The Jordan--Chevalley decomposition for G -bundles on elliptic curves
aa r X i v : . [ m a t h . AG ] J u l THE JORDAN–CHEVALLEY DECOMPOSITION FOR G -BUNDLES ON ELLIPTIC CURVES DRAGOS FRATILA, SAM GUNNINGHAM, AND PENGHUI LI
Abstract.
We study the moduli stack of degree 0 semistable G -bundles onan irreducible curve E of arithmetic genus 1, where G is a connected reduc-tive group. Our main result describes a partition of this stack indexed by acertain family of connected reductive subgroups H of G (the E -pseudo-Levisubgroups), where each stratum is computed in terms of bundles on H togetherwith the action of the relative Weyl group. We show that this result is equiva-lent to a Jordan–Chevalley theorem for such bundles equipped with a framingat a fixed basepoint. In the case where E has a single cusp (respectively, node),this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g (respectively, group G ).We also provide a Tannakian description of these moduli stacks and useit to show that if E is an ordinary elliptic curve, the collection of framedunipotent bundles on E is equivariantly isomorphic to the unipotent conein G . Finally, we classify the E -pseudo-Levi subgroups using the Borel–deSiebenthal algorithm, and compute some explicit examples.
1. Introduction 21.1. Overview and main results 21.2. Motivation: the geometric Langlands program and elliptic Springertheory 71.3. Acknowledgements 112. Preliminaries 112.1. Notation and generalities on G -bundles 112.2. Semistability 122.3. Moduli spaces and stacks 132.4. Reductions of the structure group 132.5. Framed bundles 142.6. The coarse moduli space and the characteristic polynomial map 183. The partition of semisimple bundles by Lusztig type 193.1. E -pseudo-Levi subgroups 193.2. Borel–de Siebenthal theory 203.3. E -root subsystems 213.4. Connecting E -pseudo-Levis and E -root subsystems of roots 223.5. The component group 233.6. Closure relations for the partition 243.7. Summary of section 254. The Jordan–Chevalley Theorem 254.1. The regular locus 254.2. The main results 294.3. Equivalence of Theorems 4.7 and 4.8 30 Introduction
Overview and main results.
Fix an irreducible projective curve E of arith-metic genus 1 over an algebraically closed field k . There are three possibilities:(1) E has a single ordinary cusp;(2) E has a single node;(3) E is smooth.We also fix a basepoint x in the smooth locus of E . We will refer to these casesas the cuspidal , nodal , or elliptic cases respectively. Fix also a connected reductive group G over k . We consider the moduli stack G E := Bun , ss G ( E )of degree 0, semistable G -bundles over E . We denote by G E the moduli stack ofsuch bundles P together with a framing - that is, a trivialization of the fiber P x .We will see that G E is a smooth algebraic variety with an action of G (changingthe trivialization), and that G E = G E /G . In fact, by results of Friedman–Morgan[FM01], we have: • G E ∼ = g = Lie( G ) if E is cuspidal, and • G E ∼ = G if E is nodal.In these cases, the action of G on G E corresponds to the adjoint/conjugation action.In the elliptic case, we will see that the stack G E shares many of the properties ofthe adjoint quotient stacks g /G and G/G . Note that in the first two cases, unlike the third, the pair (
E, x ) has no moduli. EOMETRY OF SEMISTABLE G -BUNDLES 3 The Jordan–Chevalley Decomposition.
One of our main results is a formof the Jordan–Chevalley decomposition for G E , which recovers the usual Jordan–Chevalley decomposition for g and G in the cuspidal and nodal cases respectively.An element p ∈ G E is called semisimple if its G -orbit is closed. We say thatan element p ∈ G E is unipotent if its orbit closure contains the trivial bundle p . We denote by G uni E the subvariety of unipotent elements of G E and by G uni E the corresponding substack in G E . We note that these definitions recover the usualnotion of semisimple and unipotent/nilpotent elements in the nodal/cuspidal cases.The Jordan–Chevalley decomposition essentially states that every framed bundle p ∈ G E can be uniquely decomposed as p = p s · p u where p s is semisimple, p u isunipotent, and p s commutes with p u . As it stands this statement is not well-formedas it is not clear what it means to “multiply” elements in G E (the usual statement ofJordan decomposition in the nodal and cuspidal cases uses multiplication in G andaddition in g ). However, it does make sense to multiply a Z ( G )-bundle P ′ and a G -bundle P : we define P ′ ·P to be the bundle induced from the external product P ′ × E P via the multiplication map Z ( G ) × G → G (which is a group homomorphism).Moreover, this construction is naturally compatible with framings, giving rise to anabelian algebraic group structure on Z ( G ) E and an action of Z ( G ) E on G E .We will also show that for a reductive subgroup H of G , the induction map onframed semistable bundles H E → G E is a closed embedding (see Proposition 2.15).Therefore we can identify H E with the corresponding closed subvariety of G E . Weare now ready to state the first result, a form of Jordan–Chevalley decomposition: Theorem 1.1. (see Theorem 4.7) Given p ∈ G E , there is a unique triple ( H, p s , p u ) ,where H is a connected reductive subgroup of G , p s ∈ Z ( H ) E with Stab G ( p s ) ◦ = H , p u ∈ H uni E , and p = p s · p u . The subgroups of G which occur as connected stabilizers of semisimple elements of G E will be called E -pseudo-Levi subgroups . In the cuspidal case, these are preciselythe Levi subgroups (centralizers of semisimple elements of g ) and in the nodal case,these are pseudo-Levi subgroups (connected centralizers of semisimple elements of G ). We give a classification of E -pseudo-Levi subgroups in Appendix A: in theelliptic case we get precisely the intersections of two pseudo-Levi subgroups. Remark 1.2.
For simply connected groups over C , a similar Jordan–Chevalley de-composition was proved in [BEG03, Theorem 5.6] using an algebraic uniformizationof G E through loop groups (see [BG96]).1.1.2. The partition according to E -pseudo-Levis. Typically, in the statements ofthe Jordan–Chevalley decomposition for G and for g , the subgroup H is not explic-itly mentioned, though it may be easily recovered as the connected centralizer ofthe semisimple element.However, in our proof of Theorem 1.1, the subgroup H will be the key player andthe decomposition in to semisimple and unipotent elements will play a subsiduaryrole. In fact, the semisimple and unipotent elements may be recovered from thesubgroup H in the following sense. DRAGOS FRATILA, SAM GUNNINGHAM, AND PENGHUI LI
Given a subgroup H of G we define the loci( Z ( H ) E ) reg = { p ∈ Z ( H ) E | Stab G ( p ) ◦ = H } ⊆ Z ( H ) E and ( H E ) reg ♥ = ( Z ( H ) E ) reg · H uni E ⊆ H E . The subgroup H is an E -pseudo-Levi precisely when ( Z ( H ) E ) reg is nonempty. Notethat the conditions in Theorem 1.1 are precisely stipulating that p ∈ ( H E ) reg ♥ . Onthe other hand, any element p ∈ ( H E ) reg ♥ may be uniquely written as a product p s · p u where p s ∈ ( Z ( H ) E ) reg and p u ∈ H uni E (see Proposition 4.9). We denote by( G E ) H , the image of ( H E ) reg ♥ in G E .In other words, Theorem 1.1 states that every element p ∈ G E lies in the imageof ( H E ) reg ♥ for a unique E -pseudo-Levi subgroup H . Thus we may rephrase Theo-rem 1.1 as follows: Theorem 1.3.
There is a locally closed partition: G E = G H ( G E ) H indexed by E -pseudo-Levi subgroups H ⊆ G . Moreover, the natural embeddings H E → G E restrict to isomorphisms ( H E ) reg ♥ ∼ −→ ( G E ) H . We will reformulate this result in one final way (in the form that will actually beproved in Section 4). Recall that G E denotes the quotient stack G E /G , and write( H E ) reg ♥ for ( H E ) reg ♥ /H . We write W G,H for the relative Weyl group N G ( H ) /H (afinite group). Theorem 1.4.
The stack G E carries a locally closed partition G E = G [ H ] ( G E ) [ H ] indexed by conjugacy classes [ H ] of E -pseudo-Levi subgroups H ⊆ G . Moreover,the natural induction maps H E → G E restrict to equivalences ( H E ) reg ♥ /W G,H ∼ −→ ( G E ) [ H ] . Our proof of Theorem 1.4 (and hence of Theorem 1.1 and Theorem 1.3) involves ageometric analysis of the induction map H E → G E (see Section 4.2). Remark 1.5.
A key difference between the statements of Theorem 1.4 and Theo-rem 1.3 is that, unlike the subvarieties ( G E ) H in G E , the substacks ( G E ) [ H ] havean a priori definition that doesn’t make reference to the induction map H E → G E .More precisely, given a G -bundle P ∈ G E , pick a framed lift p ∈ G E and then chooseany semisimple bundle in the closure of its G -orbit. The underlying G -bundle de-fines a conjugacy class [ H ] that is independent of the chosen framing and is hencecanonically associated to P . The content of Theorem 1.3 is that, in the presenceof a framing, there is a canonical choice of subgroup H within its conjugacy class,and (equivalently) a canonical choice of semisimple element p s in the orbit closureof p . EOMETRY OF SEMISTABLE G -BUNDLES 5 Unipotent bundles.
Theorem 1.4 and Theorem 1.1 allow us to reduce thestudy of degree 0, semistable G -bundles on E to semisimple and unipotent bundles.Our next result shows that, under certain hypotheses, the collection of unipotentbundles in G E is insensitive to the isomorphism type of E . We let J ( E ) denote theJacobian of E , which is either isomorphic to G a , G m , or E itself in the cuspidal,nodal, or elliptic cases respectively. We denote by G uni the unipotent cone of G (which is the same as G uni E for E a nodal curve). Theorem 1.6.
An isomorphism of formal group ˆ J ( E ) ≃ ˆ G m induces an isomor-phism of G -varieties G uni E ∼ = G uni . Moreover this isomorphism extends over a formal neighbourhood of G uni E in G E : ( G E ) ∧ uni ∼ = G ∧ uni . Remark 1.7.
In characteristic zero, there is always an isomorphism ˆ J ( E ) ≃ ˆ G m .In fact, there is also an isomorphism ˆ J ( E ) ≃ ˆ G a which gives the same result butwith the unipotent cone in G replaced by the nilpotent cone N in g .In characteristic p >
0, if E is an elliptic curve, an isomorphism ˆ J ( E ) ≃ ˆ G m existsprecisely when E is ordinary (i.e. not supersingular). Remark 1.8.
As a special case of Theorem 1.6 we recover a G -equivariant iso-morphism N ∼ = U between the unipotent and nilpotent cones for each isomorphismˆ G a ∼ = ˆ G m . The latter isomorphisms exist only in characteristic zero, and are givenby exponential maps. On the other hand, there exist G -equivariant isomorphisms(the so-called Springer isomorphisms) N ∼ = U under very mild conditions on thecharacteristic, even though ˆ G a ≇ ˆ G m in positive characteristic. It seems reasonableto expect that G uni E is isomorphic to U (and N ) under much more general condi-tions than in Theorem 1.6. From [GSB19, Theorem 3.11] one can deduce that thevarieties G uni E and G uni are smoothly equivalent for uniformizable elliptic curves un-der some restrictions on G and the characteristic. See also [GSB19, Corollary 8.8]where they show it fails for G = E , E supersingular in characteristic 2,3 or 5.1.1.4. Semisimple bundles and the classification of E -pseudo-Levi subgroups. Fixa maximal torus T of G and let Φ ⊂ X ∗ denote the corresponding space of rootssitting inside the character lattice X ∗ = X ∗ ( T ). Then T E ∼ = Hom gp ( X ∗ , J ( E )) ∼ = J ( E ) r where r is the rank of T .Note that a G -bundle P ∈ G E is semisimple if and only if it admits a reduction to T .We may understand the partition in to E -pseudo-Levi subgroups root theoreticallyas follows.First note that any character α ∈ X ∗ defines a homomorphism α ∗ : T E → J ( E ),taking a T -bundle on E to its induced line bundle via α . Given p ∈ T E , we letΣ p = { α ∈ T E | α ∗ ( p ) = J ( E ) } . The subsets Σ of Φ which occur in this way willbe called E -root subsystems of Φ. DRAGOS FRATILA, SAM GUNNINGHAM, AND PENGHUI LI
It turns out that any such E -root subsystem Σ ⊂ Φ is a closed root subsystem andso it corresponds to a connected reductive subgroup H of G (see also Section 3.2for Borel–de Siebenthal theory). In fact, we have H = Stab G ( p ) ◦ and Proposition 1.9.
There is a one-to-one correspondence between: • E -pseudo-Levi subgroups H of G containing T , and • E -root subsystems Σ ⊆ Φ .Moreover, in each of the cases cuspidal, nodal and elliptic one can characterizeprecisely the E -root subsystems of Φ (see the Appendix and Proposition A.8 for theelliptic case). Remark 1.10.
The theory of semisimple bundles becomes increasingly complicatedas one passes from the cuspidal to the nodal and then to the elliptic cases. Forexample, the centralizer of a semisimple element of g is a Levi subgroup, and inparticular connected. The centralizer of a semisimple element in G is connected(but not necessarily simply-connected) whenever G is simply connected. On theother hand, the automorphism group of a semisimple G -bundle P ∈ G E where E issmooth may be disconnected, even if G is simply connected! An example is givenfor G of type D , see Example 6.6 (this example also appears in [BEG03, p. 18]).Fortunately, our result provides control over the component groups of automor-phisms of semisimple bundles in terms of Weyl group combinatorics (just as Lusztig’sstratification does in the group case). More precisely, let p ∈ T E with Σ p = Σ forsome E -root subsystem Σ of Φ, and let P G be the induced G -bundle. Then Theo-rem 1.4 implies that π Aut( P G ) ∼ = Stab N W (Σ) ( p ) /W Σ where N W (Σ) = { w ∈ W | w (Σ) = Σ } .1.1.5. The Lusztig stratification.
Putting these results together, we can refine thepartition of G E in Theorem 1.4 as follows. Corollary 1.11.
Suppose either that char ( k ) = 0 , or that E is ordinary. There isa stratification G E = G [ H, O ] ( G E ) [ H, O ] indexed by G -conjugacy classes of pairs ( H, O ) where H is an E -pseudo-Levi sub-group of G , and O ⊆ H uni E is a unipotent H -orbit. For each such pair [ H, O ] wehave an isomorphism: ( G E ) [ H, O ] ∼ = ( Z ( H ) reg E × O ) /N G ( H ) ∼ = ( Z ( H ) reg E × O ) /W G,H . Remark 1.12.
To make the comparison with Lusztig’s work more evident, notethat to each E -pseudo-Levi subgroup H , one can associate a Levi L = L H = C G ( Z ( H ) ◦ ). This is the smallest Levi subgroup which contains H . A bundle P ∈ G E is called isolated if it is contained in ( G E ) [ H ] and H is not containedin any proper Levi. (It follows from Proposition 1.9 that there are finitely manyisomorphism classes of isolated bundles.) Instead of parameterizing the strata usingsubgroups of elliptic type and unipotent bundles, one may use Levi subgroups andisolated bundles. This is how Lusztig describes the stratification of G in [Lus84]. EOMETRY OF SEMISTABLE G -BUNDLES 7 Motivation: the geometric Langlands program and elliptic Springertheory.
Global geometric Langlands.
Recall that in the global geometric Langlandsprogram, one aims to describe the derived category D (Bun G ( C )) of D -modules orconstructible sheaves on the moduli stack of G -bundles on a smooth projective curve C in terms of the derived category of quasi-coherent (or more general Ind-coherent)sheaves on the moduli stack of L G -local systems, where L G is the Langlands dualgroup to G .Recall that for every parabolic subgroup P ⊆ G with Levi factor L , we have Eisen-stein and constant term functors:Eis
GL,P : D (Bun L ( C )) ⇆ D (Bun G ( C )) : CT GL,P
Generally speaking, these functors are defined via a pull-push construction involvingthe diagram: Bun P ( C ) p & & ▼▼▼▼▼▼▼▼▼▼ q x x qqqqqqqqqq Bun L ( C ) Bun G ( C ) (1.1)An object of D (Bun G ( C )) is called cuspidal if it is in the kernel of the constantterm functor for every proper parabolic subgroup P of G . As per Harish-Chandra’sphilosophy of cusp forms, one may think of the category D (Bun G ( C )) as built upfrom Eisenstein series of cuspidal objects in D (Bun L ( C )) as L ranges over Levisubgroups of G . In this way one can hope to understand the category D (Bun G ( C ))inductively in terms of smaller reductive groups.Typically, the Eisenstein series from different Levi subgroups will interact in a com-plicated way, and this does not lead to a straightforward description of D (Bun G ( C )).We will now describe a closely related category for which the Harish-Chandra ap-proach yields a complete description.1.2.2. Springer Theory and Character Sheaves.
