hilbert-mumford stability on algebraic stacks and applications to -bundles on...

HILBERT-MUMFORD STABILITY ON ALGEBRAIC STACKS

AND APPLICATIONS TO G-BUNDLES ON CURVES

JOCHEN HEINLOTH

Abstract In these notes we reformulate the classical Hilbert-Mumford crite-rion for GIT stability in terms of algebraic stacks this was independently done

by Halpern-Leinster [23] We also give a geometric condition that guarantees

the existence of separated coarse moduli spaces for the substack of stable ob-jects This is then applied to construct coarse moduli spaces for torsors under

parahoric group schemes over curves

Introduction

The aim of these notes is to reformulate the Hilbert-Mumford criterion from geo-metric invariant theory (GIT) in terms of algebraic stacks (Definition 12) and useit to give an existence result for separated coarse moduli spacesOur original motivation was that for various moduli problems one has been ableto guess stability criteria which have then been shown to coincide with stabilityconditions imposed by GIT constructions of the moduli stacks It seemed strangeto me that in these constructions it is often not too difficult to find a stabilitycriteria by educated guessing however in order to obtain coarse moduli spaces onethen has to prove that the guess agrees with the Hilbert-Mumford criterion fromGIT which often turns out to be a difficult and lengthy taskMany aspects of GIT have of course been reformulated in terms of stacks by Alper[3] Also Iwanari [32] gave a clear picture for pre-stable points on stacks and con-structed possibly non-separated coarse moduli spaces The analog of the numericalHilbert-Mumford criterion has been used implicitly in many places by several au-thors Most recently Halpern-Leinster [23] independently gave a formulation veryclose to ours and applied it to construct analogs of the HarderndashNarasimhan strati-fication for moduli problems under a condition he calls Θ-reductivityOur main aim is to give a criterion that guarantees that the stable points form aseparated substack (Proposition 26) Once this is available one can apply generalresults (eg the theorems of Keel and Mori [33] and Alper Hall and Rydh [4]) toobtain separated coarse moduli spaces (Proposition 28) As side effect we hopethat our formulation may serve as an introduction to the beautiful picture developed[23]The guiding examples which also provide our main applications are moduli stacksof torsors under parahoric group schemes Using our method we find a stabilitycriterion for such torsors on curves and construct separated coarse moduli spaces ofstable torsors (Theorem 319) Previously such moduli spaces had been constructedfor generically trivial group schemes in characteristic 0 by Balaji and Seshadri [7]who obtain that in these cases the spaces are schemes Also in any characteristicthe special case of moduli of parabolic bundles has been constructed in [27] butit seems that coarse moduli spaces for twisted groups had not been constructedbeforeThis problem was the starting point for the current article because in [27] most ofthe technical problems arose in the construction of coarse moduli spaces by GIT

1

2 JOCHEN HEINLOTH

that were needed to prove cohomological purity results for the moduli stack Theresults of this article allow to bypass this issue and apply to the larger class ofparahoric groupsThe structure of the article is as follows In Section 1 we state the stability criteriondepending on a line bundle L on an algebraic stack M As a consistency check wethen show that this coincides with the Hilbert Mumford criterion for global quotientstacks (Proposition 18) To illustrate the method we then consider some classicalmoduli problems and show how Ramanathanrsquos criterion for stability of G-bundleson curves can be derived form our criterion rather easily The same argumentapplies to related moduli spaces as the moduli of chains or pairsIn Section 2 we formulate the numerical condition on the pair (ML) implyingthat the stable points form a separated substack (Proposition 26) and derive anexistence result for coarse moduli spaces (Proposition 28) Again as an illustrationwe check that this criterion is satisfied in for GIT-quotient stacks and for G-bundleson curvesFinally in Section 3 we apply the method to the moduli stack of torsors under aparahoric group scheme on a curve We construct coarse moduli spaces for thesubstack of stable points of these stacks For this we also need to prove some ofthe basic results concerning stability of parahoric group schemes that could be ofindependent interestAcknowledgments This note grew out of a talk given in Chennai in 2015 Thecomments after the lecture encouraged me to finally revise of an old sketch thathad been on my desk for a very long time I am grateful for this encouragementand opportunity I thank M Olsson for pointing out the reference [23] before itappeared and J Alper P Boalch and the referees for many helpful commentsand suggestions While working on this problem discussions with V Balaji NHoffmann J Martens A Schmitt have been essential for me

Contents

Introduction 11 The Hilbert-Mumford criterion in terms of stacks 311 Motivation The classical Hilbert-Mumford criterion 312 L-stability on algebraic stacks 413 Determining very close degenerations 514 The example of GIT-quotients 615 Stability of vector bundles on curves 816 G-bundles on curves 917 The example of chains of bundles on curves 1318 Further examples 142 A criterion for separatedness of the stable locus 1421 Motivation from the valuative criterion 1422 The test space for separatedness and equivariant blow ups 1523 An existence result for coarse moduli spaces 1824 The example of GIT-quotients 1925 The example of G-bundles on curves 193 Torsors under parahoric group schemes on curves 2031 The setup 2032 Line bundles on BunG 2133 Preliminaries on parabolic subgroups of BruhatndashTits group schemes 2134 Very close degenerations of G-bundles 2435 The stability condition 2636 Canonical reduction for G-torsors 28

GIT-STABILITY FOR STACKS 3

37 Boundedness for stable G-torsors 3038 Conclusion for G-torsors 314 Appendix Fixing notations for BruhatndashTits group schemes 3241 ChevalleyndashSteinberg systems pinnings and valuations 3242 Parahoric group schemes 335 Appendix Basic results on BunG 33References 35

1 The Hilbert-Mumford criterion in terms of stacks

Throughout we will work over a fixed base field k The letter M will denote analgebraic stack over k which is locally of finite type over k and we will alwaysassume that the diagonal ∆∶M rarrM timesM is quasi-affine as this implies that ourstack is a stack for the fpqc topology ([35 Corollaire 107])We have two guiding examples in mind First global quotient stacks [XG] whereX is a proper scheme and G is an affine algebraic group acting on X and secondthe stack BunG of G-bundles on a smooth projective curve C for a semi-simplegroup G over k

11 Motivation The classical Hilbert-Mumford criterion As the numeri-cal criterion for stability from geometric invariant theory [37 Theorem 21] servesas a guideline we start by recalling this briefly To state it and in order to fix oursign conventions we need to recall the definition of weights of Gm-equivariant linebundles

111 Weights of equivariant line bundles As usual we denote the multiplicativegroup scheme by Gm ∶= Speck[t tminus1] and the affine line by A1 ∶= Speck[x] Thestandard action

act∶Gm timesA1 rarr A1

is given by tx ∶= tx ie on the level of rings act∶k[x] rarr k[x t tminus1] is given byact(x) = tx We write [A1Gm] for the quotient stack defined by this action Thisstack is called Θ in [23]By definition Gm-equivariant line bundles L on A1 are the same as line bundles on[A1Gm] For any line bundle L on A1 the global sections are H0(A1L) = k[x] sdot efor some section e which is unique up to a scalar multipleThus an equivariant line bundle L on A1 defines an integer d isin Z by act(e) = tdecalled the weight of L and we will denote it as

wt(L) ∶= d

In particular we find that

H0([A1Gm]L) =H0(A1L)Gm = k sdot xde if wt(L) = d le 00 if wt(L) = d gt 0

To compare the sign conventions in different articles the above equation is the one tokeep in mind because Mumfordrsquos construction of quotients uses invariant sectionsof line bundles Similarly a Gm-equivariant line bundle L on Speck is given by amorphism of underlying modules act∶Lrarr Lotimesk[t tminus1] with act(e) = tde for somed isin Z The integer d is again denoted by

wtGm(L) ∶= d

4 JOCHEN HEINLOTH

112 Mumfordrsquos notion of stability Mumford considers a projective scheme Xequipped with an action of a reductive group G and a G-linearized ample bundleL on X For any x isin X(k) and any cocharacter λ∶Gm rarr G Mumford definesmicroL(xλ) isin Z as follows The action of Gm on x defines a morphism λx∶Gm rarr Xwhich extends to an equivariant morphism fxλ∶A1 rarr X because X is projectiveHe defines

microL(xλ) ∶= minuswt(flowastxλL)The criterion [37 Theorem 21] then reads as follows

A geometric point x isin X(k) is stable if and only if the stabilizer of x in G is finiteand for all λ∶Gm rarr G we have

microL(xλ) gt 0 equivalently wt(flowastxλL) lt 0

12 L-stability on algebraic stacks It is easy to reformulate the numericalHilbertndashMumford criterion in terms of stacks once we fix some notation Thequotient stack [A1Gm] has two geometric points 1 and 0 which are the images ofthe points of the same name in A1 For any algebraic stackM and f ∶ [A1Gm]rarrMwe will write f(0) f(1) isinM(k) for the points given by the images of 01 isin A1(k)

Definition 11 (Very close degenerations) Let M be an algebraic stack over kand x isinM(K) a geometric point for some algebraically closed field KkA very close degeneration of x is a morphism f ∶ [A1

KGmK] rarrM with f(1) cong xand f(0) cong x

Very close degenerations have been used under different names eg in the contextof K-stability these are often called test-configurations Our terminology shouldonly emphasize that f(0) is an object that lies in the closure of a K point of MK which only happens for stacks and orbit spaces but if X =M is a scheme thenthere are no very close degenerations

Definition 12 (L-stability) Let M be an algebraic stack over k locally of finitetype with affine diagonal and L a line bundle on M A geometric point x isinM(K)is called L-stable if

(1) for all very close degenerations f ∶ [A1KGmK]rarrM of x we have

wt(flowastL) lt 0

and(2) dimK(AutM(x)) = 0

Remark 13(1) We can also introduce the notion of L-semistable points by requiring only

le in (1) and dropping condition (2)(2) The notion admits several natural extensions Since the weight of line bun-

dles extends to a elements of the groups

Pic(BGm)otimesZ R cong R

the above definition can also be applied if L isin Pic(M)otimesZ R This is oftenconvenient for classical notions of stability that depend on real parameters

In [23] Halpern-Leinster uses a cohomology class α isinH2(MkQ`) insteadof a line bundle For α = c1(L) this gives the same condition As thecondition given above is a numerical one this is sometimes more convenientand we will refer to it as α-stability

(3) As the above condition is numerical and H2d([A1Gm]Q`) = Q` for alld gt 0 one can generalize the notion further


(4) For the above definition to be nonempty it is convenient to assume thatthe automorphism groups of generic objects are finite so that (2) can befulfilled This appears to rule out the example of vector bundles on curvesbut fortunately there is a quite general method available to rigidify theproblem ie to divide out generic automorphism groups ([1 Theorem515][2 Appendix C])

(5) General representability results of [24 Theorem 16] imply that under ourconditions on M the stack of morphisms Mor([A1Gm]M) is again analgebraic stack locally of finite type As weights of Gm-actions on linebundles are locally constant in families this implies that L-stability is pre-served under extension of algebraically closed fields LK In particularif we define an S-valued point M(S) to be L-stable if the correspondingobjects are stable for all geometric points of S we get a notion that is pre-served under pull-back and therefore defines an abstract substackMs subMWe will see that in may cases L-stability turns out to be an open conditionand then Ms is again an algebraic stack but this is not true for arbitraryML

Notation Given a line bundle L on M and x isinM(K) we will denote by wtx(L)the homomorphism

wtx(L)∶Xlowast(AutM(x)) = Hom(GmAutM(x))rarr Z

which maps λ∶Gm rarr AutM(x) to wtGm((xλ)lowastL) where (xλ)∶ [SpecKGm]rarrMis the morphism defined by x and λ

Example 14 A toy example illustrating the criterion is given by the anti-diagonalaction Gm times A2 rarr A2 defined as t(x y) ∶= (tx tminus1y) The only fixed point of thisaction is the origin 0 The quotient (A2 minus0)Gm is the affine line with a doubledorigin the first example of a non-separated schemeSince the latter space is a scheme none of its points admits very close degenerationsThis changes if we look at the full quotient [A2Gm] which contains the additionalpoint [(00)Gm] The inclusions of the coordinate axes ιx ιy ∶A1 rarr A2 define veryclose degenerations of the points (01) and (10) and it will turn out (Lemma 16)that these constitute essentially the only very close degenerations in this stackA line bundle on [A2Gm] is an equivariant line bundle on A2 Since line bundleson A2 are trivial all equivariant line bundles are given by a character ()d∶Gm rarr Gmand we find wt0∶Pic([A2Gm]) cong Z Moreover for the corresponding line bundle Ldthe weights wt(ιlowastxLd) = dwt(ιlowastyLd) = minusd so for each d ne 0 only one of the points(10) and (01) can be Ld-stableFor d = 0 the points (10) (00) (01) are all semistable and these points would beidentified in the GIT quotient

13 Determining very close degenerations To apply the definition of L-stability one needs to classify all very close degenerations The next lemma showsthat these can be described by deformation theory of objects x that admit non-constant morphisms Gm rarr AutM(x) Let us fix our notation for formal discs

D ∶= Speck[[t]] D ∶= Speck((t))

Lemma 15 Let M be an algebraic stack locally of finite type over k = k withquasi-affine diagonal

(1) For any very close degeneration f ∶ [A1Gm]rarrM the induced morphism

λf ∶Gm = Aut[A1Gm](0)rarr AutM(f(0))is nontrivial

6 JOCHEN HEINLOTH

(2) The restriction functorM([A1Gm])rarr limlarrETHM([Spec(k[x]xn)Gm]) is an

equivalence of categories

Proof For part (2) elegant proofs have been given independently by Alper Halland Rydh [4 Corollary 36] and Bhatt Halpern-Leistner [10 Remark 83] Bothproofs rely on Tannaka duality one uses an underived version the other a derivedversion As the statement produces the point f(1) out of a formal datum let usexplain briefly why this is possible The composition

φ∶D = Speck[[x]]rarr A1 rarr [A1Gm]

is faithfully flat because both morphisms are flat and the map is surjective becauseboth points 10 of [A1Gm] are in the imageBy our assumptions M is a stack for the fpqc topology ([35 Corollaire 107]) wetherefore see thatM([A1Gm]) can be described as objects inM(k[[x]]) togetherwith a descent datum with respect to φ

Moreover the canonical map M(D) congETHrarr limlarrETHM(k[x](xn)) is an equivalence of

categories This follows for example because the statement holds for schemes andchoosing a smooth presentation X rarrM one can reduce to this statement [42 Tag07X8]In particular this explains already that an element of limlarrETHM([Spec(k[x]xn)Gm])will produce a k[[x]]-point of M The problem now lies in constructing a descentdatum for this morphism as

