaries of Higgs bundles and the Hitchin map associatedto real reductive algebraic groups. We put the theory in the suitable context of stacks, andexplain how the usual notions for complex groups arise naturally in this way (cf. Remark2.6).Section 3 reminds abelianization as per Donagi and Gaitsgory [D, Sco, DG], whichidentify and open subset of the Hitchin system for complex groups with a gerbe over theHitchin base, and the fibres as categories of principal torus bundles over cameral covers(cf. Theorem 3.8). We also explain how their language compares to Ngˆo’s [N].Section 4 studies a local model for the Hitchin map for quasi-split real groups. In contrastwith the complex group case, in order to obtain a gerbe (first step towards a descriptionin terms of categories of principal bundles) we need to modify either the Hitchin base(by substituting it by the orbit space of the isotropy representation, see Section 4.2) or OSCAR GARC´IA-PRADA AND ANA PE ´ON-NIETO the automorphism group (by enlarging it so the Hitchin base becomes an orbit spaceunder the suitable extension of the isotropy action, see Section 4.3). The result of theformer is a gerbe over a non separated algebraic space (see Lemma 4.10). Regarding thelatter, we study the local stack [ m reg /N G ( G R )], where m reg ⊂ m are the regular points and N G ( G R ) is the normaliser of G R inside its complexification G (which exists by algebricityof G R ). This substack of the local stack of G -Higgs bundles [ g reg /G ] is a minimal gerbecontaining the local stack of G R -Higgs bundles [ m /H ] (see Remark 4.23). It admits anatural involution whose fixed point substack contains [ m /H ] (see Proposition 4.22). Oncethe gerby structure has been proven, we move on to characterise the band of the gerbe (thatis, the automorphism group of objects in the fibre, see Remark 3.5) in terms of schemesof tori, namely, we identify schemes of regular centralisers with torus schemes. Section 4.5studies the case of semisimple Higgs bundles, case in which the fibres are easily identifiedwith categories of tori over ´etale local cameral covers. See Corollary 4.26. The followingsection extends these results to ramified local cameral covers using the results of [DG, N],which yields the main result of this section (Theorem 4.31).Section 5 studies the structure of the Hitchin map based on the local Hitchin map.We deduce that the stack of G R and N G ( G R )-Higgs bundles is a gerbe over a suitablespace (Theorem 5.2), which is moreover neutral when the degree of the bundle is even.The alternative description of the band from Theorem 4.31 is used to obtain a cocyclicdescription of the fibres of the Hitchin map for N G ( G R )-Higgs bundles in terms of cameraldata, that is, principal torus bundles over cameral covers (Theorem 5.5). Finally, wecharacterise the cameral data of fixed points under the natural involution on the stackassociated with G R in Theorem 5.9 and prove that stable fixed points are precisely stable G R -Higgs bundles (cf. Proposition 5.11).Two appendices gather the most relevant elements on Lie theory (Appendix A) and thegeometry of regular centralisers (Appendix B). Some of the results therein are original andappeared in [GPR, P1].We finish this introduction with some remarks concerning higher dimensional schemes,analytic categories, a more abstract approach `a la Donagi-Gaitsgory, and the non abelian-izable case.The formalism developed in this article can be applied to higher dimensional schemes.However, as explained in [DG], for dimension higher than 2 there may not exist any regularHiggs fields, as the subset of a scheme S on which a Higgs field is not regular is expectedto have codimension 3.Moreover, we work with the ´etale topology. However, most of the results explained arevalid in the setting of the analytic site, as long as X is smooth compact and projective.Regarding the parallelism with Donagi and Gaitsgory’s work, the notion of abstract G R -Higgs bundles also makes sense in this setting. Such an object is given by a pair ( E, σ )where E is an H -principal bundle and σ : E → H/N H ( a ) is an H -equivariant section toa scheme H/N H ( a ) parametrising the centralisers of elements in m reg (see Appendix Bfor details). Here a is a maximal abelian subalgebra of m . One can work out the wholepicture in this setting, and obtain that abstract N G ( G R )-Higgs bundles are a gerbe overthe appropriate substack of cameral covers. Most details can be found in [P1]; it would beinteresting to give an intrinsic description of the precise substack of cameral covers, in thespirit of Remark 4.33. IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 5
In this article we restrict attention to quasi-split real groups. We know by [P1, Theorem4.3.13], that the Hitchin fibration for any other real form can not be abelianized. This isa consequence of regular centralisers not being abelian for any other real groups. See [P1, § Acknowledments
The authors thank L. ´Alvarez-C´onsul, F. Beck, R. Donagi, J. Heinloth, C. Pauly, C.Simpson and especially, T. Pantev, for comments and discussions on the topics of thispaper. 2.
Higgs bundles and the Hitchin map
Let G R be a real algebraic group. We will say it is strongly reductive if it satisfiesKnapp’s definition of reductivity [Kn, § VII.2]. See Definition A.1. Namely, the groupcomes endowed with extra data ( G R , H R , θ, h · , · i ) where H R ≤ G R is a maximal compactsubgroup, θ is the Cartan involution on the Lie algebra g R of G R , and h · , · i is a θ -invariantnon degenerate bilinear form on g R . These data satisfy conditions i ) − v ) on [Kn, page446]. Consider the corresponding Cartan decomposition of the Lie algebra(1) g R = h R ⊕ m R . Then, θ is identically +1 on h R and − m R . Restriction of the adjoint action ofAd g R : G R → Aut( g R ) induces an action of H R on m R whose complexification ι : H → Aut( m ) is called the isotropy representation . Here H := H CR , m := m CR denote thecomplexifications and likewise for any other real group or (vector subspace of a) Lie algebra.Note that g R = ( g R ⊗ C ) σ for σ the involution given by ( X, Y ) ( − Y, X ) on g R ⊗ C ∼ = R g ⊕ g . Given a σ -invariant complex vector subspace V ⊂ g , we will denote by V R := V ∩ g R = V σ . Remark . Particular examples of strongly reductive real Lie groups include: complexconnected reductive Lie groups G , or their real forms G R < G . The latter are fixedpoint subgroups under an antiholomorphic involution σ . One can prove the existence ofa compact subgroup U ≤ G with defining involution τ such that στ is a holomorphicinvolution on G whose differential restricts to θ on g R . By abuse of notation, we denote θ := στ . With this, H = G θ by [Kn, Proposition 7.21].Note also that complex reductive groups G appear as real forms of G × G with associatedholomorphic involution ( g, h ) ( h, g ).According to this, a complex reductive Lie group can be seen as a particular case of areal strongly reductive group or of a real form.Given a smooth complex projective curve X , we denote by X ´ et the small ´etale site of X , and ( Sch/X ) ´ et the big ´etale site. Recall [V] that a site is a category endowed with anotion of covering satisfying some axioms. In our case, the category underlying X ´ et has asobjects ´etale morphisms S → X , and as arrows, morphisms over X . As for ( Sch/X ) ´ et , theunderlying category are schemes over X , with arrows morphisms over X . In both cases, thecoverings of the Grothendieck topology are collections of ´etale morphismos { U i → S } i ∈ I for S ∈ X ´ et / ( Sch/X ) ´ et which are jointly surjective, that is, such that F i ∈ I U i → S issurjective. OSCAR GARC´IA-PRADA AND ANA PE ´ON-NIETO
Now, let L → X be an ´etale line bundle, that we will assume to be of degree at least2 g −
2, and if deg L = 2 g − L is assumed to be the canonical bundle K of X .Note that L is also locally trivial in the analytic topology, by Oka’s Lemma and [St,Theorem 1.9]. Definition 2.2. An L -twisted G R -Higgs bundle on X is a pair ( E, φ ) where E → X isan ´etale principal H -bundle and φ ∈ H ( X, E ( m ) ⊗ L ), where E ( m ) denotes the bundleassociated to E via the isotropy representation. The section φ is called the Higgs field .We can reformulate the above as follows: on the sites X ´ et or ( Sch/X ) ´ et , we can considerthe transformation stack Higgs ( G R ) := [ m ⊗ L/H ]. Recall that for a scheme Y endowedwith an action of an algebraic group F the quotient stack [ Y /F ] parametrises principal F -bundles P together with F -equivariant maps P → Y . Thus Higgs ( G R ) is the stackthat to each scheme f : S → X associates the category of f ∗ L -twisted G R -Higgs bundlesover S . We have then the equivalent Definition 2.3. An L -twisted G R -Higgs bundle on X is a section[( P, φ )] : X → Higgs ( G R ) . Remark . Strictly speaking, if we are interested in a moduli problem over X , we shouldinstead consider the pushforward s ∗ Higgs ( G R ) by the structural morphism s : X → Spec( C ); the sections of this stack over a complex scheme S are Higgs bundles over X × C S .Let a ⊂ m be a maximal abelian subalgebra, and W ( a ) the restricted Weyl group.This can be defined as W ( a ) = N H ( a ) /Z H ( a ) where N H ( a ) denotes the normalizer of a in H and Z H ( a ) its centraliser. The well known Chevalley restriction theorem proves that C [ m ] H ∼ = C [ a ] W ( a ) . In other words, the GIT quotient m (cid:12) H is isomorphic to the quotient a /W ( a ). Thus, there exists a morphism, called the Chevalley morphism(2) χ : m → a /W ( a ) . The map (2) is C × -equivariant for the C × action on m as a vector space and the one on a /W ( a ) induced by the action on the graded ring C [ a ] W ( a ) . Recall that the latter is a ringgenerated by homogeneous polynomials p i i = 1 , . . . , a := dim a of fixed degrees d i = deg p i which are invariants of the group. The action of C × on a /W ( a ) is totally determined by t · p i = t d i p i where the RHS is the usual multiplication. Hence, it induces a morphism(3) m ⊗ L → ( a ⊗ L ) /W ( a ) . Moreover, (3) is H -invariant, so we obtain[ h ] L : [ m ⊗ L/H ] → ( a ⊗ L ) /W ( a ) . Definition 2.5.