Let us turn to the category D ( G/G )of “class sheaves” on G , i.e. conjugation equivariant constructible sheaves or D -modules on the group G . Analogous to the Eisenstein and constant term functorsabove, one has functors of parabolic induction and restriction:Ind GP,L : D ( L/L ) ⇆ D ( G/G ) : Res
GP,L defined by pull-push along the diagram:
P/P ❋❋❋❋❋❋❋❋ | | ②②②②②②②② L/L G/G. (1.2) More precisely, one must consider a fiberwise compactification of Bun P ( C ) over Bun G ( C ).One must also specify which functors (star or shriek) are employed for this process. We willignore these distinctions for the purposes of this informal discussion. DRAGOS FRATILA, SAM GUNNINGHAM, AND PENGHUI LI
Just as in Section 1.2.1 one defines the cuspidal objects as those whose parabolicrestriction to any proper Levi is zero. Again, one aims to describe the category D ( G/G ) in terms of parabolic induction from cuspidal objects in D ( L/L ) as L ranges over Levi subgroups.In his seminal paper [Lus84], Lusztig obtained a block decomposition of the categoryof equivariant perverse sheaves on the unipotent cone G uni ⊆ G :Perv G ( G uni ) = M ( L, O , E ) Rep( W G,L ) . (1.3)Here, the blocks are indexed by cuspidal data , consisting of a pair of a Levi subgroup L together with a simple cuspidal local system E on the unipotent cone of L . Theclassical Springer correspondence for representations of the Weyl group W = W G,T is recovered inside of Eq. (1.3) as one of these blocks, corresponding to the uniquecuspidal datum with L = T , a maximal torus.More recently, more general forms of the decomposition Eq. (1.3) have been ob-tained for the derived category of nilpotent orbital sheaves by Rider–Russell [RR16],the category of equivariant D -modules on the Lie algebra g by S.G. [Gun18, Gun17]and the derived category of character sheaves by P.L. [Li18].In the the setting of D -modules on g /G or sheaves supported on the unipotentcone of G , the set of cuspidal data indexing the decomposition is the same as inEq. (1.3). We will call this set the unipotent cuspidal data for G . In the caseof character sheaves on G/G , however, this set must be expanded to account forunipotent cuspidal data for pseudo-Levi subgroups H of G . More generally, onecan consider E -cuspidal data for any arithmetic genus 1 curve as we explain furthernow.1.2.3. Elliptic Springer theory.
The main object of study in this paper is the stack G E ⊆ Bun G ( E ) of semistable G -bundles on a curve of arithmetic genus 1.The stack G E which we study in the present paper forms a bridge between thesituations described in Section 1.2.1 and Section 1.2.2. On the one hand G E sitsinside Bun G ( E ) as the locus of degree 0 semistable bundles. On the other hand,when E is taken to be a nodal curve one has an isomorphism G E ∼ = G/G .Viewing the category D ( G E ) through the lens of Section 1.2.2, one defines functorsof parabolic induction and restriction using the correspondence L E ← P E → G E and study the corresponding Harish-Chandra (or generalized Springer decomposi-tion). The ordinary Springer correspondence in the elliptic setting was studied byBen-Zvi and Nadler [BN13].One can also formulate a generalized Springer correspondence for the (variousflavours of) category D ( G E ) which recovers the standard patterns for orbital andcharacter sheaves in the cuspidal and nodal cases. This will be expanded on infuture work; for now, let us just note that the indexing set of the generalizedSpringer decomposition (which we are calling E -cuspidal data) involves a choiceof an E -pseudo-Levi subgroup H of G , together with a W G,H -equivariant simpleunipotent cuspidal local system for H . EOMETRY OF SEMISTABLE G -BUNDLES 9 In this way Theorem 1.4 may be thought of as a geometric antecedent to thegeneralized Springer correspondence for D ( G E ): it expresses the geometry of thestack G E in terms of unipotent orbits for E -pseudo-Levi subgroups. We note thatthe work of P.L. with David Nadler [LN15] is another expression of the idea that G E is glued together from data indexed by E -pseudo-Levi subgroups. However,the techniques of loc. cit. involve an analytic uniformization of G E , whereas thepresent paper stays within the realm of algebraic geometry.1.2.4. Elliptic geometric Langlands.
We may view the category D ( G E ) as a sub-category (via extension by zero from the semistable locus, say) of the automorphicgeometric Langlands category D (Bun G ( E )) for an elliptic curve E . In this way,(generalized) Springer theory is embedded in to the geometric Langlands correspon-dence. This perspective is exposited in [LN15][Section 1.3.1.2].As a cautionary remark: there are now two different notions of cuspidal for an objectof D ( G E ): one using the constant term functor via diagram Eq. (1.1) (we will callthis Langlands cuspidal) and the other using the parabolic restriction functor viadiagram Eq. (1.2) (we will call this Springer cuspidal). If an object of D ( G E ) isLanglands cuspidal it is necessarily Springer cuspidal, but it is not clear if theconverse holds: the constant term may be supported on non-zero components ofBun T ( E ).Despite these difficulties, our results in this paper can be used to obtain strongrestrictions on the existence and support of Langlands cuspidals on Bun G . As asimple example, it can be shown that any Langlands cuspidal Hecke-eigensheaf in D (Bun SL ) must restrict to one of the four Springer cuspidal objects in ( SL ) E .1.2.5. Quantum geometric Langlands.
However, the relationship between Springertheory and geometric Langlands is the most direct when one considers the quan-tum deformation. Namely, one studies the category D κ (Bun G ) of L ⊗ κ -twisted D -modules on Bun G where L is the determinant line bundle (for this discussion letus restrict to the case where the ground field k has characteristic zero). Here κ canbe taken to be any complex number. The quantum geometric Langlands conjectureposits an equivalence (assuming κ is not the critical level which we normalize to κ = 0) D κ (Bun G ) ≃ D − /κ (Bun G ∨ )When κ is irrational it can be shown that any non-zero object of D κ is cleanlysupported on the semistable locus. In this way, the quantum geometric Langlandsequivalence at irrational level κ reduces to a statement about cuspidal data for G and L G . We plan to return to this in future work.There is also a Betti formulation of quantum geometric Langlands (see [BZN18]).For an elliptic curve E , this involves the category D q ( G/G ) of quantum D -moduleson G/G (see [BZBJ18]). (Here, the parameter q is roughly an exponential of the κ ∨ appearing in the de Rham formulation above.) There is a conjectural generalizedSpringer correspondence for D q ( G/G ). Interestingly, though the categories are very The Betti formulation of quantum geometric Langlands is not as symmetric as in the de Rhamsetting; here we are describing the category that naturally lives on the spectral side of geometricLanglands correspondence; the automorphic version will be described in terms of certain twistedsheaves on Bun G . different, one expects the same (discrete) cuspidal data to appear for the generalizedSpringer decomposition of D ( G E ) and D q ( G/G ). S.G. hopes to expand on thispoint in forthcoming work with David Jordan and Monica Vazirani. We note thatthe “Springer block” of such a quantum generalized Springer would involve W -equivariant modules for the algebra of q -difference modules on the torus T , whichmay explain the connection with the work of Baranovsky, Evans and Ginzburg[BEG03].1.2.6. Eisenstein Sheaves and elliptic Hall categories.
The main motivation for D.F.to study the stratification described in this paper concerns not cuspidal sheaves butrather Eisenstein sheaves. More precisely, one can define the category Q G ( E ) ofprincipal spherical Eisenstein sheaves on an elliptic curve E as the category gener-ated by Eis GT,B ( Q l ) inside D b (Bun G ( E )). This can be thought of as a categoricalversion of the space of spherical automorphic functions for the field of functions ona smooth projective curve over a finite field. The latter, at least for the groups GL n ,is nothing else but the degree n part of the spherical Hall algebra of the curve. See[Lau90, SV11, Sch12, Fra13] for more details about the relationship to automorphicfunctions. Therefore, one can think of Q G ( E ) for the groups GL n as a categoricalversion of the spherical Hall algebra. It can be actually proved [Sch12] that oneobtains in this way a categorification of the elliptic Hall algebra. The situation forhigher genus curves is not well understood.A sensible question to ask (for elliptic curves) is the classification of simple objectsof the spherical category Q G ( E ). Actually the proof of the above mentioned resulton categorification goes by first establishing such a classification. The main resultof [BN13] implies that there is an injection Irr( W ) ֒ → Irr( Q G ( E )), where W is theWeyl group of G . Previously, in [Sch12] this was shown to be a bijection for GL n .The precise expectation for simply connected groups is that the above map is abijection.Our main results in this article allow us to rule out some of the sheaves that couldappear in Irr( Q G ( E )) and that don’t arise from Irr( W ). In the case of SL n andsome groups of small rank this is enough to confirm the above sought for bijection.However, at the moment, we don’t know if it’s true for all simply connected groups.1.2.7. Affine character sheaves and local geometric Langlands.
The local geomet-ric Langlands program provides yet another interpretation of the category D ( G E ).Roughly speaking, one is motivated by local Langlands to study the category D ( LG/LG ) of class sheaves for the loop group LG of G . This may be consideredas a natural home for what one might call affine character sheaves. This categoryis technically very difficult to study (or even define). However, a slight deformation LG/ q LG of the stack LG/LG is closely related to the moduli stack Bun G ( E q ) fora certain elliptic curve E q . Thus one may consider D (Bun G ( E q )) as an avatar ofaffine class sheaves (this idea appears in [BNP13]).The most natural formulation of the relationship between LG/LG and Bun G is ana-lytic. In (unpublished) work of Looijenga, it was shown that there is an equivalenceof complex analytic stacks L hol G/ q L hol G ≃ Bun anG ( E q ) EOMETRY OF SEMISTABLE G -BUNDLES 11 Here q ∈ C × , | q | 6 = 1, E q = C × /q Z is the corresponding elliptic curve, and L hol G =Map hol ( C × , G ) is the holomorphic loop group which acts on itself by q twistedconjugation: Ad q ( g ( z )) h ( z ) = g ( qz ) h ( z ) g ( z ) − This idea of analytic unformization was generalized by P.L. with David Nadler[LN15], leading to analytic proofs of results closely related to those in this paper.1.3.
Acknowledgements.
This project grew from discussions in 2014 while D.F.and S.G. were attending the Geometric Representation Theory program at MSRI,Berkeley, and P.L. was a graduate student at U.C. Berkeley. S.G. also worked onthis project while attending the Higher Categories and Categorification program atMSRI in 2020. We thank MSRI for their support during the writing of this paper.S.G. was partially supported by Royal Society grant RGF\EA\181078, and the Eu-ropean Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation programme (grant agreement no. 637618).We would like to thank Michel Brion, Dougal Davis, Ian Grojnowski, David Nadler,Mauro Porta and Nick-Shepherd-Barron for helpful conversations.2.
Preliminaries
Notation and generalities on G -bundles. Let X be a scheme and G anaffine algebraic group over an algebraically closed field k . By a G -bundle (or prin-cipal G -bundle, or G -torsor) over a scheme X we mean a scheme π : P → X over X with a π -invariant right G -action such that, étale locally on X , π : P → X is G -equivariantly isomorphic to π : X × G → X . In other words, there exists X ′ → X étale and surjective such that the pullback of P to X ′ is G -equivariantly isomorphicto X ′ × G X ′ × G X ′ × X P X ′ π ≃ π ′ When there’s no danger of confusion we will simply write P for a G -bundle andomit the mention of π : P → X .If P is a G -bundle over a scheme X and Y is a quasi-projective variety with aleft G -action, then we can form the associated fiber space over X with fiber Y as Y P = P × G Y := ( P × Y ) /G . If moreover Y has a right H -action for some group H , then Y P is naturally endowed with an H -action.We will apply the above construction in two particular cases: • if ρ : G → H is a morphism of groups, then to a G -bundle P we associatean H -bundle: H P = P × G H . We’ll also denote it by ρ ∗ ( P ). • if V is a representation of G (viewed as an affine space with a left G -action)then to a G -bundle P we associate the vector bundle V P = ( P × V ) /G . Example 2.1.
A particularly important case of the first situation above is for thegroup morphism m : Z ( G ) × G → G . If P ′ is a Z ( G )-bundle and P is a G -bundlewe denote the induced G -bundle m ∗ ( P ′ × P ) by P ′ · P . In the case of GL n itcorresponds to tensoring a vector bundle by a line bundle. Example 2.2. (1) If G = G m then a G -bundle is simply a pair of line bundles ( L , M ). If α : G m → G m is given by α ( t, s ) = ts − then α ∗ ( L , M ) = L ⊗ M − .(2) If G = GL n then the category of G -bundles is equivalent ot the category ofvector bundles of rank n . The correspondence is given by associating to aGL n -bundle P the vector bundle P × GL n A n . We will use this correspon-dence tacitly especially in the case G = G m where we think of a G m -bundleas a line bundle using the natural representation of G m on A .(3) Consider the Lie algebra g with the adjoint action of G . For a G -bundle P we have its adjoint bundle P × G g =: g P that will play an important role inthe text. If G = GL n then the adjoint bundle is none other than the vectorbundle of endomorphisms.2.2. Semistability.
First let us recall the definition of slope: for a vector bundle V on a smooth curve X/k we put µ ( V ) := deg( V ) / rank( V ) ∈ Q and call it the slopeof V . Definition 2.3. • A vector bundle V on X is semistable (respectively, stable) if for all propersubvector bundles W ≤ V we have µ ( W ) ≤ µ ( V ) (respectively µ ( W ) <µ ( V ) ). • A G -bundle P is semistable if the induced adjoint vector bundle g P is asemistable vector bundle. Remark 2.4.
In general one doesn’t define semistability of G -bundles through theadjoint representation but rather using reduction to parabolic subgroups and a slopemap. In characteristic 0 all the definitions are equivalent, however in characteristic p they aren’t. See [Ram96, Sch14] for the definitions of semistability. Nevertheless,for an elliptic curve all these definitions coincide due to [Sun99, Thm 2.1, Cor 1.1].Therefore, for an elliptic curve, a G -bundle is semistable if and only if all associatedvector bundles P × G V are semistable, where V is a highest weight finite dimensional G -representation.It will also be useful to consider a notion of semistability for G -bundles on certainsingular curves, namely the nodal and cuspidal curve E node and E cusp consideredin the introduction Section 1.1. We caution the reader that this is only a workingdefinition for us and we don’t pretend it is the good notion of semistability over asingular curve.
Definition 2.5. A G -bundle on a singular curve X is said to be semistable if itspullback under the normalization map e X → X is a semistable G -bundle on e X . For such curves, the normalization is isomorphic to P , where the only degree 0-semistablebundle is the trivial bundle. EOMETRY OF SEMISTABLE G -BUNDLES 13 Remark 2.6.
See [Bho01, Section 2] and [Bal19, Sch05] for a more in depth dis-cussion of semistability for G -bundles on singular curves. Remark 2.7.
It follows from results of [FM01, Thm 3.3.1] that over a cuspidal ornodal curve the following two conditions are equivalent for a G -bundle P :(1) P is trivial when pulled-back to the normalization P ,(2) for any representation V of G the associated vector bundle V P is slopesemistable.This justifies our choice of calling a G -bundle on a nodal/cuspidal semistable pro-vided its pull-back to the normalization is trivial.2.3. Moduli spaces and stacks.
Let
X/k be a projective curve. For each testscheme S , we write Bun G ( X )( S ) for the groupoid of G -bundles on X S = X × S .This has an open dense substack Bun ss G ( X ) whose k -points consist of semistable G -bundles.The stack Bun G ( X ) decomposes in to connected components indexed by the alge-braic fundamental group π ( G ) (the quotient of the cocharacter lattice by the corootlattice). We write Bun G ( X ) for the component containing the trivial bundle; suchbundles are said to have degree 0.Now let E be a genus 1 curve over k with one marked smooth point which mayeither have a simple node or cusp singularity. Thus, E is either a smooth ellipticcurve E , a genus 1 curve with a single node E node or a genus 1 curve with a singlecusp E cusp .We write G E := Bun G ( E ) ss for the moduli stack of degree 0 semistable G -bundleson E . The case when E is an elliptic curve is the main object of study in thispaper. As noted in the introduction Section 1.1, the nodal and cuspidal cases maybe expressed in more concrete terms as follows. Proposition 2.8. [FM01, Thm 3.1.5, Thm 3.2.4]
There is an equivalence of stacks G E node ∼ = G/ ad G and G E cusp ∼ = g / ad G Reductions of the structure group.