D times[A1Gm] D = Spec(k[[x]]otimesk[x] k[x t tminus1]otimesk[y] k[[y]])

where the last tensor product is taken via y = xt The ring on the right handside is not complete and the formal descent data coming form an element inlimlarrETHM([Spec(k[x]xn)Gm]) only seems to induce a descent datum on the comple-

tion of the above ringHere the Tannakian argument greatly simplifies the problem as it gives a conciseway to capture the information that a Gm action induces a grading and thereforeallows to pass from power series to polynomialsLet us deduce (1) First note that this holds automatically if M is a schemebecause then f(1) is a closed point and f(0) lies in the closure of f(1)In general choose a smooth presentation p∶X rarrM If λf is trivial we can lift the

morphism f ∣0∶ [0Gm]rarr Speck rarrM to f0∶ [0Gm]rarrX Since p is smooth we caninductively lift this morphism to obtain an element in limlarrETHX([Spec(k[x]xn)Gm])Thus we reduced (1) to the case M =X

Using this lemma we can compare L-stability to classical notions of stability simplyby first identifying objects for which the automorphism group contains Gm andthen studying their deformations The next subsections illustrate this procedure inexamples

14 The example of GIT-quotients Let X be a projective variety equippedwith the action of a reductive group G and a G-linearized line bundle L Againbundles on the quotient stack [XG] are the same as equivariant bundles on X sowe will alternatively view L as a line bundle on [XG]Let us fix some standard notation Given λ∶Gm rarr G we will denote by Pλ the cor-responding parabolic subgroup Uλ its unipotent radical and Lλ the corresponding


Levi subgroup ie

Pλ(R) = g isin G(R)∣ limtrarr0

λ(t)gλ(tminus1) exists

Uλ(R) = g isin G(R)∣ limtrarr0

λ(t)gλ(tminus1) = 1

Lλ(R) = g isin G(R)∣λ(t)gλ(tminus1) = g ie Lλ = CentrG(λ)To compare L-stability on [XG] to GIT-stability on X we first observe that thetest objects appearing in the conditions coincide

Lemma 16 For any cocharacter λ∶Gm rarr G and any geometric point x isin X(K)that is not a fixed point of λ the equivariant map fλx∶A1

K rarrX defines a very close

degeneration fλx∶ [A1KGmK]rarr [XG] Moreover any very close degeneration in

the stack [XG] is of the form fλx for some xλ

Proof Since fλx(0) is a fixed point of λ and x is not we have f(0) cong f(1) thus fis a very close degenerationConversely let f ∶ [A1Gm] rarr [XG] be any very close degeneration We need tofind a Gm equivariant morphism

A1

f X

π

[A1Gm] f [XG]

Since π∶X rarr [XG] is a G-bundle the pull-back p∶Xtimes[XG] [A1Gm]rarr [A1Gm] is

a G-bundle on [A1Gm] To find f is equivalent to finding a Gm-equivariant sectionof this bundle This will follow from the known classification of Gm-equivariant G-bundles on the affine line

Lemma 17 (G-bundles on [A1Gm]) Let G be a reductive group and P a G-bundle on [A1Gm] Denote by P0 the fiber of P over 0 isin A1

(1) If there exists x0 isin P0(k) (eg this holds if k = k) Then there exists acocharacter λ∶Gm rarr G unique up to conjugation and an isomorphism ofG-bundles

P cong [(A1 timesG)(Gm (act λ))]Moreover P has a canonical reduction to Pλ

(2) Let G0 ∶= AutG(P0) and λ∶Gm rarr G0 the cocharacter defined by P ∣[0Gm]

Consider the G0 bundle PG0 ∶= IsomG(PP0) on [A1Gm] Then

PG0 cong [(A1 timesG0)(Gm (act λ))]

ie P cong IsomG0([(A1 times G0)(Gm (act λ))]P0) Moreover PG0 has acanonical reduction to P0λ sub G0

For vector bundles this result is [6 Theorem 11] where some history is given Thegeneral case can be deduced from this using the Tannaka formalism As we willneed a variant of the statement later we give a slightly different argument

Proof The second part follows from the first as theG0-bundle PG0

0 = IsomG(P0P0)has a canonical point id We added (2) because it gives an intrinsic statementindependent of choices

To prove (1) note that x0 defines an isomorphism GcongETHrarr AutG(P0) and a section

[SpeckGm] rarr P ∣[0Gm] This induces a section [SpeckGm] rarr PPλ As the

map π∶PPλ rarr [A1Gm] is smooth any section can be lifted infinitesimally to

8 JOCHEN HEINLOTH

Speck[x]xn for all n ge 0 Inductively the obstruction to the existence of a Gm-equivariant section is an element in H1(BGm T(PPλ)A10 otimes (xnminus1)(xn)) = 0 and

the choices of such liftings form a torsor under H0(BGm TPPλA10otimes(xnminus1)(xn))Now by construction Gm acts with negative weight on TPPλ0 = Lie(G)Lie(Pλ)and it also acts with negative weight on the cotangent space (x)(x2) so thereexists a canonical Gm equivariant reduction Pλ of P to PλSimilarly the vanishing of H1 implies that we can also find a compatible family ofλ-equivariant sections [(Speck[t]tn)Gm] rarr P and by Lemma 15 this defines asection over [A1Gm] ie a morphism of G-bundles [(A1 timesG)Gm λ]rarr PHere we could alternatively see this explicitly as follows Let us consider π∶Pλ rarr A1

as Gm equivariant Pλ bundle on A1 and consider the twisted action

⋆∶Gm timesPλ rarr Pλgiven by t ⋆ p ∶= (tp) sdot λ(tminus1) Our point x0 is a fixed point for this action byconstruction as we used it to identify P0 with G and e isin G is a fixed point forthe conjugation λ(t) sdot sdot λ(tminus1) Moreover λ acts with non-negative weights onLie(Pλ) and also on TA1 = ((x)(x2))or Therefore Bia lynicki-Birula decomposition[31] implies that there exists a point x1 isin πminus1(1) such that limtrarr0 t ⋆ x1 = x0 Thisdefines a λ-equivariant section A1 rarr P

This also completes the proof of Lemma 16

Proposition 18 Let XGL be a projective G-scheme together with a G-linearizedline bundle L A point x isin X(k) is GIT-stable with respect to L if and only if the

induced point x isin [XG](k) is L-stable

Proof As X is projective given x isinX(k) and a one parameter subgroup λ∶Gm rarr G

we obtain an equivariant map fλx∶A1 rarrX and thus a morphism f ∶ [A1Gm]rarrX

By Lemma 16 all very close degenerations arise in this way As wt(flowastL) = wt(flowastL)we therefore find that x satisfies the Hilbert-Mumford criterion for stability if andonly if it is L-stable

15 Stability of vector bundles on curves We want to show how the classicalnotion of stability for G-bundles arises as L-stability For the sake of clarity weinclude the case of vector bundles first Let C be a smooth projective geometricallyconnected curve over k and denote by Bundn the stack of vector bundles of rank nand degree d on C

151 The line bundle A natural line bundle on Bundn is given by the determi-nant of cohomology Ldet ie for any vector bundle E on C we have LdetE ∶=det(Hlowast(CEnd(E)))minus1 and more generally for any f ∶T rarr Bundn corresponding tofamily E on C times T one defines

flowastLdet ∶= (detRprTlowast End(E))or

Remark 19 Since any vector bundle admits Gm as central automorphisms to

apply our criterion we need to pass to the rigidified stack Bund

n ∶= Bundn(Gm ob-tained by dividing all automorphism groups by Gm ([1 Theorem 515]) To obtain

a line bundle on this stack we need a line bundle on Bundn on which the centralGm-automorphisms act trivially This is the reason why we use the determinant ofHlowast(CEnd(E)) instead of Hlowast(CE) It is known that up to multiples and bundles

pulled back from the Picard variety this is the only such line bundle on Bundn (see

eg [11] which also gives some history on the Picard group of Bundn)


152 Very close degenerations of bundles To classify maps f ∶ [A1Gm] rarr Bundnwe have as in Lemma 17

Lemma 110 There is an equivalence

Map([A1Gm]Bundn) cong ⟨(E E i)iisinZ∣E i sub E iminus1 sub E cupE i = E E i = 0 for i≫ 0⟩

ie giving a vector bundle together with a weighted filtration is equivalent to givinga morphism [A1Gm]rarr Bundn

Proof We give the reformulation to fix the signs A vector bundle E on C togetherwith a weighted filtration E i sub E iminus1 sub sdot sdot sdot sub E the Rees construction Rees(E) ∶=oplusiisinZE ixminusi defines an OC[x] module ie a family on CtimesA1 which is Gm equivariantfor the action defined on the coordinate parameter with Rees(E)∣Ctimes0 cong gr(E)For the converse we argue as in Lemma 17 Any morphism f ∶ [A1Gm] rarr Bundndefines a morphism Gm rarr AutBunn(f(0)) ie a grading on the bundle f(0) suchthat the corresponding filtration lifts canonically to the family

153 Computing the numerical invariants Given a very close degeneration [A1Gm]rarrBundn we can easily compute wt(Ldet∣f(0)) as follows We use the notation of the

preceding lemma and write Ei ∶= E i(E i+1) so that f(0) = oplusEi Note that Gm acts

with weight minusi on Ei Denoting further micro(Ei) ∶= degEirkEi

we find

wt(Ldet∣f(0)) = minuswtGm(detHlowast(CoplusHom(EiEj)))

because Ldet was defined to be the dual of the determinant of cohomology AsGm acts with weight (i minus j) on Hom(EiEj) it acts with the same weight on thecohomology groups taking determinants we find the weight (i minus j)χ(Hom(EiEj))on detHlowast(CHom(EiEj)) Thus by Riemann-Roch we find

wt(Ldet∣f(0)) =sumij

(j minus i)χ(Hom(EiEj))

=sumij

(j minus i)( rk(Ei)deg(Ej) minus rk(Ej)deg(Ei) + rk(Ei) rk(Ej)(1 minus g))

= 2sumiltj

(j minus i)(rk(Ei)deg(Ej) minus rk(Ej)deg(Ei))

As it is more common to express the condition in terms of the subbundles E l insteadof the subquotients Ei let us replace the factors (j minus i) by a summation over l withi le l lt j

wt(Ldet∣f(0)) = 2suml

(sumilel

rk(Ei))(sumjgtl

deg(Ej)) minus (sumilel

deg(Ei))(sumjgtl

rk(Ej))

= 2suml

rk(E l)(n minus rk(E l))(micro(E l) minus micro(EE l))

This is lt 0 unless micro(E i) gt micro(EE i) for some i Conversely if micro(E i) gt micro(EE i) forsome i then the two step filtration 0 sub E i sub E defines a very close degeneration ofpositive weight Thus we find the classical condition

Lemma 111 A vector bundle E is Ldet-stable if and only if for all E prime sub E we have

micro(E prime) lt micro(E)

16 G-bundles on curves Let us formulate the analog for G-bundles where Gis a semisimple group over k and we assume k = k to be algebraically closed Wedenote by BunG the stack of G-bundles on C

10 JOCHEN HEINLOTH

161 The line bundle For the stability condition we need a line bundle on BunGOne way to construct a positive line bundle is to choose the adjoint representa-tion Ad ∶ G rarr GL(Lie(G)) which defines for any G-bundle P its adjoint bundleAd(P) ∶= P timesG Lie(G) and set

Ldet∣P ∶= detHlowast(CAd(P))orIf G is simple and simply connected it is known that Pic(BunG) cong Z (eg [11]) Ingeneral Ldet will not generate the Picard group but since our stability conditiondoes not change if we replace L by a multiple of the bundle this line bundle willsuffice for us

162 Very close degenerations of G-bundles Recall from section 14 that for acocharacter λ∶Gm rarr G we denote by Pλ Uλ Lλ the corresponding parabolic sub-group its unipotent radical and the Levi subgroupTo understand very close degenerations of bundles will amount to the observationthat Lemma 17 has an extension that holds for families of bundlesThe source of degenerations is the following analog of the Rees construction Givenλ∶Gm rarr G we obtain a homomorphism of group schemes over Gm

conjλ∶Pλ timesGm rarr Pλ timesGm(p t)↦ (λ(t)pλ(t)minus1 t)

By [31 Proposition 42] this homomorphism extends to a morphism of groupschemes over A1

grλ∶Pλ timesA1 rarr Pλ timesA1

in such a way that grλ(p0) = limtrarr0 λ(t)pλ(t)minus1 isin Lλ times 0Moreover these morphisms are Gm equivariant with respect to the action (conjλact)on Pλ timesA1Given a Pλ bundle Eλ on a scheme X this morphism defines a Pλ bundle on X times[A1Gm] by

Rees(Eλ λ) ∶= [((Eλ timesA1) timesgrλA1 (Pλ timesA1))Gm]

where timesgrλA1 denotes the bundle induced via the morphism grλ ie we take the

product over A1 and divide by the diagonal action of the group scheme Pλ timesA1A1 which acts on the right factor via grλ By construction this bundle sat-isfies Rees(Eλ λ)∣Xtimes1 cong Eλ and

Rees(Eλ λ)∣Xtimes0 cong EλUλ timesLλ Pλis the analog of the associated graded bundle

Remark 112 If λprime∶Gm rarr Pλ is conjugate to λ in Pλ say by an element u isin Uλthen P ∶= Pλ = P prime

λ and

grλprime(p t) = grλ(upuminus1 t)Therefore we also have

Rees(Eλ λ) cong Rees(Eλ λprime)which tells us that the Rees construction only depends on the reduction to P andthe homomorphism

λ∶Gm rarr Z(P U) sub P UIn the case G = GL(V ) this datum is the analog of a weighted filtration on V whereas λ∶Gm rarr P sub GL(V ) would define a grading on V

Given a G-bundle E a cocharacter λ and a reduction Eλ of E to a Pλ bundle theG-bundle Rees(Eλ λ)timesPλG defines a morphism f ∶ [A1Gm]rarr BunG with f(1) = E We claim that all very close degenerations arise in this way


Lemma 113 Let G be a split reductive group over k Given a very close degenera-tion f ∶ [A1Gm]rarr BunG corresponding to a family E of G-bundles on X times[A1Gm]there exist

(1) a cocharacter λ∶Gm rarr G canonical up to conjugation(2) a reduction Eλ of the bundle E to Pλ(3) an isomorphism Eλ cong Rees(Eλ∣Xtimes1 λ)

Proof Given E we will again denote by E0 ∶= E ∣Xtimes0 and E1 ∶= E ∣Xtimes1 We define thegroup scheme GE0 ∶= AutG(P0X) = E0 timesGconj G This is a group scheme over Xthat is an inner form of G timesX And the morphism f ∣[0Gm] induces a morphism