Let A ( G R ) = tot ( a ⊗ L/W ( a )) be the total space of the bundle a ⊗ L/W ( a ).The map(4) [ h ] L : [ m ⊗ L/H ] → a ⊗ L/W ( a )is called the L -twisted Hitchin map associated to G R . The scheme A ( G R ) is called the Hitchin base scheme (associated to G R ). Remark . From Remark 2.1 we may consider
Higgs ( G ) for a complex reductive Liegroup G . In these terms we recover the usual definition, as h = m = g . In that case wehave that a is a Cartan subalgebra of g , W ( a ) is the associated Weyl group, and hence[ h ] L is the usual Hitchin map. IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 7
Remark . Note that the choice of a basis of homogeneous generators p , . . . , p a , a =rk G R of C [ m ] H induces an isomorphism H ( X, a ⊗ L/W ( a )) ∼ = L ai =1 H ( X, L d i ), where d i = deg( p i ). Remark . In relation to Remark 2.4, the condition on the degree of L ensures repre-sentability of the Hitchin base scheme s ∗ a ⊗ L/W by H ( X, a ⊗ L/W ( a )). Namely, for anycomplex scheme S , S -points of s ∗ a ⊗ L/W are simply H ( X, a ⊗ L/W ( a )).Note that the Hitchin map (4) can be defined in more generality without fixing the linebundle L . Following Ngˆo [N, §
2] we can consider the stack [ m /H × C × ], which parametrizespairs ( E × P, ψ ) where E is a principal H bundle, P is a line bundle and ψ : E × P → m is an H × C × equivariant morphism. The latter is equivalent to having an H -equivariantmorphism ψ : E → m ⊗ P , so that by considering [ m /H × C × ] we are parametrising alltwists at once.The same arguments as before imply that the Chevalley morphism induces(5) [ ˜ χ ] : (cid:2) m /H × C × (cid:3) → (cid:2) a /W ( a ) / C × (cid:3) . Furthermore, by mapping each of the above stacks to B C × via the respective forgetfulmorphisms, we obtain a commutative diagram:[ m /H × C × ] / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ [ a /W ( a ) / C × ] v v ❧❧❧❧❧❧❧❧❧❧❧❧❧ B C × × a /W ( a )where all the stacks above are seen as sheaves over the (small) ´etale site of a /W ( a ).Fixing a line bundle on X is equivalent to considering a map [ L ] : X → B C × ; so onerecovers [ h ] by looking at the restriction of [ ˜ χ ] to [ L ] : X → B C × . This implies in particularthat in order to study the Hitchin map (4), we can restrict attention to the easier map(5). The advantage to this approach is that geometric questions reduce to Lie theoreticquestions in this context, which are much easier to handle. Moreover, local triviality ofbundles implies that at a local scale, the stack is essentially the quotient [ m /H ]. So wewill first study the local case before coming back to global Higgs bundles.3. The complex group case
Before we turn to the general case, we recall in this section case of complex groups,which was studied by Donagi and Gaitsgory [DG]. We also explain the reformulation byNgˆo [N], and how both relate.Let G be a complex reductive algebraic group, and let g be its Lie algebra. Let X , X ´ et ,( Sch/X ) ´ et and L be as in Section 2.As explained there, the classifying stack of L -twisted G -Higgs bundles over X is thequotient stack Higgs ( G ) = [ g ⊗ L/G ] over X ´ et / ( Sch/X ) ´ et .As explained in Remark 2.6, the Chevalley morphism (2) induces the Hitchin map(6) [ h ] L, C : Higgs ( G ) → t ⊗ L/W ( t )where t ⊂ g is a Cartan subalgebra with associated Weyl group W ( t ). OSCAR GARC´IA-PRADA AND ANA PE ´ON-NIETO
Let g reg ⊂ g be the subset of regular elements (i.e., elements with maximal dimensionaladjoint orbits). We can consider the open substack Higgs reg ( G ) ⊂ Higgs ( G ) , consisting of Higgs bundles whose Higgs field is everywhere regular, that is, its pointwisecentraliser is minimal dimensional (of dimension rk G ).A more general object, whose relation to Higgs reg ( G ) we explore in what follows, wasstudied by Donagi and Gaitsgory [DG]. They consider the stack H iggs ( G ) of abstractHiggs bundles . To explain this, consider G/N ⊂ Gr ( r, g ) the smooth scheme param-eterising regular centralisers, namely, its points are centralisers of regular elements of g ,cf. [DG, Proposition 1.3]. In the notation, N := N G ( T ) is the normaliser of the maximaltorus T corresponding to t . The scheme G/N is a partial compactification of the quo-tient
G/N (which parameterises Cartan subalgebras). Then H iggs ( G ) parameterises pairs( E, σ ) of a principal G -bundle E → X and a G -equivariant map σ : E → G/N . With this, H iggs ( G ) ∼ = h G/N /G i .Note that there is a morphism of stacks(7) C : Higgs reg ( G ) → H iggs ( G )sending a pair ( E, φ ) ( E, σ φ ), where σ φ : E → G/N associates to p c g ( φ ( p )). Here,we consider the Higgs field as a G -equivariant map φ : E → g ⊗ L . The map σ φ is welldefined, as for any λ ∈ C × c g ( λφ ( p )) = c g ( φ ( p )). Remark . In fact, the morphism C is well defined for Higgs bundles in Higgs ( G ) withgenerically regular Higgs field (cf. [DG, § cameral cover of X is a W -covering b X → X which is ´etale locally a pullbackof t → t /W . Note that to any L -twisted Higgs bundle ( E, φ ), it corresponds a section b ( E, φ ) ∈ H ( X, t ⊗ L/W ) obtained by composing b ( E, φ ) = [ h ] L, C ◦ [( E, φ )] : X → t ⊗ L/W. where (
E, φ ) is seen as a morphism[(
E, φ )] : X → Higgs reg ( G ) . In this way, we assign to (
E, φ ) the cameral cover defined by the Cartesian diagram(8) b X b / / (cid:15) (cid:15) t ⊗ L π (cid:15) (cid:15) X b ( E,φ ) / / t ⊗ L/W.
An abstract Higgs bundle [(
E, σ )] : X → H iggs ( G ) also induces a W-Galois ramifiedcover. To see this, one defines a ramified W -Galois cover G/T → G/N , where
G/T isthe incidence variety inside
G/N × G/B for a given Borel subgroup B containing T (see[DG, Proposition 1.5]). An abstract cameral cover is a W -Galois ramified cover locallyisomorphic to G/T → G/N . These are classified by the stack C ov of abstract cameralcovers. IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 9
We may define an abstract Hitchin map (9) [ h abs ] : H iggs ( G ) → C ov as follows. Any abstract Higgs bundle ( E, σ ) induces the abstract cameral cover ˆ E = E × G/N
G/T → E . A descent argument allows to prove that it comes from a unique W -cover b X → X , the abstract cameral cover associated to ( E, σ ).Note here that there is a morphism(10) C ′ : A L ( G ) −→ C ov due to the following fact (cf. [DG, Proposition 1.5]). Fact 3.2.
Consider the Grothendieck–Springer resolution b g reg of g reg (note that the Grothendieck–Springer resolution restricts to the regular locus), i.e., the W -cover obtained by pullback(11) b g reg / / (cid:15) (cid:15) t (cid:15) (cid:15) g reg / / t /W. Let c : g reg → G/N be the map sending each element to its centraliser. Then [DG,Proposition 10.3] g reg × G/N
G/T ∼ = b g reg over g reg .Moreover, [ h abs ] ◦ C = C ′ ◦ [ h ] L . So we will henceforth refer to both kinds of covers(namely, locally isomorphic to G/T → G/N or to t → t /W ) as cameral covers. Remark . Note that over the image of C , the construction of a cameral cover is easier.Indeed, let ( E, σ ) = C ( E, φ ). Then, the datum of σ is equivalent to a section c ( φ ) of E ( G/N ). The cameral cover ˆ X is the fibreed product X × E ( G/N ) E ( G/T ).Over C ov one may consider the following sheaf of groups:(12) T ( S ) = (cid:26) s : b S → T : s ( u · w ) = s ( u ) w ∀ w ∈ W,α ( s ( x )) = − ∀ u ∈ b S : s α ( u ) = u (cid:27) where ˆ S is the cameral cover associated to S → C ov and s α ∈ W denotes the reflectionwith respect to the root α . This sheaf of groups is representable by a group scheme T ,namely, T ( S ) = Hom( S, T ) are the S -points of T . Note that its restriction to A L ( G ) has S -points for b : S → t ⊗ L/W given by(13) T L ( S ) := C ′∗ T = (cid:26) s : b S b → T : s ( u · w ) = s ( u ) w ∀ w ∈ W,α ( s ( x )) = − ∀ u ∈ b S : s α ( u ) = u (cid:27) . Before we can state the first main result concerning the structure of the Hitchin map, letus include some preliminary definition.
Definition 3.4.
Let A → Y be an abelian group scheme. An A -banded gerbe over Y is astack X which is locally (in a chosen Grothendieck topology) isomorphic to the classifyingstack BA . The group scheme A is called the band of the gerbe. Remark . We warn in here that our notion of the band differs in principle from that ofGiraud’s. Indeed, given a G -gerbe X , the band as defined by Giraud [Gi] is an Out( G )-torsor K . When G is abelian, however, K can be lifted to an Aut( G )-torsor K . In Definition3.4, the “band” rather means K ×
Aut( G ) G , which is locally isomorphic to G . Theorem 3.6 (Donagi–Gaitsgory, Ngˆo) . The Hitchin map (6) and the abstract Hitchinmap (9) induce an abelian banded gerbe structure on
Higgs reg ( G ) and H iggs ( G ) respec-tively. The respective bands are isomorphic to the group schemes T L and T . A major result in [DG] is to give a cocyclic interpretation of the stack
Higgs reg ( G ) interms of principal bundles over cameral covers, the so called cameral data . Definition 3.7.
Denote by C am (for cameral data) the stack over A L ( G ) that assignsto each b ∈ H ( S, t ⊗ f ∗ L/W ) (for f : S → X ) the category of R -twisted, N -shifted W -equivariant principal T -bundles on b S b , the cameral cover associated with b (see [DG, § P, γ, β ) where1. P is a principal T -bundle on b S b .2. A map γ fitting in the map of short exact sequences0 / / T / / (cid:15) (cid:15) N / / γ (cid:15) (cid:15) W / / id (cid:15) (cid:15) / / Hom( b S b , T ) / / Aut W ( P ) / / W / / . In the above, γ assigns to each n w ∈ N inducing w ∈ W an isomorphism γ ( n ) : P ∼ = w ( P ) ⊗ R w (that is, an object of Aut W ( P )). Here, w ( P ) = w ∗ P × w P and R w is the principal T -bundle associated to the ramification locus of w . When w = s α is the reflection associated with α , R w = ˇ α ( O ( D α )), where D α ⊂ b U b are the fixed pointsunder s α . This defines a cocycle in Z ( W, BT ), so that the R s α for simple roots determine R w for all w ∈ W . See [DG, §
5] for details.3. β = ( β i ) ri =1 is a family of isomorphisms β i : α i ( P ) | D αi ∼ = O ( D α i ) | D αi , for each simple root α i , with D α i the corresponding ramification divisor.The above data are subject to compatibility conditions for which we refer the reader to[DG, § Theorem 3.8.