Suppose we are given ρ : H → G amorphism of groups and a G -bundle P on a scheme E . A reduction of P to H is apair ( P ′ , φ ) of an H -bundle together with an isomorphism of G -bundles φ : P ′ × H G ≃ P . We say that two reduction ( P , φ ) and ( P , φ ) of P are isomorphic ifthere exists an isomorphism of H -bundles P ≃ P that intertwines φ and φ .The collection of possibles reductions of a G -bundles P to an H -bundle formsnaturally a groupoid. If moreover ρ is injective, this groupoid is equivalent to a setas can be easily checked.An alternative way of giving a reduction of P to H is to give a section s : E → P /H of the bundle P /H → E . Indeed, to such a section we associate the pullback P ′ if ρ is not injective this bundle is actually a gerbe with fiber B (ker ρ ) together with φ : P ′ → P : P ′ P E P /H φs Since
P → P /H is an H -bundle, P ′ → E is one as well. Moreover, the map φ is H -equivariant and induces a G -equivariant isomorphism P ′ × H G → P .To say that two such sections s and s are isomorphic we need to look at E and P /H as groupoids (the first one is discrete but the second one is non-discretewhen ρ is not injective). Then s and s are isomorphic if there exists a naturaltransformation η : s ⇒ s that is an isomorphism. In the case of a subgroup H ≤ G the groupoids are discrete and saying that s and s are isomorphic is the same assaying they are equal (as functions).Conversely, if ( P ′ , φ ) is a reduction of P to H then quoting out by H the composition P ′ f ( f × −→ P ′ × H G φ ≃ P we get a section E → P /H .The above correspondence is an equivalence of categories and we will use freelyeither way of looking at a reduction.The most important cases for us are H ≤ G and B ։ T . Remark 2.9.
From the above discussion we have that Bun H ( E ) → Bun G ( E ) is arepresentable morphism of stacks when H is a subgroup of G . This is used silentlythroughout the text.2.5. Framed bundles.
We start by recalling some basic notions for framed bun-dles over a pointed projective curve (
E, x ) and then consider the moduli stack G E of framed degree 0 semistable G -bundles. We show, through Lemma 2.10, thatit is a variety that is a G -bundle over the corresponding non-framed moduli stack.Then we proceed to the main point of this section, namely we show that for a closedsubgroup H ≤ G we have a closed embedding H E ֒ → G E (see Proposition 2.15).A framed G -bundle p = ( P G , θ ) is a G -bundle on E together with a G -equivariantisomorphism θ : P G | x ≃ G of the fiber of P G over x with the group G . The degreeand semistability are defined in terms of the underlying G -bundle.Sometimes, for convenience, we will omit the mention of θ and simply say that p isa framed G -bundle.It is clear that the moduli stack of framed G -bundles Bun fr G ( E ) is a G -torsor overthe moduli stack of G -bundles Bun G ( E ).Let P G be a G -bundle and θ, θ ′ two framings. Then there exists a unique g ∈ G such that θ ′ = g · θ , more precisely such that the following diagram commutes P G | x θ / / θ ′ " " ❊❊❊❊❊❊❊❊ G g · (cid:15) (cid:15) G EOMETRY OF SEMISTABLE G -BUNDLES 15 For ( P G , θ ) a framed G -bundle we have an induced map i θ : Aut( P G ) → G definedby restricting the automorphism to the fiber P G | x and using the basic fact recalledabove that any two framings differ by an element of G . If ( P G , θ ′ ) is another framingon P G , then the induced map i θ ′ : Aut( P G ) → G satisfies i θ ′ = gi θ g − , where g ∈ G is such that θ ′ = g · θ .In the case that interests us we have moreover Lemma 2.10.
Let ( P G , θ ) be a degree , semistable, framed G -bundle over anelliptic curve E . Then the induced morphism Aut( P G ) → G is injective. To prove Lemma 2.10 we will make use of a property of degree 0 semistable vectorbundles that holds in any genus. Recall that for the purposes of this paper, a linebundle on a singular curve is defined to be semistable of degree 0 if and only if itspullback to the normalization is semistable of degree 0.
Lemma 2.11.
Let X be a projective curve, and V , W semistable vector bundles ofdegree . Then any map f : V → W is of constant rank.Proof.
First let us note that we may reduce to the case when X is smooth. Indeed,given such a map f : V → W on X , we may pullback to a map e f : e V → f W on thenormalization η : e X → X , and restricting to the fiber of e f at e x ∈ e X identifies withthe fiber of η at x = η ( x ).Now suppose that X is smooth. Then ker( f ) and im( f ) must both have non-positivedegree by semistability, and as V has degree 0, we must have that both ker( f ) andim( f ) have degree 0. Now, if coker( f ) had torsion then its preimage would be apositive degree subbundle in V contradicting semistability. Thus we must have that f is of constant rank as required. (cid:3) Proof of Lemma 2.10.
Let us first start with G = GL n , so P G is equivalent to asemistable vector bundle, say V , of degree 0.We’ll show that restriction to the fiber at x gives an inclusion Aut( V ) ֒ → GL n .Suppose there is an automorphism ψ in the kernel. Then ψ − Id is an endomorphismof V which has a zero at x . Thus by Lemma 2.11, ψ − Id must be identically zeroas required.Back to the general case. Consider G ֒ → GL( V ) a faithful highest weight repre-sentation. Then we know from [Sun99, Cor 1.1, Thm 2.1] that V P is a semistable,degree 0 vector bundle.Therefore by the above, its automorphisms lie inside GL( V ).We have the following commutative diagramAut( P ) = / / Hom G - eq ( P , G ) (cid:31) (cid:127) / / (cid:15) (cid:15) Hom G - eq ( P , GL( V )) (cid:127) _ (cid:15) (cid:15) = / / Aut( V P ) G (cid:31) (cid:127) / / GL( V ) . From the injectivity of the three marked maps we deduce that the first verticalmorphism is injective as well, which is what we wanted. (cid:3)
Remark 2.12.
Lemma 2.10 is also true for higher genus curves. Given a semistable G -bundle P of degree 0 one needs to pick up a faithful representation V of G suchthat V P is semistable. For example, one of the fundamental representations will do.It is known that V P can become unstable (in characteristic p ) only if V containsa Frobenius twist and this ensures the semistability of V P for this choice of V .However, we’ll never use this result and so we don’t provide the details.When ( E, x ) is a pointed, irreducible projective curve of arithmetic genus 1 wedenote by G E the moduli stack of degree 0, semistable framed G -bundles over E where the framing is at x ∈ E . Lemma 2.10 shows that G E is a variety that is a G -torsor over the stack G E .Let now H ≤ G be a closed subgroup of G and consider the induction map from H -bundles to G -bundles. First we show that the induction preserves semistability: Lemma 2.13.
The map
Bun H ( E ) → Bun G ( E ) sends semistable bundles to semistablebundles.Proof. Let P be a semistable H -bundle of degree 0. In genus 1 we can use thedefinition of semistability through associated vector bundles (see Remark 2.4) andthis simplifies the argument.Let V be a representation of G and restrict it to H . Then ( P H × G ) G × V = P H × V is a semistable vector bundle of degree 0. Hence P H × G is a semistable G -bundleof degree 0. (cid:3) Remark 2.14.
Note that the above lemma is false without assuming degree 0.For example, for T = G m ֒ → GL(2) we have that ( O ( − , O (1)) is a semistable T -bundle of degree ( − ,
1) but the induced vector bundle O ( − ⊕ O (1) is notsemistable even if of degree 0.The following proposition, used silently in the sequel, is conceptually important inunderstanding the partition of the moduli stack G E in terms of subsgroups andit paves the way to the Jordan–Chevalley decomposition as formulated in Theo-rem 1.3. Proposition 2.15.
For any closed subgroup H ≤ G the induction map betweenmoduli spaces of framed bundles H E → G E is a closed embedding. To prove Proposition 2.15, we will make use of the following corollary of Lemma 2.11and of an additional technical lemma on equivariant embeddings:
Lemma 2.16.
Suppose X is a projective curve and V a semistable vector bundleof degree .(1) Suppose L is a degree line bundle on X . Then any injective map ofsheaves L → V is necessarily the inclusion of a subbundle.(2) Suppose L , L are line sub-bundles of V such that L | x = L | x for some x ∈ X . Then L = L . EOMETRY OF SEMISTABLE G -BUNDLES 17 Proof.
Part 1 follows immediately from Lemma 2.11, and part 2 follows by consid-ering the canonical morphism L ⊕ L → V . (cid:3) Lemma 2.17. Let H ≤ G be a closed subgroup. Then we can find a representation V of G and a line L ⊂ V such that(1) Stab G ( L ) = H ,(2) G/H ֒ → P ( V ) equivariantly,(3) the complement G/H \ ( G/H ) is empty or the support of a Cartier divisor.Proof. By Chevalley’s theorem we can find a representation W and a line L ⊂ W with properties (i), (ii) above. If H is a parabolic subgroup then we’re done. Ifnot, put X := G/H and denote by Z the complement X \ ( G/H ). Let ˜ X bethe normalization of the blow-up of X along Z . The G -action on X extends byuniversal properties to ˜ X and the complement of G/H in ˜ X is, by construction,the support of a Cartier divisor.Now we use Sumihiro’s theorem [Sum74] (or [Bri18, Theorem 5.3.3.]) to embed˜ X ֒ → P ( V ) equivariantly in the projectivization of a representation V of G . (cid:3) Proof of Proposition 2.15.
By Lemma 2.17 we may pick a representation V of G and a line L ⊆ V such that the morphism g g · L defines an equivariant embeddingin to the projective space G/H ֒ → P ( V ) and such that the complement of G/H inits closure is the support of a Cartier divisor.Now suppose P G is a G -bundle on E and put V := V P G . We have an embedding ofassociated bundles: P G /H = P G × G ( G/H ) ֒ → P G × G P ( V ) = P ( V ) . The data of an H -reduction P H of P G is equivalent to a section s : E → P G /H which we will consider as a section of the projective bundle P ( V ) via the embeddingabove. In other words, an H -reduction corresponds to a certain line subbundle L = P H × H L ֒ → V . Note that if P G and P H are semistable and of degree 0, thenso are V and L .Let us first show that H E → G E is injective. Given a framed G -bundle p = ( P G , θ ),we suppose it has two reductions to a framed H -bundle, corresponding to two linesubbundles L , L as explained above. Under the framing isomorphism V| x = P G | x × G V ∼ = G × G V = V we have that L | x and L | x both correspond to the line L ⊆ V . Thus, by Lemma 2.16,we must have L = L as required.Now let us show that H E → G E is proper. We use the valuative criterion ofproperness. Thus, let S denote the spectrum of a discrete valuation ring, U thespectrum of its generic point, and Z the spectrum of its closed point, and let E S , E U , E Z be the base change of E to S , U , and Z respectively. We thank M. Brion for providing us the proof.
Given a family ( P G,S , θ ) of framed, degree 0, semistable G -bundles on E S togetherwith a compatibly framed H -reduction on E U , we must show that it extends to acompatibly framed H -reduction on E S .As before, we have the associated vector bundle V S over E S , and we record thedata of the H -reduction as a line subbundle L U ֒ → V U . We must show that(1) L U extends to a line subbundle L S ⊆ V S over E S and(2) The corresponding section E S → P ( V S ) lands inside the subbundle P G,S /H .For the first part, we note that the line subbundle extends to a sub sheaf L S → V S .Indeed, by the properness of P ( V S ) → E S , we may extend the section s : E U → P ( V U ) over the generic point of E Z to define a line subbundle L ′ on E ′ S , where E ′ S is an open subset of E S whose complement is of codimension 2. This line subbundleis the restriction of a unique line bundle L S on E S (e.g. by taking the closure of aWeil divisor representing it), and the map L ′ → V S | E ′ S necessarily extends over thecodimension 2 locus.Now we observe that the restriction L Z of L S to E Z is a degree 0 line bundle (asit deforms to the degree 0 line bundle L U ) with a non-zero map to V Z . It followsfrom Lemma 2.16 that L Z → V Z must be a subbundle as required.To prove the second part, remember that the complement of G/H in G/H ⊂ P ( V )is the support of a Cartier divisor or empty.We will show that the section s : E S → P ( V S ) defining the line subbundle L S abovelands inside P G,S /H . If G/H is closed in P ( V ) then P G,S /H is closed in P ( V S ) andso the image of the section is contained entirely in it.In case G/H is not projective, notice that the complement of P G,S /H in its fiberwiseclosure is the support of a Cartier divisor; thus the set of points x ∈ E S for which s ( x ) / ∈ P G,S /H is possibly a divisor C in E S . As s ( x ) ∈ P G,S /H for all x ∈ E U ,we must have that C = E Z (possibly with multiplicity). But now note that theframing on P G,S defines a trivialization P ( V S ) | { x }× S ∼ = P ( V ⊗ O S )and we have that s takes the constant value [ L ] along the entire slice { x } × S of E S . In particular, the value of s is contained inside P G,S /H at at least one pointin E Z = C , and thus it must be so contained at every point, as required. (cid:3) The coarse moduli space and the characteristic polynomial map.
Thecoarse moduli space of degree 0, semistable G -bundles, which is none other thanthe GIT quotient G E //G , was identified by Laszlo [Las98] to be isomorphic to theGIT quotient T E //W . There is a natural G -invariant morphism from the framedmoduli stack to the moduli space of G -bundles that we think of as the characteristicpolynomial map (in analogy to Lie theory) χ = χ G : G E → M E ( G ) . Notice that the G -invariance of χ is equivalent to χ factorizing through the modulistack χ : G E → M E ( G ) . EOMETRY OF SEMISTABLE G -BUNDLES 19 The map χ maps a semisimple bundle P into its equivalence class in T E //W . Definition 2.18.
The moduli stack/variety of unipotent G -bundles are defined tobe the preimages of ∈ M E ( G ) : G uni E := χ − ( ) ,G uni E := χ − ( ) . The partition of semisimple bundles by Lusztig type
In this section we construct and study a certain locally closed partition of the coarsemoduli space M E ( G ).Recall that the points of the coarse moduli space M E ( G ) are in bijection withisomorphism classes of semisimple objects of G E . Let P G ∈ G E be a semisimpleobject and let p ∈ G E be a framed lift. Then the automorphism group Aut( P G ) isidentified with the reductive subgroup Stab G ( p ) of G . Different choices of framingdefine conjugate subgroups of G , and thus the conjugacy class of Stab G ( p ) in G isa well-defined invariant of the bundle P G .In this way, we may partition M E ( G ) according to the corresponding conjugacyclass in G of its automorphism group. It will be convenient to encode the data ofStab G ( p ) in two stages: • The neutral component H = Stab G ( p ) ◦ , which is a connected reductivesubgroup of G . • The component group π Stab G ( p ), which is a subgroup of the finite group N G ( H ) /H .The goal of this section is to study this partition of M E ( G ), and express it incombinatorial/root-theoretic terms.3.1. E -pseudo-Levi subgroups. Denote by E the set of conjugacy classes of con-nected reductive subgroups of G containing a maximal torus. Consider the map (ofsets) h : M E ( G ) → E which takes the isomorphism class of a semisimple bundle P G to the G -conjugacyclass of Stab G ( p ) ◦ , where p is a framed lift of P G . We denote by M E ( G ) [ H ] := h − ([ H ])the fibres of the map h . Definition 3.1.
We say that a connected reductive subgroup H of G is an E -pseudo-Levi subgroup if it is of the form Stab G ( p ) ◦ for some semisimple p ∈ G E .We write E E ⊂ E for the set of conjugacy classes of E -pseudo-Levi subgroups of E . Remark 3.2.