λ0∶Gm timesX rarr AutG(E0X) = GE0 As in Lemma 17 (2) it is convenient to replace E by the GE0-torsor E prime ∶= IsomG(E E0)We know that λ0 defines a parabolic subgroup Pλ0 sub GE0 and the canonical reduc-tion of E prime0 to Pλ0 lifts uniquely to a reduction E primeλ of E prime by the same argument usedin Lemma 17 The last step of the proof is then to consder the twisted action⋆∶Gm times E primeλ rarr E primeλ Note that the fixed points for the action are simply the points inthe Levi subgroup Lλ0 sub E prime0 = GE0 The needed analog of the Bia lynicki-Birula decomposition is a result of Hesselinck[31] By the lemma we already know that for all geometric points x of X andp isin E primeλ∣x all limit points limtrarr0 λ(t)⋆p exist On the other hand by [31 Proposition42] the functor whose S-points are morphisms StimesA1 rarr E primeλ such that the restrictionto S timesGm is given by the action of Gm on E primeλ is represented by a closed subschemeof E primeλ Thus the functor is represented by E primeλThus the twisted action ⋆ on E primeλ extends to a morphism ⋆∶A1timesE primeλ rarr E primeλ In particularthis induces A1timesEλ1 rarr Eλ and thus a morphism of Pλ0-bundles Rees(Eλ1 λ)rarr EλThis proves that the statement of the Lemma holds if we replace G by GE0 To compare this with the description given in the lemma note that for any geometricpoint x isin X(k) the choice of a trivialization E0x cong G defines an isomorphismGP0 cong G and therefore λ0∣x defines a defines a conjugacy class of cocharactersλ∶Gm rarr G This conjugacy class is locally constant (and therefore does not dependon the choice of x) because we know from [20 Expose XI Corollary 52bis] that thescheme parametrizing conjugation of cocharacters TranspG(λ0 λ) is smooth overX (This is the analog of the statement for vector bundles that a Gm action on Eallows to decompose E = oplusEi as bundles ie the dimension of the weight spaces ofthe fibers is constant over x) This defines λ To conclude we only need to recallthat reductions of E prime to Pλ0 correspond to reductions of E to Pλ

Lemma 114 Let G rarrX be a reductive group scheme λ∶GmX rarr G a cocharacterand E a G-torsor over X Then a natural bijection between

(1) Reductions Eλ of E to Pλ(2) Parabolic subgroups P sub GE = AutG(EX) that are locally conjugate to Pλ

is given by Eλ ↦ P ∶= AutPλ(EλX) sub AutG(EX)

Proof A reduction of E is a section s∶X rarr EPλ Note that GE acts on EPλ andStabGE (s) sub GE is a parabolic subgroup that is locally of the same type as Pλbecasuse this holds if E is trivial and s lifts to a section of E Locally in the smoothtopology we may assume these conditionsSimilarly given P sub GE locally the action of P on EPλ has a unique fixed pointand this defines a section

Using the lemma we find that both sections of E primePλ0 and sections of EPλ corre-spond to parabolic subgroups of AutG(E) = AutGE0 (E prime) This proves the lemma

12 JOCHEN HEINLOTH

Remark 115 Note that in the above result we assumed that G is a split group Ingeneral we saw that the natural subgroup that contains a cocharacter is AutG(E0)which is an inner form of G over X In particular it may well happen that G doesnot admit any cocharacter or any parabolic subgroupThis is apparent for example in the case k = R and G = U(n) For a G-torsor Eon C we will find a canonical reduction to a parabolic subgroup Pk sub Gk but thedescent datum will then only be given for P sub AutG(EX)

163 The numerical criterion Finally we have to compute the weight of Ldet onvery close degenerationsThe computation is the same as for vector bundles and for the criterion it is some-times convenient to reduce it to reductions for maximal parabolic subgroups Let uschoose T sub B sub G a maximal torus and a Borel subgroup and λ∶Gm rarr G a dominantcocharacter ie ⟨λα⟩ ge 0 for all roots such that gα isin Lie(B) Let us denote by Ithe set of positive simple roots with respect to (TB) and by IP ∶= αi isin I ∣λ(αi) = 0the simple roots αi for which minusαi is also a root of Pλ For j isin I let us denote byωj isinXlowast(T )R the cocharacter defined by ωj(αi) = δij And by Pj the correspondingmaximal parabolic subgroupThen λ∶Gm rarr Z(Lλ) sub Lλ sub Pλ Thus for any very close degeneration f ∶ [A1Gm]rarrBunG given by Rees(Eλ λ) the bundle Ldet defines a morphism

wtL∶Xlowast(Zλ) sub AutBunG(f(0))rarr Z

Then λ = sumjisinIminusIP ajωj for some aj gt 0

wt(Ldet∣f(0)) = wtL(λ) = sumjisinIminusIP

aj wtL(ωj)

For each j we get a decomposition Lie(G) = oplusi Lie(G)i where Lie(G)i is the sub-space of the Lie algebra on which ωj acts with weight i Each of these spaces is arepresentation of Lλ and also of the Levi subgroups Lj of Pj Using this decompo-sition we find as in the case of vector bundles

wtL(ωj) = minuswtGm(detHlowast(CoplusE0λ timesLλ Lie(G)i))=sum

i

i sdot χ(E0λ timesLλ Lie(G)i)

=sumi

i(deg(E0λ timesLλ Lie(G)i) + dim(Lie(G)i)(1 minus g))

= 2sumigt0

i(deg(E0λ timesLλ Lie(G)i)

Now deg(E0λtimesLλLie(G)i) = (deg(det(E0λtimesLλLie(G)i)) Since the Levi subgroupsof maximal parabolics have only a one dimensional space of characters all of thesedegrees are positive multiples of det(Lie(Pj)) Thus we find the classical stabilitycriterion

Corollary 116 A G-bundle E is Ldet-stable if and only if for all reductions EPto maximal parabolic subgroups P sub G we have deg(EP timesP Lie(P )) lt 0

164 Parabolic structures Parabolic G-bundles are G-bundles equipped with areduction of structure group at a finite set of closed pointsLet us fix notation for these We keep our reductive group G the curve C and a fi-nite set of rational points x1 xn sub C(k) and parabolic subgroups P1 Pn subG


BunGP x(S) ∶= ⟨(E s1 sn)∣E isin BunG(S) si∶S rarr ExitimesSPi sections ⟩The forgetful map BunGP x rarr BunG is a smooth proper morphism with fibersisomorphic to prodiGPiIn particular very close degenerations of a parabolic G bundle are uniquely definedby a very close degenerations of the underlying G-bundleThere are more line bundles on BunGP x namely any dominant character ofχi∶Pi rarr Gm defines a positive line bundle on GPi and this induces a line bun-dle on BunGP xThe weight of a this line bundle on a very close degeneration is given by the pairingof χi with the one parameter subgroup in AutPi(Exi)We will come back to this in the section on parahoric bundles (Section 35)

17 The example of chains of bundles on curves We briefly include the ex-ample of chains of bundles as an easy example of a stability condition that dependson a parameterAgain we fix a curve C A holomorphic chain of length r and rank n isin Nr+1 isthe datum (Ei φi) where E0 Er are vector bundles of rank ni and φi∶Ei rarr Eiminus1

are morphisms of OC-modules The stack of chains is denoted Chainn It is analgebraic stack locally of finite type One way to see this is to show that theforgetful map Chainn rarr prodri=0 Bunni is representable As for the stack Bundn allchains admit scalar automorphisms Gm so we will need to look for line bundles onwhich these automorphisms act triviallyThe forgetful map to prodri=0 Bunni already gives a many line bundles on Chainn aswe can take products of the pull backs of the line bundles Ldet on the stacks Bunni Somewhat surprisingly these are only used in [40] whereas the standard stabilityconditions (eg [5]) arise from the following bundles

(1) Ldet ∶= det(Hlowast(CEnd(oplusEi)))or(2) Fix any point x isin C and i = 1 r Set Li ∶= det(Hom(oplusjgeiEjxopluslltiElx))

Remark 117 Note that on all of these bundles the central automorphism groupGm of a chain acts trivially and one can check that up to the multiple [k(x) ∶ k]the Chern classes of the bundles Li do not depend on x We will not use this factThe choice of the bundles Li isin Pic(Chainn) is made to simplify our computationsFrom a more conceptual point of view the lines Lni ∶= det(Eix) define bundles onBunni which are of weight ni with respect to the central automorphism group GmThe pull backs of these bundles generate a subgroup of Pic(Chainn) and the Li area basis for the bundles of weight 0 in this subgroup

To classify maps [A1Gm] rarr Chainn note that composing with forget ∶Chainn rarrprodBunni such a morphism induces morphisms [A1Gm]rarr Bunni which we alreadyknow to correspond to weighted filtrations of the bundles Ei and a lifting of amorphism [A1Gm]rarrprodBunni to Chainn is given by homomorphisms φi∶Ei rarr Eiminus1

that respect the filtrationThus we find that a very close degeneration of a chain E is a weighted filtrationE i sub E and we already computed

wt(Ldet∣gr(E)) = 2sumi

rk(E i)(n minus rk(E i))(micro(EE i) minus micro(E i))

= 2sumi

rk(E i)n(micro(E) minus micro(E i))

Further we have

wt(Lj ∣gr(E)) =sumi

(sumlgej

rk(E il )n) minus ni(sumlgej

rk(E i))

14 JOCHEN HEINLOTH

Thus we find that a chain E is Ldet otimes L2mii -stable if and only if for all subchains

E prime sub E we have

sumi deg(E primei) +sumjmj rk(opluslgejE primel)sumi rk(E primei)

lt sumideg(Ei) +sumjmj rk(opluslgejEl)

sumi rk(Ei)

This is equivalent to the notion of αminusstability used in [5 Section 21]

Remark 118 Also for the moduli problem of coherent systems on C ie pairs(E V ) where E is a vector bundle of rank n on C and V sub H0(CE) is a subspaceof dimension r one recovers the stability condition quite easily Families over Sare pairs (E V φ∶V otimes OC rarr E) where E is a family of vector bundles on X times SV is a vector bundle on S and we drop the condition that φ corresponds to aninjective map V rarr prSlowastE We denote this stack by CohSysnr There are naturalforgetful maps CohSysnr rarr Bunn and CohSysnr rarr BGLn induced respectivelyby the bundles E and V As above for any point x isin X we obtain a line bundledet(V)n otimes det(Ex)minusr on CohSysnr and together with Ldet one then recovers theclassical stabilty condition that one finds for example in [15 Definition 22]

18 Further examples Other more advanced examples can be found in thearticle [23 Section 42] For example this contains an argument how the Futakiinvariant introduced by Donaldson arises from the point of view of algebraic stacks

2 A criterion for separatedness of the stable locus

We now want to give a criterion which guarantees that the set of L-stable pointsis a separated substack if the stack M and the line bundle L satisfy suitable localconditions (Proposition 26) The article [36] by Martens and Thaddeus was animportant help to find the criterion Again the proofs turn out to be quite closeto arguments that already appear in Mumfordrsquos book

21 Motivation from the valuative criterion Let us first sketch the basicidea LetM be an algebraic stack For the valuative criterion for separatedness oneconsiders pairs f g∶D = Speck[[t]]rarrM together with an isomorphism f ∣Speck((t)) congg∣Speck((t)) and tries to prove that for such pairs f cong g The basic datum is thereforea morphism

f cup g∶D cupD DrarrM

Note that the scheme D cupD D is a completed neighborhood of the origins in theaffine line with doubled origin

A1 cupGm A1 cong [A2 minus 0Gm (t tminus1)] sub [A2Gm]Thus also the union of two copies of D along their generic point is naturally anopen subscheme of a larger stack

D cupD D sub [(A2 timesA1 D)Gm] = [Spec(k[x y]otimesk[π]π=xy k[[π]])Gm (t tminus1)]where the right hand side inserts a single point [0Gm]Further the coordinate axes Speck[x]Speck[y] sub Speck[x y] = A2 define closedembeddings [A1Gm] rarr [(A2 timesA1 D)Gm] that intersect in the origin [0Gm] Asthe weights of the Gm action on the two axes are 1 and minus1 we again see that forany line bundle L on this stack the weights of the restriction to the two copies of[A1Gm] differ by a signIn terms of L-stability onM this means that whenever a morphism f g∶DcupDDrarrMextends to [(A2 timesA1 D)Gm] only one of the two origins can map to an L-stablepointAs the complement of the origin [0Gm] is of codimension 2 in the above stack onecould expect that a morphism f cup g∶D cupD D rarrM extends to a morphism of some


blow up of [(A2 timesA1 D)Gm] centered at the origin This will be the assumptionthat we will impose on MThe basic observation then is that the exceptional fiber of such a blow up can bechosen to be a chain of equivariant projective lines and it will turn out that theweight argument indicated above still works for such chains if the line bundle Lsatisfies a numerical positivity condition

22 The test space for separatedness and equivariant blow ups Let R bea discrete valuation ring together with a local parameter π isin R K ∶= R[πminus1] thefraction field and k = R(π) the residue fieldAs for the affine line the scheme SpecR has a version with a doubled special pointSTR ∶= SpecR cupSpecK SpecR the test scheme for separateness The analog of[A2Gm] is given as follows The multiplicative group Gm acts on R[x y](xy minusπ)via tx ∶= tx ty ∶= tminus1y Let us denote

STR ∶= [Spec(R[x y](xy minus π))Gm]As before we have

(1) two open embeddings

jx∶SpecR

cong

STR

[SpecR[xxminus1]Gm] cong [SpecR[xxminus1 y](xy minus π)Gm]

OO

jy ∶SpecR

cong

STR

[SpecR[y yminus1]Gm] cong [SpecR[x y yminus1](xy minus π)Gm]

OO

that coincide on SpecK(2) two closed embeddings

ix∶ [A1kGm] cong [Speck[x]Gm] cong [SpecR[x y](y xy minus π)Gm] sub STR

iy ∶ [A1kGm] cong [Speck[y]Gm] cong [SpecR[x y](xxy minus π)Gm] sub STR

and the intersection of these is [SpeckGm] =∶ [0Gm]We will need to understand blow ups of STR supported in [SpeckGm] For this letus introduce some notation A chain of projective lines is a scheme E = E1cupsdot sdot sdotcupEnwhere Ei sub E are closed subschemes together with isomorphisms φi∶Ei

congETHrarr P1

such that Ei cap Ei+1 = xi is a reduced point with φi(xi) = infin φi+1(xi) = 0 Anequivariant chain of projective lines is a chain of projective lines together with anaction of Gm such that for each i the action induces the standard action of someweight wi on P1 = Projk[x y] ie this is given by tx = twi+dx ty = tdy for somedWe say that an equivariant chain is of negative weight if all the wi are negativeIn this case for all i the points φminus1

i (0) are the repellent fixed points in Ei for trarr 0

Lemma 21 Let I sub R[x y](xyminusπ) be a Gm invariant ideal supported in Speck =SpecR[x y](x y π) Then there exists an invariant ideal I and a blow up

p∶BlI(STR) = [BlI(SpecR[x y](xy minus π))Gm]rarr STR

dominating BlI such that pminus1([0Gm]) cong [EGm] where E is a chain of projectivelines of negative weight

