ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaRi i{Corre ted Derivativesand Invariant Di�erential OperatorsDavid M. J. CalderbankTammo DiemerVladimir Sou� ek

Vienna, Preprint ESI 1496 (2004) July 5, 2004Supported by the Austrian Federal Ministry of Edu ation, S ien e and CultureAvailable via http://www.esi.a .at

RICCI-CORRECTED DERIVATIVESAND INVARIANT DIFFERENTIAL OPERATORSDAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKAbstra t. We introdu e the notion of Ri i- orre ted di�erentiation in paraboli ge-ometry, whi h is a modi� ation of ovariant di�erentiation with better transformationproperties. This enables us to simplify the expli it formulae for standard invariant oper-ators given in [16℄, and at the same time extend these formulae from the ontext of AHSstru tures (whi h in lude onformal and proje tive stru tures) to the more general lassof all paraboli stru tures (in luding CR stru tures).Introdu tionA fundamental part of di�erential geometry is the study of di�erential invariants of geo-metri stru tures. Our on ern in this paper is the expli it onstru tion of su h invariants.More spe i� ally, we wish to onstru t invariant di�erential operators for a lass of stru -tures known as paraboli geometries. These geometries have attra ted attention in re entyears for at least two reasons: �rst, they in lude examples of long-standing interest indi�erential geometry, su h as onformal stru tures, proje tive stru tures and CR stru -tures; se ond, they have a ri h algebrai theory, due to their intimate relation with therepresentation theory of paraboli subgroups P of semisimple Lie groups G.A great deal of progress in our understanding of invariant di�erential operators in para-boli geometry has been made through the e�orts of many people. The key idea, pioneeredby Eastwood{Ri e [20℄ and Baston [3℄ is that (generalized) Bernstein{Gelfand{Gelfand(BGG) omplexes of paraboli Verma module homomorphisms are dual to omplexes of in-variant linear di�erential operators on generalized ag varieties G=P , and these omplexesshould admit ` urved analogues', that is, there should exist sequen es of invariant lineardi�erential operators on urved manifolds modelled on these homogeneous spa es. This isnow known to hold for all regular BGG omplexes [17℄.The prototypi al paraboli geometry is onformal geometry, whose model is the gener-alized ag variety SO0(n + 1; 1)=CO(n)n Rn�, namely the n-sphere Sn with its standard at onformal stru ture. In this ase another, more expli it, approa h to the study ofdi�erential invariants has been fruitfully applied: one �rst hooses a representative Rie-mannian metri (or, more generally, a ompatible Weyl stru ture); then one invokes thewell-known results of invariant theory in Riemannian (or Weyl) geometry; �nally one stud-ies how the di�erential invariants depend on the hoi e of Riemannian metri (or Weylstru ture). The di�erential invariants whi h are independent of the hoi e are invariantsof the onformal stru ture. The advantage of these methods is that they give expli it for-mulae for the di�erential onformal invariants in familiar Riemannian terms. They haveDate: July 2004.The �rst author is grateful for support from the Leverhulme Trust, the William Gordan Seggie BrownTrust and an EPSRC Advan ed Resear h Fellowship. The third author is supported by GA�CR, grant Nr.201/02/1390 and MSM 113 200 007. We also thank the Erwin S hr�odinger Institute for its hospitality.1

2 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKbeen parti ularly su essful in the onstru tion of �rst and se ond order linear di�erentialoperators [7, 22, 24℄, but have been extended to higher order operators and ertain otherparaboli stru tures (variously known as the j1j-graded, abelian or AHS stru tures) in [16℄.In the �rst order ase, the approa h of Fegan [22℄ has also been extended to arbitraryparaboli geometries [32℄.In this paper we build upon these expli it onstru tions of invariant di�erential operatorsin terms of a ompatible Weyl stru ture. The natural geometries for su h onstru tions areparaboli geometries, be ause we have a good notion of Weyl stru ture [13℄, similar to the onformal ase, and we an be sure than many invariant operators exist be ause we have theBGG sequen es [17℄. Our results are three-fold: �rst, we simplify the formulae for standardlinear di�erential operators given in [16℄; se ond, we extend these formulae to arbitraryparaboli stru tures; third, we un over a fundamental obje t in paraboli geometry, theWeyl jet operator, and its omponents, the Ri i- orre ted Weyl derivatives of the title ofour paper. Sin e it will take us a little while in the body of the text to rea h these results,we shall spend some time now explaining what the Ri i- orre ted Weyl derivatives are, inthe ase of onformal geometry.Let M be an n-dimensional manifold with a onformal stru ture . A ompatible Weyl onne tion D on M is a torsion-free onformal onne tion on the tangent bundle of M . Ittherefore indu es a onne tion on the onformal frame bundle of M , and hen e ovariantderivatives on any ve tor bundle V asso iated to the frame bundle via a representation �of the onformal group CO(n) on a ve tor spa e V.The most familiar Weyl onne tions are the Levi-Civita onne tions of representativeRiemannian metri s for . However, the broader ontext of Weyl onne tions has a fewadvantages in onformal geometry:� Weyl onne tions form an aÆne spa e, modelled on the spa e of 1-forms and we writeD 7! D + for this aÆne stru ture;� the onstru tion of the Levi-Civita onne tion from a Riemannian metri involves takinga derivative, so that di�erential invariants have one order higher in the metri than inthe onne tion;� a hoi e of Riemannian metri redu es the stru ture group to SO(n), making it easyto forget the 1-dimensional representations of the onformal group, an omission whi h omes ba k with a vengean e in the form of onformal weights.The aÆne stru ture of the spa e of Weyl onne tions provides a straightforward for-mulation of the well-known folklore that onformal invarian e only needs to be he kedin�nitesimally: we regard a di�erential invariant F onstru ted using a Weyl onne tionas a fun tion F (D); F is said to be a onformal invariant if it is independent of D; bythe fundamental theorem of al ulus, this amounts to he king that � F (D) = 0 for allWeyl onne tions D and 1-forms . More generally, if F (D) is polynomial in D thenthis dependen e an be omputed using Taylor's Theorem. In parti ular, if s is a se -tion of an asso iated bundle V and X is a ve tor �eld then � DXs = [ ;X ℄ �s where[ ;X ℄ = (X)id + ^X 2 o(TM) = R idTM � so(TM) and � denotes the natural a tionof o(TM) on V (indu ed by the representation � of o(n) on V).The Ri i- orre ted derivatives have their origins in the observation that the expli itformulae for onformally invariant di�erential operators in terms of a Weyl onne tionappear to have a systemati form, essentially depending only on the order of the operator.