The stacks C am and Higgs reg ( G ) are isomorphic. In particular, the fibreof the Hitchin map over b ∈ A L ( G ) is the category of R -twisted, N -shifted W -equivariantprincipal T -bundles on b X b .Sketch of proof. Since we need to refer to the proof in the sequel, we will recall the mainelements. For details, we refer the reader to [DG, Theorem 6.4].Firstly, we have a universal cameral datum(14) ( P univ , γ univ , β univ ) → G/T , given by:
IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 11 a) The bundle P univ := G/T × G/B
G/U is given by the pullback of the principal T bundle G/U → G/B onto the incidence variety
G/T ⊂ G/N × G/B .b) The datum γ univ is obtained by observing that for reflections under simple roots s α ,the isomorphism P univ | G/T ∼ = s ∗ α P univ | G/T extends meromorphically with associated divisor − ˇ α ( D α ).c) The datum β univ is checked from the above by reducing the statement to suitableLevi subgroups.Given a Higgs bundle ( E, φ ), let C ( E, φ ) = (
E, σ φ ) where C is as in (7). Since σ φ is G -equivariant E σ φ −→ G/N , then of ˆ E = E × G/N
G/T −→ E is also G -equivariant, and so descends to a uniqueˆ X −→ X. The same applies to γ univ , β univ , which yields a cameral datum ( P E , γ E , β E ) on ˆ X . Sincethe assignment ( E, φ ) ( P E , γ E , β E ) is functorial, it defines a morphism Higgs reg ( G ) −→ C am. By Theorem 3.6 and the discussion on page 38 of [DG]), this is a morphism of of T L -gerbes,which must therefore be an isomorphism. (cid:3) Remark . Note that according to [DG, Corollary 17.6], an element of
Higgs reg ( G ) isequivalent to an element of H iggs together with a W -equivariant embedding ˆ X → t ⊗ L of schemes over X . The way we have defined cameral covers for L -valued Higgs bundlesdirectly determines this embedding.4. The local situation: untwisted G R -Higgs bundles Quasi-split real groups and the local Hitchin map.
The purpose of this sectionis to analyse the local Hitchin map(15) [ χ ] : [ m /H ] → a /W ( a ) , constructed by H -equivariance of the Chevalley morphism (2).From now on we will assume that G R is a quasi-split strongly reductive algebraic group.Amongst simple groups, quasi-split forms include split groups and those whose Lie algebrais su ( n, n ), su ( n, n + 1), so ( n, n + 2) and e .We note in here that algebraic groups admit a complexification. We let G := G R , andassume it to be connected.Recall that a real group is quasi-split if any of the following equivalent conditions hold:(QS1) The centraliser c g ( a ) of a inside h is abelian. In this case c g ( a ) is a Cartansubalgebra.(QS2) There exists a θ invariant Borel subgroup B < G such that B θ = B op is theopposed Borel subgroup. Borel subgroups satisfying this condition are called θ -anisotropicBorel subgroups.We fix once and for all an anisotropic Borel subgroup B < G , a θ invariant maximaltorus T < B which we assume to be the complexification of T R = T ∩ G R . Assumethat the Lie algebra of T satisfies t = d ⊕ a with a = t ∩ m maximal and d = t ∩ h .We let S ⊂ ∆ = ∆( g , t ) be the associated sets of (simple) roots, W the Weyl group. Similarly, we can define Σ( a ) the set of restricted roots associated to a (which is the imageof the restriction map res | a : ∆ → a ∗ ). This is a root system (possibly non reduced).Reflection with respect to simple such roots generates a group W ( a ) called the restrictedWeyl group. This group is also isomorphic to N H ( a ) /C H ( a ) and N H R ( a R ) /C H R ( a R ), where a R = a ∩ g R . Moreover, any root α can be expressed as α = λ + iβ with λ ∈ a ∗ R , β ∈ d ∗ R . ABorel subalgebra is θ -anisotropic if and only if it induces an ordering like this. Moreover,quasi-split real forms have no purely imaginary roots (a necessary condition to admit a θ -anisotropic Borel subgroup).The following lemma tells us that up to isogeny, strongly reductive real group are realforms. Lemma 4.1.
Let G R be a strongly reductive real algebraic group and G its complexification.Then there exists a real form G σ of G such that:(1) g R = g σ , namely, G R and G σ are isogenous.(2) N G ( G R ) = N G ( G σ ) .Proof. To see (1), note that g R is a real form of g . Let σ be the involution defining it, andlet θ = στ be the Cartan involution compatible with a compact form u = g τ of g . Let U = exp( u ) < G be the corresponding compact subgroup. Then by [Kn, § V.II] G ∼ = U × i u ,and the involution on G defined by U × i u ∋ ( e Y , X ) σ ( e σY , σX )defines a real form G σ . By construction, its Lie algebra is that of G R .To prove (2), note that strong reductivity of G implies thats N G ( G R ) ⊂ N G ( g R ) (as G acts by inner automorphisms on g and so does N G ( G R ) on g R ), so equality holds. Onthe other hand, real forms of strongly reductive complex groups are strongly reductive by[GPR, Proposition 3.6], so also N G ( G σ ) = N G ( g σ ). We may conclude from (1). (cid:3) Returning to the study of (15), one cannot expect to have any “nice” (gerby) structureof the local Hitchin map (15) as a whole. The reason is that the inertia stack, which is thestack that classifies automorphisms of objects, is far from being flat. This is a necessarycondition in order to have a gerbe structure [AOV, Appedix A].Indeed, consider the following group scheme on m : Definition 4.2.
We let C θ → m be the group scheme over m defined by C θ = { ( m, h ) ∈ m × H | h · m = m } . Remark . The superscrip θ in the notation is related with the fact that H = G θ for aholomorphic involution θ on G lifting the extension of the Cartan involution by complexlinearity (cf. Remark 2.1). Thus, if C ⊂ m × G is the centraliser group scheme, C θ is thefixed point set of C by ( x, g ) ( x, g θ ).Note that there is an action of H on C θ (namely, the adjoint action) lifting the isotropyaction on m . This means that the inertia stack of [ m /H ] is induced from the latter sheaf.Indeed: Lemma 4.4.
The sheaf represented by the group C θ → m descends to the inertia stack on [ m /H ] . IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 13
Proof.
This is standard: sections of [ m /H ] are, up to covering, trivial bundles together withequivariant maps to m . Automorphisms of a pair ( P, φ ) over S ∈ ( Sch/ C ) ´ et are sections ofthe adjoint bundle Ad( P ) centralising φ , which locally are just equivariant maps to C θ . (cid:3) Consider the regular locus:
Definition 4.5.
We denote by m reg = { x ∈ m : dim C H ( x ) = dim a } . This is the set of regular elements, and it can be proved it corresponds to elements of m with minimal dimensional centraliser, or maximal dimensional (isotropy) orbits [KR,Proposition 7]. Remark . An equivalent definition of quasi-splitness is to assume that m reg = m ∩ g reg where g reg ⊂ g is the subset of (adjoint) regular elements. This implies in particular that c m ( x ) has dimension equal to the dimension of a , so in particular, if x ∈ a is regular, then c m ( x ) = a . Definition 4.7.
We will call the stack [ m reg /H ] the stack of everywhere regular local G R -Higgs bundles. Lemma 4.8. C θ → m reg is smooth.Proof. This proof is similar to the one of [DG, Proposition 11.2].Given a complex point ( x, h ) ∈ C θ ( C ), we have that the tangent space T ( x,h ) C θ ( C ) isdefined inside T ( x,h ) m reg × H = m × h by the equation d ( x,h ) f ( y, ξ ) = 0 where f ( x, h ) = Ad( h )( x ) − x Now: ∂∂h | ( h,x ) f ( y, ξ ) = ∂∂h | ( h,x ) Ad( h ) ◦ ev x ( y, ξ ) = h · [ ξ, x ]Hence d ( x,h ) f ( y, ξ ) = Ad( h )([ ξ, x ])+ h · y − y . Clearly, the differential of the map C θ → m reg sends ( y, ξ ) y . So all we need to check is that { y ∈ m | y − h − ( y ) ∈ ad( x )( h ) } = m One inclusion is clear, so lets see that any z ∈ m satisfies the condition. First note that g ∼ = [ x, g ] ⊕ c g ( x ) ∼ = [ x, h ] ⊕ c m ( x ) ⊕ [ x, m ] ⊕ c h ( x )so that m ∼ = [ x, h ] ⊕ c m ( x )Since the action of any h ∈ C H ( x ) respects the direct sum, it is enough to check that c m ( x ) ⊆ { y ∈ m | y − Ad( h − )( y ) ∈ ad( x )( h ) } First, suppose that C H ( x ) is connected. Then, by quasi splitness, c h ( x ) = c h ( c m ( x )) (as thecentraliser of a regular element is abelian), so that for any z ∈ c m ( x ), Ad h ( z ) − z = 0 ∈ ad( x )( h ).Now, for the non connected case, since fibres are algebraic groups in characteristic zerothey are smooth. Thus independently of the component h is in, the dimension of thetangent bundle will not vary. (cid:3) In order to obtain a gerbe, we need to “quotient” the stack [ m reg /H ] by the inertiastack parameterizing automorphisms of objects, thus obtaining a gerbe over the sheaf ofisomorphism classes of objects. This process is called rigidification [AOV]. In the complexgroup case, the sheaf of isomorphism classes is the GIT quotient g (cid:12) G . The situation inthe real group case is somewhat different, as [ m reg /H ] fails to be locally connected overthe GIT quotient a /W ( a ) ∼ = m (cid:12) H . This is due to the fact that a /W ( a ) does not alwaysparameterise H -orbits, but rather orbits of the larger group N G ( G R ). So in the forthcomingsections we analyse both possible gerby structures, first by rigidifying (Section 4.2) andsecondly by increasing the automorphism group (Section 4.3). While the first approach isvalid for any strongly reductive group by Lemma 4.8 and [AOV, Theorem A.1], the secondis totally determined by real forms (by Lemmas 4.1 and 4.13).4.2. The case of real forms.
When G R is a real form, the following example shows thatwe cannot expect for (15) to induce a gerbe structure. Example 4.9.