As mentioned in the introduction (see also Proposition 2.8), if E iscuspidal (respectively, nodal) then G E ∼ = g (respectively G E ∼ = G ). Thus E -pseudo-Levi subgroups correspond to connected centralizers of semisimple elements of g (respectively G ). These are precisely the Levi (respectively, pseudo-Levi) subgroupsof G (see Appendix A). In Section 3.6 we will give an alternative description of this partition, which allowsus to understand the closure relations. In particular, at the end of Section 3.4 wewill establish the following result (which may also be deduced from the theory ofLuna stratifications; see Remark 3.4).
Proposition 3.3.
For each E -pseudo-Levi H , the subset M E ( G ) [ H ] is a locallyclosed subvariety of M E ( G ) . We may record the finer partition according to the conjugacy class of the full auto-morphism group as follows. If P G is a representative of a point in M E ( G ) [ H ] and p a framed lift, then Stab G ( p ) ⊆ N G ( H ) (as every algebraic group normalizes itsown neutral component). Thus the component group of Stab G ( p ) is naturally asubgroup of the relative Weyl group: π Stab G ( p ) = Stab G ( p ) /H ⊆ W G,H := N G ( H ) /H. Given a subgroup A of W G,H , we write M E ( G ) [ H,A ] for the subset of M E ( G ) [ H ] corresponding to semisimple bundles P G such that the component group of Aut( P G )is identified with a conjugate of A as above. (The pair [ H, A ] is defined up tosimultaneous conjugation in G .)In particular, the subset M E ( G ) [ H, consists of isomorphism classes of semisimplebundles P G whose full automorphism group Aut( P G ) is identified with the con-nected group H . Remark 3.4.
Suppose we are given a G -variety Y such that every point has a G -invariant affine chart. Then we have the categorical quotient Y //G whose pointsare in bijection with closed G -orbits on Y . We may partition Y //G according to theconjugacy class in G of the (necessarily reductive) stabilizer of the correspondingclosed orbits. One can show that this defines a locally closed partition ([Lun73],see e.g. [KR08] for an overview). In the case of G acting on the framed modulispace G E , this reproduces the partition M E ( G ) [ H,A ] as described above. For ourpurposes it is convenient to focus mainly on the coarser partition according toconnected reductive subgroups H , hence the choice of notation.3.2. Borel–de Siebenthal theory.
Fix a maximal torus T of G , and let Φ denotethe corresponding set of roots. A subset Σ ⊆ Φ is called closed if Z Σ ∩ Φ = Σ. Wedenote by A the set of closed subsets of Φ.The connection between closed subsets of roots and connected reductive subgroupsis given by the following theorem of Borel–de Siebenthal: Theorem 3.5. [BDS49]
The collection of connected reductive subgroups H ≤ G that contain the maximal torus T is in bijection with A . The correspondence isgiven by associating to H its root system and conversely, to Σ ∈ A the subgroup C G ( Z (Σ)) ◦ .Moreover, under this correspondence we have G (Σ) = C G ( Z (Σ)) ◦ and Z ( G (Σ)) = Z (Σ) = Z ( C G ( Z (Σ))) (3.1) Proof.
Apart the last equality in Eq. (3.1), all is part of Borel–de Siebenthal theory[BDS49, Théorème 4 and Section 6, page 213].
EOMETRY OF SEMISTABLE G -BUNDLES 21 For the last equality, it is enough to show that Z ( C G ( Z (Σ))) is contained in theneutral component C G ( Z (Σ)) ◦ . If h is in Z ( C G ( Z (Σ))), then h ∈ C G ( T ) = T ⊂ C G ( Z (Σ)) ◦ . Since Z ( C G ( Z (Σ)) ◦ ) = Z (Σ) we deduce Z ( C G ( Z (Σ))) ⊂ C G ( Z (Σ)) ◦ which moreover implies Z ( C G ( Z (Σ))) ⊂ Z (Σ). (cid:3) E -root subsystems. In this subsection, we will present an alternative ap-proach to the theory of E -pseudo-Levi subgroups in terms of their associated rootdata.Recall that T E denotes the algebraic group parameterizing framed T -bundles onthe fixed curve E . Thus T E is isomorphic to either t or T in the cuspidal and nodalcases respectively. In general T E ∼ = Hom( X ∗ ( T ) , J ( E ))where J ( E ) denotes the Jacobian variety of E .Note that each character α ∈ X ∗ ( T ) gives rise to a homomorphism α ∗ : T E → J ( E ),taking a T -bundle to its associated line bundle. For p ∈ T E , we setΣ p := { α ∈ Φ | p ∈ ker( α ∗ ) } . We write A E for the subset of A consisting of subsets Σ p ⊂ Φ which occur in thisway. The elements of A E are called E -root subsystems of Φ.Thus we have a map κ : T E → A which assigns to a point p ∈ T E the set Σ p . The image is A E by definition. Lemma 3.6.
The map κ : T E → A E is continuous with respect to the topology on A E induced by the partial order given by inclusion. In other words the partition T E = G Σ ∈A E ( T E ) Σ is locally closed.Proof. It suffices to show that the preimage( T E ) ≥ Σ = κ − ( { Σ ′ | Σ ′ ⊇ Σ } )is closed. But this subset is just the intersection of root hyperplanes ker( α ∗ ) ⊆ T E for α ∈ Σ. (cid:3) As we will see in Section 3.4, the partition of T E according to E -root subsystemsrecords the conjugacy class of the neutral component the stabilizer of G acting on T E . We may also record the component group of the stabilizer as follows.Let W = W G,T be the Weyl group. Let Σ ⊆ Φ be an E -root subsystem. Wewrite N W (Σ) ⊆ W for the normalizer of Σ. Let W Σ denote the Weyl group of Σ(considered as a root system in its own right). We have that N W (Σ) = N W ( W Σ )and so W Σ is a normal subgroup in N W (Σ). Lemma 3.7.
Let p ∈ T E and set Σ = Σ p . Then W Σ ⊆ Stab W ( p ) ⊆ N W (Σ) . We write A p for the corresponding subgroup of W G, Σ := N W (Σ) /W Σ . Thus to each p ∈ T E we have a pair (Σ p , A p ) consisting of an E -root subsystem Σ p and a subgroup A p of W G, Σ . Proof.
First note that, by definition of Σ, p is contained in each of the root hyper-planes corresponding to roots in Σ. Thus p is fixed by the corresponding reflectionsin W Σ and thus by all of W Σ . This proves the inclusion on the left.Now let w ∈ Stab G ( p ), and suppose α ∈ Σ. Then w ( α ) ∗ ( p ) = α ∗ ( w − p ) = α ∗ ( p ) = J ( E ) Thus w ∈ N W (Σ), establishing the inclusion on the right. (cid:3) Connecting E -pseudo-Levis and E -root subsystems of roots. Recallthat we have fixed a maximal torus T ⊆ G , with associated roots Φ and Weylgroup W .It follows from Borel-de-Siebenthal theory (see Section 3.2) that the assignmentΣ [ G (Σ)] defines an order preserving bijection between E and A /W . The follow-ing result states that Σ is an E -root subsystem if and only if G (Σ) is an E -pseudo-Levi subgroup. Proposition 3.8.
The assignment Σ G (Σ) defines an order-preserving bijection A E /W ∼ −→ E E . This follows immediately from the following result:
Lemma 3.9.
Let p ∈ T E and view it inside G E . Then Stab G ( p ) ◦ = G (Σ p ) .Proof. To show that these two connected subgroups of G are equal, it suffices toshow that their corresponding Lie algebras are equal inside g . Put p = ( P G , θ ). TheLie algebra of Stab G ( p ) ◦ ∼ = Aut( P G ) is given by the global sections of the adjointbundle H ( E ; g P ). As P G is induced from a T -bundle P , the adjoint bundle g P G splits as a direct sum g P G = g P = t P ⊕ M α ∈ Φ g α, P . For each α ∈ Φ, the line bundle g α, P is trivial precisely when α ∈ Σ p . In this case,the framing provides a canonical identification H ( E ; g α, P ) = g α . Similarly, we have H ( E ; t P ) = t . On the other hand, if α / ∈ Σ p , then H ( E ; g α, P ) =0 because a line bundle of degree 0 on a curve has a section if an only if it’s trivial.Thus we have that H ( E ; g P ) = g (Σ p )as required. (cid:3) EOMETRY OF SEMISTABLE G -BUNDLES 23 Now recall that the map q : T E → M E ( G )identifies M E ( G ) with the categorical quotient T E //W . Putting all this together,we have Lemma 3.10.
The following diagram commutes: T E κ / / q (cid:15) (cid:15) A E (cid:15) (cid:15) M E ( G ) h / / E E This proves Proposition 3.3 in view of Lemma 3.6.3.5.
The component group.
Given a semisimple bundle in P G ∈ G E , we haveshown that the neutral component Aut( P G ) ◦ may be identified with G (Σ p ) whereΣ p is the set of roots which annihilate a given framed lift p ∈ T E of P G .We will now refine this to give a combinatorial description (i.e. in terms of the W -action on T E ) of the component group of Stab G ( p ) ( ∼ = Aut( P G )).Recall that for a connected reductive subgroup T ⊆ H ⊆ G with root subsystemΣ ⊆ Φ, we have an isomorphism W G,H := N G ( H ) /H ∼ = N W (Σ) /W Σ := W G, Σ . This finite group is referred to as the relative Weyl group of H (or of Σ) in G . Itnaturally acts on the algebraic group Z ( H ) E (also written Z (Σ) E ). The followingresult identifies the component group of a semisimple bundle in M E ( G ) [ H ] withthe stabilizer in the relative Weyl group of a corresponding lift to p ∈ Z ( H ) Σ . Lemma 3.11.
Let p ∈ T E , and let H = G (Σ p ) . Then there are compatible identi-fications Stab W G, Σ ( p ) ≀ (cid:15) (cid:15) (cid:31) (cid:127) / / N W (Σ) /W Σ ≀ (cid:15) (cid:15) π Stab G ( p ) Stab G ( p ) /H ∼ o o (cid:31) (cid:127) / / N G ( H ) /H. Proof.
Note that we have a commutative diagram with exact rows1 / / N H ( T ) (cid:15) (cid:15) / / Stab N G ( H ) ∩ N G ( T ) ( p ) (cid:15) (cid:15) / / Stab N W (Σ) ( p ) /W Σ (cid:15) (cid:15) / / / / H / / Stab G ( p ) /H / / Stab G ( p ) /H / / (cid:3) Closure relations for the partition.
We shall see presently that the closureof the variety ( T E ) Σ is a union of varieties ( T E ) Σ ′ . However, the Σ ′ that appearin the closure relation are determined by a slightly modified partial order relationwhich we determine below.Given Σ ∈ A E we consider the subgroup L (Σ) := C G ( Z (Σ) ◦ ) where Z (Σ) = ∩ α ∈ Σ ker( α ). It is a Levi subgroup of G (as it is the centralizer of a torus) whichcontains G (Σ) (as every element of G (Σ) centralizes Z (Σ) ◦ ). In fact, it is thesmallest such subgroup. We write Σ ◦ for the set of roots of L (Σ). We may nowdefine a new partial order on A . Definition 3.12.
Given Σ , Σ ∈ A , we define the partial order (cid:23) by Σ (cid:23) Σ if Σ ⊇ Σ and Σ = Σ ∩ Σ ◦ . We say that a closed subset is isolated if it is maximal with respect to this partialorder.
In other words, maximal subsets Σ are characterized by the fact that Z (Σ) ◦ = Z ( G ) ◦ . This notion is useful in order to relate our partition/stratification to theone of Lusztig [Lus84, 3.1] Proposition 3.13.
We have the closure relation ( T E ) Σ = G Σ ′ (cid:23) Σ ( T E ) Σ ′ . Proof.
First we claim that we have the following description of the closure: { ( T E ) Σ } − = ( Z (Σ) ◦ ) E · ( T E ) Σ . (3.2)Indeed, note that ( T E ) Σ is open in Z (Σ) E = ( T E ) ≥ Σ , and thus its closure is neces-sarily a union of connected components of Z (Σ) E . In particular, the closure mustbe a union of orbits for the neutral component ( Z (Σ) E ) ◦ = ( Z (Σ) ◦ ) E , from whichthe claim follows.Now suppose p ∈ T E is in the closure of ( T E ) Σ . We must show thatΣ = Σ p ∩ Σ ◦ . As ( T E ) ≥ Σ is closed and contains ( T E ) Σ , it must also contain ( T E ) Σ and thus p .In other words, we have Σ ⊆ Σ p . By construction we have Σ ⊆ Σ ◦ and henceΣ ⊆ Σ p ∩ Σ ◦ .It remains to show the other inclusion. Let α ∈ Σ p ∩ Σ ◦ . By Eq. (3.2) we may write p = q · r where q ∈ Z (Σ) ◦ E and r ∈ ( T E ) Σ . As α ∈ Σ p , α ∗ ( p ) = J ( E ) ; as α ∈ Σ ◦ , α ∗ ( q ) = J ( E ) . Thus we have that α ∗ ( r ) = J ( E ) which implies α ∈ Σ as required. (cid:3)
EOMETRY OF SEMISTABLE G -BUNDLES 25 Summary of section.
Recall that we have defined a locally closed partitionin two ways M E ( G ) = G [Σ] ∈A E /W M E ( G ) [Σ] (3.3)= G [ H ] ∈E E M E ( G ) [ H ] We thus obtain a locally closed partition G E = G [Σ] ∈A E /W ( G E ) [Σ] (3.4)= G [ H ] ∈E E ( G E ) [ H ] by pulling back via the characteristic polynomial map χ : G E → M E ( G ). Similarlyfor the framed version G E using χ : G E → M E ( G ).A stratum that plays a special role for us is the one corresponding to H = G . Weput ( G E ) ♥ := ( G E ) [ G ] . One should think of this locus as those G -bundles whose semisimplification "iscentral" (i.e. has a reduction to the center Z ( G ) of G ).Note that the set E carries partial orders ≤ , (cid:22) induced from the same named orderson A (see Definition 3.12). Proposition 3.14.
The partitions from (3.3) , (3.4) of M E ( G ) and G E have theclosure relations determined by the partial order (cid:22) .Proof. Follows from Lemma 3.10 and Proposition 3.13 (cid:3)
Remark 3.15.
The upshot of this section is that there are two approaches todetermining the type of a semisimple bundle: either compute its automorphismgroup, or choose a reduction to T and compute the subset of roots on which thebundle vanishes. 4. The Jordan–Chevalley Theorem
In this section we will prove Theorem 1.1 and Theorem 1.4 from the introduction.A key concept here is the notion of regularity which we define in Section 4.1. Thenwe will establish the equivalence of the two main theorems in Section 4.3. Finallywe will prove Theorem 1.4 over Section 4.4 and Section 4.5.4.1.
The regular locus.
Fix H an E -pseudo-Levi subgroup of G . Recall thatthis means that there exists a semisimple framed G -bundle q ∈ G E such thatStab G ( q ) ◦ = H . Definition 4.1.
We say that p ∈ H E is(1) regular if Stab G ( p ) ◦ ⊆ H ,(2) strongly regular if Stab G ( p ) ⊆ N G ( H ) , (3) maximally regular if Stab G ( p ) ⊆ H . One may immediately check that for a given p ∈ H E we have implicationsmaximally regular = ⇒ strongly regular = ⇒ regular(the second implication uses that N G ( H ) ◦ = H ).If the stabilizers of semisimple elements of G E are always connected, then all threenotions above coincide. For example this happens when E is cuspidal (so G E = g )or when E is nodal and G = G E is simply connected. However, in general, thethree notions are all distinct as illustrated by the following example. Example 4.2.
Consider the case where E is a nodal curve, and thus we mayidentify G E = G . Assume also that char ( k ) = 0.(1) Let G = P GL and T the maximal torus represented by the classes ofdiagonal matrices. Consider the matrix X = (cid:18) − (cid:19) One can check that Stab G ([ A ]) = N G ( T ). Thus [ X ] is a strongly regular(and thus regular) element of T , but it is not maximally regular .(2) Now let G = P GL and H the Levi subgroup consisting of classes of blockmatrices of the form: ∗ ∗∗ ∗
000 0 ∗ We write T for the diagonal maximal torus again.Consider the matrix Y = ζ
00 0 ζ where ζ is a primitive third root of unity. Then Stab G ([ Y ]) is the preimagein N G ( T ) of the cyclic subgroup group A ⊆ S ∼ = N G ( T ) /T .In particular Stab G ([ Y ]) ◦ = T ⊆ H and thus [ Y ] is a regular element of H . However,Stab G ([ Y ]) * H = N G ( H ) , so [ Y ] is not a strongly regular element of H (it is however, a strongly butnot maximally regular element of T ).We write H reg E (respectively H str - reg E , respectively H max - reg E ) for the locus of regular(respectively strongly regular, respectively maximally regular) elements. As theseloci are manifestly H -invariant (in fact, N G ( H )-invariant) we have correspondingloci H reg E , H str - reg E , H max - reg E in the stack H E . EOMETRY OF SEMISTABLE G -BUNDLES 27 Remark 4.3.