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 3For �rst order operators, this is quite straightforward [22, 24℄: onformally invariant �rstorder linear operators are all of the form � ÆD : C1(M;V ) ! C1(M;W ), where V andW are asso iated bundles, D the ovariant derivative on V indu ed by a Weyl onne tion,and � is indu ed by an equivariant map Rn�V! W. Evidently � �(Ds) = ��[ ; �℄ �s�,so we obtain onformally invariant operators by letting � be the proje tion onto the zeroeigenspa e of the operator 2 End(T �M V ) de�ned by ( s) = [ ; �℄ �s. Sin e theseeigenvalues an be shifted, by tensoring Vwith a one dimensional representation of CO(n),a large number of �rst order operators are obtained.Ri i orre tions make their �rst appearan e at the level of se ond order operators. Hereone �nds that many onformally invariant operators are of the form �(D2s+ rD s) where� is indu ed by an equivariant map Rn�Rn�V! S2Rn�V!W (the �rst map beingsymmetrization), and rD is the normalized Ri i urvature of the Weyl onne tion, whi his a ove tor-valued 1-form on M onstru ted from the urvature of the Weyl onne tion.For present purposes, all we need to know about rD is its dependen e on D: � rD = �D .Now ompare this with the variation of the se ond derivative:� D2X;Y s = [ ;X ℄ �DY s�D[ ;X℄�Y s +DX�[ ; Y ℄ �s�� [ ;DXY ℄ �s= [DX ; Y ℄ �s + [ ;X ℄ �DY s+ [ ; Y ℄ �DXs �D[ ;X℄�Y s:This formula means that we an use the Ri i urvature to make the se ond derivativealgebrai in D.De�nition. The Ri i- orre ted se ond derivative on se tions s of an asso iated bundle Vis de�ned by D(2)X;Y s = D2X;Y s + [rD(X); Y ℄ �s.Hen e � D(2)s = �1Ds where( �1 �)X;Y = [ ;X ℄ ��Y + [ ; Y ℄ ��X � �[ ;X℄�Y= ([ ;X℄ ��)Y + [ ; Y ℄ ��X :It is now a purely algebrai matter to �nd proje tions � su h that ��D(2)s� is a on-formal invariant of s. As it turns out, these proje tions often have the property that��[rD(�); �℄ �s� = �(rD s). The simplest example is the onformal hessian sym0D(2)s =sym0(D2s + rDs) where s is a se tion of the weight 1 line bundle L.The same ideas apply to higher order operators: we want to write these operators as�ÆD(k) for some proje tions indu ed by an equivariant map (kRn�)V! SkRn�V!W,where D(k) is a Ri i- orre ted kth power of the Weyl onne tion. Again we make theobservation that if � (� ÆD(k)) = 0 then � ÆD(k) must ertainly be algebrai in D, and we an in fa t arrange for D(k) itself to be algebrai in D.De�nition. The Ri i- orre ted powers of the Weyl onne tion on an asso iated bundle Vare de�ned indu tively by D(0)s = s, D(1)s = Ds and�XD(k+1)s = DXD(k)s + rD(X) �1D(k�1)s:where ( �1 �)X1;:::Xk = kXj=1([ ;Xi℄ ��X1;:::Xi�1)Xi+1;:::Xk :A al ulation shows that � D(k)s = �1 D(k�1)s.

4 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKThe indu tive formula an easily be summed to giveD(k) = X`+m=k X16i1<���<i`6mDm�i` Æ (rD(�)�1) ÆDi`�i`�1�1 Æ � � �� � � Æ (rD(�)�1) ÆDi2�i1�1 Æ (rD(�)�1) ÆDi1�1:Thus the sear h for expli it invariant operators redu es to an algebrai problem, and we shall�nd that a large lass of proje tions � annihilating �1D(k�1)s produ e universal numberswhen applied to the terms of D(k), essentially be ause the a tion �1 on (jT �M) V forj < k is losely related to the a tion on (kT �M) V .This theory of Ri i- orre ted derivatives in onformal geometry was developed over sev-eral years by the �rst two authors, and des ribed, in part, in [19℄. In the homogeneous ase,i.e., on Sn, the se ond author explained the a tion �1 in terms of the se ond order partof the a tion of a onformal ve tor �eld on se tions of a homogeneous ve tor bundle [19℄.It be ame lear however, that there was a systemati underlying prin iple behind theseformulae, even in the urved ase, whi h should also generalize to arbitrary paraboli ge-ometries. More pre isely, the a tion �1 is related to part of the a tion of the nilradi alof the paraboli subalgebra p on ertain semiholonomi jet modules Jk0V asso iated to amodule V [21, 31℄. This a tion is onsiderably more ompli ated in general than it is inthe onformal ase, and we are for ed to onsider the representation theory of the entireparaboli subgroup P , not just its Levi fa tor P0 as we did in the onformal ase (whereP0 = CO(n) and P = CO(n) n Rn�). Hen e we must work with P -modules V, and the orresponding ve tor bundles V are asso iated to a larger prin ipal P -bundle, whi h in the onformal ase is the Cartan bundle with its normal Cartan onne tion. This developmentultimately provides a simple on eptual explanation for the formulae we obtain.The stru ture of the paper is as follows. We begin by de�ning Cartan geometries andinvariant di�erentiation in se tion 1: this is a standard way to treat paraboli geometries [9,11, 31℄, although in pra ti e a geometry is de�ned by more primitive data, whi h mustbe di�erentiated to obtain the Cartan onne tion [12℄. The key point is that a Cartan onne tion determines semiholonomi jet operators taking values in bundles asso iated tosemiholonomi jet modules.In se tions 2{3 we begin the study of paraboli geometries and Weyl stru tures. Weadopt a novel approa h to Weyl stru tures (whi h we relate to the approa h of �Cap andSlovak [13℄ in Appendix A) in order to emphasise the relationship between the geometry ofWeyl stru tures and some elementary representation theory whi h we exploit throughoutour treatment. At the algebrai level, a Weyl stru ture is a lift " of a ertain `gradingelement' "0 in the Levi fa tor p0 to the paraboli Lie algebra p. The grading element indu esa �ltration of a P -module Vand a lift " splits this �ltration, i.e., determines an isomorphism"Vof Vwith its asso iated graded module gr V. Geometri Weyl stru tures for paraboli geometries are given simply by applying the same pro edure pointwise on the underlyingmanifold: a geometri Weyl stru ture E then determines a splitting EV : V ! gr V of anyasso iated �ltered P -bundle V . Now if V is a �ltered P -bundle, so is its semiholonomi k-jetbundle JkV , and splitting this bundle allows us to proje t out Ri i orre ted derivativesas omponents of the k-jet. This simple onstru tion, whi h we present in se tion 4, makesRi i orre ted di�erentiation easy to study from a theoreti al point of view. On the otherhand, it is also easily related to ovariant di�erentiation: expli it formulae are obtained assoon as one understands the a tion of the nilradi al of p on jet modules.

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 5In se tion 5, we pave the way for the onstru tion of invariant operators by studyingspe ial types of proje tions from jet modules, whi h have the e�e t of killing most of the ompli ated terms in the jet module a tion. For irredu ible modules, the remaining termsof the jet module a tion redu e to the proje tion of a s alar a tion, whi h we omputeusing Casimirs. In se tion 6 we give our onstru tion of a large lass of invariant operators,and write out the formulae for operators up to order 8. We illustrate the s ope of the onstru tions in se tion 7 and give some examples in onformal geometry.Finally let us mention further potential appli ations of Ri i- orre ted di�erentiation.Although in this paper we have applied Ri i- orre ted derivatives to the onstru tion ofinvariant linear di�erential operators, the same ideas an be expe ted to yield expli itformulae for multilinear di�erential operators, su h as the operators of [9℄. Indeed, this wasour original motivation to study Ri i- orre ted di�erentiation in onformal geometry: oneapproa h to onstru t (say) bilinear di�erential operators is to ombine terms onstru tedfrom pairs of noninvariant linear di�erential operators; to do this one needs noninvariantoperators whi h nevertheless depend on the hoi e of Weyl stru ture in a simple way|proje tions of Ri i- orre ted derivatives onto irredu ible omponents have this property.A knowledgements. We would like to thank Fran Burstall and Jan Slovak for helpfuldis ussions on erning Weyl stru tures.1. Invariant derivatives in Cartan geometryParaboli geometries are geometries modelled on a generalized ag variety G=P , i.e., Gis a semisimple Lie group and P is a paraboli subgroup. A standard way to de�ne ` urvedversions' of homogeneous spa es is as Cartan geometries. In this se tion, we re all the basi al ulus of su h geometries, following [9, 11, 21, 30, 31℄.Fix a Lie algebra g with a Lie group P a ting by automorphisms su h that p is a P -equivariant subalgebra of g, and the derivative of the P -a tion on g is the adjoint a tion ofp on g. (These te hni al onditions are simply those that arise when P is a subgroup of aLie group G with Lie algebra g.)1.1. De�nition. Let M be a manifold with dimension dimM = dim g � dim p. A Car-tan onne tion of type (g; P ) on M is a prin ipal P -bundle � : G ! M , together witha P -invariant g-valued 1-form � : TG ! g su h that for ea h y 2 G, �y : TyG ! g is anisomorphism whi h sends ea h generator �� of the P -a tion to the orresponding � 2 p,i.e., ��;y = ��1y (�). Here P -invarian e means that Ad(p) �r�p� = � for any p 2 P , where rpdenotes the right P -a tion on G. We refer to (M;G; �) as a Cartan geometry.Note that Cartan onne tions form an open subset of an aÆne spa e modelled on thespa e of horizontal P -invariant g-valued 1-forms. However, one an freely add horizontalP -invariant p-valued 1-forms to a Cartan onne tion, without losing invertibility.Asso iated to any P -module V is a ve tor bundle V = G�P V, de�ned to be the quotientof G �Vby the a tion (y; v) 7! (yp�1; p �v). This indu es an a tion (p � f)(y) = p � f(yp) onfun tions f 2 C1(G;V) whi h identi�es se tions ' of V = G �P VoverM with P -invariantfun tions f : G ! V: C1(M;V ) = C1(G;V)P :Similarly P -invariant horizontal V-valued forms on G are identi�ed with forms on M withvalues in V . In parti ular the 1-form TG ! g=p indu ed by the Cartan onne tion � is