16 JOCHEN HEINLOTH

Proof As I is supported in (x y) there exists n such that (x y)n sub I and as it isGm invariant it is homogeneous with respect to the grading for which x has weight1 and y has weight minus1 Let P (x y) = sumNi=l aixd+iyi be a homogeneous generator ofweight d ge 0 with al ne 0 that is not a monomial We may assume ai isin Rlowast as π = xyWe claim that then xd+lyl isin I As (x y)n sub I we may assume that d + 2N lt nBut then P (x y) minus (al+1al)xyP (x y) isin I is an element for which the coefficient ofxd+l+1yl+1 vanishes Inductively this shows that xd+lyl isin I so that I is monomialWrite I = (xn ym xniymi)i=1N with ni lt nmi ltm This ideal becomes principalafter sucessively blowing up 0 and then blowing up 0 or infin in the exceptional P1rsquosBlowing up (x y) we get charts with coordinates (x y) ↦ (xprimey y) and (x y) ↦(xxyprime) Since x has weight 1 and y has weight minus1 we see that the weights of (xprime y)are (2minus1) and the weights of (x yprime) are (1minus2)In the first chart the proper transform of I is (xprimenyn ym xprimeniymi+ni)i=1N Thisideal is principal if m = 1 and otherwise equal to an ideal of the form

yk(ymminusk mixed monomials of lower total degree)A similar computation works in the other chart By induction this shows that theideal will become principal after finitely many blow ups and that in each chart thecoordinates (x(i) y(i)) have weights (wi vi) with wi gt vi

Remark 22 Let L be a line bundle on [P1(Gmactw)] then

deg(L∣P1) = 1

w(wtGm(L∣infin) minuswtGm(L∣0))

Proof Write d = deg(L∣P1) Let A10 = Speck[x]A1

infin = Speck[y] be the two coor-dinate charts of P1 and ex isin L(A1

0) ey isin L(A1infin) two generating sections such that

ey = xdex on Speck[xxminus1]Then act(ey) = twtGm(L∣infin)ey and act(xdex) = twdtwtGm(L∣0)xdex Thus we find

twd = twtGm(L∣infin)minuswtGm(L∣0)

Combining the above computations we propose the following definitions

Definition 23 Let M be an algebraic stack locally of finite type with affinediagonal We say that M is almost proper if

(1) For all valuation rings R with field of fractions K and fK ∶SpecK rarr Mthere exists a finite extension RprimeR and a morphism f ∶SpecRprime rarrM suchthat f ∣SpecKprime cong fK ∣SpecKprime and

(2) for all complete discrete valuation rings R with algebraically closed residue

field and all morphisms f ∶STR rarrM there exists a blow up BlI(STR) sup-

ported at 0 such that f extends to a morphism f ∶BlI(STR)rarrM

Given a line bundle L on an almost proper algebraic stack M we say that L is nefon exceptional lines if

for all f ∶STR rarrM the extension f from (2) can be chosen such that forall equivariant projective lines Ei in the exceptional fiber of the blow up wehave deg(L∣Ei) ge 0

Remark 24 Note that for schemes (and also algebraic spaces) of finite typethe above definition reduces to the usual valuative criterion for properness Wealready saw in Lemma 15 that in case X =M is a scheme any morphism from anequivariant projective line to X must be constant and more generally if Gm actson SpecR[x y](xy minus π)) with positive weight on x and negative weight on y themorphism [(SpecR[x y](xy minus π))Gm] rarr SpecR is a good coarse moduli space


([3]) so for any algebraic space X any morphism BlI(STR) rarr X factors throughSpec(R) Therefore the existence of a morphism as in (2) implies the valuativecriterion for properness ([42 Tag 0ARL])Similarly (2) above could be used to define a notion of almost separatedness forstacks

Remark 25 In the above definition we could have replaced condition (2) by thecondition

(2rsquo) For all discrete valuation rings R and all morphisms f ∶STR rarr M there

exists a finite extension RprimeR and a blow up BlI(STRprime) supported at 0

such that f extends to a morphism f ∶BlI(STRprime)rarrM

These two conditions are equivalent for stacks locally of finite type we put (2)because it is sometimes slightly more convenient to check

Proof We only need to show that (2) implies (2rsquo) The standard argument forschemes can be adapted here with some extra care taking into account automor-phisms of objects Note first that because M is locally of finite type it suffices toprove (2rsquo) in the case where the DVR R has an algebraically closed residue field In

this case we denote by R the completion of R and by fR the restriction of f to STR

This map has an extension R∶BlI(STR)rarrM and we have seen in Lemma 21 that

we may assume that I is already defined over R ie that BlI(STR) = BlI(STR)R

As the extension RR is faithfully flat to define an extension f ∶BlI(STR) rarrM of f we need to define a descent datum for f ie for the two projections

p12∶BlI(STR)RotimesRR rarr BlI(STR)R we need an element φ isin IsomM(plowast1(f) plowast2(f))that over STRotimesRR coincides with the one given by f

Now note that if we denote by K the fraction field of R there is a cartesian diagramof rings (see eg [28 Lemma 5])

RotimesR R

mult

K otimesR K

mult

R K

Also by our assumptions on the diagonal of M we know that Isom(plowast1 f plowast2 f) rarrBlI(STR)RotimesRR is affine and we have canonical sections of this morphism over

the diagonal ∆∶BlI(STR) rarr BlI(STR)RotimesRR and over BlI(STR)KotimesRK= STKotimesRK

we have the section defined by the descent datum for f As these agree on theintersections ∆ ∶ Spec K rarr Spec K otimesR K the cartesian diagram implies that thesedefine φ This map is a cocycle because this holds over the open subscheme STR

Recall from Remark 13 that L-stable points define an abstract substackMs subMfor these points Definition 23 implies the valuative criterion

Proposition 26 (Separatedness of stable points) Let M be an algebraic stacklocally of finite type over k with affine diagonal and L a line bundle onM Supposethat M is almost proper and L is nef on exceptional lines Then the stack of stablepoints Ms sube M satisfies the valuative criterion for separatedness ie for anycomplete discrete valuation ring R with fraction field K and algebraically closedresidue field any morphism STR rarrMs factors through SpecR

Proof We need to show that for any morphism f ∶STR rarrM such that all pointsin the image of f are stable we have jx f cong jy f

18 JOCHEN HEINLOTH

Let us first show that conditions imply that the morphisms coincide on closedpoints ie f(jx((π))) cong f(jy((π))) isinM

If f extends to a morphism f ∶STR rarrM we have wt(ilowastxflowastL) = minuswt(ilowastyflowast(L)) Aswe assumed that both closed points are stable this shows that neither ix f noriy f can be a very close degeneration ie f(jx((π))) cong f(ix(0)) = f(iy(0)) congf(jy((π)))If f does not extend then by assumption there exists an extension f ∶BlI(STR) rarrM such that deg(L∣Ei) ge 0 on all equivariant P1rsquos contained in the exceptional fiberof the blow upNow for line bundles Lprime on [P1Gmactd] we saw that deg(Lprime) = 1

d(wtGm(L∣infin) minus

wtGm(L∣0)) Thus if we order the fixed points x0 xn of the Gm-action on thechain Ei such that x0 is the point in the proper transform of the xminusaxis and xi xi+1

correspond to 0infin in Ei we find that xn corresponds to the proper transform of

the y-axis Note that x0 being the repellent fixed point of E1 we have wt(ilowastxflowastL) =

wtGm(flowastL∣x0) and similarly wtGm(flowastL∣xk) = minuswt(ilowastyflowastL) Finally the condition

deg(L∣Ei) ge 0 implies wtGm(flowastL∣xi) ge wtGm(flowastL∣xi+1) for all i as the exceptionaldivisors are of negative weight Thus we find

wt(ilowastxflowastL) = wtGm(flowastL∣x0) ge sdot sdot sdot ge wtGm(flowastL∣xn) = minuswt(ilowastyf

lowastL)

Now if f(jx((π))) cong f(x0) is stable we know that wtGm(flowastL∣x0) lt 0 so that

wtGm(flowastL∣xn) = minuswt(ilowastyflowast(L)) lt 0 contradicting stability of f(jy((π)))Thus we find that deg(L∣Ei) = 0 for all i and f(jx((π))) cong f(x0) Then f(x0) is

stable so that f ∣E1minusx1 must be constant and we inductively find that f(x0) cong sdot sdot sdot congf(xn) cong f(jy((π))) and that f is constant on the exceptional divisor so f does

extend to STTo conclude that this implies jx f cong jy f choose an affine scheme of finite type

p∶Spec(A)rarrM such that p is smooth and f(jx((π))) = f(0) = f(jy((π))) isin Im(p)Choose moreover a lift x isin Spec(A) of f([0Gm]) By smoothness we can induc-

tively lift f ∣[(SpecR[xy](πnxyminusπ))Gm] to a morphism fn∶ [(SpecR[x y](πn xy minusπ))Gm] rarr SpecA All of these maps factor through their coarse moduli space[(SpecR[x y](πn xyminusπ))Gm]rarr SpecR(πn)rarr SpecA defining a map SpecR rarrSpecA and thus f ∶STR rarr SpecR rarr SpecA that lifts both jx f and jy f so thesemaps coincide

Remark 27 For semistable points that are not stable the above computation alsosuggests to define a notion of S-equivalence as it shows that in an almost properstack any two semistable degenerations could be joined by a chain of projective linesIf L is nef on exceptional lines the line bundle would restrict to the trivial bundleon such a chain In the examples this reproduces the usual notion of S-equivalence

Before giving examples let us note that the Keel-Mori theorem now implies theexistence of coarse moduli spaces for Ms in many situations

23 An existence result for coarse moduli spaces

Proposition 28 Let (ML) be an almost proper algebraic stack with a line bundleL that is nef on exceptional lines and suppose thatMs subM is open Then the stackMs admits a coarse moduli spaceMs rarrM where M is a separated algebraic space

Proof By Proposition 26 we know that ∆∶Ms rarr Ms timesMs is proper and weassumed it to be affine so it is finite Therefore by the Keel-Mori theorem [33][19Theorem 11] the stackMs admits a coarse moduli space in the category of algebraicspaces


24 The example of GIT-quotients

Proposition 29 Let X be a proper scheme with an action of a reductive groupG and let L be a G-linearized bundle that is numerically effective then ([XG]L)is an almost proper stack and L is nef on exceptional lines

Proof As any morphism SpecK rarr [XG] can after passing to a finite extensionK primeK be lifted to X the stack [XG] satisfies the first part of the valuative criterionLet f ∶STR = SpecR cupSpecK SpecR rarr [XG] be a morphism Since R is completewith algebraically closed residue field and X rarr [XG] is smooth we can lift jx fand jy f to morphisms fx fy ∶SpecR rarrX Now since the morphism f defines an

isomorphim φK ∶ fx∣K cong fy ∣K there exists gK isin G(K) such that fy ∣K = gK fxUsing the Cartan decomposition G(K) = G(R)T (K)G(R) we write gK = kyλ(π)kywith ky kx isin G(R) and some cocharacter λ∶Gm rarr G Replacing fx fy by kxfx k

minus1y fx

respectively we obtain may assume that fy ∣k = λ(π)fx ie for this choice the iso-morphism φK is defined by the element λ(π) isin G(K)This defines a (Gm λ)-equivariant morphism F ∶Spec (R[x y](xy minus π)) minus 0rarrX

that we can describe explicitly by λtimesfx∶GmR = Spec (R[xxminus1])rarrX and similarly

by λminus1timesfy on Spec (R[y yminus1]) which glues because λ(x)fx = λ(yminus1π)fx = λminus1(y)fyon the intersectionThis morphism is a lift of f ie fits into a commutative diagram

Spec (R[x y](xy minus π)) minus 0 F

X

[Spec (R[x y](xy minus π)) minus 0Gm] = STRf [XG]

because taking the standard sections sx sy ∶SpecR rarr Spec (R[x y](xy minus π)) minus 0given by x = 1 and y = 1 the identification of STR with the quotient appearing inthe above diagram is induced by the morphism of groupoids

[ SpecK SpecR∐SpecR ]rarr [Spec (R[x y](xy minus π)) minus 0Gm]

given by π isin Gm(K) sx sy and by construction F maps π to λ(π) = φK Since X is proper the morphism F extends after an equivariant blow up and sinceL is nef the numerical condition will automatically be satisfied

25 The example of G-bundles on curves

Proposition 210 Let G be a reductive group and C a smooth projective geomet-rically connected curve The stack BunG is almost proper and the line bundle Ldet

is nef on exceptional lines

Proof We follow the same strategy as for GIT-quotients replacing the projectiveatlas X by the Beilinson-Drinfeld Grassmannian p∶GRG rarr BunG ie

GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR

E isin BunG(S)D isin C(d)(S) for some d

φ∶E ∣CtimesSminusDcongETHrarr G times (C times S minusD)

⎫⎪⎪⎬⎪⎪⎭

It is known that GRG is the inductive limit of projective schemes that Ldet definesa line bundle on GRG that is relatively ample with respect to the morphism to

∐dC(d) and that the forgetful map GRG rarr BunG is formally smooth Moreoverthis morphism admits sections locally in the flat topology ([9 Section 53] [21])To show that BunG is almost proper take f ∶STR rarr BunG After extending k wemay assume that R = k[[π]] This defines bundles ExEy on C times SpecR togetherwith an isomorphism Ex∣CtimesSpecK cong Ex∣CtimesSpecK By the properties of GRG we can

20 JOCHEN HEINLOTH

find lifts fx fy ∶SpecR rarr GRG In particular we find a divisor D = Dx cupDy sub CRsuch that Ex∣CRminusD cong G times (CR minus D) cong Ey ∣CRminusD Through fx fy the isomorphismEx∣CK cong Ey ∣CK therefore defines an element g isin G(CK minus D) sub G(K(C)) As inProposition 29 we would like to apply Cartan decomposition now for the fieldK(C) that comes equipped with a discrete valuation induced by the valuation ofR ie the valuation given by the codimension 1 point Speck(C) isin CR Its ring ofintegers OK are meromorphic functions on CK that extend to an open subset onthe special fiber CkUsing this we can write g = k1λ(π)k2 with ki isin G(OK) Now each of the ki definesan element ki isin G(U) for some open subset U sub CR that is dense in the specialfiber so after enlarging D we may assume ki isin G(CR minusD)The elements ki allow us to modify the lifts fx fy such that with respect to thesenew maps we find g = λ(π)As before this datum defines a Gm λ-equivariant morphism

Spec(R[x y](xy minus π)) minus 0rarr GRG

and by ind-projectivity this can be extended after a suitable blow up to a Gm λ-equivariant morphism which defines BlI(ST) rarr BunG Finally as the Gm-action

preserves the forgetful map GRG rarr C(d) and Ldet is relatively ample with respectto this morphism we see that Ldet is nef on exceptional lines