Whereas the loci ( G E ) [ H ] and ( G E ) [ H,A ] introduced in Section 3 areintrinsic to G , the regular locus ( H E ) reg and its relatives are defined in terms ofhow H sits as a subgroup of G (i.e. are not intrinsic to H ). In what follows, wewill often need to consider both the intrinsic loci of H E (such as ( H E ) [ K ] for some E -pseudo-Levi K of H ) alongside the regular loci. To keep track of these notions,we will always label intrinsic loci in the subscript and regularity conditions in thesuperscript.It will be useful to have a few other characterizations of the regularity condition.To state the result we will need to recall some notation.Let P H ∈ H E . Choose an element p ∈ T E lifting χ ( P H ) under the map T E → M E ( H ) = T E //W H . This element determines a closed root subsystem Σ p ⊆ Φ (see Section 3.3). Notethat we consider Σ p as a subset of the roots Φ of G and it might not be containedin the root subsystem Σ H corresponding to H .Recall also that there is a unique-up-to-isomorphism semisimple bundle P ss H with χ ( P ss H ) = χ ( P H ). One can construct P ss H by choosing a reduction P B H to a Borel B H of H , and taking the induced bundle P T under the projection B H → T , theninducing P T back up to an H -bundle. Moreover, we may choose the B H -reductioncompatibly with the lift p ∈ T E above, so that p is a framed lift (unique-up-to-isomorphism) of P T . Proposition 4.4.
Let P H ∈ H E , and choose p ∈ T E and P ss H as described above.The following are equivalent:(1) P H is regular,(2) P ss H is regular,(3) The morphism π : H E → G E is étale at P H ,(4) H • ( E ; ( g / h ) P H ) = 0 ,(5) Σ p ⊆ Σ H . Remark 4.5.
The equivalence of (2) and (1) means that the condition of P H beingregular depends only on the "characteristic polynomial" χ ( P H ). Thus we have alocus M E ( H ) reg ⊆ M E ( H )such that P H is regular if and only if χ ( P H ) ∈ M E ( H ) reg . According to (5), thelocus M E ( H ) reg is equal to the image of ( T E ) ≤ Σ H under the map T E → M E ( H ).In particular, it follows from Proposition 4.4 that M E ( H ) reg (respectively, H reg E ) isopen and dense in M E ( H ) (respectively, H E ). Proof of Proposition 4.4.
First, let us show that conditions (4) and (3) are equiv-alent. The map π is étale precisely at the points where its differential is a quasi-isomorphism of tangent complexes. Recall that the cohomology of the tangentcomplex of H E at a bundle P H is given by the cohomology of the adjoint bundleof P H , that is by H i ( E ; h P H ). The differential of π at P H ∈ H E is the map T H E , P H → π ∗ T G E ,π ( P H ) which upon taking cohomology groups becomesH i ( E ; h P H ) → H i ( E ; g P G ) , i = 0 , . The cone of this map of complexes is given by H • ( E ; ( g / h ) P H ). Thus we have thatH • ( E ; ( g / h ) P H ) = 0 if and only if π is étale at P H as required.Let us show the equivalence of (4) and (5). Fix P B H a reduction of P H to aBorel subgroup of H such that the induced T -bundle is precisely P T such that P T T × H = P ss H . (See paragraph above Proposition 4.4.) The vector bundle( g / h ) P H = ( g / h ) P BH carries a filtration whose associated graded is ( g / h ) P T . Thislatter bundle is a direct sum of line bundles ( g α ) P T corresponding to roots α ∈ Φ \ Σ.By definition, the bundle ( g α ) P T is trivial if and only if α ∈ Σ P T . Noting that thecohomology of a degree 0 line bundle on E vanishes if and only if it is trivial, wededuce that (4) is equivalent to (5). The same argument shows the equivalence of(2) and (5).Notice that the bundle P H is regular if and only if the inclusion Aut( P H ) ◦ → Aut( P G ) ◦ is an isomorphism, where we put P G = P H H × G . Taking the correspond-ing Lie algebras, we see that P H is regular if and only if the map H ( E ; h P H ) → H ( E ; g P H )is an isomorphism. Using the long exact sequence associated to the short exactsequence of H -modules 0 → h → g → g / h → (cid:3) We write Z ( H ) reg E for the intersection Z ( H ) E ∩ H reg E (similarly for Z ( H ) str - reg E , Z H ( E ) max - reg ). The various notions of regularity are somewhat simpler here. Lemma 4.6.
Fix an element p ∈ Z ( H ) E .(1) The element p is regular if and only if it is strongly regular.(2) Assume p is regular. It is maximally regular if and only if the relative Weylgroup W G,H acts freely on the orbit of p .Proof. (1) Suppose p is regular, i.e. Stab G ( p ) ◦ ⊆ H . But p ∈ Z ( H ) E , so H ⊆ Stab G ( p ) and thus Stab G ( p ) ◦ = H . As the neutral component of any algebraicgroup is a normal subgroup, we have Stab G ( p ) ⊆ N G ( H ) as required.(2) By regularity of p , Stab G ( p ) ◦ = H as explained above. Thus p is maximallyregular if and only if Stab G ( p ) is connected. According to Lemma 3.11, the compo-nent group of Stab G ( p ) is precisely the stabilizer of p in W G,H , hence the claimedresult. (cid:3)
EOMETRY OF SEMISTABLE G -BUNDLES 29 The main results.
For convenience, we remind the reader of the statementsof Theorem 1.1 and Theorem 1.4, to be proved in this section.To state the first result, recall that for an E -pseudo-Levi subgroup H of G , wedefined the regular locus ( H E ) reg in Section 4.1. We denote by Z ( H ) reg the inter-section Z ( H ) E ∩ H reg E . The locus of unipotent bundles H uni E is defined to be fibre χ − H ( ) of the characteristic polynomial map χ H : H E → M E ( H ). Theorem 4.7 (Jordan decomposition) . Given a semistable, degree , framed G -bundle p ∈ G E , there is a unique triple ( H, p s , p u ) where H is an E -pseudo-Levi, p s ∈ Z ( H ) reg E , p u ∈ H uni E , and p = p s · p u where the multiplication is defined via the group morphism m : Z ( H ) × H → H (seeExample 2.1). To state the next result recall that we have defined a partition of G E (and also of G E ) in Section 3.7 G E = G [ H ] ∈E E ( G E ) [ H ] and we defined the heart locus to be ( G E ) ♥ := ( G E ) [ G ] . Theorem 4.8 (Galois theorem) . For H an E -pseudo-Levi subgroup of G the mor-phism ( H E ) reg ♥ → G E is a W G,H -Galois covering onto ( G E ) [ H ] . To begin with we will establish the following result, which expresses the Jordan–Chevalley decomposition on the heart locus (where, in fact, it is simply a directproduct). This may be thought of as a baby version of Theorem 4.7.
Proposition 4.9.
There is an equivalence of stacks ( G E ) ♥ ≃ Z ( G ) E × G uni E . Proof.
The following is the cartesian diagram defining ( G E ) ♥ and G uni E : G uni E ( G E ) ♥ G E { } Z ( G ) E M E ( G ) χ The group Z ( G ) E acts on both G E and M E ( G ) and the map χ is equivariant. Thisreadily implies the required isomorphism from the statement.One could also argue as follows: the natural product map Z ( G ) E × G uni E → ( G E ) ♥ is Z ( G ) E -equivariant and this enables us to define its inverse by the followingformula P 7→ ( χ ( P ) , χ ( P ) − · P ) . (cid:3) Corollary 4.10.
We have an equivalences of stacks ( H E ) reg ♥ ≃ Z ( H ) reg E × H uni E ≃ ( H E ) str - reg ♥ Proof.
The first equivalence follows from Proposition 4.9 and from the fact thatregularity of an H -bundle is governed by it’s semisimple part, i.e. by the partin Z ( H ) E (see Proposition 4.4). The analogous equivalence also holds for thestrongly regular locus. The second equivalence then follows from the fact that Z ( H ) reg E = Z ( H ) str - reg E (Lemma 4.6). (cid:3) Equivalence of Theorems 4.7 and 4.8.
Our next step will be to show thatthe two main results are mutually equivalent.Let us first recast Theorem 4.8 in terms of framed bundles. It states that the map e π reg ♥ : G × N G ( H ) ( H E ) reg ♥ → G E (4.1)is a ( G -equivariant) isomorphism onto ( G E ) [ H ] .To prove Theorem 4.8, we must show the following two statements:(1) (“Surjectivity”) The image of e π reg ♥ is precisely ( G E ) [ H ] .(2) (“Injectivity”) The map e π reg ♥ is injective.Analogously, to prove Theorem 4.7, we must show the following two statements:(1) (“Existence”) Every p ∈ G E has a Jordan datum ( H, p s , p u ) with p = p s · p u ∈ ( H E ) reg ♥ .(2) (“Uniqueness”) Given p ∈ G E with two Jordan data ( H, p s , p u ) and ( H ′ , p ′ s , p ′ u )we have ( H, p s , p u ) = ( H ′ , p ′ s , p ′ u ).We will show that the “existence” (respectively, “uniqueness”) part of Theorem 4.7is equivalent to the “surjectivity” (respectively, “injectivity”) in Theorem 4.8.4.3.1. Existence implies surjectivity.
Suppose p ∈ ( G E ) [ H ] with associated Jordandata H ′ , p s , p u . It follows that p ∈ ( H ′ E ) reg ♥ ⊆ ( G E ) [ H ] and thus H ′ = Ad ( g ) H forsome g ∈ G . Then p is the image of( g, g − · p ) ∈ G × N G ( H ) ( H E ) reg ♥ as required.4.3.2. Surjectivity implies existence.
Given p ∈ ( G E ), we let [ H ] denote the uniqueconjugacy class of E -pseudo-Levi subgroups such that p is in the locus ( G E ) [ H ] .Surjectivity means that there exists( g, p ′ ) ∈ G × ( H E ) reg ♥ such that g · p ′ = p . By replacing p ′ with g − · p ′ and H with Ad ( g − ) H , we mayassume that p ∈ ( H E ) reg ♥ . Recall (Corollary 4.10) that ( H E ) reg ♥ ∼ = Z ( H ) reg E × H uni E .Thus p has a Jordan decomposition p = p s · p u as required. EOMETRY OF SEMISTABLE G -BUNDLES 31 Uniqueness implies injectivity.
We must show that if p, p ′ ∈ ( H E ) reg ♥ and g ∈ G such that g · p = p ′ , then g ∈ N G ( H ). As we have assumed p, p ′ ∈ ( H E ) reg ♥ ,we have Jordan decompositions p = p s · p u and p ′ = p ′ s · p ′ u . Note that H =Stab G ( p s ) ◦ = Stab G ( p ′ s ) ◦ . By the uniqueness of the Jordan decomposition, wemust have that g · p s = p ′ s . Thus Ad ( g ) H = H , so g ∈ N G ( H ) as required.4.3.4. Injectivity implies uniqueness.
Let p ∈ G E and suppose we have two Jordandata ( H, p s , p u ) and ( H ′ , p ′ s , p ′ u ). To prove the uniqueness of the Jordan decomposi-tion it suffices to show that H = H ′ (as within ( H E ) reg ♥ = Z ( H ) reg E × ( H E ) uni , everyelement has a unique Jordan decomposition).First observe that the Lusztig type of p is well-defined, so [ H ] = [ H ′ ], i.e. thereexists g ∈ G such that Ad ( g ) H = H ′ .Now we have p ∈ ( H ′ E ) reg ♥ , and thus g − · p ∈ ( H E ) reg ♥ . It follows that p is the imageof both (1 , p ) and ( g, g − · p ) under the map e π reg ♥ : G × N G ( H ) ( H E ) reg ♥ → G E By injectivity, we must have that g ∈ N G ( H ). But then H ′ = H as required.The remainder of this section will be taken up with the proof of Theorem 4.8 (andthus Theorem 4.7).4.4. Proof of surjectivity.
In this subsection we give a proof of the surjectivitypart of Theorem 4.8. Specifically, we show the following:
Proposition 4.11.
The restriction π reg ♥ : ( H E ) reg ♥ → G E maps surjectively onto ( G E ) [ H ] . The statement splits into two parts:(1) The image of π reg ♥ : ( H E ) reg ♥ → G E is contained in ( G E ) [ H ] .(2) The image of π reg ♥ contains all of ( G E ) [ H ] .4.4.1. Proof of (1) . Let P H ∈ H E and let P G = π ( P H ) be the induced bundle.Then we will show(a) if P H ∈ H reg E then P G ∈ ( G E ) ≤ [ H ] ;(b) P H ∈ ( H E ) ♥ if and only if P G ∈ ( G E ) ≥ [ H ] . Remark 4.12.
The converse of part (a) above is false. For example, suppose weare in the group case E = E node with G = G E = GL . Let H ⊆ G denote thesubgroup consisting of matrices of the form ∗ ∗∗ ∗
000 0 ∗ Now let p ∈ H denote the matrix: Then p ∈ G [ H ] as Stab G ( p ) is conjugate to H , but p / ∈ H reg as Stab G ( p ) * H . Proof of (a) First suppose that P H ∈ H E is semisimple. Let p ∈ H E denote a liftto a framed bundle.Then P H is regular if and only if Aut( P G ) ◦ ∼ = Stab G ( p ) ◦ ⊆ H . On the otherhand, P G is contained in ( G E ) ≤ [ H ] if and only if Stab G ( p ) is contained in some G -conjugate of H .Thus we have the required implication in case P H is semisimple. In general, theresult follows from the fact that the conditions P H ∈ H reg E and P G ∈ ( G E ) ≤ [ H ] onlydepend on the characteristic polynomial of P H (respectively P G ) and thus onlydepend on the isomorphism class of the semisimplification. Proof of (b) ) Again, suppose P H is semisimple. Then P H is contained in ( H E ) ♥ if and only if H = Stab H ( p ) = Stab G ( p ) ∩ H . This in turn is equivalent to P G ∈ ( G E ) ≥ [ H ] . As before, the general case follows from the fact that the conditions onlydepend on the characteristic polynomial.4.4.2. Proof of (2) . Suppose P G is contained in ( G E ) ≤ [ H ] (respectively, ( G E ) [ H ] ).Then we will show that there exists P H in ( H E ) reg (respectively, ( H E ) reg ♥ ) such that π ( P H ) = P G .Recall that by Proposition 4.4, the substack H reg E is precisely the locus on whichthe map π is étale. In particular, the image π ( H reg E ) is an open substack of G E . Wemust show that this open substack contains all of ( G E ) ≤ [ H ] .First suppose P G ∈ ( G E ) ≤ [ H ] is semisimple. Let T be a maximal torus of H (andthus of G ). Then, by Lemma 3.9, there is a framed reduction p ∈ T E such thatStab G ( p ) ⊆ H . Thus the induced H -bundle P H is a regular H -reduction of P G asrequired.Now let us drop the assumption that P G is semisimple, and let P ss G denote a semisim-plification of P G . Then P ss G is semisimple and contained in ( G E ) ≤ [ H ] , thus by theabove argument P ss G is contained in π ( H reg E ). But P ss G is contained in the closure ofthe point P G ; thus any open neighbourhood of P ss G in G E contains P G . As π ( G reg E )is such an open neighbourhood, we must have P G = π ( P H ) for some P H ∈ H reg E asrequired.It remains to show that if moreover P G ∈ ( G E ) [ H ] then the above constructed P H belongs to ( H E ) ♥ . If p is a framed lift of P H such that Stab G ( p ) ◦ ⊂ H then thecondition P G ∈ ( G E ) ≥ [ H ] means H ⊂ Stab G ( p ) ◦ . We deduce that H = Stab G ( p ) ◦ which in turn implies Stab H ( p ) = H , or in other words P H ∈ ( H E ) ♥ . EOMETRY OF SEMISTABLE G -BUNDLES 33 Proof of injectivity.