6 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKP -invariant and horizontal, orresponding to a bundle map TM ! G �P g=p. The open ondition on the Cartan onne tion means that this is an isomorphism and hen eforth weidentify TM with G �P g=p in this way. We let gM = G �P g and observe that there is asurje tive bundle map from gM to TM , with kernel pM = G �P p.Cartan onne tions do not in general indu e ovariant derivatives on asso iated bundles,but there is a way of di�erentiating se tions of su h bundles using gM instead of TM .1.2. De�nition. Let (G; �) be a Cartan onne tion of type (g; P ) on M , and let V be aP -module with asso iated ve tor bundle V = G �P V. Then the linear map de�ned byr� : C1(G;V)! C1(G; g�V)r��f = df���1(�)�(for all � in g) is P -equivariant. The restri tion to C1(G;V)P , or equivalently the indu edlinear map r� : C1(M;V )! C1(M; g�M V ), is alled the invariant derivative on V .The urvature K : ^2TG ! g of a Cartan geometry is de�ned byK(X; Y ) = d�(X; Y ) + [�(X); �(Y )℄:It indu es a urvature fun tion � : G ! ^2g� g via�y(�; �) = Ky���1(�); ��1(�)� = [�; �℄� �y [��1(�); ��1(�)℄;where y 2 G and the latter bra ket is the Lie bra ket of ve tor �elds on G.The p-invarian e of � 7! ��1(�) for � 2 g means that [��1(�); ��1(�)℄ = ��1[�; �℄ forany � 2 p, and hen e that �(�; �) = 0 for � 2 p so that � 2 C1(G;^2(g=p)� g)P . In otherwords K is a horizontal 2-form and indu es KM 2 C1(M;^2T �M gM ).1.3. Lemma. Let (G; �) be a Cartan geometry of type (g; P ) on M .(i) For f 2 C1(G;V)P , we have (r��f)(y) + � � �f(y)� = 0 for all � 2 p and y 2 G.(ii) We also have r��(r��f)� r��(r��f) = r�[�;�℄f � r��(�;�)f for all �; � 2 p.Proof. (i) Di�erentiate the P -invarian e ondition p � (f(yp)) = f(y).(ii) Both sides are equal to df([��1(�); ��1(�)℄). �These fa ts enable us to de�ne a semiholonomi jet operator jk� identifying the semi-holonomi jet bundle JkV with an asso iated bundle G �P Jk0V [9, 14, 31℄. Re all thatthe semiholonomi jet bundles are de�ned indu tively by J1V = J1V and Jk+1V is thesubbundle of J1JkV on whi h the two natural maps to J1Jk�1V agree. The advantage ofsemiholonomi jets is that they depend only on the 1-jet fun tor, the natural transformationJ1V ! V and some abstra t nonsense.1.4. Proposition. Let (G; �) be a Cartan geometry of type (g; P ) onM and Va P -module.(i) The map j1� : C1(M;V ) ! C1�M;V � (g�M V )� sending ' to (';r�') de�nes aninje tive bundle map, from the 1-jet bundle J1V to V ��g�MV �, whose image is G�P J10Vwhere J10V= f(�0; �1) 2 V� �g�V� : �1(�) + � ��0 = 0 for all � 2 pg.(ii) Similarly the map jk� sending a se tion ' to �';r�'; (r�)2'; : : :(r�)k'� de�nes anisomorphism between the semiholonomi jet bundle JkV and the subbundle G �P Jk0V of

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 7Lkj=0�(jg�M ) V �, where Jk0V is the set of all (�0; �1; : : :�k) in Lkj=0�(jg�)V� sat-isfying (for 1 6 i < j 6 k) the equations�j(�1; : : : �i; �i+1; : : : �j)� �j(�1; : : : �i+1; �i; : : : �j) = �j�1(�1; : : : [�i; �i+1℄; : : :�j)�i(�1; : : : �i) + �i � ��i�1(�1; : : :�i�1)� = 0for all �1; : : : �j 2 g with �i 2 p.Proof. (i) Certainly the map on smooth se tions only depends on the 1-jet at ea h point,and it is inje tive sin e the symbol of r� is the in lusion T �M V ! g�M V . It mapsinto G �P J10Vby verti al triviality, but this has the same rank as J1V .(ii) Similar: the equations are those given by the verti al triviality and the Ri i identity,bearing in mind that � is horizontal. The (semiholonomi ) symbols of the iterated invariantderivatives are still given by in lusions (jT �M) V ! (jg�M ) V . �2. Paraboli subalgebras and algebrai Weyl stru turesParaboli geometries are Cartan geometries of type (g; P ) where g is semisimple and theLie algebra p of P is a paraboli subalgebra. In this se tion we develop a few basi fa tsabout paraboli subalgebras and their representations, emphasising the relation between�ltered and graded modules. The key feature of paraboli subalgebras is the presen eof `algebrai Weyl stru tures' whi h split �ltered modules. Su h splittings, arried outpointwise, equip paraboli geometries with ovariant derivatives on asso iated bundles.A paraboli subalgebra of a semisimple Lie algebra is a subalgebra ontaining a Borel (i.e.,maximal solvable) subalgebra. However, to keep our treatment as self- ontained as possible,with minimal use of stru ture theory, we �nd the following equivalent and elementaryde�nition more onvenient. We refer to [5, 13, 31℄ for an alternative approa h.2.1. De�nition. Let g be a semisimple Lie algebra. For a subspa e u of g we let u? bethe orthogonal subspa e with respe t to the Killing form (� ; �). Then a subalgebra p of gis paraboli i� p? is the nilradi al of p, i.e., its maximal nilpotent ideal. It follows that thequotient p0 := p=p? is a redu tive Lie algebra, alled the Levi fa tor.Let p?; (p?)2 = [p?; p?℄; : : :(p?)j+1 = [p?; (p?)j℄; : : : be the des ending entral seriesof p?. Sin e p? is nilpotent there is an integer k > 0, alled the depth of p, su h that(p?)k+1 = 0 but (p?)k 6= 0. Thus p? has a k-step �ltration (k = 0 is the trivial ase p? = 0and p = g). We obtain from this a �ltration of g by setting g(�j) = (p?)j and g(j�1) = g?(�j)for j > 1 so thatg = g(k) � g(k�1) � � � � � g(1) � g(0) = p � p? = g(�1) � � � � � g(�k) � g(�k�1) = 0:It is easily veri�ed that [g(i); g(j)℄ � g(i+j), so that g is a �ltered Lie algebra. The asso iatedgraded Lie algebra is gr g =Lkj=�k gj where gj = g(j)=g(j�1) and is said to be jkj-graded.Note in parti ular that p0 = g0.An important fa t about paraboli subalgebras is that this �ltration of g is split.2.2. Lemma. There are (non- anoni al) splittings of the exa t sequen es(2.1) 0! g(j�1) ! g(j) ! gj ! 0whi h indu e a Lie algebra isomorphism between g and gr g.