3 Torsors under parahoric group schemes on curves

In this section we give our main application to moduli of torsors under Bruhat-Titsgroup schemes on curves as introduced by Pappas and Rapoport [38] It will turnout that the notion of stability we find is a variant of the one introduced by Balajiand Seshadri in the case of generically split group schemes We will then applythis to construct coarse moduli spaces for stable torsors over fields of arbitrarycharacteristic

31 The setup We fix a smooth projective geometrically connected curve Ckand G rarr C a parahoric Bruhat-Tits group scheme in the sense of [38] ie G is asmooth affine group scheme with geometrically connected fibers such that there isan open dense subset U sub C such that G∣U is reductive and such that for all p isin CminusUthe restriction G∣SpecOCp is a parahoric group scheme as in [17] (see Appendix 4for details) We will denote by Ram(G) sub C the finite set of closed points for whichthe fiber Gx is not a reductive groupWe will denote by BunG the moduli stack of G-torsors on C As usual we willoften denote base extensions by an index ie for a k-scheme X we abbreviateXC ∶=X timesC

Example 31 It may be helpful to keep the following examples in mind

(1) (Parabolic structures) Let Gk be a reductive group B sub G be a Borelsubgroup and p1 pn isin C(k) rational points We define GpB to be

the smooth groupscheme over C that comes equipped with a morphismGpB rarr GC such that for all i the image GpB(OCpi) = g isin G(OCpi)∣gmod pi isin B is the Iwahori subgroup Since GpB(OCpi) is the subgroup of

automorphism group of the trivial G torsor that fixes the Borel subgroupsB sub Gtimes pi torsors under this group scheme are G-bundles equipped with areduction to B at the points pi

(2) (The unitary group) Suppose char(k) ne 2 and let π∶ C rarr C is a possiblyramified Z2Z-covering then the group scheme πlowastGLnC admits an auto-

morphism given by the ()tminus1 on the group and the natural action on thecoefficients OC The invariants with respect to this action is called the


unitary group for the covering Torsors under this group scheme can beviewed as vector bundles on C that under the involution become isomor-phic to their dual

32 Line bundles on BunG As observed by Pappas and Rapoport ([38][28])there are many natural line bundles on BunG

(1) We define Ldet to be the determinant line bundle given by

Ldet∣E ∶= det (Hlowast(CAd(E))minus1

(2) For every x isin Ram(G) we have a homomorphism Xlowast(Gx) Pic(BunG)induced from the pull back via the canonical map BunG rarr BGx given byE ↦ E ∣x and the canonical morphism

Xlowast(Gx) = Hom(GxGm)rarrMor(BGxBGm) cong Pic(BGx)which is surjective on isomorphism classes We write Lχx for the line bundlecorresponding to χx isinXlowast(Gx)

(3) We will abbreviate

Ldetχ ∶= Ldet otimes otimesxisinRam(G)

Lχx

As before positivity of L will be checked on affine Grassmannians Let us fixthe notation For a point x isin X we denote by GrGx the ind-projective scheme

classifying G-torsors on X together with a trivialization on C minus x Its k-points

are G(Kx)G(Ox) It comes with a forgetful map

gluex∶GrGx rarr BunGx

By definition the bundles obtained from gluex are canonically trivial outside x sothe bundles Lχx pull back to the trivial line bundle on GrGy for y ne xTo check that L is nef on exceptional lines we will need a line bundle L = Ldetχ

such that for all x isin X the bundle pulls back to a positive line bundle on thecorresponding affine Grassmannian

Remark 32 If Gη is simply connected absolutely almost simple and splits overa tame extension the positivity condition can be given explicitly as for examplecomputed in [44 Section 4] As this requires some more notation we only notethat Ldet always satisfies this numerical condition as this is the pull back of adeterminant line bundle on a Grassmanian GrGLN x

The proof of Proposition 210 now applies to G as the proof only uses a grouptheoretic decomposition at the generic point of C where G is reductive We thereforefind

Proposition 33 Let G be a parahoric BruhatndashTits group on C and let Ldetχ be

chosen such that for all x isin Ram(G) the bundle gluelowastxLdetχ is nef on GrGx Then

the pair (BunG Ldetχ) satisfies the valuative criterion (⋆)

To obtain coarse moduli spaces we now have to show that the stable locus is anopen subset of finite type For this we will need analogs of the basic results onstability for G-torsors To do this we first need to rephrase Ldetχ-stability in terms

of reductions of structure groups

33 Preliminaries on parabolic subgroups of BruhatndashTits group schemesAs before very close degenerations of G-bundles will give us cocharacters GmC rarrAutGC(E) =∶ GE In order to describe these in terms of reductions of structuregroup we first need some general results on cocharacters and analogs parabolicsubgroups of BruhatndashTits group schemes

22 JOCHEN HEINLOTH

Let us first consider the local situation Let R be a discrete valuation ring withfraction field K π isin R a uniformizer and k = R(π) the residue field Let G rarr SpecRparahoric Bruhat-Tits group schemeGiven λ∶GmR rarr G we denote by

(1) Pλ(S) ∶= g isin G(S)∣ limtrarr0 λ(t)gλ(t)minus1 exists in G(S) the concentratorscheme of the action of Gm

(2) Lλ(S) ∶= g isin G(S)∣λ(t)gλ(t)minus1 = g the centralizer of λ(3) Uλ(S) ∶= g isin G(S)∣ limtrarr0 λ(t)gλ(t)minus1 = 1

These are analogs of parabolic subgroups and Levi subgroups of G Alternativelyone could define such analogs by taking the closure of parabolics in the generic fiberGK The following Lemma shows that this leads to an equivalent notion

Lemma 34 Let R be a discrete valuation ring and GR a parahoric BruhatndashTitsgroup scheme over SpecR

(1) Given a 1-parameter subgroup λ∶GmR rarr G the group Pλ is the closureof PKλ sub GK Lλ is the closure of the Levi subgroup LKλ sub GK and Uλis the closure of the unipotent radical of PKλ The group Lλ is again aBruhatndashTits group scheme

(2) Let PK sub GK be a parabolic subgroup and denote by P sub G the closure ofPK in G Then there exists a 1-parameter subgroup λ∶GmR rarr GR such thatP = Pλ

(3) Let PK sub GK be a parabolic subgroup and λK ∶GmK rarr GK be a cocharactersuch that PK = PλK Then there exist an element u isin U(K) such thatλuK ∶= uλKu

minus1 extends to a morphism λu∶Gm rarr P The class of u inU(K)U(R) is uniquely determined by λK

Before proving the lemma let us note that part (3) will be useful to define a Reesconstruction for G-bundles In the global setup of a group scheme GC on a curvewe cannot expect that every parabolic subgroup Pk(C) sub Gk(C) can be defined bya globally defined cocharacter GmC rarr G as G may not admit any non-trivialcocharacters However given λk(C) part (3) will give us a canonical inner form ofG for which λ extends

Remark 35 In the setup of the above lemma given two parabolic subrgoupsPK P primeK sub GK that are conjugate over GK their closures PP prime sub G need not beconjugate Roughly this is because the closure contains information about therelative position of the generic parabolic and the parahoric structure in the specialfiber More precisely the Iwasawa decomposition we know that for any Borelsubgroup BK sub GK we have G(K) = B(K)WG(R) ([16 Propositions 443 731])This implies that there are only finitely many conjugacy classes P of closures ofgeneric parabolic subgroups

Proof of Lemma 34 To show the first part of (1) we have to show that the genericfibers of LλUλ and Pλ are dense As G is smooth over R the fixed point schemeLλ and the concentrator scheme Pλ are both regular ([31 Theorem 58] ) Letx isin Lλ(R(π)) sub G(R(π)) be a closed point of the special fiber then TGx rarr TSpecR0

is surjective and equivariant so there exists an invariant tangent vector lifting thetangent direction in 0 Thus Lλ is smooth over R and therefore the generic fiberis dense The morphism Pλ rarr Lλ is an affine bundle ([31 Theorem 58]) so thegeneric fiber of Pλ must also be dense and Uλ is even an affine bundle over SpecRLet us prove (2) and (3) Any parabolic subgroup PK of the reductive group GK isof the form PK = PK(λ) for some λ∶GmK rarr GK [41 Lemma 1512] The image ofλ is contained in a maximal split torus of GK and these are all conjugate over K [41Theorem 1526] Fixing a maximal split torus TR sub GR we therefore find g isin G(K)


such that gλgminus1∶Gm rarr GK factors through T By Iwasawa decomposition we canwrite g = kwu with k isin G(R)w isin N(T ) p isin U(K) Thus we can conjugate λ by anelement of U(K) such that it extends to GR Applying (1) to this subgroup we find(2) It also shows the existence statement in (3) To show uniqueness assume thatλ∶GmR rarr Pλ is given and that u isin U(K) is such that uλuminus1 still defines a morphismover R Recall that UK has a canonical filtration UKger sub sdot sdot sdot sub UKge1 = UK such thatUKiUKi+ congprodα∣αλ=i Uα Write Ui ∶=prodα∣αλ=i Uα and decompose u = u1 sdot sdot sdot sdot sdot urWe know that uλ(a)uminus1 isin P (R) for all a isin Rlowast ie U(R) ni uλ(a)uminus1λminus1(a) Theimage of this element in Uge1Uge2 is u1au

minus11 aminus1 which can only be in U1(R) for all a

if u1 isin U(R) but then we can replace u by u2 ur and conclude by inductionFinally we need to show that the group scheme Lλ in (1) is a BruhatndashTits groupscheme By construction it suffices to show this after an etale base change SpecRprime rarrSpecR so we may assume that GR is quasi-split ie that GK contains a maximaltorus TK and that G contains the connected Neron model T of TK As conjugation by elements of G(R) produces isomorphic group schemes we mayassume as above that λ∶GmR rarr T sub G The scheme GR is given by a valued rootsystem and the restriction of this to the roots of LλK defines a BruhatndashTits groupscheme with generic fiber LλK contained in Lλ Finally by definition the specialfiber of Lλ is the centralizer of a torus so it is connected Thus the smooth schemeLλ has to be equal to this BruhatndashTits scheme

Let us translate this back to our global situation As before let GC be a BruhatndashTits group scheme over our curve C and denote by η isin C the generic point ofC

Lemma 36 Let λη ∶Gmη rarr Gη be a cocharacter and Pηλ the corresponding par-abolic Uηλ its unipotent radical and UλPλ the closures of UηλPηλ in G Then

(1) PλUλ are smooth group schemes over C The quotient PλUλ =∶ Lλ is aBruhatndashTits group scheme

(2) The morphism λη extends to λ∶Gm timesC rarr Z(Lλ) sub Lλ(3) There exist a canonical Uλ-torsor Uu together with an isomorphism Uu∣k(C) congUk(C) such that λ extends to λ∶Gm rarr Uu timesUconj G

Proof First note that λη extends canonically to a Zariski open subset U sub C (eg[20 Expose XI Proposition 312 (2)]) Let us denote this morphism λU ∶GmU rarr GU Over U the subgroup PλU sub GU is a parabolic subgroup of the reductive groupscheme GU the groups UλU LλU are the corresponding unipotent radical andLevi subgroup Part (1) and (2) can thus be checked locally around all pointsx isin C minus U and there Lemma 34 gives the result Since a U-torsor together withan isomorphism Uu∣k(C) cong Uk(C) is given by a finite collection of elements ux isinU(k(C))U(OCx) for some x isin C the last part also follows from Lemma 34 (3)

This lemma allows us to generalize the Rees construction Given λη ∶Gmη rarr Gη theabove lemma constructs an inner form Pu sub Gu such that λ∶GmC rarr Pu extendsAs we proved that fiberwise conjugation by λ contracts Pu to Lλ we again obtainthe morphism of group schemes over C timesGm

conjλ∶Pu timesGm rarr Pu timesGm(p t)↦ (λ(t)pλ(t)minus1 t)

By [31 Proposition 42] this homomorphism extends to

grλ∶Pu timesA1 rarr Pu timesA1

24 JOCHEN HEINLOTH

in such a way that gr(p0) = limtrarr0 λ(t)pλ(t)minus1 isin Lλ times 0Thus given a Pu torsor Eu we can define the Rees construction

Rees(Eu λ) ∶= (Eu timesA1) timesgrλ (Pu timesA1)Given a G torsor E together with a reduction EP to a parabolic subgroup P anda cocharacter λη ∶Gm rarr Pη defining P we take the associated Pu-torsor Eu ∶=IsomPC(EP Uu timesU P) and define

Rees(EP λη) ∶= IsomPuC(Rees(Eu λ)Uu timesU P)

As before this construction only depends on P and the composition λ∶Gmη rarr Lη

Remark 37 Note that the adjoint bundle of Rees(EP λη) is the vector bundle onC times [A1Gm] that on the generic fiber corresponds to the very close degenerationgiven by Ad(EP)η and the cocharacter given by Ad(λ) As we know that thisalready defines the very close degeneration the passage to Pu-torsors was onlyneeded in order to give a formula for the very close degeneration in terms of Gtorsors

34 Very close degenerations of G-bundles We can now classify very closedegenerations f ∶ [A1Gm] rarr BunG Such a morphism defines a G-torsor E overC times [A1Gm] and f(0) defines a non-trivial cocharacter λ∶Gm timesC rarr AutGC(E0) =∶GE0 This defines LλPλ sub GE0 As Lλ is the concentrator scheme of the Gm actionwe again get that the morphism


extends to


in such a way that gr(p0) = limtrarr0 λ(t)pλ(t)minus1 isin Lλ times 0 And so we can define theRees construction for Pλ-bundles Fλ

Rees(Fλ λ) ∶= [((Fλ timesA1) timesgrλ (Pλ timesA1))Gm]

The G torsor E is determined by the GE0 -torsor E prime ∶= Isom(E E0) As beforeIsom(E0E0) being the trivial GE0 torsor E prime0 comes equipped with a canonical reduc-tion to Lλ and as in Lemma 17 the corresponding reduction to Pλ lifts canonicallyto C times [A1Gm] We denote this reduction by E primeλ Now we can argue as in Lemma113 to identify

E primeλ cong Rees(E primeλ1 λ)and thus

E cong IsomGE0 (E primeE0) cong IsomGE0 (Rees(E primeλ1 λ) timesPλ GE0 E0)Formulating stability in these terms would have the annoying aspect that all possi-ble group schemes GE0 would appear in the formulation This can be avoided thisby restricting to the generic fiber as followsBy the description of G-bundles over [A1Gm] (Lemma 17) we have

Isom(E E0)ηtimes[A1Gm] = E prime∣ηtimes[A1Gm] cong [A1 times GE0η Gm]In particular E1η cong E0η Therefore the canonical reduction of E prime1 to Pλ correspondsto a reduction of E1k(C) to a parabolic PλK sub GE1

Thus given the cocharacter λ∶Gm rarr GE0 we can find a cocharacter λK ∶Gm rarr GE1such that the canonical reduction of E prime1 to Pλ sub GE0 defines a reduction EP prime