Let G R = SL(2 , R ). By suitably choosing the involutions, we may identifySO(2 , C ) with diagonal matrices, and sym (2 , C ) (the subspace of zero traced symmetricmatrices) with off diagonal matrices.The Chevalley morphism is det : sym (2 , C ) → C . Nilpotent elements are preciselydet − (0). Now, all elements of sym (2 , C ) \ , C ) on sym (2 , C ) is given by:(16) λ · (cid:18) βγ (cid:19) = (cid:18) λ βλ − γ (cid:19) . So we see that for nilpotent elements there are two maximal open orbits (with 0 inthe closure of both): upper and lower triangular matrices. These however become onevia conjugation by (cid:18) ii (cid:19) , which together with SL(2 , R ) generates the group N := N SL(2 , C ) (SL(2 , R )), which thus fits into an exact sequence1 → SL(2 , R ) → N → Z → . Now, semisimple elements in sym (2 , C ) are elements with both off diagonal entries differentto zero; hence by suitably choosing λ ∈ C × in (16), we see that for any such element x , N · x = SO(2 , C ) · x .Hence, the Hitchin map does not induce a gerbe structure, but we may replace theHitchin base by the space of orbits as follows.Consider the sheaf A associated to the following presheaf:(17) A ′ : Sch −→ Sets S
7→ { isomorphism classes in [ m reg /H ] ( S ) } . There is a surjective morphism [ m reg /H ] → A . Moreover: Lemma 4.10. [ m reg /H ] → A is a gerbe banded by the space fibreed in abelian groups J θ −→ S obtained by faithfully flat descent from C θ .Proof. This is just [AOV, Theorem A.1], where the group stack by which one rigidifiesit the full inertia stack, which applies by Lemma 4.8. Note this applies to any stronglyreductive quasi-split real group. (cid:3)
IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 15
Remark . Note that A is the orbit space of the isotropy representation. In the par-ticular case of SL(2 , R ) this is the non separated scheme consisting of a line with a doubleorigin. In general, it will be a non separated algebraic space.4.3. The associated gerbe of N G ( G R ) -Higgs bundles. The Hitchin map (15) fails tobe a gerbe even when restricted to the regular locus, but by enlarging the automorphismsslightly we obtain a nicer structure.
Definition 4.12.
Let θ be the holomorphic involution on G associated to the real form G R , in such a way that G θ = H . Consider the algebraic group G θ = { g ∈ G : g − g θ ∈ Z ( G ) } . We will see that the stack [ m reg /G θ ] is a gerbe under the local Hitchin map(18) [ χ ] : [ m /G θ ] → a /W ( a ) , obtained by G θ equivariance of (2). Lemma 4.13.
Let N G ( G R ) be the normaliser inside G of the real form G R . Then [ m reg /G θ ] is the stack of local N G ( G R ) -Higgs bundles.Proof. Since n g ( g R ) = g R + z ( g ) ∩ i g R , all we need to check is that the maximal compactsubgroup of N G ( G R ) complexifies to G θ .Let G R = G σ , and let U < G be maximal compact subgroup defined by an involution τ commuting with σ . Then θ = στ and N G ( G R ) = { g ∈ G : g − g σ ∈ Z G ( G R ) } . Thus its maximal compact subgroup is N U ( G R ) = { g ∈ U : g − g σ ∈ Z G ( G R ) } = { g ∈ U : g − g θ ∈ Z G ( G R ) } . Hence, if we check that Z G ( G R ) = Z ( G ), the result follows, as G θ ∩ U = { g ∈ U : g − g θ ∈ Z ( G ) } . First, G is strongly reductive (reductive in the sense of [Kn, § VII.2]), so by definition G acts on its Lie algebra by inner automorphisms, and so Z G ( G R ) centralises g R , and so also g . But then it centralises G = G . (cid:3) Remark . In fact, we may obtain a gerbe in different ways from the local Hitchin map(15). Independently of whether G is strongly reductive or not, the stack of everywhereregular N G ( G R )-Higgs bundles defines a gerbe over the same space as [ m reg /G θ ], since thespace of orbits in m reg under G θ and N U ( G R ) C is the same in virtue of Propositions A.3and [GPR, Proposition 3.21].Now, on m reg , we consider the following group scheme:(19) C θ = { ( m, h ) ∈ m reg × G θ | h · m = m } . Lemma 4.15.
The group scheme C θ → m reg is a smooth abelian group scheme.Proof. Commutativity follows from quasi-splitness, as for any x ∈ g reg C G ( x ) is abelian(cf. Remark 4.6) and ( C θ ) x ⊂ C G ( x ).For smoothness of the morphism, it is enough to observe that g θ = h ⊕ z m ( g ), so theproof follows from the same argumenst used in Lemma 4.8 (cid:3) Lemma 4.16.
There is an exact sequence (20) 1 → H → G θ → F → , where F = { a ∈ exp( a ) : a ∈ Z ( G ) } and the map G θ → F is given by g g − g θ .Moreover, G θ = F H .Proof.
By Lemma A.5 and connectivity of G , its maximal compact subgroup U has theform HA u H . Now, let g ∈ G θ , and let g = ue iV be its polar decomposition, with u ∈ U , V ∈ u . By uniqueness of the latter and reductivity of Z ( G ) (see [Kn, Corollary 7.26]), g − g θ ∈ Z ( G ) ⇐⇒ u − u θ ∈ Z U ( G ) , e − iV e iθV = e − i ( V − θV ) ∈ e i z ( u ) . But now u = h ah for h , h ∈ H , a ∈ A u , so u − u θ = Ad( h − )( a − ) ∈ Z U ( G ) ⇐⇒ Ad( h − )( a − ) = a ∈ Z U ( G ) . On the other hand, given that i u = i h ⊕ m , we have i ( V − θV ) ∈ m , so i ( V − θV ) =Ad( h )( iX ) for some h ∈ H , X ∈ i a by [Kn, Lemma 7.29]. So clearly, e i ( V − θV ) ∈ e i z ( u ) ⇐⇒ e i ( V − θV ) = e iX ∈ ie z ( u ) ∩ m ⊂ e a where the last inclusion follows by maximality of a . Since clearly F ⊂ G θ , exactness of thesequence follows.To see that G θ = F H , first note that g = h ah e iV for h i ∈ H , a ∈ F . So all we need tocheck is e iV ∈ F H . Write V = V h + V m , V h ∈ h , V m ∈ i m . Since V − θV = 2 V m ∈ i a ∩ z ( u ),it follows that e iV = e iV j e iV m , with e iV j ∈ H, e iV m ∈ F . (cid:3) Corollary 4.17.
The scheme C θ fits into an exact sequence → C θ → C θ p → F → . where F ⊂ m reg × F is the group scheme defined as follows:(1) F | m reg,ss = m reg,ss × F .(2) F x = { y : y ∈ Z ( C G θ (Ad( h )( x s ))) ∩ F } , where x = x s + x n is the Jordandecomposition and h ∈ H is such that Ad( h )( x s ) ∈ a .Proof. From Lemma 4.16, we have an exact sequence0 → m reg × H → m reg × G θ → m reg × F . So we need to characterise the image.Statement (1) is clear when restricted to a reg := a ∩ m reg . Indeed, C θ | a reg = a reg × T θ ,where T θ = C G θ ( a ), so that T θ ⊃ A ∩ T θ . Then, from Lemma 4.16 we may conclude.Suppose that x ∈ m reg,ss . Then, by [KR, Theorem 1], there exists some h ∈ H such that x ′ = Ad( h )( x ) ∈ a . Thus, given that p is H -equivariant, and that F ⊂ Z ( G ), we mayconclude that (1) holds.To see (2), for any x ∈ m , if x = x s + x n is its Jordan decomposition, it follows that x s , x n ∈ m by [KR, Proposition 3]. Also, C G θ ( x ) = C G θ ( x s ) ∩ C G θ ( x n ). Since g R is quasi-split, C G θ ( x n ) ∩ C G θ ( x s ) is unipotent within C G θ ( x s ), so the intersection is the center of C G θ ( x s ) (since centralisers of nilpotent elements are unipotent). By the same argumentsas used in the semisimple case, the image of this into F is independent of the conjugacyclass. (cid:3) IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 17
Remark . Given x = x s + x n ∈ m reg such that x s ∈ a , it follows that for all roots α such that α ( x s ) = 0 F α = F ∩ Ker( α ) ⊂ Z ( C θ ( x s )). The latter is in fact generated by Z ( G ) and F α for roots vanishing on x s .Note also that the semisimple part of F acts on m reg by permuting different H or-bits within the same G θ orbit. This happens only for non semisimple elements by [KR,Proposition 3]. Proposition 4.19.
The scheme C θ descends to an affine abelian group scheme J θ over a /W ( a ) .Proof. First of all, note that G θ normalises m , which induces an action on C θ making C θ → m reg G θ -equivariant. From this point on we can adapt the proof of Lemma 2.1.1 in[N] to our context.Since for x ∈ m reg , ( C θ ) x is abelian, hence, we can define the fibre over χ ( x ) ∈ a /W ( a )to be ( C θ ) x itself. Any other choice will be canonically isomorphic over a /W ( a ) by [KR,Theorem 11] and commutativity of the centraliser. As for the sheaf itself, it can be definedby descent of C θ along the flat morphism m reg → a /W ( a ).For C θ to descend, both pullbacks to m reg × a /W ( a ) m reg must be isomorphic. Considerboth projections p , p : m reg × a /W ( a ) m reg → m reg , and let C i = p ∗ i C θ . Consider f : G θ × m reg → m reg × a /W ( a ) m reg given by ( h, x ) ( x, h · x ).We will proceed by proving that there exists an isomorphism f ∗ C ∼ = f ∗ C over G θ × m reg ,and then check it descends to an isomorphism over m reg × a /W ( a ) m reg (since the above mapis smooth and therefore flat).Consider F : f ∗ C / / f ∗ C (( m, h ) , g ) ✤ / / (( m, h ) , Ad h g ) . It defines an isomorphism over G θ × m reg . To see whether F descends to m reg × a /W ( a ) m reg ,we need to check that F ( m, h, g ) depends only on ( m, h · m, g ) and not on the particularelement h ∈ G θ . To see that, it must happen that the pulbacks of F to S := ( G θ × m reg ) × m reg × a /W ( a ) m reg ( G θ × m reg )by each of the projections π , π : S → G θ × m reg fit into the commutative square π ∗ f ∗ C / / (cid:15) (cid:15) π ∗ f ∗ C (cid:15) (cid:15) π ∗ f ∗ C / / π ∗ f ∗ C . To do this, note that S ∼ = G θ × C by the map(( m, h ) , ( m, h ′ )) ( h, ( m, hm ) , h − h ′ ) . Then, in these terms, π : ( h, ( m, hm ) , z ) ( m, h ) π : ( h, ( m, hm ) , z ) ( m, hz ) F : [( h, ( m, hm ) , z ) , g ] [( h, ( m, hm ) , z ) , Ad h g ] , so that the above square reads[( h, ( m, hm ) , z ) , g ] ✤ / / ❴ (cid:15) (cid:15) [( h, ( m, hm ) , z ) , Ad h g ] ❴ (cid:15) (cid:15) [( h, ( m, hm ) , z ) , Ad h g ] ( ∗ ) [( hz, ( m, hm ) , z ) , g ] ✤ / / [( h, ( m, hm ) , z ) , Ad hz g ] , where ( ∗ ) follows from commutativity of ( C θ ) m . (cid:3) Corollary 4.20.