8 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKProof. Any semisimple Lie algebra admits a Cartan involution, i.e., an automorphism� : g ! g su h that �2 = id and h(�; �) := (�(�); �) is positive de�nite. We split (2.1)for ea h j by identifying gj with the h-orthogonal omplement to g(j�1) in g(j). Suppose� 2 g(i) is h-orthogonal to g(i�1), i.e., �(�) 2 g?(i�1) = g(�i), and � 2 g(j) is h-orthogonal tog(j�1). Then [�; �℄ 2 g(i+j) and �[�; �℄ = [�(�); �(�)℄ 2 g(�i�j) so [�; �℄ is h-orthogonal tog(i+j�1). Hen e the splittings de�ned by � indu e a Lie algebra isomorphism. �We refer to su h a splitting of g as an algebrai Weyl stru ture: it is not unique, but we an obtain very good ontrol over the possible splittings thanks to the following.2.3. Lemma. There is a unique element "0 in the entre of p0 = p=p? su h that ["0; �℄ = j�for all � 2 gj and all j.Proof. Sin e gr g is semisimple, the derivation de�ned by � 7! j� for � 2 gj must be inner,i.e., equal ad "0 for "0 2 gr g (whi h is unique sin e Z(g) = 0). Now ["0; "0℄ = 0 and["0; �℄ = 0 for all � 2 g0, so "0 is in the entre of g0 = p0. �2.4. De�nition. The element "0 is alled the grading element. Let w = f" 2 p : �0(") = "0gbe the set of all lifts of "0 to p with respe t to the exa t sequen e(2.2) 0! p? ! p �0�! p0 ! 0:The elements of w are pre isely the algebrai Weyl stru tures: the isomorphism of g withgr g is given by the eigenspa e de omposition of ad " for a lift of "0 to " 2 p � g. The spa eof algebrai Weyl stru tures is therefore w, an aÆne spa e modelled on p?.Let P be a Lie group a ting on g with Lie algebra p as in the previous se tion, andsuppose additionally that the quotient group P0 = P= exp p? stabilizes "0 2 p0 (whi h isautomati if P0 is onne ted) so that the adjoint a tion of P on p preserves w.2.5. Lemma. exp p? 6 P a ts freely and transitively on w.Proof. If 2 p?, (Ad exp )" = exp(ad )" = " + [ ; "℄ + � � � . The result follows be ausead is nilpotent on p, and 7! [ ; "℄ is a bije tion on p?. �The stabilizer of " is thus a subgroup of P proje ting isomorphi ally onto P0, so that analgebrai Weyl stru ture splits the quotient group homomorphism �0 : P ! P0.The fundamental ve tor �elds � ( 2 p?) generating the a tion of exp p? on w give riseto a Maurer{Cartan form � : Tw ! p? with �(� ) = . If we identify Tw with w � p?using the aÆne spa e stru ture then � ;" = [ ; "℄, so �" is the inverse of 7! [ ; "℄ on p?.2.6. Filtered and graded modules. We say that a P0-module is semisimple if it is ompletely redu ible and the grading element "0 a ts by a s alar on irredu ible omponents(the latter ondition is automati for omplex modules). The eigenvalues of "0 will be alledthe geometri weights of the module.There is a one to one orresponden e between P0-modules and P -modules on whi hexp p? a ts trivially. We say su h a P -module is semisimple if the orresponding P0-moduleis. More generally, we shall onsider P -modules Vwith a P -invariant �ltration(2.3) V= V(�) � V(��1) � V(��2) � � � � � V(��`) � 0(for a s alar � and an integer `) su h that the asso iated graded module gr Vis a semisimpleP -module graded by geometri weight. We refer to su h modules as �ltered P -modules. Weextend the de�nition in a straightforward way to dire t sums of su h modules.

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 9An algebrai Weyl stru ture " splits any �ltered P -module V into the eigenspa es of ",giving a ve tor spa e isomorphism "V: V! gr V. This isomorphism is not P -equivariant(though it is tautologi ally P0-equivariant using the splitting of �0 : P ! P0 de�ned by ").However, by the naturality of the onstru tion, the map " 7! "V is P -equivariant. Morepre isely, if ~" = (Ad p)" is any other algebrai Weyl stru ture (with p 2 P ), then(2.4) ~"V(v) = p �"V(p�1 � v):In parti ular, if p 2 exp p?, we have ~"V= "VÆ (q�1 � ). Sin e the a tion of exp p? on w isfree and transitive, this dependen e implies that if we have a smooth family " = "(s) ofalgebrai Weyl stru tures then d"V(X)(v) = �"V(�(d"(X)) �v).2.7. Lemma. For a smooth map " : S ! w and a �ltered P -module V, the EndV-valued1-form "�1V d"Von S is given by the natural a tion of �"�� on V.2.8. The tangent module. We end this se tion by onsidering the module m := g=p,whi h is the �ltered P -module dual to p?. More pre isely, the Killing form of g gives anondegenerate pairing between g=p and p?, whi h will also be denoted by m�. This dualitydepends on the normalization of the Killing form, whi h we do not spe ify at present.We have seen that g is a �ltered P -module, with m and m� as quotient and sub- modulesrespe tively. The asso iated graded modules gr m and gr m� are graded nilpotent subalge-bras of gr g. In parti ular, as semisimple P -modules, gr g = gr m � p0 � gr m�, althoughthe Lie bra ket is not ompatible with this dire t sum de omposition. An algebrai Weylstru ture " therefore determines a ve tor spa e isomorphism "� : m� p0 �m�! g.Observe that f := g1 is the lowest geometri weight subspa e of gr m, and so is a P -submodule of m; the dual f� is naturally a quotient P -module of p? = m�.3. Paraboli geometries and Weyl stru tures3.1. De�nition. A paraboli geometry on M is a Cartan geometry (G; �) of type (g; P )with g semisimple and p paraboli , satisfying the onditions of the previous two se tions.We de�ne G0 to be the prin ipal P0-bundle G= exp p? and we let �0 also denote theproje tion G ! G0, so that �0(yp) = �0(y)�0(p) for y 2 G and p 2 P .The tangent bundle TM = G �P m has a natural �ltration indu ed by the �ltration ofm, the smallest nontrivial distribution in the �ltration being fM = G �P f. The otangentbundle T �M = G �P m� = G �P p? is a bundle of nilpotent Lie algebras, the nilradi albundle of pM = G �P p. The quotient pM=T �M is a redu tive Lie algebra bundle, namelypM;0 := G �P p0. Observe that pM;0 has a anoni al grading se tion E0, indu ed by thegrading element "0 of p0, whi h is P -invariant.3.2. De�nition. Let (G; �) be a paraboli geometry on M . Then a (geometri ) Weylstru ture E on M is a smooth lift of the grading se tion E0 to a se tion of pM .Thus a Weyl stru ture amounts to a smooth hoi e of algebrai Weyl stru ture at ea hpoint. Sin e algebrai Weyl stru tures form an aÆne spa e, a Weyl stru ture is a se tionof an aÆne bundle, the bundle of Weyl geometries wM = G �P w. In parti ular, Weylstru tures always exist, and form an aÆne spa e modelled on the spa e of 1-forms on M .3.3. Remark (Key observation). Any onstru tion with algebrai Weyl stru tures an be arried out with geometri Weyl stru tures. We an either work with asso iated bundlesor on the prin ipal bundle G, and both points of view are useful.