In this section we prove the injectivity required (see(4.1)) in the proof of Theorem 4.8. In other words, we must show that π reg ♥ : ( H E ) reg ♥ /W G,H → G E (4.2)is an embedding.Let us first sketch an outline of the strategy of proof. We wish to apply the followinggeneral principle: Proposition 4.13.
If an étale morphism of schemes X → Y is an embedding overa dense open subset of Y and X is separated, then it is an open embedding.Proof. We reduce it to [Gro67, Thm 17.9.1]. Namely, according to loc.cit., a mor-phism of schemes U → V is an open immersion if and only if it is flat, locally offinite presentation and a monomorphism in the category of schemes.In our case we only need to check that the morphism is a monomorphism whichfollows at once from birationality and separatedness. (cid:3) Remark 4.14.
Since being an open immersion is a property that is smooth-localon the target, the proposition can be applied to a morphism of finite type stacksthat is representable by separated schemes and this is how it will be used below.While π is étale over the locus ( H E ) reg , we cannot apply Proposition 4.13 directlyto the morphism π as it is not generically an embedding - its generic fiber hascardinality | W | / | N W ( W H )) | (see Lemma 4.22).However, the failure of π to be generically an embedding is precisely accounted forby the corresponding map ρ : M E ( H ) //W G,H → M E ( G )on the level of coarse moduli spaces, which also has generic degree | W | / | N W ( W H ) | (Lemma 4.22). The idea is thus to replace the target G E with the base-change g G E : H E /W G,H π ' ' ν / / χ H ' ' PPPPPPPPPPPP g G E e ρ / / (cid:15) (cid:15) (cid:3) G Eχ G (cid:15) (cid:15) M E ( H ) //W G,H ρ / / M E ( G )In this way, we obtain a morphism ν which is generically an embedding (Lemma 4.16).Moreover, we will show that ν is étale when restricted to the locus H str - reg E /W G,H (see Lemma 4.15), and thus an open embedding on this locus by Proposition 4.13(we can apply it in this situation since ν comes from an obvious G -equivariant mor-phism between varieties). As this locus contains the desired substack ( H E ) reg ♥ /W G,H (recall that ( H E ) str - reg ♥ = ( H E ) reg ♥ by Corollary 4.10), the morphism ν reg ♥ : ( H E ) reg ♥ /W G,H → g G E is an embedding. Finally, we show that the restriction of the base change morphism e ρ reg ♥ : ( g G E ) reg ♥ → G E is an embedding (Corollary 4.19). Thus the composite( H E ) reg ♥ /W G,H (cid:31) (cid:127) / / ( g G E ) reg ♥ (cid:31) (cid:127) / / G E is an embedding as required in Eq. (4.2).Now let us go through the steps in the proof one by one. We start with the twolemmas below, whose proof will be given in Section 4.6. Lemma 4.15.
The following restriction of ν is étale ν str - reg : ( H E ) str - reg /W G,H → g G E . Lemma 4.16.
There is an open dense subset of H E on which ν is an open embed-ding. Assuming Lemma 4.15 and Lemma 4.16, we may now establish:
Lemma 4.17.
The restriction ν reg ♥ : ( H E ) reg ♥ /W G,H → g G E is an embedding.Proof. By Lemma 4.15 ν str - reg is étale, and by Lemma 4.16 it is generically anembedding. Thus by Proposition 4.13, ν str - reg is an open embedding. The claim thenfollows immediately from the fact that ( H E ) reg ♥ = ( H E ) str - reg ♥ (Corollary 4.10). (cid:3) The following lemma is an analogue on the level of coarse moduli spaces of thedesired injectivity part of Theorem 4.8. Its proof is given in Section 4.6.
Lemma 4.18.
The morphism ρ reg ♥ : M E ( H ) reg ♥ //W G,H −→ M E ( G ) is an embedding. It follows immediately that the base change is also an embedding:
Corollary 4.19.
The morphism e ρ reg ♥ : ( g G E ) reg ♥ −→ G E is an embedding. Putting Lemma 4.17 and Corollary 4.19 together we obtain that the composite( H E ) reg ♥ /W G,H (cid:31) (cid:127) / / ( g G E ) reg ♥ (cid:31) (cid:127) / / G E is an embedding. This completes the proof of Theorem 4.8 (modulo the proofs inthe following subsection). EOMETRY OF SEMISTABLE G -BUNDLES 35 Proofs of Lemma 4.16, Lemma 4.15, and Lemma 4.18.
For this sub-section we will fix a maximal torus T of our fixed E -pseudo-Levi subgroup H (andthus T is also a maximal torus of G ).The following lemma (which is essentially Theorem 4.8 for the special case H = T )forms a key step in the proof of Lemma 4.16. Proposition 4.20.
The induction map T reg E → G E is an unramified W -Galois cover onto ( G E ) [ T ] .Proof. The claim is equivalent to the statement that G × N G ( T ) T reg E → ( G E ) [ T ] is an isomorphism. We have already seen that the map is onto and étale (Propo-sitions 4.4 and 4.11). For injectivity, we must show that for all p ∈ T reg E we haveStab G ( p ) ⊆ N G ( T ). But by definition Stab G ( p ) ◦ = T , so the required statementfollows from the fact that Stab G ( T ) normalizes its own neutral component. (cid:3) Definition 4.21.
We say that a morphism f : X → Y is generically a coveringof degree d if there are dense open subsets U of X and V = f ( U ) of Y such that f | U : U → V is a covering (i.e. finite and étale) of degree d . Lemma 4.22.
The following morphisms are generically covering maps of degree | W | / | N W ( W H ) | :(1) π : H E /W G,H → G E ,(2) ρ : M E ( H ) //W G,H → M E ( G ) .Proof. (1) Consider the following diagram: T E β / / $ $ ❏❏❏❏❏❏❏❏❏❏ α $ $ H E γ / / δ (cid:15) (cid:15) G E H E /W G,H π : : tttttttttt By Proposition 4.20, we have that α is generically a covering of degree | W | and β is generically a covering of degree | W H | . Thus γ is generically a covering of degree | W | / | W H | . By construction, δ is a covering of degree | W G,H | . Thus π is genericallya covering of degree | W | / | W H || W G,H | = | W | / | N W ( W H ) | as required.(2) Now consider the diagram: T E β ′ / / α ′ ' ' T E //N W ( W H ) / / ≀ (cid:15) (cid:15) T E //W ≀ (cid:15) (cid:15) M E ( H ) //W G,H ρ / / M E ( G ) As W acts freely on a dense open subset of T E (namely T max - reg E ; see Definition 4.1),we have that α ′ is generically a covering of degree | W | and β ′ is generically a coveringof degree | N W ( W H ) | . Thus ρ is generically a covering of degree | W | / | N W ( W H ) | asrequired. (cid:3) Proof of Lemma 4.16.
Consider again the diagram: H E /W G,H π ' ' ν / / χ H ' ' PPPPPPPPPPPP g G E e ρ / / (cid:15) (cid:15) (cid:3) G Eχ G (cid:15) (cid:15) M E ( H ) //W G,H ρ / / M E ( G )By Lemma 4.22, π and ρ are both generically coverings of degree | W | / | N W ( W H ) | .Thus, by base change, e ρ is generically a covering of degree | W | / | N W ( W H ) | . But as π = e ρ ◦ ν we must have that ν is generically a covering of degree 1, i.e. genericallyan embedding as required. (cid:3) Lemma 4.23.
The morphism M E ( H ) //W G,H → M E ( G ) is étale on the locus M E ( H ) str - reg //W G,H .Proof.
Recall that M E ( G ) ≃ T E //W and M E ( H ) //W G,H ≃ T E //N W (Σ)where Σ is the root system of H . Using [Gro71, Prop V.2.2] we deduce that if N W (Σ) contains the stabilizer Stab W ( p ) then the map T E //N W (Σ) → T E //W is étale at p . Thus the claim reduces to Lemma 4.24 below. (cid:3) Lemma 4.24.
Let p ∈ T E ∩ H str - reg E . Then Stab W ( p ) ⊆ N W (Σ) .Proof. By assumption Stab G ( p ) ◦ ⊆ H . Suppose w ∈ Stab W ( p ) and choose a liftto e w ∈ N G ( T ). Then e w ∈ N G ( H ) by assumption, and thus w preserves the rootsystem Σ of H as required. (cid:3) We may now proceed with:
EOMETRY OF SEMISTABLE G -BUNDLES 37 Proof of Lemma 4.15.
Consider once more the diagram: H E /W G,H π ' ' ν / / χ H ' ' PPPPPPPPPPPP g G E e ρ / / (cid:15) (cid:15) (cid:3) G Eχ G (cid:15) (cid:15) M E ( H ) //W G,H ρ / / M E ( G )By Lemma 4.23, ρ is étale on M E ( H ) str - reg //W G,H . Therefore its base change, e ρ is étale on g G E str - reg . We also know that H str - reg E ⊆ H reg E is étale over G E byProposition 4.4. Thus both the source and target of ν str - reg is étale over G E , andhence ν str - reg is itself étale as required. (cid:3) Finally we come to
Proof of Lemma 4.18.
We have a commutative diagram T E { { ✈✈✈✈✈✈✈✈✈ ❍❍❍❍❍❍❍❍❍ M E ( H ) / / M E ( G )which exhibits M E ( G ) as T E //W and M E ( H ) as T E //W H . The further quo-tient M E ( H ) //W G,H is thus identified with T E //N W ( W H ) (recall that W G,H ∼ = N W ( W H ) /W H ).Let us denote by Σ = Σ H ⊆ Φ the roots of H with respect to T . The locus M E ( G ) [ H ] (respectively, M E ( H ) reg ♥ ) is precisely the image of ( T E ) Σ (see Lemma 3.10).Thus M E ( G ) [ H ] is identified with the quotient of ( T E ) Σ by the subgroup of W which preserves the locus ( T E ) Σ . But this subgroup is precisely N W ( W H ). Thusboth M E ( G ) [ H ] and M E ( H ) reg ♥ //W G,H are identified with ( T E ) Σ //N W ( W H ). Inparticular, the map M E ( H ) reg ♥ //W G,H → M E ( G ) [ H ] is an isomorphism as required. (cid:3) The Tannakian approach and unipotent bundles
The goal of this section is to understand the geometry of the locus of unipotentbundles G uni E in G E . Let ( G E ) ∧ uni denote the formal neighbourhood of the unipotentlocus. Similarly, we have the unipotent cone G uni in G and its formal neighbourhood G ∧ uni . We will prove: Theorem 5.1.
Let
E, E ′ be two pointed curves of arithmetic genus 1. Then anyisomorphism of formal groups [ J ( E ) ∼ = \ J ( E ′ ) defines G -equivariant isomorphisms ( G E ′ ) ∧ uni ∼ / / ( G E ) ∧ uni G uni E ′ ∼ / / ?(cid:31) O O G uni E ?(cid:31) O O Corollary 5.2. If E is an ordinary elliptic curve over k (in particular, if char ( k ) =0 ) then we have isomorphisms G ∧ uni ∼ / / ( G E ) ∧ uni G uni ∼ / / ?(cid:31) O O G uni E ?(cid:31) O O In order to prove Theorem 5.1 we will use the following result.
Theorem 5.3.
Given a genus 1 marked curve E as above, there is an equivalenceof stacks G E ≃ Fun ⊗ (Rep k ( G ) , Tor( J ( E ))) . Theorem 5.3 is a combination of a Tannaka duality statement, expressing G -bundlesin terms of their associated vector bundles, and a Fourier-Mukai duality, relatingvector bundles on E and torsion sheaves on J ( E ).The idea of the proof of Theorem 5.1 is to use the isomorphism of formal groups c E ′ ∼ = b E to identify torsion sheaves on E and E ′ which are supported in a formalneighborhood of the identity. Remark 5.4.
Note that in characteristic zero, we can identify b E ∼ = d G m ∼ = c G a .Thus, under these conditions we obtain isomorphisms between the nilpotent, unipo-tent, and elliptic unipotent cones (and their formal neighbourhoods). These isomor-phisms arise from exponential maps in characteristic 0.In characteristic p >
0, there is no isomorphism of formal groups c G a ∼ = d G m . Nev-ertheless, under very mild conditions on the characteristic, there are G -equivariantisomorphisms (the so-called Springer isomorphisms) between the nilpotent cone in g and the unipotent cone in G . It is natural to conjecture that there are also Springerisomorphisms between G uni E and G uni (and g uni ). We plan to return to this in futurework.5.1. Tannaka duality.
In this subsection, we will use Tannaka duality to expressthe stack G E in terms of its associated vector bundles. The primary reference willbe the paper [Lur04], however see also [Nor76] for the original approach.Let S be a k -scheme. We denote by Rep k ( G ) denote the symmetric monoidalcategory of finite dimensional representations of G over k . Given a G -bundle P G on S , the associated vector bundle construction affords a symmetric monoidal functorass( P G ) : Rep( G ) → Vect( S ) V
7→ P G × G V EOMETRY OF SEMISTABLE G -BUNDLES 39 where the right hand side denotes the exact category of vector bundles on S (withmonoidal structure given by tensor product). Moreover, ass( P G ) is continuous(meaning it preserves all small colimits), exact, and sends finite dimensional repre-sentations to vector bundles on S .The basic idea of Tannakian reconstruction is that the bundle P G can be recoveredfrom the functor ass( P G ).More precisely, let Fun ⊗ (Rep k ( G ) , Vect( S )) denote the groupoid of exact tensorfunctors (note that Vect( S ) is an exact category, so this makes sense). Theorem 5.5.
The associated bundle construction defines an equivalence of groupoids ass: Bun G ( S ) ∼ −→ Fun ⊗ (Rep k ( G ) , Vect( S )) Proof.
By [Lur04, Theorem 5.11] the associated bundle construction defines anequivalence of groupoidsass : Bun G ( S ) ∼ −→ Fun ⊗ tame (QC( BG ) , QC( S )) (5.1)where the right hand side denotes the groupoid of continuous (i.e. colimit preserv-ing), tame tensor functors. By definition, a functor is tame (see [Lur04, Definition5.9]) if it preserves flat objects, and short exact sequences of flat objects.As every object of QC( BG ) (which is identified with the category of O ( G )-comodules)is flat, the right hand side consists of exact functors which take values in flat ob-jects of QC( S ). Moreover, QC( BG ) is the Ind-completion of Rep k ( G ), and thusthe data of an exact, continuous functor from QC( G ) is equivalent to specifyingan exact functor from Rep k ( G ). By continuity of the tensor product, this equiv-alence preserves symmetric monoidal structures. Finally, by construction, everyass( P G )( V ) is a vector bundle for every finite dimensional representation V . Thuswe can identify the groupoid of tensor functors in (5.1) with those in the statementof the theorem, as required. (cid:3) Let E be a curve of arithmetic genus 1 as usual, and let E S = E × S denote thebase-change to an arbitrary test scheme S .Note that a G -bundle on E S is semistable if and only if all its associated vectorbundles are semistable and of degree 0 (see Remark 2.4). Thus we may identify thesublocus G E = Bun G ( E ) ss in terms of Tannaka duality: Corollary 5.6.
For each test scheme S , the associated bundle construction definesan equivalence of groupoids G E ( S ) ∼ −→ Fun ⊗ (Rep k ( G ) , Vect ss , ( E S )) where the right hand side denotes the groupoid of exact tensor functors. Fourier-Mukai transform.
Let J be a one-dimensional commutative groupscheme, for e.g. J ( E ) for E elliptic curve, G m , or G a . Given a test scheme S , wedefine the category of S -families of torsion sheaves on J :Tor( J )( S ) := (cid:26) P ∈
Coh( J S ) (cid:12)(cid:12)(cid:12)(cid:12) P flat over S Supp( P ) → S is finite (cid:27) . The category Tor( J )( S ) carries a monoidal structure given by convolution: P ∗ P = m ∗ ( P ⊠ P )where m : J S × S J S → J S is the multiplication map. This construction defines apresheaf of tensor categories on Sch k . Theorem 5.7.
Let E be an irreducible projective curve of arithmetic genus one.The assignment L 7→ O [ L ] taking a degree line bundle on E to the corresponding skyscraper sheaf on J ( E ) extends to an equivalence of symmetric monoidal categories (cid:0) Vect , ss ( E S ) , ⊗ (cid:1) ≃ (Tor( J ( E ))( S ) , ∗ ) Proof.