The Hitchin map [ m reg /G θ ] → a /W ( a ) is a neutral gerbe banded by J θ .Proof. Since the inertia stack descends to J θ → a /W ( a ), it follows that the stack is locallyconnected. We need to prove that it admits a section, which yields local non emptinessand neutrality at once. Let s KR be the Kostant–Rallis section, that is, the section of theChevalley morphism (2) constructed in [KR, Theorem 11] for groups of the adjoint typeand adapted in [GPR, Theorem 4.6] to strongly reductive real groups. This induces auniversal object of our gerbe by assigning to each scheme f : S → a /W ( a ) the trivial G θ bundle together with the section φ : S × G θ → m reg sending ( x, g ) g · s KR ( f ( x )) (cid:3) Remark . The Kostant–Rallis section factors through the atlas m reg and thus factorsthrough the image of [ m reg /H ] −→ [ m reg /G θ ].4.4. Involutions on the local stack.
A way to retrieve local G R -Higgs bundles from[ m reg /G θ ] is by studying fixed points by involutions. Indeed, we note that the substack of G R -Higgs bundles is contained in [ m reg /G θ ] Θ , where Θ is the involution sendingΘ : ( E, φ ) ( E × θ G θ , φ ) . Proposition 4.22.
The involution on C θ given by Θ( x, g ) = ( x, g θ ) descends to an invo-lution Θ on J θ making the isomorphism [ m reg /G θ ] → BJ θ Θ -equivariant. In particular, the image of [ m reg /H ] → [ m reg /G θ ] is contained inside ( BJ θ ) Θ = { P ∈ BJ θ : P × Θ J θ ∼ = P } . Proof.
Descent of the involution follows from G θ -equivariance; the induced involution on J θ (that we will also denote by Θ) is given by s Θ ( x ) = θ ( s ( x )) for all s ∈ J θ ( S ).The second statement follows by definition of the inertia stack, as sending ( E, φ ) to( E × θ H, φ ) translates into changing the action of an automorphism by θ . Since theimage of [ m reg /H ] is fixed by the involution, it follows that the image will be contained in( BJ θ ) Θ . (cid:3) Remark . Note that the essential image of [ m reg /H ], i.e., the minimal stack containingall objects of [ m reg /H ] and those isomorphic to them, is the whole [ m reg /G θ ]. This isimmediate from commutativity of the local Hitchin map with extension of the structuregroup, and the facts that [ m reg /G θ ] is a gerbe over the Hitchin base and that the latterclassifies isomorphism classes. IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 19
Note that the scheme J θ is very twisted and hard to work with in concrete examples.We next give an alternative description of it, also keeping an eye on BJ Θ θ . Since the rightobjects are much more clearly understood when looking at the semisimple locus, we willrestrict to semisimple Higgs bundles first, then go back to the general case.4.5. An alternative description of the band: the semisimple locus.
In this sectionwe focus on bundles (
E, φ ) such that φ takes only semisimple values. Namely, sections ofthe stack [ m reg,ss /G θ ], where m reg,ss ⊂ m reg denotes semisimple elements of m reg . This willallow us to find alternative descriptions of J Θ θ , J θ over a suitable sublocus of a /W ( a ).Let a reg = a ∩ m reg,ss . We have a commutative diagram(21) a reg i / / π a % % ❑❑❑❑❑❑❑❑❑❑ m reg,ssπ m (cid:15) (cid:15) a reg /W ( a ) . Note that(22) i ∗ C θ = a reg × T θ where T θ = T ∩ G θ . Proposition 4.24.
Let N θ ( a ) be the normaliser of a inside G θ . The embedding a reg ֒ → m reg,ss induces an isomorphism [ a reg /N θ ( a )] ∼ = [ m reg,ss /G θ ] of stacks over a reg /W ( a ) .Proof. Let [ χ ] a : [ a reg /N θ ( a )] → a reg /W ( a ) be the restriction of [ χ ] definde in (18). Weclaim that [ χ ] a is a subgerbe of [ χ ] with the same band.To see it is a gerbe, we check local connectedness and non emptiness.Local non emptiness of [ χ ] a follows from the fact that given a principal N θ ( a )-bundle P → S , locally P × a reg /W ( a ) a reg ∼ = S × a reg /W ( a ). Indeed, this is a consequence of W ( a ) = N H ( a ) /C H ( a ) and Corollary 4.17 (1).Local connectedness follows from local connectedness of [ m reg,ss /G θ ] and the fact thattwo elements x, y ∈ a reg conjugate by G θ must be conjugate by elements in N θ ( a ), as byregularity c a ( x ) = a (cf. Definition 4.6).Commutativity of diagram (21) also implies that both gerbes are locally isomorphic over a reg /W ( a ). Indeed, their bands descend from C θ | m reg,ss and a reg × T θ (by Proposition 4.19and similar arguments for the substack [ a reg /N ( a )]), which in turn descend to the inertiastacks; since by commutativity of diagram (21) these descended schemes are isomorphicto J θ | a reg /W ( a ) , it follows that both stacks are locally isomorphic over a reg /W ( a ).But since the band is abelian and any such gerbe is a torsor over a lift of the band, sincethe local isomorphism of gerbes is globally defined it must be a global isomorphism. (cid:3) Through Proposition 4.24 we get a clearer picture of the sheaf J θ : Corollary 4.25.
Let J θ be the group scheme defined in Proposition 4.19. Then, its re-striction to a reg /W ( a ) has S points (for b : S → a reg /W ( a ) ) J θ ( S ) = Hom W ( a ) ( S b , T θ ) , where (23) S b = S × a /W ( a ) a Moreover, J Θ θ | a reg /W ( a ) descends from C θ | m reg,ss for C θ defined in (4.2) or, equivalently,from J θ | A × a /W ( a ) a reg /W ( a ) (cf. Lemma 4.10).Proof. From Proposition 4.24 we have that J θ ( U ) = Hom N θ ( a ) ( U b , T θ ) . Since W ( a ) = N θ ( a ) /T θ (by [Kn, Proposition 7.49], quasi-splitness of G R and the fact that T θ is the trivial extension of T ∩ H by a central subgroup of G ), it follows that the actionof N θ ( a ) on U b and T θ factors through the quotient.Descent of C θ | m reg,ss follows from Lemma 4.17, which implies that F acts trivially onsemisimple orbits. Equivalence with descent of J θ | A × a /W ( a ) a reg /W ( a ) is a consequence ofLemma 4.10. (cid:3) Corollary 4.26.
The fibre of the gerbe [ m reg,ss /G θ ] over b : S → a reg /W ( a ) is the categoryof T θ principal bundles P over S b as in (23) satisfying w ∗ P × w T θ ∼ = P for all w ∈ W ( a ) .Likewise [ m reg,ss /H ] is a gerbe over a reg /W ( a ) , whose fibre over b : X → a reg /W ( a ) is thecategory of principal D -bundles P over X b satisfying w ∗ P × w D ∼ = P for all w ∈ W ( a ) . An alternative description of the band: back to arbitrary Higgs fields.
Inorder to extend the results of Section 4.5, we will compare it with the complex case studiedin [DG, N]. This will allow us to characterise J θ as a group of tori over the whole a /W ( a )(compare with Corollary 4.25) and obtain a cocyclic description of the Hitchin fibres.By quasi-splitness, m reg ⊂ g reg ; this embedding is H -equivariant, and hence it inducesa morphism on the level of stacks(24) [ m reg /H ] κ −→ [ g reg /G ] . This morphism factors through [ m reg /G θ ] (and [ m reg /G θ ] Θ ), which, as Example 4.9 illus-trates (see also Remark 4.23) a minimal subgerbe containing the image of [ m reg /H ]. Due tothis, the set of isomorphism classes of the image of [ m reg /H ] inside [ m reg /G θ ] embeds intothe set of isomorphism classes of objects in [ g reg /G ], unlike what happens for [ m reg /H ]. Lemma 4.27.
Let G R ≤ G be a quasi-split real form. Then:1. There is an equality N G ( m reg ) = G θ ⊂ N G ( H ) .2. We have an embedding (25) ι : a /W ( a ) ֒ → t /W.
3. Given x, y ∈ m reg , if for some g ∈ G Ad g x = y , then there is h ∈ G θ such that Ad h x = y . If x, y ∈ m reg,ss , then h can be taken inside of H .Proof. 1. Note that g ∈ N G ( m reg ) if and only if g − g θ ∈ C G ( x ) for all x ∈ m reg . Butsince m reg contains both semisimple and nilpotent elements, the intersection of all suchcentralisers is the center Z ( G ). So N G ( m reg ) ⊂ G θ ⊂ N G ( H ) and the statement follows. By [KR, Theorem 11], the choice of a principal normal triple { e, f, x } , with e, f ∈ m reg nilpotent, establishes an isomorphism a /W ( a ) ∼ = f + c m ( e ). By quasi-splitness, e, f ∈ g reg ,and so t /W ∼ = f + c g ( e ) by [K, Theorem 7], so we have the desired embedding. IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 21 Follows from above and [KR, Theorem 7], which implies that a /W ( a ) parametrizes G θ -orbits, while t /W parametrizes G -orbits, by [K, Theorem 2]. The statement aboutsemisimple elements follows from [KR, Theorem 1]. (cid:3) With respect to the local Hitchin maps, there is a commutative diagram(26) [ m reg /H ] κ / / [ χ ] (cid:15) (cid:15) [ g reg /G ] [ χ ] C (cid:15) (cid:15) a /W ( a ) (cid:31) (cid:127) ι / / t /W. In the above, t = d ⊕ a is a maximal θ -anisotropic Cartan subalgebra of g , that is, a θ -invariant Cartan subalgebra containing a , and d = t θ . The lower horizontal arrow isan embedding by Lemma 4.27. Recall from Lemma 2.1.1. in [N], that the scheme ofcentralisers C ⊂ g reg × G defined analogously to C θ (cf. Definition 4.2) descends to ascheme of abelian groups J → t /W . Using adjunction, one sees that its S points are givenby(27) J ( S ) = Hom G ( S × t /W g reg , C ) . The group scheme J is isomorphic to another group scheme whose S points are given by(28) T ( S ) = (cid:26) f : ˆ S → T (cid:12)(cid:12)(cid:12)(cid:12) W − equivariant α ( f ( x )) = 1 if s α ( x ) = x, α ∈ ∆( g , t ) (cid:27) for S → t /W and ˆ S := S × t /W t .On a /W ( a ) we consider the following sheaf of groups:(29) T θ ( S ) = (cid:26) f : S → T θ (cid:12)(cid:12)(cid:12)(cid:12) W ( a ) − equivariantsatisfying ( † ) , (cid:27) where( † ) w ( x ) = x ⇒ f ( x ) ∈ ( T wθ ) In the above S := S × a /W ( a ) a is considered as a subscheme of ˆ S := S × t /W t and ( T θ ) isthe identity component of T θ . Proposition 4.28.