λof E to

P primeλ ∶= PλK sub GE1


Given the reduction of E to P primeλ we already defined the corresponding Rees con-struction Thus we find

Lemma 38 Let G rarr C be a BruhatndashTits group scheme E isin BunG and GE ∶=AutGC(E) the corresponding inner form of GThen any morphism f ∶ [AGm] rarr BunG with f(1) = E can be obtained from theRees construction applied to a generic cocharacter λη ∶Gm rarr GEk(C)

In the above we described reductions by cocharacters λ∶Gm rarr GEk(C)because this

description works over any field As in [8] this is more suitable to study rationalityproblems for canonical reductions of G-bundles Over algebraically closed fields wecan also reformulate this in terms of reductions to parabolic subgroups of Gk(C) asfollows

Lemma 39 Let k = k be an algebraically closed field G rarr C be a BruhatndashTitsgroup scheme E isin BunG(k) Then any morphism f ∶ [AGm]rarr BunG with f(1) = Ecan be obtained from the Rees construction applied to a reduction of E to a subgroupPλ sub G which is defined by a generic cocharacter λ∶Gm rarr Gk(C)

Proof By Lemma 38 we know that f can be defined by applying the Rees con-struction to a cocharacter λη ∶Gm rarr GEk(C)

Let PEλ sub GE be the closure of the

parabolic subgroup defined by λη As explained in the appendix (Proposition 51)any G bundle on Ck can be trivialized over an open subset U sub Ck which containsRam(G) sub U Choose such a trivialization ψ∶E ∣U cong G∣U This induces an isomor-phism GEk(C)

cong Gk(C) and this defines λ∶Gm rarr Gk(C) Denote by Pλ sub G the closure

of the parabolic subgroup defined by λ By construction we know that PEλ ∣U cong Pλ∣Uand since G is a reductive group scheme over C minusU the groups Pλ and PEλ are alsoisomorphic in a neighborhood of the points in C minus U Thus the reduction of E toPEλ defines a reduction of E to Pλ This proves our claim

Remark 310 In the case of G-bundles there are only finitely many conjugacyclasses of parabolic subgroups P sub G and therefore any very close degenerationof a bundles E is induced from viewing E as lying in the image of a morphismBunP rarr BunG To show that the semistable points of BunG form an open substackthe fact that one needs only to consider finitely many such P is helpfulFrom Lemma 39 we can now conclude that the analogous result also holds forG-bundles Suppose PP prime sub G are closures of parabolic subgroups in the genericfiber If P and P prime happen to be conjugate in G at the generic point of C theyare also conjugate locally on C minusRam(G) as parabolic subgroups of the same typeare conjugate in reductive groups Also locally around any point x isin Ram(G) wesaw in Remark 35 that there are only finitely many conjugacy classes of closuresof parabolic subgroups P sub G Thus up to local conjugation in G there are onlyfinitely many closures of parabolic subgroups P sub GAs in the case ofG-bundles if PP prime sub G are locally conjugate over C the transporterTranspG(PP prime) of elemnents of G that conjugate P into P prime is a P minus P prime bi-torsor(because parabolic subgroups are equal to their normalizer over C minus Ram(G) anda local section of G normalizes P if and only it normalizes the generic fiber) Thisdefines a commutative diagram

BunPcong

$$

BunP prime

zz

BunG

identifying reductions to the structure groups P and P prime

26 JOCHEN HEINLOTH

35 The stability condition Let us fix our group scheme G and a line bundleL ∶= Ldetχ as in Section 32 In this section we want to describe L-stability condition

in terms of degrees of bundles Lemma 39 and the definition of L-stability (12)imply

Remark 311 A G-bundle E on Ck is L-stable if and only if for all parabolicsubgroups P sub Gk(C)

with closure P sub G all dominant cocharacters λ∶Gm rarr P and

all reductions of E to P we have

wtL(Rees(EP λ)) gt 0

As in the case of G-bundles we want to express the above weight in terms degrees ofline bundles attached to reductions of E We start out with the intrinsic formulationof reductions as in Lemma 38 but in the end this reduces to a computation on theadjoint bundle Ad(E)Fix E a G-bundle EB a reduction to a Borel subgroup B sub AutG(E) As before letus denote by U sub B the closure of the unipotent radical over the generic point η isin Cand T = BU the maximal torus quotient Fix S sub Tη sub Bη a maximal split torus ina lifting of the maximal torus at η We will denote by Φ = Φ(Gη S) the roots of Gηand Φ+

B will be the roots that are positive with respect to BFor any point x isin Ram(G) we obtain a character χBx ∶Bx rarr Gm as composition

χBx ∶BEx rarr GExχETHrarr Gm

This morphism factors through T Ex For any λ∶Gm rarr T we will write ⟨χBx λ⟩ ∶=⟨χBx λ∣x⟩The Rees construction applied to a generic dominant 1-parameter subgroup givesus a bundle E0 that is induced from the T bundle EBU =∶ ET To compute the weights we can decompose the adjoint bundle of E0 into weightspaces

ad(E0) = ad(ET )oplusaisinΦ(Gη) uE0a

and the Rees construction induces a filtration of ad(E) such that the associatedgraded pieces are uEa cong uE0a For any 1-parameter subgroup λ∶Gm rarr GEη which is dominant with respect to B wethen have

wtEB(λ) ∶= wtE(λ) = sumaisinΦ

χ(uEa)⟨a λ⟩ + sumxisinRam(G)

⟨χBx λ⟩

As rk(ua) = rk(uminusa) we can further compute

wtEB(λ) = sumaisinΦ


⟨χBx λx⟩

= sumaisinΦ

deg(uEa)⟨a λ⟩ + sumxisinRam(G)

⟨χBx λx⟩

In the case of unramified constant group schemes the degree deg(ua) is a linearfunction in the root a This does no longer hold for parahoric group schemes butwe still have relationsAt the generic point η of C we get a canonical isomorphism

ka∶uEminusaηcongETHrarr (uEaη)or

from the Killing form and this extends to an isomorphism at all points c isin C whereGc is reductive Therefore the determinant of ka defines a divisor

Da = sumxisinRam(G)

fBaxx


With this notation we have

deg(uEminusa) = minusdeg(uEa) minus sumxisinRam(G)

fBax

Here we denote the coefficients by the letter f because for a global torus T sub Gwith valuated root systems fax (see Section 4) these numbers are minus(fax + fminusax)As any two tori are conjugate we always find

∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+


⟨χBx + sumaisinΦ+

B

fBaxa λ⟩(31)

To compare this to the usual (parabolic)-degree let us fix a norm on the set of all1-parameter subgroups A convenient choice for us will be to fix for any maximaltorus containing a maximal split torus Sη sub Tη sub Gη the canonical invariant bilinearform on Xlowast(Tη)

( ) ∶= sumαisinΦ(Gη)

⟨ α⟩⟨ α⟩

We will denote the restriction of ( ) to Xlowast(T ) by the same symbol and we will

denote by ∣∣ sdot ∣∣ ∶=radic

(sdot sdot) the induced norm

Remark 312 The bilinear form on Xlowast(Tη) also induces a form on the cocharactergroups Xlowast(Tx) for all x isin C closed

Remark 313(1) If G0 = G timesX is a split group scheme we remarked above that deg(uEa) is a

linear function in a denoted deg as this is the usual degree of the T -bundle

ET = EBU Then the above formula reads

wtEB(λ) = (deg λ)This expresses the weight in terms of the degree that is classically used todefine stability for G-torsors

(2) If G rarr G0 = GtimesX is G is obtained as the parahoric subgroup defined by thechoice of parabolic subgroups Px sub G0x = G the numbers fax are 01minus1depending on the relative position of Bx and the image of GEx rarr (GE0 )x Bythe previous point this then again gives the same relation of the weight andthe parabolic degree

(3) As the difference deg(uEa) minus deg(uEminusa) is always an integer the formula alsoshows that as in the case of parabolic bundles the weight cannot be 0 iffor at least one x isin Ram(G) the group scheme G∣Ox is an Iwahori groupscheme and χx is chosen generically eg such that the numbers ⟨χx αi⟩for some basis αi of one parameter subgroups of T are rational numberswith sufficiently large denominators

(4) For generically split groups G in [7 Definition 634] Balaji and Seshadridescribe stability in terms of (ΓG) bundles In their setup the reductionsEP are computed from taking invariants of an invariant parabolic reductionof a G-bundle on a covering C rarr C ([7 Proof of Proposition 631]) Thenumerical invariants defining stability are then expressed as the parabolicdegrees defined by characters of P attached to a reduction of EP To com-pare this with our condition one can proceed as in (1) by using (31) In

28 JOCHEN HEINLOTH

this case as G is generically split all uEa are line bundles which are theinvariant direct images of the corresponding bundles of the reduction toP on C The parabolic weight then turns out to be related to the con-tribution form the ramification points which depends only on the relativeposition of the generic parabolic and the valuation of the root system defin-ing G As the precise relation requires a careful recollection on the relationof (ΓG)-bundles and valued root systems we leave this to a later time

36 Canonical reduction for G-torsors In this section we want to check thatthe canonical reduction of G-bundles introduced by Behrend also exists for G-bundles We will then use this to deduce that the stack of stable and semistableG-bundles are open substacks of finite type of BunG In [23] Halpern-Leistner gives general criteria for the existence and uniqueness ofcanonical reductions for θ-reductive stacks Unfortunately BunG does not satisfythis condition so that we have to give a separate argument It will turn out thatonce we formulate the classical approach for G-bundles (see [25][8]) in a suitableway most of the arguments generalize to this framework This was also explainedby Gaitsgory and Lurie in [22 Section 10] for a notion of stability induced fromG-bundles Note that Harder and Stuhler also introduced the concept of canonicalreductions for BruhatndashTits groups in the adelic description of the points of themoduli stack [26]Let us fix our group scheme G and a line bundle L ∶= Ldetχ as in Section 32 In

general to define canonical destabilizing 1-parameter subgroups one needs to fix anorm on the set of all such subgroups We will simply use the invariant form ( )on Xlowast(Tη) from the previous sectionAs for parabolic bundles we will need the following assumption on χ We will call

L admissible if 2⟨χx a⟩ le rkua⟨a a⟩ for all roots a of GxAs in [23] a canonical reduction of a G-torsor should be a 1-parameter subgroup

λ isinXlowast(GEη ) such that wt(λ)∣∣λ∣∣

is maximal First of all this number is bounded

Lemma 314 For every G torsor E there exists cE gt 0 such that wt(λ)∣∣λ∣∣

le c for all

λ∶Gmη rarr GEη

Proof As in the classical case for every reduction EP of E to a parabolic subgroupwe have that H0(ad(EP)) sub H0(ad(E)) By Riemann-Roch this implies that thedegree of the unipotent radical deg(EP timesP uP) is uniformly bounded above for allreductions In turn this gives for every reduction to a Borel subgroup B sub GE anupper bound for the weight wt(ωi) for all dominant coweights ωi As any dominantλ is a positive linear combination of these this gives the required bound

Notation For any G-torsor E over an algebraically closed field we define

micromax(E) ∶= supλ

wtE(λ)∣∣λ∣∣

361 Comparing weights of different reductions To characterize the canonical re-duction we need to compare the weights of different reductionsSuppose BBprime sub GE are two Borel subgroups Any two such subgroups share amaximal split torus S sub Tη sub Bη cap Bprimeη ([13 Proposition 44])Let a1 an be the positive simple roots of BWe say that BBprime are neighboring reductions if there exists i0 such that minusai0 is apositive simple root of Bprime and ai are positive roots of Bprime for all i ne i0In this case BBprime generate a parabolic P1 that is minimal among the parabolicsubgroups that are not Borel subgroups Denote by L1 the corresponding Leviquotient


Lemma 315 Let BBprime be neighboring parabolic subgroups Then we have

(1) wtB(λ) = wtBprime(λ) for all λ isin Z(LI)(2) wtB(αi) +wtBprime(minusαi) le 0

Proof As the weight only depends on the Rees construction for E we may replaceE by the Rees construction applied to any λ that is dominant for B and Bprime Thuswe may assume that E = EL1 timesL1 GE is induced from a L1 torsor In this case (1)is immediate from the definition as the Rees construction for λ isin Z(L1) does notchange EL1 Moreover L1 is of semisimple rank 1 and B1 ∶= BcapL1 and Bprime1capL1 are Borel subgroupsthat are opposite over the generic point η isin C Let us denote by uuprime the Lie algebrasof the corresponding unipotent radicals in L1 and by uminus ∶= Lie(L1)Lie(B1) andsimilarly uprimeminus Then we get an injective homomorphism

EB1 timesB1 urarr ad(EL1)rarr ad(EL1)ad(EBprime1) cong EBprime1 timesBprime1 uminusprime

If a is a multipliable root then the homomorphism respects the filtration on uuprimeminus

given by the roots a2a So we find

deg(uBa ) le deg(uBprime

a ) = minusdeg(uBprime

minusa) minus sumxisinRam(G)

fBprime

minusax

minusdeg(uBa ) minus sumxisinRam(G)

fBax = deg(uBminusa) ge deg(uBprime

minusa)

an the same hods for u2a if 2a is also a rootThus

2(deg(uBa ) + deg(uBprime

minusa)) le sumxisinRam(G)

fBprime

minusax + fBax

Moreover the map uBa rarr uBprime

a is an isomorphism at x if and only if the groups BExand BprimeEx are opposite in GEx And in this case ⟨χB1

x αi0⟩ = minus⟨χBprime1x minusαi0⟩ If the

morphism is not an isomorphism then the Borel subgroups are parallel in x so

that ⟨χB1x αi0⟩ = ⟨χB

prime1x minusαi0⟩ Thus if 2⟨χx a⟩ le rkua⟨a a⟩ for all roots a we find


minusa))⟨a a⟩ le ⟨ sumxisinRam(G)

fBprime

minusax + fBax + χB1x minus χB

prime1x a⟩

And this means

wtB1(αi0) +wtBprime1(minusαi0) le 0

if χ is admissible

To compute wtB(αi0)+wtBprime(minusαi0) we note that EP1 cong EB1 timesB1 P1 cong EBprime1 timesBprime1 P1 and

the unipotent radical of Lie(P1) has a filtration by L-invariant subspaces such thatover the generic point the associated graded pieces are isomorphic to the unipotent

groups ublowastη = oplusc=b+nauBcη = oplusc=bminusnauBprime

cη Moreover since a is a positive root for B1

and a negative root for Bprime1 over C the isomorphic bundles EB1timesB1 ublowast and EBprime1timesBprime1 ublowast

come with canonical filtrations that are opposite at the generic point Thereforewe find again that

sumc=b+na

(deg(uBcη) minus deg(uBprime

cη))⟨c αi0⟩ le 0

Summing over all b we obtain the result

30 JOCHEN HEINLOTH

We define L minus deg(EB) by the formula

(L minus deg λ) ∶= minuswtEB(λ)Then the previous lemma shows that L minus deg defines a complementary polyhedronas defined by Behrend [8] As in [27 Section 43] we therefore obtain