The equivalence of abelian categories is classical in the case of a smoothelliptic curve; in our situation where E is an integral curve of arithmetic genus 1,it may be deduced from [Teo99], Theorem 1.3 (for the absolute case), Theorem1.9 (for the relative case). More generally, it is shown in loc. cit. that there is acanonical equivalence between degree 0 semistable torsion-free sheaves on E andtorsion sheaves on E . It follows readily from the construction (see Proposition1.8 of loc. cit. ) that when the torsion-free sheaf happens to be a vector bundle,the corresponding torsion sheaf is supported on the smooth locus of E , which isidentified with J ( E ) (using the section x ).In fact, this construction is an example of a Fourier-Mukai transform. We have acanonical Poincaré line bundle P on E × J ( E ) (normalized at x ∈ E ) with thedefining property that P| E ×{L} ∼ = L for any degree zero line bundle L ∈ J ( E ). Thiskernel extends to a torsion-free sheaf e P on E × E . Taking e P as a Fourier-Mukaikernel defines a functor F : D b ( E ) → D b ( E )which according to Burban–Kreussler [BK05], Theorem 2.21 agrees with the functorof [Teo99] when restricted to semistable torsion-free sheaves of degree zero, and thusfurther restricts to the desired functor on semistable degree 0 vector bundles.From this perspective, the claim about symmetric monoidal structures is a specialcase of a more general claim that the Fourier-Mukai transform D b ( E ) → D b ( J ( E ))induced by the Poincaré bundle intertwines the tensor product on E with convolu-tion (also known as Pontryagin product) induced by the group operation on J ( E ).See e.g. [BBR09] for the case of abelian varieties, which applies to our setting when E is a smooth elliptic curve.Alternatively, in the cuspidal and nodal cases, one may directly apply the results of[FM01], Corollary 2.1.4, 2.2.4 which give an equivalence between degree 0 semistablevector bundles on E of rank n and conjugacy classes of n × n matrices (respectively,invertible matrices). It may be readily checked that, taking all ranks n together,these equivalences define a symmetric monoidal equivalence between all degree zerosemistable vector bundles on E , and torsion modules for k [ t ] (respectively k [ t, t − ]). (cid:3) In particular, we obtain a description of the moduli stack of GL n -bundles in termsof torsion sheaves. Each S -family of torsion sheaves has a well defined length: the EOMETRY OF SEMISTABLE G -BUNDLES 41 degree of Supp( P ) → S . For each test scheme S we let Tor n ( J )( S ) denote themaximal subgroupoid of Tor( J )( S ) whose objects are torsion sheaves of length n . Corollary 5.8.
There are equivalences of stacks GL n,E ∼ = Vect , ss n ( E ) ∼ = Tor n ( J ( E ))Putting together the Tannakian statement Corollary 5.6 with the Fourier-Mukaistatement Theorem 5.7 we obtain: Corollary 5.9.
There is an equivalence of stacks G E ≃ Fun ⊗ (Rep k ( G ) , Tor( J ( E ))) . Torsion sheaves and effective divisors.
We wish to identify the loci ofunipotent bundles in terms of their associated torsion sheaves. This will be ex-pressed in terms of the
Norm map which associates a Cartier divisor to a torsionsheaf.First let us recall that for a smooth curve J , we have isomorphismsDiv n ( J ) ∼ = Hilb n ( J ) ∼ = Sym n ( J )where: • Div n ( J ) denotes the moduli of effective Cartier divisors of degree n , • Hilb n ( J ) is the Hilbert scheme of points of length n on J , • Sym n ( J ) is the quotient J n / S n .(See e.g. Milne [Mil86][Theorem 3.13].) Moreover, these are all smooth varieties ofdimension n .Given an object P of Tor n ( J )( T ) we will associate a relative effective Cartier divisor Div n ( P ) on J T which measures the support of P with multiplicity.We will sketch one construction of Div ( P ) from [MFK94][Section 5.3] (see also[KM76]). Suppose we are given P an S -family of torsion sheaves on J . In particular, P is a coherent sheaf on J S which is flat over S and with no support in depth 0.We consider the determinant line det( P ), which may be defined locally in terms ofa free resolution. As P has no generic support, det( P ) carries a canonical section O J S → det( P ) defined away from the support of P . This rational section definesthe Cartier divisor Div ( P ).This construction gives rise to a morphism of stacks: Div : Tor n ( J ) → Div n ( J ) ∼ = Sym n ( J ) . Note that Sym n ( J ( E )) is isomorphic to the base of the characteristic polynomialmap M E ( GL n ). In fact, the following lemma explains that the morphism Div is arealization of the characteristic polynomial map via the equivalence of GL n -bundlesand torsion sheaves of length n . Lemma 5.10.
The equivalence of Corollary 5.8 fits in to a commutative square: GL n,E (cid:15) (cid:15) ∼ / / Tor n ( J ( E )) ν (cid:15) (cid:15) M E ( GL n ) ∼ / / Sym n ( J ( E )) Proof.
We have already constructed all the arrows in the diagram. To see thatthe diagram commutes, it is sufficient to check the commutativity on the densesubstacks consisting of semisimple objects. This in turn reduces to checking theassertion for n = 1 where it is clear. (cid:3) Unipotent bundles.
Recall that ∈ J denotes the unit for the group struc-ture. Let n ∈ Sym n ( J ) correspond to the effective Cartier divisor on J given bythe unique length n subscheme of J whose underlying reduced scheme is × S . Wewill also use the notation n to denote the corresponding length n subscheme of J .The space of unipotent torsion sheaves on J , denoted Tor n ( J ) uni is defined to bethe fiber of Tor n ( J ) over the point n in Sym n ( J ). In other words, an S -family oftorsion sheaves is called unipotent if the corresponding divisor is equal to n,T .We write Tor n ( J ) ∧ uni for the completion of Tor n ( J ) along the substack Tor n ( J ) uni .The goal of the remainder of this subsection is to show that these stacks only dependon the formal neighborhood of ∈ J .Just as above, we may define the presheaf of categories Tor n ( b J ) and the presheafof sets Sym n ( b J ) parameterizing S -families of torsion sheaves (respectively effectiveCartier divisors) in b J . Again, there is an associated divisor map denoted abusivelyalso by Div : Div : Tor n ( b J ) → Sym n ( b J ) . We denote by Tor n ( b J ) uni the fiber over n . Lemma 5.11.
We have natural isomorphisms giving rise to a commutative diagram
Tor n ( b J ) ∼ / / (cid:15) (cid:15) Tor n ( J ) ∧ uni (cid:15) (cid:15) Sym n ( b J ) ∼ / / Sym n ( J ) ∧ n . In particular, there is an equivalence
Tor n ( J ) uni ∼ = Tor n ( b J ) uni . Proof.
First, we claim that for a test scheme S , the image of Tor n ( b J )( S ) in Tor n ( J )( S )consists of S -families of torsion sheaves on J which are set-theoretically supportedon the closed subset S ⊆ J S . Indeed, note that an S -family of torsion sheaves on b J is, by definition, given by an S -family of torsion sheaves on some infinitesimalthickening of ∈ J . Giving such a family is indeed equivalent to an S -family oftorsion sheaves on J which is set-theoretically supported on S ⊆ J S . Similarly, EOMETRY OF SEMISTABLE G -BUNDLES 43 one shows that the image of Sym n ( b J )( S ) in Sym n ( J )( S ) consists of S -families ofdivisors which are set-theoretically supported in S .To show that Sym n ( b J ) ∼ = Sym n ( J ) ∧ n observe that an S -point of the completionSym n ( J ) ∧ n corresponds to an S -family of degree n effective divisors in Sym n ( J )( S )whose restriction to S red is equal to the divisor n,S red . This is equivalent to sayingthat the set-theoretic support of the divisor is contained in S as required.Finally, to show that Tor n ( b J ) ∼ = Tor n ( J ) ∧ uni , it suffices to prove that the diagramTor n ( b J ) (cid:15) (cid:15) / / Tor n ( J ) (cid:15) (cid:15) Sym n ( b J ) / / Sym n ( J )is cartesian. This follows from the observation that an S -family of torsion sheaveson J is set-theoretically supported on S if and only if the associated divisor isset-theoretically supported on S . (cid:3) Remark 5.12.
In fact, an S -family of torsion sheaves is unipotent if and onlyif its (scheme-theoretic) support is contained in the subscheme n,S . This is aconsequence of the Cayley-Hamilton theorem (see Example 5.13). Example 5.13.
Let J = G a = Spec( k [ t ]), and let S = Spec( R ) for a Noetherianlocal ring R . Then the groupoid Tor n ( G a )( R ) consists of R [ t ]-modules M whichare free over R of rank n .Given such a module M and choosing an R -basis, the action of t is expressed bya matrix A M . Then the associated divisor is given by the function det( t − A M )(a polynomial of degree n with coefficients in R ). The coefficients of χ A ( t ) =det( t − A M ) define the corresponding element of R n = A n ( R ) ∼ = Sym n ( G a )( R ).Note that, by the Cayley-Hamilton theorem, M A is scheme-theoretically supportedin the divisor χ A ( t ). An R -point is nilpotent (respectively, formally nilpotent) if χ A ( t ) = t n (respectively, the non-leading coefficients of χ A ( t ) are nilpotent in R ).5.5. Unipotent cones.
The next result explains that the subfunctors of G E corre-sponding to unipotent (respectively infinitesimally unipotent) bundles can be under-stood via the equivalence of Corollary 5.9 as those functors which factor through thesubcategories of unipotent (respectively, infinitesimally unipotent) torsion sheaves. Proposition 5.14.
Let P denote an object of G E ( S ) with associated functor F : Rep( G ) → QC( J ( E ) × S ) . Then P is contained in the sub-groupoid of unipotent (respectively infinitesimallyunipotent) bundles if and only if, for each V ∈ Rep( G ) , F ( V ) is a unipotent (re-spectively, infinitesimally unipotent) family of torsion sheaves. Proof.
First note that for any morphism of groups G → G ′ , there is a commutativediagram: G E (cid:15) (cid:15) / / G ′ E (cid:15) (cid:15) M E ( G ) / / M E ( G ′ )It follows that if an object of G E ( S ) is unipotent (respectively infinitesimally unipo-tent) then its image in G ′ E ( S ) is unipotent (respectively infinitesimally unipotent).In particular, we obtain such a diagram with G ′ = GL ( V ) for each representation V of G . Thus (noting Lemma 5.10) it follows that if P G is (infinitesimally) unipotentthen all the associated torsion sheaves F ( V ) are (infinitesimally) unipotent.Note that if V is a faithful representation, then the map M E ( G ) → M E ( GL n ) ∼ = Sym n ( J ( E ))has the property that the set-theoretic fiber of the basepoint n consists only ofthe basepoint G of M E ( G ) (i.e. a semisimple G -bundle is trivial if and only if theassociated vector bundle is trivial, at a set-theoretic level). It follows then that if F ( V ) is infinitesimally unipotent, then P is infinitesimally unipotent.This argument doesn’t quite work for the non-infinitesimal case, as the map M E ( G ) → M E ( GL n ) ∼ = Sym n ( J ( E ))is not injective on S -points for a general (possibly non-reduced) scheme S .However, Lemma 5.15 below implies that it is enough to take a sufficiently large col-lection of representations V , . . . , V m (for example the fundamental representations)to obtain that the map M E ( G ) → Y i M E ( GL d i ) ∼ = Sym d ( J ( E )) × . . . × Sym d m ( J ( E ))is a closed embedding in a formal neighbourhood of the basepoint (and thus injectiveon T -points set-theoretically supported on the basepoint). (cid:3) Lemma 5.15.
Let V , . . . , V m be such that their classes generate R ( G ) as a ring .For example, we can take the collection of fundamental representations. Then thecorresponding map M E ( G ) → Sym d ( J ( E )) × . . . × Sym d m ( J ( E )) (5.2) is a closed embedding in a formal neighbourhood of the basepoint G ∈ M E ( G ) .Proof. By assumption, the map of representation rings R ( GL ( V )) ⊗ . . . ⊗ R ( GL ( V r )) → R ( G ) (5.3)is surjective. Thus the corresponding map of varieties T //W → Sym d ( G m ) × . . . × Sym d r ( G m )is a closed embedding. In particular it is a closed embedding after completing atthe basepoint of T //W . EOMETRY OF SEMISTABLE G -BUNDLES 45 Choosing an isomorphism of formal schemes (not necessarily respecting the groupstructure) d G m ∼ = [ J ( E ) gives an identification of the maps in (5.2) and (5.3) in aformal neighbourhood of the basepoint as required. (cid:3) We are now ready to prove Theorem 5.1 (and consequently Theorem 1.6 from theintroduction).
Theorem 5.16.
Any equivalence of formal groups \ J ( E ) ∼ = \ J ( E ) defines equiva-lences: ( G E ) ∧ uni ∼ / / ( G E ) ∧ uni G uni E ∼ / / O O G uni E . O O Proof of Theorem 5.16.
The first part of the theorem says that there is an equiva-lence of stacks of infinitesimally unipotent bundles:( G E ) ∧ uni ∼ / / ( G E ) ∧ uni . By Corollary 5.9, we have an identification G E i ∼ = Fun ⊗ (Rep( G ) , Tor( \ J ( E i ))) . The identification of formal groups \ J ( E ) ∼ = \ J ( E ) defines, for each test scheme S ,an equivalence of symmetric monoidal categories Tor( \ J ( E ))( S ) ≃ Tor( \ J ( E ))( S ),and thus we obtain the required equivalence.The second part of the theorem means that this equivalence preserves the substacksof unipotent bundles. But according to Proposition 5.14, the unipotent bundles maybe recognized as those functors whose corresponding tensor functors factor throughthe subcategory of unipotent torsion sheaves. As the subcategory of unipotenttorsion sheaves is preserved under the equivalenceTor( \ J ( E )) ≃ Tor( \ J ( E )) . we obtain the required result. (cid:3) Examples
In this section we compute (partially) some examples. The reference and notationfor root systems that we used is from the appendix of [Bou81].We recall that G E stands for the stack of semistable G -bundles of degree 0 on E .Denote by J the Jacobian of E . Example 6.1.
Take G = PGL(2). We have two possible closed sets: empty set,full set. The Weyl group acts on ( G m ) E = J by L 7→ L − .We have a decomposition G E = N /G G ( J \ {O} ) /T ⋊ S . Observe that the PGL(2)-bundles
O ⊕ L where
L ∈ J [2] and L 6 = O , have auto-morphism group T ⋊ S which is disconnected. This is to be expected because thecentralizer of a semisimple element in a non-simply-connected group doesn’t haveto be connected. Example 6.2.
Take G = SL(2). Then we have G E = ( J [2] × N /G ) G ( J \ J [2]) /T ⋊ S and here all the bundles have connected automorphism groups. Example 6.3.
Take G = GL(2), T = G m . Then we have G E = J × N /G G ( T E \ diag ) / ( T ⋊ S )and here also all the automorphism groups are connected. Example 6.4.
Let G be a group of type G . The root system isΦ = ±{ α, β, α + β, α + β, α + β, α + 2 β } where α is the short root.The closed subsets not contained in any proper Levi are Φ andΣ = ±{ β, α + β, α + 2 β } . We have G (Σ) = SL(3) which is a pseudo-Levi subgroup. The other closed sets(up to conjugation by W ) are { α } , { β } and ∅ and one can easily see that G ( α ) ≃ GL(2) ≃ G ( β ) and G ( ∅ ) = T ≃ G m . It is also an exercise to check that N G ( G ( α )) = G ( α ) and similarly for G ( β ).The roots give us an isomorphism T E ≃ J where the first coordinate correspondsto α and the second to β . The partition of G E is therefore G E = N G /G ⊔ (cid:0) J [3] × N SL(3) (cid:1) / SL(3) ⊔ ( J − J [3]) × N G ( α ) /G ( α ) ⊔ J × N G ( β ) /G ( β ) ⊔ ( T E − coord ) /T ⋊ W. where coord ≃ J × {O E } ∪ {O E } × J ⊂ J × J corresponds to the coordinate axes.Now a more involved example: EOMETRY OF SEMISTABLE G -BUNDLES 47 Example 6.5.