The sheaf T θ is the intersection of the Weil restriction of the torus a × T θ along the finite flat morphism a −→ a /W ( a ) and ι ∗ T , where ι is defined in (25) .In particular, it is representable by a scheme of tori over a /W ( a ) .Proof. Representability of the Weil restriction follows by [BLR, § ι ∗ T by[N, Proposition 2.4.7], hence, the intersection is also representable. So representabilityfollows by proving that T θ is the intersection of both.Now, by adjunction ι ∗ T ( U ) consists of W -equivariant morphisms U × a /W ( a ) t → T satisfying ( † ) for reflections along roots. By Theorem 3.8, the same must be true for anyelement of the Weyl group, by the way the ramification divisors are defined. Thus, theintersection with the Weil restriction of a × T θ consists of morphisms U × a /W ( a ) t → T satisfying ( † ) arising from W ( a )-equivariant morphisms. Hence, all we need to prove isthat extension of sections of T θ by W -equivariance is well defined and injective. Given w ∈ W , let w · x ∈ a for some x ∈ a . Then, there exists some w ′ ∈ W ( a ) suchthat w ′ x = wx , namely w − w ′ · x = x. The condition ( † ) ensures that s ( x ) ∈ ( T θ ) w − w ′ , thus we may unambiguoulsy define s ( w · x ) = w · s ( x )for any w ∈ W and x ∈ a . This forces to any two distinct sections to extend differently,so injectivity follows. (cid:3) Proposition 4.29.
We have an embedding J θ ⊂ ι ∗ J , where ι is defined in (25) .Proof. Consider the commutative diagram m reg ˜ ι / / χ (cid:15) (cid:15) g regχ (cid:15) (cid:15) a /W ( a ) ι / / t /W. Then C | m reg := ˜ ι ∗ C = ˜ ι ∗ χ ∗ J = χ ∗ ι ∗ J . So given that C θ ⊂ C | m reg , adjunction gives amorphism J θ → χ ∗ ˜ ι ∗ C . Using the fact that adjunction yields isomorphisms between triplecombinations, we have that on the level of S -points the morphism sends a section of J θ ( S )to a section of ι ∗ J ( S ) by establishing s ( g · x ) = g · s ( x ). This is well defined, as if g · x ∈ m reg for x ∈ m reg then there exists some g ′ ∈ G θ such that g ′ · x = y , so that s ( y ) = g ′ · s ( x ),and by commutativity of the centralisers, s ( y ) = g · s ( x ), so extensions are well defined.Injectivity follows from left exactness of pullback, or by tracking all morphisms involved,which are injective. (cid:3) On g × G we define the involution Θ by(30) Θ : ( x, g ) ( − θx, g θ ) . The subscheme of fixed points is m × H . The restriction of Θ to t × T also induces aninvolution whose fixed point set is a × D . Proposition 4.30. (1) The involution Θ descends to an involution on J → t /W (de-noted also by Θ ) such that ι ∗ J Θ ∼ = J Θ θ .(2) The restriction of Θ to t × T descends to an involution (also denoted by Θ ) on T → t /W such that T Θ θ ∼ = T Θ .(3) The isomorphism J ∼ = T is Θ equivariant. In particular J Θ ∼ = T Θ . .Proof. (1) Note that the source and target maps s, t : g reg × G → g reg are (Θ , − θ )equivariant. So Θ induces an action on J (see equation (27)) such that J → t /W is(Θ , − θ )-equivariant (the action on t /W follows from stability of orbits by the action of θ ). By Proposition 4.29 and compatibility of the involutions, we have J Θ θ ⊂ ι ∗ J Θ . Forthe inverse, by seeing ι ∗ J ( S ) ⊂ Hom G θ ( S × a /W ( a ) g , C ), we have that pullback by theembedding i S : S × a /W ( a ) m reg ֒ → S × t /W g reg (cf. Lemma 4.27) induces a morphism i ∗ S : i ∗ J ( S ) Θ → J Θ θ ( S )which follows by definition of the involutions taking into account that a /W ( a ) = ( t /W ) − θ . IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 23
Both maps are clearly inverse, and so the conclusion follows.(2) The same arguments as in (1) (see also Remark ?? ) imply that Θ | t × T induces an in-volution Θ on T . Likewise, for each S → a /W ( a ), there is an embedding i S : S × a /W ( a ) a ֒ → S × t /W t by Lemma 4.27, and restriction by i S induces a morphism i ∗ S : ι ∗ T Θ ( S ) → T Θ θ ( S ).The conditions imposed on ramification points ensure that any section f ∈ T ( S ) Θ θ ( S ) canbe uniquely extended to a section of ι ∗ T Θ ( S ) by setting f ( wx ) = wf ( x ) for each w ∈ W .Injectivity follows because T Θ ( S ) is contained in the subset { s ∈ T ( S ) : w ( s ( x )) = s ( x ) ∀ w ∈ C W ( x ) , x ∈ a } . (3) As for checking that the isomorphism J → T respects the involutions, it followsfrom θ -equivariance of the morphisms C → B and B → ˆ g reg × T where B → ˆ g reg is thesheaf of Borel subgroups determined by the choice of a θ anisotropic Borel subgroup B containing T . Note that although B is not θ invariant, B/ [ B, B ] is and B/ [ B, B ] ∼ = T is θ -equivariant. See [N, Proposition 2.4.2] for details. The second statement is immediatefrom this. (cid:3) The following theorem is the key towards a cocyclic description of BJ θ (and thus, of thefibres of the local Hitchin map (15)): Theorem 4.31.
We have T θ ∼ = J θ .Proof. From Proposition 4.30 (1) and (3), it is enough to prove that the isomorphism J ∼ = T takes J θ to T θ . The isomorphism is Θ equivariant by Proposition 4.30 (3), whichimplies that that C θ | m reg descends to the same group scheme as χ ∗ ι ∗ T ∩ G θ . By tracingback the construction of T (see the proof of [N, Proposition 2.4.2]), we see that this isequivalent to having that the restriction of C θ to ˆ m reg := m reg × a /W ( a ) t being mapped tothe intersection of ˆ m reg × T θ and T | ˆ m reg . But by Proposition 4.28, the latter group schemeand T θ match, and so the isomorphism follows. (cid:3) Remark . If G R is such that C W ( x ) ⊂ W ( a ) for all x ∈ a , then we can alternativelydefine T θ as T θ ( S ) = f : S × a /W ( a ) a → T θ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) W ( a ) − equivariant α ( f ( x )) = 1if s λ ( x ) = x, λ = α | a ∈ Σ( a ) . There are two main differences with respect to (22): on the one hand, condition ( † ) issubstituted by a condition involving only restricted roots. The second difference, a conse-quence of the former, is that the cover S need not be seen as a subscheme of ˆ S . It thereforebecomes a statement about restricted cameral covers only. Simple groups satisfying theformer condition are all the split forms and SU( p, p ). They can be characterised by thefact that the set a sing of points on which more than one restricted root λ ∈ Σ( a ) vanishesis the intersection t sing ∩ a of points of a on which more than one root vanishes. In termsof spectral curves, these are the ones with generically smooth spectral curves. Remark . The content of Proposition 4.30 can be reformulated in terms of descentto the stack [ a /W ( a ) / Z ], where the action of Z on a /W ( a ) is trivial. Indeed, on t wecan consider the action of the group W Θ = W ⋊ Θ, where the action of Θ on W is givenby w Θ ( x ) = θ ( w ( − θ ( x ))). Then j : [ a /W ( a ) / Z ] ֒ → [ t /W/ Z ]. The schemes J θ and J descend further to J θ → [ a /W ( a ) / Z ] and J → t /W Θ respectively, and Proposition 4.30 says precisely that j ∗ J ( θ ) ∼ = J ( θ ) . Note that [ t /W/ Z ] parametrizes pairs of cameral coversrelated by an order two morphism, while [ a /W ( a ) / Z ] can be identified with the set ofcameral covers together with involutions.5. The global situation: twisted Higgs bundles
In this section we study the structure of the Hitchin map [ h ] L defined in (4) from thatof [ χ ] from Equation (15) analysed in Section 4.Let us recall the notation: we fix X a smooth complex projective curve of genus g ≥ L → X an ´etale line bundle, that we will assume to be of degree at least 2 g − K . Let Higgs reg ( G R ) := [ m reg ⊗ L/H ] and
Higgs reg,θ ( G R ) :=[ m reg ⊗ L/G θ ]. Recall the Hitchin map [ h ] L defined in (4) and denote by [ h ] L,θ its naturalextension to
Higgs reg,θ ( G R ) (also induced by [ χ ] as in (15) by noticing it is G θ equivariant).Let J θ → A and J θ → a /W ( a ) be the regular centraliser schemes defined through Lemma4.10 and Proposition 4.19. Similarly, we have sheaves of tori T θ → a /W ( a ) and T → t /W defined in (29) and (28) Lemma 5.1.
The group scheme J θ (resp. J θ ) descends to group schemes J θ C (resp. J θ, C )over [ A / C × ] (resp. [ a /W ( a ) / C × ] ) whose respective pullbacks by [ L ] : X → B C × we denoteby J θL and J θ,L . Similar statements hold for T and T θ , which yield group schemes T L , T θ,L over A L ( G R ) .Proof. Descent follows from C × -equivariance of C θ → m reg , C θ → m reg , a × T → a and a × T θ → a . (cid:3) A consequence of the above is that the Hitchin map induces a gerbe structure on thecorresponding stack:
Theorem 5.2.
Let ( G R , H, θ, B ) < ( G, U, τ, B ) be a quasi-split real form of a connectedcomplex reductive algebraic group. Let X be a smooth complex projective curve, and let L → X be an ´etale line bundle. Then,(1) The Hitchin map (31) [ h ] L,θ : Higgs reg,θ ( G R ) → a ⊗ L/W ( a ) is a gerbe banded by the abelian group scheme J θ,L .(2) Moreover, Higgs reg,θ ( G R ) ∼ = BJ θ,L when the degree of L is even.(3) Likewise [ h ] L : Higgs reg ( G R ) → A L := A × [ L ] B C × is a J θL -banded gerbe which isneutral whenever L has even degree.Proof. For (1), note that [ m reg /G θ × C × ] is a gerbe banded by J θ, C . Indeed, it is locallynon empty over [ a /G θ × C × ], since any covering of a /W ( a ) over which [ m reg /H ] is non-empty is a cover for [ a /W ( a ) /G θ × C × ] over which [ m reg /G θ × C × ] is non-empty. Localconnectedness follows in the same way. Clearly, inertia descends to J θ, C , by C × equivarianceof C θ → m reg , which implies that the band is J θ, C . IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 25
On the other hand, the diagram
Higgs reg,θ ( G ) / / ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ [ m /G θ × C × ] ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ ! ! ❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈❈ a ⊗ L/W ( a ) (cid:15) (cid:15) / / [ a /W ( a ) / C × ] (cid:15) (cid:15) X [ L ] / / B C × is Cartesian, so for (31) to define a gerbe it is enough to check non-emptiness, as theremaining structure is preserved by pullback. Local non emptiness is a consequence of localtriviality of L and Corollary 4.20. The statement about the band follows from Lemma 5.1.Statement (2) follows from Theorem 6.13 in [GPR], where the existence of a section ofthe Hitchin map is proved. Note that the construction in [GPR] goes through to the stackysetting.Finally, (3) follows by the same arguments as (1). Neutrality is a consequence of thefact that the Hitchin–Kostant–Rallis section from [GPR] factors through the common atlas( m ⊗ L ) reg . (cid:3) Cameral data.