10 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEK(i) If V = G�PVis bundle asso iated to a �ltered P -moduleV, with graded bundle gr V =G�P gr V= G0�P0 gr V, then a Weyl stru ture E provides an isomorphism EV : V ! gr V ,simply by applying the onstru tion of the previous se tion pointwise. We also obtain abundle isomorphism E� : TM � pM;0 � T �M ! gM .(ii) A Weyl stru ture may equally be regarded as a P -invariant fun tion E : G ! w. Forany �ltered P -module V, we then have a P -equivariant isomorphism EV: G �V! G �gr Vwhose �bre at y 2 G is E (y)V; the indu ed isomorphism of asso iated bundles is EV .Similarly we get a P -equivariant isomorphism E� : G �(m� p0 �m�)! G �g indu ing E�.If we �x an algebrai Weyl stru ture then any geometri Weyl stru ture may be writtenE = (Ad q)", where q : G ! exp p? is P -invariant in the sense that pq(yp)�0(p�1) = q(y):here P0 a ts on P via the lift de�ned by ". As we dis uss in Appendix A, this allows us torelate our approa h to Weyl stru tures to the original approa h of �Cap{Slov�ak [13℄.4. Ri i- orre ted Weyl differentiationThe main diÆ ulty in the study of invariant di�erential operators on paraboli geometriesis that there is no natural ovariant derivative on asso iated bundles: we only have theinvariant derivative r� : C1(M;V ) ! C1(M; g�M V ) in general. There is no anoni alproje tion g�M ! T �M ; equivalently, the restri tion map g�! m�= p? is not P -equivariant.Weyl stru tures provide two solutions to this problem, one well known (Weyl onne -tions), the other impli itly known, but not properly formalized (Ri i- orre ted Weyl on-ne tions). In our theory, both an be de�ned straightforwardly using Remark 3.3(i).4.1. De�nition. Let (G !M; �) be a paraboli geometry and E be a Weyl stru ture onM . Let V = G �P Vbe a �ltered P -bundle (i.e., asso iated to a �ltered P -module).(i) The Ri i- orre ted Weyl onne tion D(1) : C1(M;V ) ! C1(M;T �M V ) is givenby D(1)X ' = r�E�X' for all ve tor �elds X and se tions ' of V . In other words D(1)obtained by restri ting the invariant derivative to tangent ve tors using the isomorphismE� : TM � pM � T �M ! gM indu ed by E .(ii) The Weyl onne tion D : C1(M;V ) ! C1(M;T �M V ) is D' = E�1V D(1)(EV'),i.e., the onne tion on V indu ed by D(1) on gr V via the isomorphism EV : V ! gr V .By de�nition, D(1) and D agree on bundles asso iated to semisimple P -modules (whenexp p? a ts trivially and V and gr V are anoni ally isomorphi ). In the notation wesuppress their dependen e on the Weyl stru ture E . A priori they also depend on the hosen P -module V. This latter dependen e is straightforward as they are asso iated toprin ipal P - onne tions. To see this, we use the isomorphism E� : G�(m� p0�m�)! G�gde�ned by the Weyl stru ture to de ompose the Cartan onne tion � : TG ! g as(4.1) � = E��m + �p; �p = E��p0 + �m�;where �m := (� mod p) : TG ! m is the solder form, indu ed by proje ting � onto m = g=pand similarly �p0 := (�p mod p?) : TG ! p0 = p=p?. Thus(4.2) E�1� Æ � = �m + �p0 + �m�and we an write the p-part on eptually as(4.3) �p = (E��p0 � E��) + (�m�+ E��);where � is the Maurer-Cartan form on w. This leads to the following proposition.

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 114.2. Proposition. Let (G !M; �) be a paraboli geometry with Weyl stru ture E . Then:(i) �p is a prin ipal P - onne tion on G indu ing D(1) on asso iated bundles ;(ii) �E = E��p0 � E�� is a prin ipal P - onne tion on G indu ing D on asso iated bundles ;(iii) � = �m�+E�� is a horizontal, P -invariant p?-valued 1-form on G; if rD is the indu edT �M -valued 1-form on M and � is the natural a tion of T �M = p?M on V , then(4.4) D(1)X ' = DX'+ rD(X) �':Proof. (i) Clearly �p is P -invariant and p-valued, and so, sin e � is a Cartan onne tion, �pis a prin ipal P - onne tion. Let X be a ve tor �eld and ' a se tion of V , and let � : G ! mand f : G !Vbe the orresponding P -invariant fun tions. Sin e the identi� ation of TMwith G �P m is via the solder form, � = �m(X) for any P -invariant lift X of X to G.As a P -invariant fun tion on G, D(1)X ' is thenr�E��f = r�E��m(X)f = r��(X)f � r��p(X)f = df(X) + �p(X) � f;whi h is pre isely the P -invariant fun tion on G orresponding to the ovariant derivativeof ' along X indu ed by �p.(ii) We now mirror the onstru tion of D from D(1) on the prin ipal bundle level, usingRemark 3.3(ii): if V is a �ltered P -module and f : G !V is P -invariant, orresponding toa se tion ' of V = G �P V, then D' orresponds to the P -invariant horizontal 1-formE�1V (d+ �p)(EVf) = df + E�(�p0) � f + E�1V (dEV)f = df + �E � fwith �E = E�(�p0)� E��, by Lemma 2.7, as required.(iii) This follows immediately be ause �p = �E + �. (One an also easily see dire tly that�m�+ E�� is a P -invariant p?-valued horizontal 1-form on G.) �In onformal geometry rD is the normalized Ri i urvature (aka. the S houten or Rhotensor) of D. This is the origin of the term Ri i- orre ted Weyl onne tion.We wish to see how the obje ts we have onstru ted depend on the hoi e of Weylstru ture. We an either do this on M , or for the orresponding P -invariant obje ts on G.4.3. Proposition. Let E and ~E = (Ad q�1)E be Weyl stru tures, with q : M ! G�P exp p?(asso iated to the adjoint a tion), and let V be a �ltered P -bundle. Then ~EV = EV Æ (q �).This is immediate from equation (2.4). In pra ti e it suÆ es to understand in�nitesimalvariations. Let qt be a urve of se tions of pM with q0 = id and _q0 = for a 1-form (equivalently a P -invariant fun tion G ! p?). Then for any obje t F (E) depending on E ,de�ne (� F )(E) to be the t-derivative of F ((Ad q�1t )E) at t = 0 (so that � E = � ). Bythe fundamental theorem of al ulus, F is independent of the Weyl stru ture if and onlyif (� F )(E) = 0 for all Weyl stru tures E and all 1-forms . Proposition 4.3 implies that� EV = EV Æ ( �). It is now straightforward to di�erentiate the de�nition of D(1).4.4. Proposition. For a Weyl stru ture E , a 1-form , and a ve tor �eldX, let [ ;X ℄EpM =[ ; E�X ℄� E�( �X). Then for any se tion ' of a �ltered P -module V ,(4.5) � D(1)X ' = [ ;X ℄EpM �'Proof. For X 2 TM , � E�X = E�( �X)� [ ; E�X ℄ = �[ ;X ℄EpM . This is pM -valued (it isthe pM omponent of the Lie bra ket [ ;X ℄, where the lift of X to gM and the proje tionto pM are de�ned using E), and so � r�E�X' = r�� E�X' = [ ;X ℄EpM �'. �

12 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKThe Ri i- orre ted �rst derivative D(1) agrees with the Weyl onne tion on semisimplemodules. Ri i orre tions start to play a more important role when higher derivativesare onsidered, be ause jet modules are not semisimple. The following de eptively simplede�nition learly generalizes the previous de�nition of D(1).4.5. De�nition. Let E be a Weyl stru ture. Then we de�ne an isomorphism fromJk� V 6 kMj=0(jg�M) V to JkEV := kMj=0(jT �M) Vby sending � = (�0; �1; : : :�k) to = ( 0; 1; : : : k) where j = (�j ÆE�)jjTM .If ' 2 C1(M;V ), then the se tion of JkEV orresponding to jk�' is denoted jkD' =(';D(1)';D(2)'; : : :D(k)'). We all jkD the Weyl jet operator, and its omponents D(j) theRi i- orre ted powers of the Weyl onne tion.An alternative des ription of the Weyl jet operator is obtained from the obvious naturalisomorphism between gr Jk� V and gr JkEV . Then jkD = E�1JkEVEJk� V jk� . It follows that(4.6) � jkD' = E�1JkEVEJk� V � jk�'� �E�1JkEVEJk� V jk�' = � jkD'� � jkD';where � is the a tion of T �M on JkEV indu ed by the isomorphism E�1JkEVEJk� V with Jk� V .The omputation of � is a straightforward exer ise in algebra: it suÆ es to des ribethe jet ~� = �� for � 2 Jk0V, 2 p?, in terms of the elements ~ and of Jk"V :=Lkj=0(jm�) V orresponding to ~� and � using an algebrai Weyl stru ture ". This p-module stru ture on Jk"V is omputed in [13, 31℄: let us sket h brie y the omputation.We write = ( 0; 1; : : : k) with j 2 (jm�)Vand similarly for ~ .First, note that the existen e of natural proje tions Jk0V! J0V(for k > `) implies thatthe j omponent ontributes only to ~ j+s for s > 0, independently of k > j + s. We maytherefore write ~ ` = ( � )` = Pj+s=` �s j for any ` 6 k, where �s j denotes the ontribution from j . Clearly �0 j = � j is the ordinary a tion of p? 6 p.Se ondly, observe that the natural in lusions Jk+10 V! J10 Jk0Vmean that we an ompute �s j indu tively using the de�nition of J10V. Re all that this onsists of the pairs (�0; �1)in V� (g�V) with �1(�) + � ��0 = 0 for � 2 p. The identi� ation with V� (m�V) isobtained by restri ting �1 to m, using ". The indu ed p-module stru ture on V� (m�V)is given by � � (�0; �1jm) = �� ��0; (� ��1)jm�, and one easily omputes that (� ��1)jm =� � (�1jm) + [�; �℄"p ��0, where " is used to split the natural maps p! g ! m. Therefore � ( 0; 1) = ( � 0; � 1 + �1 0) with �1 0 = [ ; �℄"p � 0:(Applying this pointwise on M , we rederive equation (4.5).) Iterating this a tion we have �s 0 = �[: : : [ ; �℄"p; : : :℄"p; ��"p � 0. The formula for �s j is also omputed indu tively,by onsidering the a tion of on J10 Jj+s�10 V: apply this a tion to (0; : : :0; �j; 0; : : :0) 2Jj+s�10 V to obtain �s j as the m� (j+s�1m�)V omponent with respe t to ".Passing from the algebra to asso iated bundles, we obtain the following result.4.6. Proposition. The a tion of 2 T �M on 2 JkEV indu ed by the identi� ation withJk� V is given by ( � )` = Xj+s=` �s j

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 13where, if we suppose that j = A1 � � � Aj v for Ai 2 T �M and v 2 V , we have �0 j = � j �1 j = X06i6j A1 A2 � � � Ai [ ; �℄EpM � (Ai+1 � � � Aj v) �s j = X06i16i26���6is6ja1;a2;:::as A1 � � � Ai1 ea1 Ai1+1 � � � Ai2 ea2 Ai2+1 � � �� � � � � � Ais�1 eas�1 Ais�1+1 � � � Ais eas�[: : : [[ ; ea1℄EpM ; ea2℄EpM ; : : :eas�1 ℄EpM ; eas�EpM � (Ais+1 � � � Aj v):In these formulae ea is a frame of TM with dual frame ea.This a tion not only gives an expli it formula for � jD: it also provides an expli itindu tive formula for the Ri i- orre ted powers D(k) of the Weyl onne tion D. Indeed,sin e the order k + 1 part of j1� jk�' is the same as that of jk+1� ', we obtain:�XD(k+1)' = DXD(k)'+ proj(kT �M)V �rD(X) � jkD'�= DXD(k)'+ Xj+s=k rD(X) �s D(j)':(4.7)Expli it formulae for invariant di�erential operators will follow by omputing proje tionsof D(k) using this indu tive expression and some representation theory.5. Strongly invariant operators5.1. Jet module homomorphisms. Our goal is to explain how Ri i- orre ted Weyldi�erentiation leads to expli it formulae for a lass of invariant di�erential operators. Theseare the strongly invariant operators of [16, 17, 21℄, de�ned as follows. Let V and W beP -modules and let �: Jk0V! W be a P -homomorphism. Then � indu es a bundle mapF : G �P Jk0V! G �P W, and hen e an invariant di�erential operator F Æ jk� from V to W .In pra ti e, su h P -homomorphisms are onstru ted by lifting a P0-homomorphismgr �: gr Jk0V! gr W using an algebrai Weyl stru ture " to give �" = "�1W Æ gr � Æ "V.Sin e �(Ad q)"v = q ��"(q�1 � v), it follows that � = �" is a P -homomorphism if and onlyif it is independent of the algebrai Weyl stru ture ".If we mirror this onstru tion on asso iated bundles, for any Weyl stru ture E , a bundlemap gr F : gr JkV ! grW (asso iated to gr �: gr Jk0V! gr W) indu es a di�erentialoperator E�1W Æ gr F ÆEJkV Æ jk� from V to W , whi h will be invariant if it is independent ofthe hoi e of Weyl stru ture E . An obvious suÆ ient ondition is that E�1W Æ gr F ÆEJkV isindependent of the hoi e of Weyl stru ture and these are the strongly invariant operators.(The ondition is not ne essary be ause jk�' will satisfy some Bian hi identities not satis�edby general se tions of JkV .)The method we shall adopt for onstru ting strongly invariant operators is to onstru ta P -homomorphism �: Jk"V! W, indu ing a bundle map F : JkEV ! W and hen e, forany Weyl stru ture E , a di�erential operator F Æ jkD from V toW . Sin e Jk"Vhas the sameasso iated graded module as Jk0V, it is straightforward to obtain onditions that � indu esa P -homomorphism Jk0V! W, and hen e a strongly invariant operator. The expressionF Æ jkD then gives an expli it formula for this operator in terms of a Weyl stru ture.The P -homomorphisms we onstru t here all fa tor through the proje tions Jk"V !km�V!kf�V, where the se ond proje tion is indu ed by the restri tion map m�! f�.

14 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKOur �rst task is to apply these proje tions to the jet module a tion �. Then we apply furtherproje tions (kf�)V!W to obtain the P -homomorphisms we seek. In geometri terms,operators obtained from homomorphisms fa toring through these proje tions are given byapplying a bundle map (kf�M) V !W to the restri tion of D(k) to kfM .5.2. Restri ting the jet module a tion. We �rst restri t � to f. For notationalsimpli ity we give the result algebrai ally: the formulae on asso iated bundles easily follow.5.3. Proposition. Suppose V is a semisimple P -module. Let �f denote the restri tionmaps jm�! jf� and let ea, ea be dual bases for f, f�. Then ( � )` = Pj+s=` �s j,where for ea h j we have, supposing j = A1 � � � Aj v as before,�f( �0 j) = 0�f( �1 j) = X06i6j �f(A1 A2 � � � Ai) � �f(Ai+1 � � � Aj v)��f( �s j) = X06i16i26���6is6ja1;a2;:::as�1 �f(A1 � � � Ai1) ea1 �f(Ai1+1 � � � Ai2) ea2 � � �� � � � � � eas�1 �f(Ais�1+1 � � � Ais)��: : : [[ ; ea1℄"p; ea2℄"p; : : :eas�1�"f� �f(Ais+1 � � � Aj v)�:Here, for any semisimple P -module ~V, we de�ne : f� ~V! f� ~V by(5.1) (A ~v)(�) = [A; �℄ � ~vfor � 2 f, whi h is well de�ned sin e ~V is semisimple. In the above formulae for �1 and �s,we have ~V= (j�if�)V and (j�isf�)V respe tively.Proof. This is immediate from (the algebrai version of) Proposition 4.6, and the equality�f��[: : : [[ ; ea1℄"p; ea2 ℄"p; : : :eas�1℄"p ; eas�"p � (Ais+1 � � � Aj v)�= �[: : : [[ ; ea1℄"p; ea2℄"p ; : : :eas�1 ℄"f�; eas�"p0 � ��f(Ais+1 � � � Aj) v�;whi h holds be ause the proje tion of the Lie bra ket p? p? ! p? ! f� vanishes and themodule (j�isf�)V is semisimple. �The formulae of this proposition may not seem simpler than the full formulae of Propo-sition 4.6, but they are easier to handle, sin e is an operator on semisimple modules.5.4. Spe ial types of proje tions. To progress further, we need to understand the map: f�V! f�V; (A v) =Xa ea ([A; ea℄ � v);where V is a semisimple P -module and ea and ea are dual bases of f and f� respe tively.Sin e is a P -homomorphism, it a ts by a s alar on every irredu ible omponent of f�Vand these s alars an be omputed expli itly using Casimirs. Note now that all weights ofthe p0-module m� have multipli ity one (they are just positive roots of g). Hen e resultsfrom [6, 28, 32℄ show that all irredu ible omponents of m�Vare multipli ity free.We an write f�Vas a sum of well-de�ned irredu ible P0- omponentsVb. Consequently,(2f�)V= f� (Lb1Vb1) an be again written as a sum of invariant subspa es labelled bya ouple (b1; b2) of indi es indi ating that it is the isotypi omponent with label b2 insidef�Vb1. Indu tively, we get a well de�ned de omposition of (kf�) V into P0-invariantsubspa es labelled by paths b = (b1; : : : bk) of indi es showing their positions in onse utive