Take G = Sp(6) which is a group of type C . The simple roots are { α , α , α } with α the long root. The longest root is 2 α + 2 α + α .We put α := − (2 α + 2 α + α ). The affine Dynkin diagram is >α α α α < We use the Borel-de Siebenthal algorithm to produce closed sets (see Section 3.2).If we remove only α or only α we get back the group G . If we remove the affinevertex and some other vertices we get all the Levi subgroups of G .If we remove one vertex different from α or α we get α α α < Σ = ± (cid:26) α , α , α + α , α + α , α + 2 α + α (cid:27) SL(2) × Sp(4) >α α α Σ = ± (cid:26) α , α , α + 2 α + α ,α + 2 α + α , α + α (cid:27) Sp(4) × SL(2)One can check easily that s s takes Σ into Σ , hence also the group G (Σ ) into G (Σ ).By applying once more the above algorithm we get (discarding the Levi subgroups)the following root system: α α ′ α Σ = ± { α , α + 2 α + α , α + α } SL(2) where α ′ = − (2 α + α ) = 2 α + α .The groups G (Σ) are computed by inspecting the root/coroot system and by lookingat the center and the fundamental group.We can also compute the relative Weyl groups and find W G, Σ = 1 and W G, Σ = S .Actually the Weyl group of Sp(6) is W = ( Z / Z ) ⋊ S and one can check (or seefrom the diagram) that the Weyl group of Σ is W Σ ≃ Z .If we iterate once more the algorithm we only get Levi subgroups of Φ, Σ or Σ .For the partial order (cid:23) the closed sets Φ , Σ , Σ are the maximal ones (up topermutation by W ).Hence the closed pieces in the partition of G E are J [2] × N Sp(6) / Sp(6) (cid:0) J [2] − diag (cid:1) × N SL(2) × Sp(4) / SL(2) × Sp(4) (cid:0) ( J [2] − diags ) × N SL(2) / SL(2) (cid:1) / S . Here also, all the bundles have connected automorphism group even though wequotient by S because its action is free on J [2] \ diags . Example 6.6.
The last example we compute (in detail) is a simply connected groupof type D . We would like to provide an example to show that the automorphismgroup of a semisimple bundle can be disconnected. Let G = Spin(8) be the simply connected group of type D and denote by T amaximal torus. The simple roots are denoted by α i , i = 1 , , , G is isomorphic to µ × µ .For convenience, let us spell out the root datum that we used for the computations(for more details one should consult [Bou81, Planche IV, p.256]): • the character lattice and the root lattice are X ∗ ( T ) = h ω , ω , ω , ω i ⊃ h α , α , α , α i where the roots are given in terms of fundamental characters as: α = − ω + ω α = − ω + 2 ω − ω − ω α = − ω + 2 ω α = − ω + 2 ω • the cocharacter lattice which equals the coroot lattice (because simply con-nected) is X ∗ ( T ) = h ˇ α , ˇ α , ˇ α , ˇ α i• the longest root is α := α + 2 α + α + α .Notice that the simple coroots give us an isomorphism (ˇ α , ˇ α , ˇ α , ˇ α ) : G m → T .It is useful ot think of the simple coroots (and fundamental characters) as beingthe coordinates of T .The affine Dynkin diagram is α α α α α where α = − ( α + 2 α + α + α ).If we remove a vertex different from α we get the whole Φ. Removing α we obtainthe diagram of a group of type A × A × A × A .Therefore (using again Borel–de Siebenthal algorithm, Section 3.2), up to conjugacy,there are only two maximal closed sets, namely Φ andΣ := ±{ α , α , α , α + 2 α + α + α } which is a root system of type A . All the other closed subsets that we can obtainiterating the algorithm are Levi subgroups of G or of G (Σ).The group G (Σ) can be computed to be(SL(2) × SL(2) × SL(2) × SL(2)) /µ where µ is the diagonal central subgroup. (This is achieved by computing the rootdatum for G (Σ).) Notice that this is not simply connected! EOMETRY OF SEMISTABLE G -BUNDLES 49 Let us recall that the Weyl group of G is isomorphic to S ⋉ P where P ≤ ( Z / Z ) isthe hyperplane Q x i = 1 and where the symmetric group acts on it by permutations.One sees best the action of S ⋉ P on characters/cocharacters by introducingadditional variables ε i , i = 1 , , , ω = ε , ω = ε + ε , ω = ( ε + ε + ε + ε ) and ω = ( ε + ε + ε − ε ). Using these coordinates the action of S is by permuting the ε i and the action of P is by multiplication (i.e. changingsigns).The Weyl group W Σ of G (Σ) is generated by the permutations (12), (34) togetherwith ( − , − , , , , − , − ∈ P where we think of Z / Z = {± } multiplica-tively.Another computation shows that the normalizer of W Σ in W is generated by W Σ , P and the permutation (13)(24) . The relative Weyl group is W G, Σ ≃ h (13)(24) i × h ( − , , − , i ≃ S × Z / Z . There are two closed strata in G E , namely J [2] × N G /G (cid:0) ( J [2] − J [2] ) × N G (Σ) (cid:1) /G (Σ) ⋊ W G, Σ . (6.1)Let us be more explicit about the semisimple parts J [2] and J [2] .The center of Spin(8) is { t ∈ T | α i ( t ) = 1 } which, using the identification G m ≃ T given by cocharacters becomes { ( z , z , z , z ) ∈ T | z = z = z = z and z = z z z } which can be rewritten as { ( z , z , z , z ) ∈ T | z = z = z = z = z z z } ≃ µ . Similarly, the center of G (Σ) is Z ( G (Σ)) = { ( z , z , z , z ) ∈ T | z = z = z = z } ≃ µ . So in Eq. (6.1) the J [2] corresponds to Z (Spin(8)) ≃ µ and J [2] corresponds to Z ( G (Σ)) ≃ µ . The complement can also be made explicit J [2] − J [2] = { ( L , O , L , L ) | L i ∈ J [2] and L L L
6≃ O} . One can check that the relative Weyl group W G, Σ acts on the above locus withoutfixed points. Hence we won’t find a semisimple bundle whose automorphism group(as a Spin(8)-bundle) contains G (Σ) and is disconnected.However, we’ll produce a semisimple bundle whose automorphism group (as aSpin(8)-bundle) is disconnected with connected component precisely the maximaltorus. Using the identification G m ≃ T given by the simple coroots, a T -bundle isa quadruple of line bundles P T = ( L , L , L , L ).To simplify the analysis we impose furthermore L i ≃ O for i = 1 , , , P T T × Spin(8)) ◦ to be T we must have α ∗ ( P T )
6≃ O for all roots α (there are 12 positive roots). Given the simplifying assumption we’ve made L i ≃ O , i = 1 , , , L and O 6≃ L L L . Let L , L ∈ J [2] \ {O} be two non-isomorphic line bundles (this is possible if we’renot in characteristic 2). Then the T -bundle P T := ( O , L , O , L ) satisfies the aboveconditions and hence the automorphism group of P := P T T × Spin(8) has connectedcomponent equal to T .The following element in the Weyl group σ := σ α σ α σ α σ α stabilizes P T (as a T -bundle!). More precisely, in general we have σ ( L , L , L , L ) = ( L − , L − , L − , L − ).Hence σ ∈ Aut( P ) which implies that Aut( P ) is a disconnected group (with con-nected component equal to T ). EOMETRY OF SEMISTABLE G -BUNDLES 51 Appendix A. Classification of elliptic closed subsets
The focus of this section is on the case when E is an elliptic curve but we’ll quicklyreview the cusp (rational) and nodal (trigonometric) cases. In Section 3.3 we gavea general recipe to produce closed subsets of the root system Φ of G as centralizersof elements in t , T and T E . More precisely, we put x ∈ t Σ x := { α ∈ Φ | α ( x ) = 0 } (A.1) t ∈ T Σ t := { α ∈ Φ | α ( t ) = 1 } (A.2) P ∈ T E Σ P := { α ∈ Φ | α ∗ ( P ) ≃ triv } . (A.3)The collection of these subsets will be denoted (in this section) by A rat , A trig re-spectively A ell .Over the complex numbers, using the analytic uniformizations ( X ∗ ( T ) ⊗ Z C ) / X ∗ ( T ) ≃ T and ( X ∗ ( T ) ⊗ Z ( R )) / X ∗ ( T ) ⊕ ≃ E , the proof of Proposition A.6 is all that isneeded. However, to deal also with the positive characteristic, one needs to do alittle combinatorics of root systems and diagonalizable groups. The key ingredientis a Lemma from [Ste68, 5.1] that we record as Lemma A.1 and its elliptic analogLemma A.3.Let a ∈ t and consider Φ a ∈ A rat . Then Φ a is the root system of the Levi subgroup C G ( k · a ) (the centralizer of a subtorus is always a Levi subgroup). If L is a Levisubgroup of G then for a generic element a in the center of the Lie algebra l , theclosed set Φ a is the set of roots of L .The trigonometric situation is a bit more complicated. For t ∈ T the closed subsetΦ t is the root system of the connected centralizer C G ( t ) ◦ which might not be a Levisubgroup if the order of t is finite. The reductive subgroups of G thus obtained arecalled pseudo-Levi subgroups and they are well known in the theory of reductivegroups.Going further to the elliptic case, it turns out that the subsets Φ P are the rootsystems of intersections of two pseudo-Levi subgroups of G . We will sketch theproofs below after fixing some notation.Fix a Borel subgroup B containing the maximal torus T ⊂ B ⊂ G , and let Φ ⊃ ∆be the set of roots and of simple roots respectively.A prime number p is said to be good for G if p doesn’t divide any coefficient ofthe highest root (w.r.t. ∆) of Φ. This is a very tiny restriction on p and G . Forexample, p > p ≥ X = X ∗ ( T ) the character lattice of T and by Y its dual lattice. Onecan construct, in a functorial way, the compact abelian Lie group T c = ( Y ⊗ Z R ) / Y such that Hom gr ( T c , C × ) = X .For t ∈ T we put X t := { λ ∈ X | λ ( t ) = 1 } and similarly for x ∈ T c we put X x = { λ ∈ X | λ ( x ) = 1 } . We define analogously Φ t and Φ x .We start with a preparation lemma from [Ste68, 5.1] that is needed to pass froman arbitrary field to R . if the root system is not irreducible, consider all the highest roots Lemma A.1. (1) For any t ∈ T there exists x ∈ T c such that X t = X x .(2) If x ∈ T c is of finite order prime to p = char( k ) then there exists t ∈ T such that X t = X x .Proof. We sketch the idea of the proof.The abelian group X / X t is finitely generated and injects into k × through the map λ λ ( t ). Since a finite subgroup of k × is cyclic we deduce that the torsion part ofthe abelian group X / X t is cyclic. To X / X t corresponds a sublattice of Y and hencea subgroup T ′ c ⊂ T c which is the product of a compact torus and a cyclic group. Assuch it has a topological generator, say x ∈ T ′ c . By construction we have X x = X t .Conversely, the finite cyclic group X / X x corresponds to a cyclic subgroup µ of T of order prime to p . Hence µ has a generator t of the same order as x and byconstruction we have X t = X x . (cid:3) Remark A.2.
The reason we need the order to be prime to p = char( k ) is that in G m there are no points of order p , i.e. µ p is infinitesimal.If the field k has elements of infinite order then G m has a Zariski generator andhence any torus has a Zariski generator. Therefore in (2) above we can replace theassumption x of finite order prime to p by the component group of h x i has orderprime to p .For P ∈ T E put X P = { λ ∈ X | λ ∗ ( P ) ≃ O} . Similarly one can prove Lemma A.3.
For any
P ∈ T E there exist x , x ∈ T c such that X P = X x ∩ X x .Conversely, for x , x ∈ T c of finite order prime to p there exists P ∈ T E such that X P = X x ∩ X x .Proof. In the proof of Lemma A.1 we used that a finite subgroup of k × must becyclic. We also used that k × has a primitive r th root of unity if and only if r isprime to char( k ).The analog for an elliptic curve is: a finite subgroup of J ( E ) is a product of twocyclic groups. The group of torsion points J ( E )[ r ] is isomorphic to Z /r × Z /r ifand only if r is prime to char( k ).Hence the quotient X / X P is a product of a free abelian group and two cyclic groups.The rest of the proof is the same. (cid:3) We can now easily deduce
Corollary A.4. (1) For any t ∈ T there exists x ∈ T c such that Φ x = Φ t .(2) For any P ∈ T E there exist x , x ∈ T c such that Φ P = Φ x ∩ Φ x . Conversely, by [MS03, Prop. 30,32] we have
Proposition A.5.
Assume char( k ) is good for G . EOMETRY OF SEMISTABLE G -BUNDLES 53 (1) For any x ∈ T c there exists t ∈ T such that Φ t = Φ x .(2) For any x , x ∈ T c there exists P ∈ T E such that Φ P = Φ x ∩ Φ x . By construction, Φ S is a closed subset of Φ. Let e ∆ := ∆ ⊔ { α ( l )0 : l a connected component of Φ } be the set of simple roots of the corresponding affine root system, where α ( l )0 is thenegative of the longest root in the corresponding connected component Φ ( l )+ . (If theDynkin diagram is not connected there are several longest roots corresponding toeach connected component and we want to add the negative of each of them to ∆.)Given a subset S ⊂ e ∆ put Φ S := Z S ∩ Φ.Just as Levi subgroups correspond, up to conjugation, to subsets of the simple roots,pseudo-Levi subgroups admit a similar characterization.
Proposition A.6. [MS03, Lemma 29] , [Lus95, Lemma 5.4] The set { Φ x | x ∈ T c } /W consists of subsets of the form Φ S for some proper subset S of e ∆ as definedabove.Proof. We recall the proof from loc.cit. for the convenience of the reader. Put π : t R = Y ⊗ Z R → T c the projection.Let x ∈ T c . If the closure of the subgroup generated by x is a torus, its centralizeris a Levi subgroup and we’re done.Otherwise, write x = x ′ s with s of finite order and x ′ that generates a torus. Wehave Φ x ′ s = Φ x ′ ∩ Φ s and since Φ x ′ corresponds to a Levi subgroup we are left todeal with Φ s . So we can suppose x is of finite order in T c .The affine Weyl group X ∗ ( T ) ⋉ W acts on t R and, up to conjugating x by someelement of W , we can assume x lies in the image of the fundamental alcove throughthe map π : t R → T c . Hence we we can take ˜ x ∈ π − ( x ) in the fundamental alcove.Looking at the roots as linear functions on t R we haveΦ x = { α ∈ Φ | α (˜ x ) ∈ Z } . Define S := { α ∈ e ∆ } | α (˜ x ) ∈ { , }} . By construction Φ S ⊂ Φ x . Let us prove thatΦ S = Φ x .Recall that the fundamental alcove in t R is defined by the inequalities0 ≤ α ( y ) ≤ , for all α ∈ Φ + . Denoting by γ ( l ) = − α ( l )0 the highest root of Φ ( l ) , the fundamental alcove can berewritten as 0 ≤ α i ( y ) ≤ α i ∈ ∆0 ≤ γ ( l ) ( y ) ≤ l of Φ . For α ∈ Φ ( l ) , + x we have 0 ≤ α (˜ x ) ≤ γ ( l ) (˜ x ) ≤ α (˜ x ) ∈ { , } . If α (˜ x ) = 0, then α , being a positive sum of simple roots, is a sum of simple rootsall of which must belong to S , hence α ∈ Φ S .If α (˜ x ) = 1 then γ (˜ x ) − α (˜ x ) = 0 hence all the simple roots appearing in γ ( l ) − α must be in S . Thus γ ( l ) − α ∈ Φ S and since γ ( l ) (˜ x ) = 1 we also have γ ( l ) ∈ Φ S . Wededuce α ∈ Φ S and the proof is finished. (cid:3) Remark A.7.
One can think of the above construction in the following way. Takethe extended Dynkin diagram whose nodes are indexed by e ∆ and remove somenon-zero number of nodes (at least one from each connected component). This willthen be the (non-extended) Dynkin diagram corresponding to some pseudo-Levisubgroup of G . In fact it is known that the Dynkin diagrams corresponding toclosed subsets Σ ⊂ Φ can be obtained by repeatedly applying this procedure, see[BDS49] for more details.
Proposition A.8.
Assume that char( k ) is good for G . The collection of closedsubsets A ell consists precisely of intersections of two elements of A trig . In otherwords, the G (Σ) for Σ ∈ A ell are precisely the neutral component of intersectionsof two pseudo-Levi subgroups.Proof. It follows from Proposition A.6 together with Proposition A.5 (cid:3)
Remark A.9.
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