We next proceed to the description of the fibres of the Hitchin map(31) in terms of cameral data.
Definition 5.3.
We define the cameral cover associated with b ∈ H ( X, a ⊗ L/W ( a )) tobe the ramified W -Galois cover of X fitting in the Cartesian diagram(32) ˆ X b / / p b (cid:15) (cid:15) t ⊗ L π (cid:15) (cid:15) X ι ◦ b / / t ⊗ L/W.
We denote by X b := X × b a ⊗ L , that is, the subscheme of ˆ X b fitting in the Cartesiandiagram(33) X b / / q b (cid:15) (cid:15) a ⊗ L (cid:15) (cid:15) X b / / a ⊗ L/W ( a ) Remark . Note that X b is only a W ( a )-subcover if ramification is determined by W ( a ),namely, if C W ( x ) ⊂ W ( a ) for all x ∈ a . By Remark 4.32, in the simple group case thishappens for split groups and SU( p, p ). Theorem 5.5.
The fibre of the Hitchin map [ h ] L over b ∈ H ( X, a ⊗ L/W ( a )) is given bythe subcategory C am θ of C am consisting of weakly R -twisted, N -shifted principal T -bundlesover ˆ X b admitting a reduction of the structure group to T θ over X b .Proof. Since
Higgs reg,θ ( G R ) is a subgerbe of Higgs reg ( G ), by Theorem 3.8 it is enoughto identify the points of the stack of C am corresponding to N ( G R )-Higgs bundles. In order to do this, by Theorem 5.2 (1) and Theorem 4.31, it is enough to prove that thesubcategory C am θ is a T θ,L banded gerbe that intersects the image of Higgs reg,θ ( G R ) insideof Higgs reg ( G ).We will check non emptiness of C am θ and that the intersection with Higgs reg,θ ( G R ) isnon empty at once, by directly checking that a cameral datum of an element in Higgs reg,θ ( G R )satisfies the required condition.In order to do this, we need a different approach to the cameral cover X b , which is givenby Proposition B.9. According to this, a cameral cover is, ´etale locally, the cover H/N H ( a ) → H/C H ( a ) , where H/C H ( a ) parametrises regular centralisers and H/N H ( a ) ⊂ H/C H ( a ) × G θ /T θ isthe incidence variety. We recall that by Proposition B.6 G θ /T θ parametrises θ -anisotropicBorel subgroups.Recall now [DG] that locally, the cameral datum is constructed by pullback to G/T ofthe torus bundle
G/U → G/B.
Over the subvariety G θ /T θ , we have the principal T θ -bundle G θ → G θ /T θ . This fits in a commutative diagram G θ (cid:31) (cid:127) / / (cid:15) (cid:15) G/U (cid:15) (cid:15) G θ /T θ (cid:31) (cid:127) j / / G/B.
Thus G θ ( T ) ∼ = j ∗ G/U . Since the universal cameral datum is pullbacked from
G/U , Propo-sition B.9 allows us to conclude that
Higgs reg,θ ( G R ) ∩ C am θ = ∅ .Now, by local connectedness of C am it is enough to compute the automorphism sheaf ofelements in C am θ . These are clearly sections of T L which over the smaller cameral coverreduce to automorphisms of the T θ -bundle, namely, sections of T θ,L . (cid:3) Cameral data for fixed points under involutions.
We have already observed(cf. Proposition 4.22) that a natural way to obtain candidates for G R -Higgs bundles interms of cameral data is to understand the involution on C am induced from the involutionon Higgs reg ( G ) sending(34) Θ : ( E, φ ) ( E × θ G, − θ ( φ )) . We start by some Lie theoretic preliminaries. Let
B < G be a θ -anisotropic Borelsubgroup (cf. Definition B.5). Note that B is associated to an ordering of the roots suchthat a ∗ > i d ∗ .The action of − θ on g is induces an action: − θ : G/B −→ G/B, b b − θ (35) − θ : G/N −→ G/N , c c − θ (36) − θ : G/T −→ G/T , ( c , b ) ( c − θ , b − θ ) , (37)which makes the universal cameral cover G/T → G/N − θ equivariant. IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 27
Lemma 5.6.
We have
G/B − θ ∼ = G θ /T θ ∼ = H/C H ( a ) . In particular: m reg × H/N H ( a ) G/T − θ ∼ = m reg × H/N H ( a ) H/C H ( a ) . Proof.
Note that the second statement is automatic from the first one, θ -equivariance ofthe morphism g reg → G/N and Proposition B.6.Now, the involution θ acts on the fixed set of roots by inverting the order. By expressing b = t ⊕ M λ ∈ Λ( a ) + g λ , we may identify the action of − θ on G/B by(38) − θ : gB g θ B. Indeed, this follows from the fact that θ exchanges t root spaces by g α g θα . Thusfixed points of G/B consist of Borel subalgebras corresponding to cosets gB such that g θ B = gB , namely, such that g − g θ ∈ B . We need to check that this is equivalent tosaying that ( gB ) θ = ( gB ) op , i.e., that g θ B θ ∩ gB = gT. Now, B θ = B op , thus g θ B op ∩ gB = gT ⇐⇒ g − g θ B op ∩ B = T ⇐⇒ g − g θ ∈ B . (cid:3) Remark . From the above proof we deduce that g − g θ B = B ⇐⇒ g − g θ ∈ T . Lemma 5.8.
Let ( E, φ ) ∈ Higgs reg ( G ) have associated cameral cover ˆ X b for some b ∈ H ( X, t ⊗ L/W ) . Then, the associated cameral cover to Θ( E, φ ) is ˆ X − θ ( b ) = X × b ( − θ ) ∗ t ⊗ L, where − θ : t ⊗ L/W → t ⊗ L/W is defined by − θπ ( x ) := π ( − θx ) for x ∈ t ⊗ L , π as in (32) . In particular, if b ∈ H ( X, a ⊗ L/W ( a )) then we have an involutive isomorphism of W -covers ˆ X b ∼ = X × b ( − θ ) ∗ t ⊗ L. Equivalently, if ( E, σ ) is the associated abstract Higgs bundle associated with ( E, φ ) , with σ : E → G/N given by σ ( x, e ) = c ( φ ( x )) , then ˆ X σ − θ := X × σ − θ G/T ∼ = X × σ ( − θ ) ∗ G/T .
Proof.
By Fact 3.2, cameral covers can be defined in either of these ways. By Lemma 5.6,the choice of the involutions is compatible, and so both statements are equivalent, so it isenough to prove one of them, for instance, the first, which follows by definition. (cid:3)
In a similar way, on the stack of cameral data C am over A L ( G ) (see Definition 3.7), weconsider the following involution: given an R -twisted, N -shifted W -equivariant principal T -bundles ( P, γ, β ) over b S b , we assign to it(39) Θ : ( P, γ, β ) (( − ˜ θ ) ∗ b P, ( − ˜ θ ) ∗ b γ, ( − ˜ θ ) ∗ b β ) . In the above, ( − ˜ θ ) b is locally given by universal version in the Cartesian diagram(40) ( − θ ) ∗ G/T − ˜ θ / / (cid:15) (cid:15) G/T (cid:15) (cid:15)
G/N − θ / / G/N .
Theorem 5.9.
The equivalence
Higgs reg ( G ) ∼ = C am is Θ -equivariant. In particular, theinvolution (39) is well defined and induces an equivalence Higgs reg ( G ) Θ ∼ = C am Θ . Wemay describe (41) C am Θ ∼ = { ( P, γ, β ) : P × θ T ∼ = η ∗ P } , where η : ˆ X b → ˆ X b is the involution naturally induced from the isomorphism ( E, φ ) ∼ =Θ( E, φ ) .Proof. In order to check Θ-equivariance, note first that the involution is well defined.Indeed, it is enough to check this on the universal cameral datum (14). The fact that theprincipal T bundle transforms in the stipulated way follows by definition. To see the way γ and β transform, we note that − ˜ θ is induced from ( − θ, id ) y G/N × G/B , and thus itis W -equivariant and exchanges the action of g and g θ on the first factor. Hence ( − ˜ θ ) ∗ γ induces an isomorphismΘ( P ) ∼ = ( − ˜ θ ) ∗ ( w ∗ P × w T ) ⊗ ( − ˜ θ ) ∗ R w = w ∗ ( − ˜ θ ) ∗ P × w T ⊗ R w . Finally, let β := ( β i ) with β i ( n i ) : α i ( P ) | D αi ∼ = O ( D α i ) ) | D αi for all n i ∈ N a lift of thesimple root α i . Then, W -equivariance (induced from G -equivariance on the G/B factor of
G/T ) allows us to conclude.Now, since cameral data are fully determined by the universal version once an abstractcameral cover has been assigned, equivariance is clear from Lemma 5.8.Let us now prove (41). Note that the cameral cover associated to a fixed point can beobtained in two ways as illustrated by the following diagram:(42) ˆ X b ˜ σ θ / / (cid:15) (cid:15) η (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ G/T (cid:15) (cid:15) ( − θ, − θ ) " " ❋❋❋❋❋❋❋❋ ˆ X b (cid:15) (cid:15) ˜ σ / / G/T (cid:15) (cid:15) X ❄❄❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄❄❄ − θφ / / E θ ( g ) ⊗ L − θ & & ▼▼▼▼▼▼▼▼▼▼▼ c / / G/N − θ " " ❋❋❋❋❋❋❋❋ X φ / / E ( g ) ⊗ L c / / G/N .
The fact that the rightmost top arrow is given by ( − θ, − θ ) follows from the fact that it isan isomorphism of universal cameral covers making the diagram commute.Now, we note that P univ = ( − θ, − θ ) ∗ P univ × θ T . Indeed, this is a consequence of( G × W, G θ × W θ ) equivariance of ( − θ, − θ ) y G/N × G/B . Thus, assume Θ(
E, φ ) ∼ = ( E, φ ). IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 29
Then, on the one hand, the cameral datum for Θ(
E, φ ) is η ∗ ˜ σ ∗ P univ (because it is a fixedpoint). On the other hand, it is simply ˜ σ θ, ∗ P univ = ˜ σ ( − θ, − θ ) ∗ P univ × θ T which bycommutativity of the top square in (42) equals η ∗ ˜ σ θ, ∗ P univ × θ T . Hence the statementfollows. (cid:3) As a corollary we obtain some simple consequences:
Corollary 5.10.