Proposition 316 Suppose Ldetχ is an admissible line bundle and E a G-torsor

Then there exists a reduction λ∶Gm rarr GEη such that EPλ is a reduction for whichwt(λ)∣∣λ∣∣

is maximal and such that for every other such reduction to a parabolic sub-

group Qλprime we have Qλprime sub Pλ

Lemma 317 (Semicontinuity of instability) Let Ldetχ be an admissible line bun-

dle on BunG Let Rk be a discrete valuation ring with fraction field K and residuefield κ and let ER be a G-torsor on CR

(1) If EK is unstable and the canonical reduction is defined over K then Eκ isunstable and

micromax(EK) le micromax(Eκ)The equality is strict unless the canonical reduction of EK extends to R

(2) If EK is semistable but not stable then Eκ is also not stable

Proof The first part follows as in [27 Lemma 442] This also shows that if EKadmits a reduction of weight 0 then Eκ also cannot be stable

Finally suppose dim Aut(EK) gt 0 We know that GER pETHrarr CR is an affine groupscheme of finite type over CR The group of global automorphisms of this groupscheme is Specplowast(OGER ) so the generic fiber of this is not a finite K-algebra Butthen by semi continuity also the special fiber will not be finite

37 Boundedness for stable G-torsors With the canonical reduction at handwe can now deduce

Proposition 318 Let Ldetχ be an admissible line bundle on BunG Then the

stacks BunstG sub BunsstG sub BunG of Ldetχ-(semi)-stable G-torsors are open substacks

of finite type

Proof By Lemma 317 instability and strict semistability are stable under special-ization Therefore we only need to show that BunsstG is constructible and containedin a substack of finite type Again we argue as in [8] First we show that for anyc ge 0 the stack of G-torsors of fixed degree satisfying micromax(E) le c is of finite typeTo prove this we may suppose that χ = 0 as the linear function λ ↦ ⟨χλ⟩ only

changes wt(λ)∣∣λ∣∣

by a finite constant

By [8] the claim holds if G = G0 is a reductive group scheme over C Moreoverif Gprime rarr G0 is a parahoric group scheme mapping to G0 such that Gprime is an Iwahorigroup scheme at all x isin Ram(Gprime) then the morphism BunGprime rarr BunG0 is a smoothmorphism with fibers isomorphic to a product of flag varieties G0xBx Moreoverthe cokernel of Lie(Gprime) rarr Lie(G0) is of finite length Thus there exists a constant

d such that for any E is a Gprime torsor we have that if E timesGprime G0 admits a reduction

of slope micro(E timesGprime G0) gt c + d then this reduction induces a reduction of E of slopemicro(E) gt cTherefore the result also holds for Gprime Now any parahoric group scheme contains anIwahori group scheme so that by the same reasoning the result also holds if G is anyparahoric BruhatndashTits group scheme such that G∣CminusRam(G) admits an unramifiedextensionNow choose π∶ C rarr C a finite Galois covering with group Γ such that πlowastG∣CminusRam(G)

is a generically split reductive group scheme on C minusπminus1(Ram(G)) equipped with a


Γ-action We can extend this group scheme to a BruhatndashTits group scheme G thatis Γ equivariant and admits an morphism πlowast(G) rarr G that is an isomorphism over

C minus πminus1(Ram(G)) We already know the result for BunG

Now if E is a G-torsor

then E ∶= πlowastE timesπlowastG G is a Γ-equivariant G torsor Now if E admits a canonicalreduction to a parabolic subgroup P this will define an equivariant reduction ofπlowastE ∣CminusRam(G) and therefore a reduction of E Again we can compare the weights of the reductions because πlowast ad(πlowast(E)) = ad(E)otimesπlowast(OC) Therefore the determinant of the cohomology det(Hlowast(Cad(πlowastE)) =detHlowast(Cad(E) otimes πlowastOC) defines a power of Ldet on BunG This implies thatthe weight of the reduction of πlowastE is a just deg(π)-times the weight of the induced

reduction of E So again a very destabilizing canonical reduction of E induces adestabilizing reduction of E We are left to show that the stable and semistable loci are constructible We sawthat unstable bundles admit a canonical reduction to some P sub G ie they lie inthe image of the natural morphism BunP rarr BunG From Remark 310 we knowthat it suffices to consider the image of this morphism for finitely many PTo conclude we need to see that for every substack of finite type U sub BunG onlyfinitely many connected components of BunP contain canonical reductions thatmap to U We just proved that on U the slope micromax is bounded aboveWe claim that for the canonical reduction of a G-bundle with micromax le c the degree ofthe corresponding P-bundle deg isin (Xlowast(P))or lies in a finite set By construction thecanonical reduction was obtained from a complementary polyhedron and thereforethe L-degree of the reduction that was defined from the weight of the reductionthat is bounded below when evaluated at fundamental weights As we bounded themicromax this will also be bounded above Now we saw that the from equation 31 thatthe weight of the reduction can be computed from the degree of the reduction anda local term only depending on the group P Therefore we find obtain our boundfor the degree of the P-bundleTo conclude we use that the number of connected components of BunP of a fixeddegree is finite This is known for G-bundles (see Proposition 52 ) From this wecan deduce our statement by looking at the Levi quotient P rarr L and the morphismBunP rarr BunL which is smooth with connected fibers because the kernel U rarr Phas a filtration by additive groupsFor the stack of stable points we can argue in the same way oserving that strictlysemistable bundles admit a reduction to a parabolic with maximal slope equal to0

38 Conclusion for G-torsors

Theorem 319 Let G be a parahoric BruhatndashTits group scheme that splits over

a tamely ramified extension ˜k(C)k(C) and let L = Ldetχ be an admissible line

bundle on BunG Then the stack of L-stable G bundles BunstG admits a separatedcoarse moduli space of finite type over k

Proof This now follows from Proposition 28 because the stack of stable G-torsorsBunstG is an open substack by Proposition 318 and it satisfies the conditions ofProposition 26 by Proposition 33

Remark 320 For admissible line bundles L = Ldetχ the results on the existence

of canonical reductions for G-bundles allow to copy the proof of the semistablereduction theorem using Langtonrsquos algorithm from [29] and [30] In particular inthose cases where L-semistability is equivalent to L-stability this then implies thatthe coarse moduli spaces are proper

32 JOCHEN HEINLOTH

4 Appendix Fixing notations for BruhatndashTits group schemes

In this appendix we collect the results from [17] on the structure of BruhatndashTitsgroup schemes that we use As the definition is local let us fix a discrete valuationring R with fraction field K and residue field k and GK a reductive group overK In general the groups are defined by descent starting from the quasi-split caseover an unramified extension of R In our applications we can always extend thebase field and if k = k the group GK is quasi-split by the theorem of Steinberg [12Section 86] We will therefore assume that GK is quasi-splitWe choose a maximal split torus SK sub GK and denote TK ∶= Z(SK) the centralizerof S which is a maximal torus of GK because GK was quasi-split1

To construct models of GK over R BruhatndashTits first extend the torus TK to ascheme over R and then the root subgroups of GK using a pinning of GK that theyupgrade to a Chevalley-Steinberg valuation ϕ ([17] Section 421 and 413) Letus recall these notions

41 ChevalleyndashSteinberg systems pinnings and valuations If GK is splita ChevalleyndashSteinberg system is simply a pinning of our group ie an identifica-

tion xα∶GacongETHrarr Uα which is compatible for αminusα in the sense that it comes from

an embedding of ζα∶SL2K rarr GK identifying Ga with the strict upper and lowertriangular matrices [17 Paragraph 321] and is compatible with reflections [17Paragraph 322 (Ch1)(Ch2)]

IfGK is not split we can split it over a Galois extension K and choose an equivariantpinning [17 Paragraph 413] Let Γ ∶= Gal(KK) and denote by Φ the roots withrespect to TK of GK and Φ the roots for S Then Xlowast(S) = Xlowast(SK) and the

elements of Φ are the restrictions of elements of Φ to SThe root subgroups Ua can then be described as follows For any root ray a isin Φdenote ∆a sub Φ the set of simple roots that restrict to a The analog of the SL2

defined by a root is a morphism ζa∶GaK rarr GK If a is not a multiple root we have

GaK cong ResLαK SL2

where Lα sub K is the field obtained by the stabilizer of any α isin ∆a In this case apinning is an isomorphism

ResLαK GacongETHrarr Ua

which is again assumed to be compatible for a and minusa ( [17 Paragraph 417418])If a is a multiple root ray Ga is the Weil restriction of a unitary group2 In this casefor any pair ααprime isin ∆a such that α+αprime is a root let Lα = KStabα which is a quadraticextension of Lα+αprime =∶ L2 This extension defines the unitary group SU3(LαL2))over L2 (with respect to the standard hermition form) Then

Ga = ResL2K SU3(LαL2)In this case the root subgroups Ua Uminusa are of the form

Ua(L2) = xa(u v) =⎛⎜⎝

1 minusuσ minusv0 1 u0 0 1

⎞⎟⎠ Uminusa(L2) = xminusa(u v) =

⎛⎜⎝

1 0 0u 1 0minusv minusuσ 1

⎞⎟⎠

with v+vσ = uuσ This has a filtration U2a cong v isin LαL2∣ tr(v) = 0 and UaU2a cong LαThe Chevalley pinning induces valuations on the groups Ua For non multipliableroots one sets φα(xα(u)) ∶= ∣u∣ and for multiple roots one defines φa(xa(u v)) ∶= 1

2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣

1In [7] Balaji and Seshadri study the case where SK = TK is a split maximal torus ie thecase where GK is a split reductive group which already shows many interesting features

2This happens if the Galois group interchanges neighboring roots in the Dynkin diagram


(These choices define a valued root system and identify the standard appartmentA congXlowast(S)R in the BruhatndashTits building of G)

42 Parahoric group schemes With this notation we can recall the construc-tion of an open part of parhoric group schemesFor the torus TK Bruhat and Tits choose a version of the Neron model as extensionIn order to be consistent with the conventions from [38] we choose the connectedNeron model TR as an extension over SpecRFor the unipotent groups Ua the valuations introduced above can be used to defineextensions of Ua to group schemes Uak over SpecR for any k isin R For non-multipleroots the pinning identifies the abstract group Uak ∶= u isin Ua(K)∣φa(u) le k congu isin La∣∣u∣ le k and this can be equipped with the structure of a groups schemeisomorphic to ResRaRGaFor multiple roots this is slightly more complicated to spell out but again thesegroup schemes always correspond to free R modules [17 439]The open subset of a parahoric group scheme will be of the form ([17 Theorem381])

prodaisinΦminusUaf(a) times T times prod

aisinΦ+Uaf(a)

Now a facet Ω sub A defines a valuation of the root system (4626)

f(a) ∶= infk isin R∣a(x) + k ge 0foralla isin Ωhere we used our Chevalley-Steinberg valuation which defines an isomorphismminusφ∶A congXlowast(S)RThe product described above carries a birational group law ([34 Propostion 512])and thus one can use [14 Theorem 5] to construct a the group scheme GΩ (denotedby GΩ in [17]) containing the product as an open neighborhood of the identity Forour computations this is sufficient as these only use the Lie-algebra of G

5 Appendix Basic results on BunG

As in Section 3 we fix a smooth projective geometrically connected curve C overan algebraically closed field k and G rarr C a parahoric Bruhat-Tits group schemeIn this appendix we collect some results on the stack of G-bundles BunG for whichwe could not find a referenceThe basic tool will be the BeilinsonndashDrinfeld Grassmannian GRG ie the ind-projective scheme representing the functor of G-bundles together with a trivializa-tion outside of a finite divisor on C

GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR

E isin BunG(T )D isin C(d)(T ) for some d

φ∶E ∣CtimesTminusDcongETHrarr G timesC (C times T minusD)

⎫⎪⎪⎬⎪⎪⎭

It comes equipped with a natural forgetful maps

p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)

For Bruhat-Tits groups it will be useful to consider the open subfunctor where thedivisor D does not intesect some fixed finite subset of C eg the ramification pointsof the group G Given S sub C finite we will denote by GRGCminusS ∶= suppminus1 (∐(C minusS)(d)) ie the ind-scheme parametrizing G bundles together with a trivializationoutside of a divisor D that is disjoint form S Similarly we denote by GrGx ∶=suppminus1(x) the space parametrizing G-bundles equipped with a trivialization onC minus x

34 JOCHEN HEINLOTH

The following is a probably well-known geometric version of a weak approximationtheorem

Proposition 51 For any Bruhat-Tits scheme G rarr C and any finite subschemeS sub C the natural forgetful map pCminusS ∶GRGCminusS rarr BunG is surjective ie afterpossibly passing to a flat extension any G-bundle admits a trivialization on anopen neighborhood of S

Proof For S = empty this follows from a theorem of Steinberg and Borel-Springer [12]

stating that for any algebraically closed field K any G-bundle on CK is trivial over

the generic point K(C) As any such trivialization is defined over an open subsetthis allows to deduce that p∶GRG rarr BunG is surjective in the fppf-topologyTo deduce the result for pCminusS we can argue as in [28 Theorem 5] As GRGCminusS subGRG is a dense open subfunctor it follows that the generic point of any connectedcomponent of BunG is in the image of pCminusS Let E isin BunG be any bundle For theinner form GE ∶= AutG(E) of G we can apply the same reasoning and find that theimage of the map GRGE CminusS rarr BunGE cong BunG also contains an open subset of everyconnected component In particular there exist G-bundles E prime in the image that arealso in the image of pCminusS For such any such bundle E prime there exist divisors D1D2

on C minus S such that E prime∣CminusD1 cong G∣CminusD1 and E prime∣CminusD2 cong E ∣CminusD2 Composing theseisomorphisms we find that E ∣Cminus(D1cupD2) cong G∣Cminus(D1cupD2) ie E is also in the image ofpCminusS Thus pCminusS is surjective on geometric points As it is the restriction of a flatmorphism to an open subfunctor this implies thst it is again fppf surjective

To formulate properties of the set of connected components of BunG we will needsome more notation The generic point of C will be denoted by η = Speck(C) andη will be a geometric generic point From [38 Theorem 01] we know that for any

x isin C there is a natural isomorphism π0(GrGx) cong π1(Gη)I where I = Gal(KxKx)is the inertia group at x and π1(Gη) is the algebraic fundamental group ie thequotient of the coweight lattice by the coroot lattice of GηWe denote by Xlowast(G) = Hom(GGmC) the group of characters of G As G is a

smooth group scheme with connected fibers Xlowast(G) congXlowast(Gη) =Xlowast(Gη)Gal(ηη) Asany character χ∶G rarr Gm defines a morphism BunG rarr PicX given by E ↦ E(χ) ∶=E timesG Gm it defines a degree d∶BunG rarr Xlowast(G)or by E ↦ deg(E(χ)) and we will

denote by BundGsub BunG the substack of bundles of degree d which is open and

closed because the degree of line bundles is locally constant in families

Proposition 52(1) The natural map π1(Gη)Gal(ηη) rarr π0(BunG) is surjective

(2) For any d isin Hom(Xlowast(G)Z) the stack BundG

has finitely many connectedcomponents

Proof The first part follows from the surjectivity of GRG rarr BunG and the descrip-tion of the connected components π0(GrGx) cong π1(Gη)I from [38] as follows Let