Let θ be the involution produced from the split real form. Then C am Θ are the order two points of C am .Proof. Note that for split real forms one has a = t , so that η = id (by Proposition B.9)and θ is − T . Hence, fixed points are principal bundles on ˆ X b (for all b ) such that P − = P × θ T = P. (cid:3) Proposition 5.11.
Let G be a simple Lie group. Let ( E, φ ) ∈ Higgs reg ( G ) Θ be a stable Higgs bundle. Then, ( E, φ ) ∈ Higgs reg ( G R ) .Proof. Let (
E, φ ) ∈ Higgs reg ( G ) Θ be a stable bundle. By stability , it is simple and by[GR, Proposition 7.5] the structure group reduces to a real form with involution in theInt( G ) orbit of θ . By regularity, the form must be quasi-split. But there are at most twonon-isomorphic quasi-split real forms, corresponding to an outer involution (the split realform, which always exists) and an inner involution (which may or may not exist). Sinceboth cases are exclusive, the result follows. (cid:3) Appendix A. Lie theory
In this section we summarize the main results about real reductive groups. The referencesare [Kn, KR, GPR].A reductive real Lie group G R is a Lie group in the sense of [Kn, § VII.2, p.446], that is,a tuple ( G R , H R , θ, h · , · i ), where H R ⊂ G R is a maximal compact subgroup, θ : g R → g R isa Cartan involution and h · , · i is a non-degenerate bilinear form on g R , which is Ad( G R )-and θ -invariant, satisfying natural compatibility conditions. Definition A.1. A real reductive group is a 4-tuple ( G R , H R , θ, h · , · i ) where(1) G R is a real Lie group with reductive Lie algebra g R .(2) H R < G R is a maximal compact subgroup.(3) θ is a Lie algebra involution of g R inducing an eigenspace decomposition g R = h R ⊕ m R where h R = Lie( H R ) is the (+1)-eigenspace for the action of θ , and m R is the( − h · , · i is a θ - and Ad( G R )-invariant non-degenerate bilinear form, with respect towhich h R ⊥ h · , · i m R and h · , · i is negative definite on h R and positive definite on m R .(5) The multiplication map H R × exp( m R ) → G R is a diffeomorphism.(6) G R acts by inner automorphisms on the complexification g of its Lie algebra viathe adjoint representation The group H R acts linearly on m R through the adjoint representation of G R — this isthe isotropy representation that we complexify to obtain a representation (also referred asisotropy representation) ρ i : H → GL( m ) . Let g := g CR , and similarly for m and h . Let m reg denote the set of regular elements of m , namely, elements with maximal isotropy orbits,and by m reg,ss the set of regular semisimple elements.Let t = d ⊕ a be a θ -invariant Cartan subalgebra of g (where θ denotes the extension bycomplex linarization of the Cartan involution of g R ), with a = t ∩ m maximal and d = t ∩ h . Proposition A.2. [KR, Theorem 1]
Let x ∈ m be semisimple. Then x is H -conjugate toan element of a . The above proposition fails to be true for non semisimple elements of m . Proposition A.3. [KR, Theorem 9]
The space a /W ( a ) classifies (Ad( G )) θ orbits in m reg . Proposition A.4.
There exists a section of m reg → a /W ( a ) intersecting each G θ -orbit atexactly one point.Proof. This is Theorem 11 in [KR] (adjoint group case) and a consequence of the formerand Theorem 4.6 in [GPR] for the general case. (cid:3)
We include a here more careful study of N G ( G R ). Lemma A.5.
Let ( G R , H, θ, B ) < ( G, U, τ, B ) be a real form of a complex reductive Liegroup. Let A u = e i a ⊂ U . Then, we have a short exact sequence → H A u H → U → π ( U ) → where H denotes the connected component of H and π ( U ) is the group of connectedcomponents of U .Proof. To see that H A u H = U , we note that both subgroups have the same Lie algebra,as u = h ⊕ i m and m = ∪ h ∈ H Ad( h ) a by Proposition 7.29 in [Kn], and clearly H A u H ⊆ U . (cid:3) Appendix B. The geometry of regular centralisers
In this section we explain the main features about regular centraliser schemes for realgroups. This yields to a natural description of cameral covers in terms of anisotropic Borelsubgroups (cf. Proposition B.9).Let g R be a real reductive Lie subalgebra with complexification g . Let θ denote theCartan subalgebra of g R or its extension by complex linearization. Let t = d ⊕ a ⊂ g be a θ -invariant Cartan subalgebra of g , with a = t ∩ m maximal and d = t ∩ h . We let S ⊂ ∆ = ∆( g , t ) be the associated sets of (simple) roots, W the Weyl group. Similarly,we can define Σ( a ) the set of restricted roots associated to a (which is the image of therestriction map res | a : ∆ → a ∗ ). This is a root system (possibly non reduced). Reflectionwith respect to simple such roots generates a group W ( a ) called the restricted Weyl group.This group is also isomorphic to N H ( a ) /C H ( a ) and N H R ( a R ) /C H R ( a R ), where a R = a ∩ g R .Moreover, any root α can be expressed as α = λ + iβ with λ ∈ a ∗ R , β ∈ d ∗ R .Let a = dim a . Denote by Ab a ( m ) the closed subvariety of Gr ( a, m ) whose points areabelian subalgebras of m . Define the incidence variety(43) µ reg = { ( x, c ) ∈ m reg × Ab a ( m ) : x ∈ c } . We have the following.
IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 31
Proposition B.1.
The map ψ : m reg → Ab a ( m ) x z m ( x ) is smooth with smooth image and its graph is µ reg .Proof. First of all, note that the map is well defined: indeed, it is clear for regular andsemsimimple elements in m . By Theorem 20 and Lemma 21 of [KR], it extends to thewhole of m reg . As for smoothness, the proof of [DG, Proposition 1.3] adapts as follows.We check that ψ is well defined and has graph µ reg by proving that µ reg → m reg is anembedding (hence, by properness and surjectivity, an isomorphism). To see this, as m reg isreduced and irreducible (being a dense open set of a vector space), if the fibres are reducedpoints we will be done. We have that T ( x, b m ) ( µ reg ∩ { x } × Ab a ( m )) ∼ = (cid:26) T : b → m / b (cid:12)(cid:12)(cid:12)(cid:12) [ T ( y ) , x ] = 0 for any y ∈ b T [ y, z ] = [ T y, z ] + [ y, T z ] for any y, z ∈ b (cid:27) . By definition, the only T satisfying those conditions is T ≡
0, so the map is well defined.For smoothness, given a closed point x ∈ m reg , as m reg ⊂ m is open and dense, it fol-lows that T x m reg ∼ = m . Consider T x m reg ∼ = m d x ψ / / T z m Ab a ( m ) ev x / / m / z m ( x ) y / / { T : [ T ( z ) , x ] = [ − z, y ] } / / T ( x ) = [ y ] . Namely, d x ψ sends y to the only map satisfying [ T ( z ) , x ] = [ − z, y ]. Now, clearly ev x ◦ d x ψ is the projection map m → m / z m ( x ). Also, ev x is surjective. We will prove it is injective,so it will follow that Im ( ψ ) is contained in the smooth locus of Ab a ( m ). The same factproves that d x ψ must be surjective, and so we will be done.Suppose T ( x ) = T ′ ( x ) for some T, T ′ ∈ T z m ( x ) Ab a ( m ). Then:0 = [ T ( x ) − T ′ ( x ) , y ] = [ − x, T ( y )] − [ − x, T ′ ( y )] = [ − x, T − T ′ ( y )]for all y ∈ z m ( x ), and hence ev is injective. (cid:3) Definition B.2.
We will call the image of ψ the variety of regular centralisers, and denoteit by H/N H ( a ). Remark
B.3 . H/N H ( a ) ⊂ H/N H ( a ) is an open subvariety consisting of the image of m reg,ss .This coincides with G θ /N G θ ( a ), where N G θ ( a ) denotes the normaliser of a in G θ , since thelatter group normalizes m reg and semisimple orbits are the same for H and G θ . Remark
B.4 . Note that ψ is G θ -equivariant for the isotropy representation on m reg andconjugation on Ab a ( m reg ).In what follows we present an alternative approach to cameral covers, following [DG]. Definition B.5.
A parabolic subalgebra p ⊂ g is called minimal θ -anisotropic if it isopposed to θ ( p ) and contains a θ -invariant Cartan subalgebra with maximal intersectionwith m . Namely, such objects are G θ or H conjugate to parabolic subalgebras of the form p = a ⊕ c h ( a ) ⊕ M λ ∈ Λ( a ) + g λ , where Λ( a ) + denotes a set of positive restricted roots. When g R ⊂ g is quasi-split, sinceminimal θ -anisotropic parabolic subalgebras are Borel subalgebras. Proposition B.6.
The variety
H/C H ( a ) ∼ = G θ /C H ( a ) θ parametrises minimal θ -anisotropicparabolic subalgebras of g .Proof. The fact that
H/C H ( a ) parametrises minimal θ -anisotropic parabolic subalgebrasof g follows from [Vu, Proposition 5]. The isomorphism is a consequence of the morphismexact sequences defined by the following commutative diagram: C H ( a ) / / (cid:15) (cid:15) C H ( a ) θ / / (cid:15) (cid:15) F H / / G θ / / F (cid:3) Proposition B.7.
Any element x ∈ m reg satisfies that c m ( x ) ⊂ p for some p ∈ H/C H ( a ) .Proof. By [KR], any such element belongs to ˜ g , where ˜ g ⊂ g is a maximal split subalgebra,inside of which its centraliser remains the same. Thus, for some Borel subalgebra b ⊂ ˜ g ,which may be easily chosen to be θ -invariant, c m ( x ) ⊂ b . Since any such can be promotedto a minimal θ -anisotropic parabolic subalgebra, we are done. (cid:3) Definition B.8.
Define the group scheme
H/C H ( a ) to be the incidence variety inside H/N H ( a ) × H/C H ( a ). Proposition B.9.
Let G R < G be a quasi-split real form. Then, the choice of a θ -anisotropic Borel subalgebra a ⊂ b determines an isomorphism m reg × H/N H ( a ) H/C H ( a ) ∼ = m reg × a /W ( a ) a . Proof.
By regularity of m reg it is enough to prove that both schemes admit a morphismwhich makes them isomorphic over a dense open set. Define the morphism m reg × H/N H ( a ) H/C H ( a ) ∋ ( x, b ) ( x, π b ( x )) ∈ m reg × a /W ( a ) a where π b ( x ) ∈ a is the image of the class of x in b / [ b , b ] under the canonical isomorphism b / [ b , b ] ∼ = a ⊕ c h ( a ), which is θ equivariant and so well defined.By Remark B.3, over H/N H ( a ), the above morphism is an isomorphism, as both schemesare W ( a )-principal bundles over m reg,ss . (cid:3) IGGS BUNDLES, ABELIAN GERBES AND CAMERAL DATA 33
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