GRdG denote the components of GRG mapping to C(d) As the projection GRd

G

is ind-projective every connected component intersects the fibers over the diago-nal C sub C(d) The preimage of the diagonal is isomorphic to GR1

G For GR1G the

fiber wise isomorphism π0(GrGx) cong π1(Gη)I is given by the Kottwitz homomor-phism which by construction is induced from a Galois-equivariant map π1(Gη) tothe sheaf of connected components of the fibers of p Thus this induces a surjectionπ1(Gη)Gal(ηη) rarr π0(GR1

G)rarr π0(BunG) which proves (1)

This implies (2) because the map Hom(Xlowast(G)Z)I rarr Hom(Xlowast(G)Z)I induces anisomorphism up to torsion


References

[1] D Abramovich A Corti A Vistoli Twisted bundles and admissible covers Special issue in

honor of Steven L Kleiman Comm Algebra 31 (2003) no 8 3547ndash3618

[2] D Abramovich T Graber A Vistoli Gromov-Witten theory of Deligne-Mumford stacksAmer J Math 130 (2008) no 5 1337ndash1398

[3] J Alper Good moduli spaces for Artin stacks Ann Inst Fourier (Grenoble) 63 no6 (2013)

2349-2402[4] J Alper J Hall and D Rydh A Luna etale slice theorem for algebraic stacks

arXiv150406467v1

[5] L Alvarez-Consul O Garcıa-Prada A Schmitt On the geometry of moduli spaces of holo-morphic chains over compact Riemann surfaces Int Math Res Papers (2006) 1ndash82

[6] A Asok Equivariant vector bundles on certain affine G-varieties Pure Appl Math Q2

(2006) no 4 part 2 1085ndash1102[7] V Balaji CS Seshadri Moduli of parahoric G-torsors on a compact Riemann surface J

Algebraic Geom 24 (2015) no 1 1ndash49[8] KA Behrend Semi-stability of reductive group schemes over curves Math Ann 301 (1995)

281-305

[9] A Beilinson V Drinfelrsquod Quantization of Hitchinrsquos integrable system and Hecke eigensheavesPreprint available under httpwwwmautexasedubenzvi

[10] B Bhatt and D Halpern-Leistner Tannaka duality revisited arXiv150701925v1

[11] I Biswas and N Hoffmann The line bundles on moduli stacks of principal bundles on acurve Documenta Math 15 (2010) 35ndash72

[12] A Borel TA Springer Rationality properties of linear algebraic groups II Tohoku Math

J (2)20 (1968) 443ndash497[13] A Borel J Tits Groupes reductifs Publ Math IHES 27 (1965) 55ndash150

[14] S Bosch W Lutkebohmert M Raynaud Neron models Ergebnisse der Mathematik und

ihrer Grenzgebiete Springer-Verlag Berlin 1990[15] S Bradlow Coherent systems A brief survey With an appendix by H Lange London Math

Soc Lecture Note Ser 359 Moduli spaces and vector bundles 229ndash264 Cambridge Univ

Press Cambridge 2009[16] F Bruhat J Tits Groupes reductifs sur un corps local Publ Math IHES 41 (1972) 5ndash251

[17] F Bruhat J Tits Groupes reductifs sur un corps local II Schemas en groupes Existencedrsquoune donnee radicielle valuee Publ Math IHES 60 (1984) 197ndash376

[18] P-H Chaudouard G Laumon Le lemme fondamental pondee I Constructions geometriques

Compositio Math 146 (6) 1416ndash1506 2010[19] B Conrad The KeelndashMori theorem via stacks Nov 2005 Preprint

httpmathstanfordedu conradpaperscoarsespacepdf

[20] M Demazure Schemas en groupes II Seminaire de Geometrie Algebrique du Bois Marie196264 (SGA 3) Dirige par M Demazure et A Grothendieck Lecture Notes in Mathematics

153 Springer-Verlag BerlinndashNew York 1970 viii+529 pp

[21] G Faltings Algebraic loopgroups and moduli spaces of bundles J Eur Math Soc 5 41ndash682003

[22] D Gaitsgory J Lurie Weilrsquos conjecture over function fields Preprint httpwwwmath

harvardedu~luriepaperstamagawapdf

[23] D Halpern-Leistner On the structure of instability in moduli theory arXiv14110627

[24] D Halpern-Leistner APreygel Mapping stacks and categorical notions of propernessarXiv14023204

[25] G Harder Halbeinfache Gruppenschemata uber vollstandigen Kurven Invent Math 6

(1968) 107ndash149[26] G Harder U Stuhler Canonical parabolic subgroups for Arakelov group schemes unpub-

lished manuscript[27] J Heinloth A Schmitt The cohomology rings of moduli stacks of principal bundles over

curves Doc Math 15 (2010) 423ndash488

[28] J Heinloth Uniformization of G-bundles Math Ann 347 (2010) no 3 499ndash528

[29] J Heinloth Semistable reduction for G-bundles on curves J Algebraic Geom 17 (2008)no 1 167ndash183

[30] J Heinloth Addendum to rdquoSemistable reduction of G-bundles on curvesrdquo J Algebraic Geom19 (2010) no 1 193ndash197

[31] W Hesselink Concentration under actions of algebraic groups Paul Dubreil and Marie-Paul

Malliavin Algebra Seminar 33rd year Lecture Notes in Mathematics 867 Springer-VerlagBerlin 1981

[32] I Iwanari Stable points on algebraic stacks Adv Math 223 (2010) 257ndash299

36 JOCHEN HEINLOTH

[33] S Keel S Mori Quotients by groupoids Ann of Math (2) 145 (1997) no 1 193ndash213

[34] E Landvogt A compactification of the Bruhat-Tits building Lecture Notes in Mathematics

1619 Springer-Verlag 1996[35] G Laumon L Moret-Bailly Champs algebriques Ergebnisse der Mathematik und ihrer

Grenzgebiete 3 Folge 39 Springer-Verlag Berlin 2000[36] J Martens M Thaddeus Compactifications of reductive groups as moduli stacks of bundles

To appear in Compositio Math (2015)

[37] D Mumford J Fogarty F Kirwan Geometric invariant theory (third edition) Ergebnisseder Mathematik und ihrer Grenzgebiete (2) 34 Springer-Verlag Berlin 1994

[38] G Pappas M Rapoport Some questions about G-bundles on curves Algebraic and arith-

metic structures of moduli spaces (Sapporo 2007) 159ndash171 Adv Stud Pure Math 58 MathSoc Japan Tokyo 2010

[39] D Rydh Existence and properties of geometric quotients J Alg Geom 22 (2013) 629ndash669

[40] A Schmitt Moduli problems of sheaves associated with oriented trees Algebr RepresentTheory 6 (2003) no 1 1ndash32

[41] T A Springer Linear algebraic groups Modern Brikhauser Classics Birkhauser Boston

2009[42] The Stacks Project Authors The stacks project httpstacksmathcolumbiaedu

[43] R Steinberg Regular elements of semisimple algebraic groups Publ Math IHES No 251965 49ndash80

[44] X Zhu On the coherence conjecture of Pappas and Rapoport Ann of Math (2) 180 (2014)

no 1 1ndash85

Universitat DuisburgndashEssen Fachbereich Mathematik Universitatsstrasse 2 45117 Es-sen Germany

E-mail address JochenHeinlothuni-duede

Introduction


11 Motivation The classical Hilbert-Mumford criterion

12 L-stability on algebraic stacks

13 Determining very close degenerations


15 Stability of vector bundles on curves

16 G-bundles on curves

17 The example of chains of bundles on curves

18 Further examples


21 Motivation from the valuative criterion

22 The test space for separatedness and equivariant blow ups





31 The setup

32 Line bundles on `39`42`613A``45`47`603ABunG

33 Preliminaries on parabolic subgroups of BruhatndashTits group schemes

34 Very close degenerations of G-bundles

35 The stability condition

36 Canonical reduction for G-torsors

37 Boundedness for stable G-torsors



41 ChevalleyndashSteinberg systems pinnings and valuations

42 Parahoric group schemes

5 Appendix Basic results on `39`42`613A``45`47`603ABunG

References

2 JOCHEN HEINLOTH

that were needed to prove cohomological purity results for the moduli stack Theresults of this article allow to bypass this issue and apply to the larger class ofparahoric groupsThe structure of the article is as follows In Section 1 we state the stability criteriondepending on a line bundle L on an algebraic stack M As a consistency check wethen show that this coincides with the Hilbert Mumford criterion for global quotientstacks (Proposition 18) To illustrate the method we then consider some classicalmoduli problems and show how Ramanathanrsquos criterion for stability of G-bundleson curves can be derived form our criterion rather easily The same argumentapplies to related moduli spaces as the moduli of chains or pairsIn Section 2 we formulate the numerical condition on the pair (ML) implyingthat the stable points form a separated substack (Proposition 26) and derive anexistence result for coarse moduli spaces (Proposition 28) Again as an illustrationwe check that this criterion is satisfied in for GIT-quotient stacks and for G-bundleson curvesFinally in Section 3 we apply the method to the moduli stack of torsors under aparahoric group scheme on a curve We construct coarse moduli spaces for thesubstack of stable points of these stacks For this we also need to prove some ofthe basic results concerning stability of parahoric group schemes that could be ofindependent interestAcknowledgments This note grew out of a talk given in Chennai in 2015 Thecomments after the lecture encouraged me to finally revise of an old sketch thathad been on my desk for a very long time I am grateful for this encouragementand opportunity I thank M Olsson for pointing out the reference [23] before itappeared and J Alper P Boalch and the referees for many helpful commentsand suggestions While working on this problem discussions with V Balaji NHoffmann J Martens A Schmitt have been essential for me

Contents

Introduction 11 The Hilbert-Mumford criterion in terms of stacks 311 Motivation The classical Hilbert-Mumford criterion 312 L-stability on algebraic stacks 413 Determining very close degenerations 514 The example of GIT-quotients 615 Stability of vector bundles on curves 816 G-bundles on curves 917 The example of chains of bundles on curves 1318 Further examples 142 A criterion for separatedness of the stable locus 1421 Motivation from the valuative criterion 1422 The test space for separatedness and equivariant blow ups 1523 An existence result for coarse moduli spaces 1824 The example of GIT-quotients 1925 The example of G-bundles on curves 193 Torsors under parahoric group schemes on curves 2031 The setup 2032 Line bundles on BunG 2133 Preliminaries on parabolic subgroups of BruhatndashTits group schemes 2134 Very close degenerations of G-bundles 2435 The stability condition 2636 Canonical reduction for G-torsors 28









wt(L) ∶= d




wtGm(L) ∶= d

4 JOCHEN HEINLOTH











wt(flowastL) lt 0





















6 JOCHEN HEINLOTH















Levi subgroup ie











A1

f X

π

[A1Gm] f [XG]
















8 JOCHEN HEINLOTH





























we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References









wt(L) ∶= d




wtGm(L) ∶= d

4 JOCHEN HEINLOTH











wt(flowastL) lt 0





















6 JOCHEN HEINLOTH















Levi subgroup ie











A1

f X

π

[A1Gm] f [XG]
















8 JOCHEN HEINLOTH





























we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

4 JOCHEN HEINLOTH











wt(flowastL) lt 0





















6 JOCHEN HEINLOTH















Levi subgroup ie











A1

f X

π

[A1Gm] f [XG]
















8 JOCHEN HEINLOTH





























we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References













6 JOCHEN HEINLOTH















Levi subgroup ie











A1

f X

π

[A1Gm] f [XG]
















8 JOCHEN HEINLOTH





























we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References


Levi subgroup ie











A1

f X

π

[A1Gm] f [XG]
















8 JOCHEN HEINLOTH





























we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

8 JOCHEN HEINLOTH





























we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References










we find





=sumij


= 2sumiltj




(sumilel

rk(Ei))(sumjgtl


deg(Ei))(sumjgtl

rk(Ej))

= 2suml






10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

10 JOCHEN HEINLOTH













λ and















12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References











12 JOCHEN HEINLOTH






aj wtL(ωj)



i


=sumi


= 2sumigt0















= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References











= 2sumi


Further we have


(sumlgej


rk(E i))

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

14 JOCHEN HEINLOTH





sumi rk(Ei)
















jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References






jx∶SpecR

cong

STR


OO

jy ∶SpecR

cong

STR


OO





congETHrarr P1






16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

16 JOCHEN HEINLOTH




deg(L∣P1) = 1






























RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References















RotimesR R

mult

K otimesR K

mult

R K







18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

18 JOCHEN HEINLOTH










lowastL)

















minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References






minus1y fx





X









GRG(S) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭



20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

20 JOCHEN HEINLOTH























Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References










Lχx















22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

22 JOCHEN HEINLOTH



























24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References














24 JOCHEN HEINLOTH








extends to









λof E to














BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References













BunPcong

$$

BunP prime

zz

BunG


26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

26 JOCHEN HEINLOTH















⟨χBx λ⟩




⟨χBx λx⟩

= sumaisinΦ


⟨χBx λx⟩




Da = sumxisinRam(G)

fBaxx




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References




fBax


∣fBax

rk(ua)∣ le 1

Thus we find



⟨χBx λ⟩

= 2 sumaisinΦ+



B

fBaxa λ⟩(31)



⟨ α⟩⟨ α⟩












28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

28 JOCHEN HEINLOTH








le c for all

λ∶Gmη rarr GEη
















fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References











fBprime

minusax



minusa)




fBprime

minusax + fBax









fBprime


prime1x a⟩

And this means


if χ is admissible







sumc=b+na




30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

30 JOCHEN HEINLOTH

















of finite type





















32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References














32 JOCHEN HEINLOTH















Ua(L2) = xa(u v) =⎛⎜⎝



⎛⎜⎝


⎞⎟⎠


2∣v∣

and φ2a(xa(0 v)) ∶= ∣v∣







aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References





aisinΦ+Uaf(a)





GRG(T ) =⎧⎪⎪⎨⎪⎪⎩(E Dφ)

RRRRRRRRRRR



⎫⎪⎪⎬⎪⎪⎭


p∶GRG rarr BunG

supp∶GRG rarr∐d

C(d)


34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

34 JOCHEN HEINLOTH

















G


G For GR1G the





References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References


References






arXiv150406467v1





281-305































36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

36 JOCHEN HEINLOTH















no 1 1ndash85



Introduction









18 Further examples








31 The setup












References

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