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 15de ompositions. Note that the full isotypi omponent of (kf�) Vwith label bk is thedire t sum of all subspa es Vb labelled by paths b ending with bk.We now suppose that our omponent Vb is in the symmetri tensor produ t Skf�V.5.5. Proposition. Let � be a proje tion of Jk"Vto an invariant subspa e in (Skf�V)\Vbfor some Vb in the de omposition des ribed above. Then for any element of Jk"V, theonly ontribution to �( � ) is from k�1 and if = A1 � � � Ak�1 v we have�( � ) = X06i6k�1 ��A1 A2 � � � Ai ��f( ) �f(Ai+1 � � � Ak�1 v)��:Proof. Note �rst that �( � ) =Pj+s=k �( �s j) and that the s = 0 term vanishes (sin eV is semisimple). Hen e it remains to show that the terms with s > 2 are zero. To dothis is suÆ es to show that terms in whi h is applied to a Lie bra ket are killed by theproje tion. Consider therefore an expression of the formPa ea ([ ; ea℄"f� v)and suppose we apply a proje tion � to (2f�)Vwhi h fa tors through S2f�V and isof the form �2 Æ id �1 where �1 : f�V!V1 is a proje tion onto an isotypi omponent.Sin e a ts by a s alar on su h a omponent, the result is a multiple of � applied toPa ea [ ; ea℄"f� v:The proje tion � fa torizes through the symmetri produ t, so it suÆ es to note that�Pa ea [ ; ea℄"f��(�1; �2) = [ ; �1℄"f�(�2) = (["��1; "��2℄ mod p);whi h is learly antisymmetri in �1; �2 2 f. The terms appearing in the a tion for s > 2 areall of this form, with and v repla ed by iterated bra kets and suitable tensor produ ts. �5.6. Casimir omputations. In this se tion we ompute the eigenvalues of and hen eprove that a ts by a s alar on ea h isotypi omponent. This is the �rst point at whi h weneed detailed information from representation theory, so we set up the ne essary notation.Let g be a semisimple Lie algebra with omplexi� ation gC . We hoose a Cartan subal-gebra hC in gC , a set �+ of positive roots, and its subset S = f�1; : : :�rg of simple roots.Using the Killing form (� ; �), with any normalization, the fundamental weights !1; : : :!r arede�ned by (�_i ; !j) = Æij , where �_i = 2�i=(�i; �i).The dominant Weyl hamber C is given by linear ombinations of fundamental weightswith nonnegative oeÆ ients. Finite dimensional omplex irredu ible representations of gC(as well as of g) are hara terized by their highest weights � 2 C, whi h lie in the weightlatti e fP�i!i : �i 2Zg. The orresponding representation will be denoted by V�.A redu tive algebra is a dire t sum of a ommutative and a semisimple algebra (eitherof whi h an be trivial). Its irredu ible ( omplex) representations are tensor produ ts ofirredu ible representations of the summands, where irredu ible representations of a om-mutative Lie algebra a are one dimensional, hara terized by an element of Hom(a; C).5.7. Remark. For simpli ity, we fo us on omplex representations of the Lie algebras inquestion. In pra ti e, we may well be more interested in real representations. For this, itis suÆ ient to use the following des ription: a real or quaternioni stru ture on a omplexg-module Vis a onjugate-linear g-map J : V!Vwith J2 = 1 or J2 = �1 respe tively. Anirredu ible real representation of a Lie algebra an be identi�ed either with an irredu ible

16 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEK omplex representations, or with su h a representation endowed with a real or quaternioni stru ture. We note also that if V is a omplex g-module and U is a real g-module with omplexi� ation U , then UVand U C Vare equivalent as omplex modules.We now spe ialize to the situation where g is a semisimple Lie algebra with paraboli subalgebra p and Levi fa tor p0. The set S of simple roots for gC an be hosen in su h away that all positive root spa es are ontained in pC . This �xes an algebrai Weyl stru ture,and the positive root spa es lying in pC0 orrespond to roots in the span of the subset S0of `un rossed' simple roots|we write S = S� [ S0 for the de omposition into rossed andun rossed simple roots. In this situation, we shall say that a weight � (integral for g) isdominant for p, if its restri tion to hCss = hC \ pC0;ss is dominant for pC0;ss. Su h a weightspe i�es uniquely an irredu ible P -module.Let us denote by Æ the half sum of positive roots for gC and by Æ0 the half sum of thosepositive roots for gC for whi h the orresponding root spa e belongs to gC0 .5.8. Proposition. Let V� be an irredu ible representation of p0 with highest weight � 2(hC )�. Let f�V� =L�2AV� be the de omposition of the tensor produ t into the sum ofisotypi omponents with highest weight � and let �� be the proje tion to V�. Let Æ denotethe half sum of positive roots for the Lie algebra g. Then =X ���; � = 12 �j�+ Æj2 � j�+ Æj2�withand j�j2 = (�; �). [Note that depends upon the normalization of the Killing form, sin ewe used it to identify p? with (g=p)�, and hen e the bra ket of f� with f depends on (� ; �).℄Proof. We �rst give a formula for in terms of the Casimir operator C of p0, as in [22, 32℄.Let Ei and Ei be bases for p0 whi h are dual with respe t to (� ; �). Then( v) =Pa ea [ ; ea℄ � v =Pi[Ei; ℄Ei � v =Pi[Ei; ℄ Ei � v2( v) =PiEi �Ei � ( v)� �PiEi �Ei � v�� �PiEi �Ei � � vand so = C( v)� C(v)� C( ) v:It is well known that on an irredu ible representation with highest weight �, C a ts by thes alar (�; �+ 2Æ0): with the de�nition of Æ0 above, this holds even though p0 is redu tiverather than semisimple|see [32℄.Let us write S� = f�i : i = 1; : : :r� 6 rg for the rossed simple roots of g. We know that��i are pre isely the highest weights of the irredu ible omponents f�i of the P0-module f�,so that the irredu ible omponents of f�i V� have highest weights of the form �� �i + ,where is an integral linear ombination of simple roots for pC0 . Hen e for any isotypi omponent V� there is an i so that V� is an invariant subspa e of f�i V�.Sin e the a tion of the Casimir depends only on the highest weight, we dedu e, follow-ing [32℄, that a ts on the entire isotypi omponent V� by the s alar = 12�(�; �+ 2Æ0)� (�; �+ 2Æ0)� (��i;��i + 2Æ0)�:It remains to identify with the onstant � above. For any simple root �, we have2(Æ; �) = (Æ; 2�=j�j2) j�j2 = nXj=1(!j ; �_)j�j2 = j�j2:

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 17Hen e (�; �+ 2Æ0)� (�; �+ 2Æ0)� (��i;��i + 2Æ0) is given by(�; �+ 2Æ)� (�; �+ 2Æ)� 2(Æ � Æ0; �� �)� 2(�i; Æ � Æ0)= j�+ Æj2 � j�+ Æj2 + 2(Æ � Æ0;��i � � + �):We know that ��� is a weight of f�i; hen e ��i� (���) =P�2S0 n��. But for all � 2 S0;we have (�; Æ � Æ0) = 0, so the last term vanishes. �6. Expli it onstru tions of invariant operatorsIn this se tion, we are going to onstru t a large lass of invariant di�erential operators.Most of them belong to the lass of standard regular operators, but a ertain sub lass arestandard singular operators. The lass of operators onstru ted here does not over allstandard regular operators, but we shall see in examples that it overs many of them.Various spe ial ases of the main results of this se tion an be found in [7, 16, 17, 22, 32℄.6.1. Theorem. Let � 2 (hC )� be a positive root with g� � f�. In the ase that g has rootsof di�erent lengths, we shall suppose that � is a long root. Let �; � be two integral dominantweights of p0 with the property� + Æ = ��(�+ Æ) = �+ Æ � (�+ Æ; �_)�:Inter hanging � and � if ne essary, suppose that k := �(�+ Æ; �_) is positive.(i) There is a unique irredu ible omponent V� with highest weight � in (km�) V�.Furthermore, V� belongs to Skf� V� and is of the form Vb, where Vbj = V�+j� forb = (b1; : : : bk).(ii) If � : Jk"V�!V� is the orresponding proje tion, then � indu es a P -homomorphismJk0V� ! V� and hen e a strongly invariant di�erential operator � Æ D(k) of order k fromse tions of V� to se tions of V�.Proof. (i) To prove uniqueness, let us note �rst that all weights of m� are positive roots.Weights of km� are hen e sums of k positive roots. The highest weight of any irredu ible omponents of (km�) V� is of a form �+ �; where � is a weight of km�: The uni ity laim is therefore true by the triangle inequality, � being a long root.We now show that there is su h a omponent V�. By assumption, both � and � = �+k�are P -dominant, and j� is an extremal weight of �j f�i for all j = 1; : : :k. The so- alledParthasarathy{Ranga-Rao{Varadarajan onje ture (proved in [29℄) states that if �; � arehighest weights of two irredu ible ( omplex) p0-modules V�;V�, and if � is an extremalweight of V�, then an irredu ible omponent V�+� with extremal weight �+ � will appearwith multipli ity at least one in V�C V�. (In on rete ases, more elementary argumentsare available.) It follows that there is an irredu ible omponent of (�j f�i)V� having �+j�as its highest weight. Hen e ne essarily V� � (�kf�i) V� � Skf�i V�. Furthermore, thesame holds with � repla ed by �+ j 0� for all j 0 = 1; : : :k � 1. Hen e, by uniqueness, theproje tion fa tors through (�jf�i)V�.(ii) Let us prove now that � indu es a P -homomorphism. The a tion of m� onV� is trivial,hen e we must show that �( � ) vanishes for any 2 Jk"V� and any 2 m�.Sin e � is the proje tion to an irredu ible pie e of Skf� V� lying in a omponent ofthe form Vb, the a tion is given by Proposition 5.5. Using the Casimir omputation of

18 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKProposition 5.8, the proje tion of the a tion by on an element 2 JkV is given by�( � ) = �( k�1)2 = kXj=1� j�+ j�+ Æj2 � j�+ (j � 1)�+ Æj2 � = j�+ Æj2 � j�+ Æj2;withwhi h is zero be ause �+ Æ = ��(�+ Æ) and �� is an isometry. �We turn now the formulae for these operators in terms of a Weyl stru ture. Expli itformulae for the oeÆ ients of various urvature terms for standard operators were �rstfound in the onformal ase [4, 25℄, and later extended to the j1j-graded ase [3, 16℄. A verysurprising fa t was that the formulae were quite universal and did not depend on the spe i� paraboli stru ture or on the highest weights of the representations involved. The generalstru ture of oeÆ ients des ribed in [16℄ was quite ompli ated. Here we noti e that theorganization of the terms produ ed by the Ri i- orre ted derivative leads to mu h simpler oeÆ ients. At the same time, the formulae are extended from the j1j-graded ase to thebroad lass of standard operators in all paraboli geometries with no extra ompli ations:the form of the operator depends only on its order.6.2. Theorem. Let a positive integer k and a long root � with g� � f� be given. De�nedi�erential operators Dk;j (of order j = 0; : : :k), a ting on se tions of any asso iated bundle,by the re urren e relation(6.1) �X Dk;j+1 = DX Æ Dk;j + j(k� j)�(X)Dk;j�1with Dk;0 = id ; Dk;1 = D. Here D be the ovariant derivative given by the hoi e of theWeyl stru ture and � = �12 j�j2rD. Let Dk = Dk;k. Then any invariant operator of order k onstru ted in Theorem 6.1, mapping se tions of V� to se tions of V�, � = �+ k� is givenby � Æ Dk where � is the proje tion onto V�.Proof. We know that the invariant operator is given by � Æ D(k) and we have given are urren e formula for D(k) in se tion 4. Hen e we only have to ompute the proje tionof the a tion of rD on Sjf�V�, whi h is straightforward using the results of x5.2-5.6: we�nd that the a tion is the tensor produ t with rD; where 2 = j�+ j�+ Æj2 � j�+ Æj2.Now, sin e k = �(�+ Æ; �_), we have2 = j�+ j�+ Æj2 � j�+ Æj2 = (2�+ j�+ 2Æ; j�) = j�j2j(�k+ j):Substituting this into the proje tion of the re urren e formula for D(j+1) gives (6.1). �Hen e the universal nature of the expli it formulae for invariant operators arises fromthe fa t that Dk only depends upon k. It is straightforward to ompute Dk for small k.Omitting the tensor produ t sign when tensoring with �j = � � � � �, we haveD1s = DsD2s = D2s + �sD3s = D3s + 2D(�s) + 2�DsD4s = D4s + 3D2(�s) + 4D(�Ds) + 3�D2s+ 9�2s

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 19D5s = D5s+ 4D3(�s) + 6D2(�Ds) + 6D(�D2s) + 4�D3s+ 24D(�2s) + 16�D(�s) + 24�2DsD6s = D6s+ 5D4(�s) + 8D3(�Ds) + 9D2(�D2s) + 8D(�D3s) + 5�D4s+ 45D2(�2s) + 40D(�D(�s)) + 25�D2(�s)+ 64D(�2Ds) + 40�D(�Ds) + 45�2D2s+ 225�3sD7s = D7s+ 6D5(�s) + 10D4(�Ds) + 12D3(�D2s)+ 12D2(�D3s) + 10D(�D4s) + 6�D5s+ 72D3(�2s) + 72D2(�D(�s)) + 120D2(�2Ds) + 60D(�D2(�s))+ 100D(�D(�Ds)) + 36�D3(�s)+ 60�D2(�Ds) + 120D(�2D2s) + 72�D(�D2s) + 72�2D3s+ 720D(�3s) + 432�D(�2s) + 432�2D(�s) + 720�3DsD8s = D8s+ 7D6(�s) + 12D5(�Ds) + 15D4(�D2s)+ 16D3(�D3s) + 15D2(�D4s) + 12D(�D5s) + 7�D6s+ 105D4(�2s) + 112D3(�D(�s)) + 192D3(�2Ds) + 105D2(�D2(�s))+ 180D2(�D(�Ds)) + 84D(�D3(�s)) + 225D2(�2D2s)+ 144D(�D2(�s)) + 49�D4(�s) + 84�D3(�Ds) + 180D(�D(�D2s))+ 105�D2(�D2s) + 192D(�2D3s) + 112�D(�D3s) + 105�2D4s+ 1575D2(�3s) + 1260D(�D(�2s)) + 1344D(�2D(�s)) + 735�D2(�2s)+ 2304D(�3Ds) + 784�D(�D(�s))+ 735�2D2(�s) + 1344�D(�2Ds) + 1260�2D(�Ds) + 1575�3D2s+ 11025�4s:The ombinatori s of the oeÆ ients are simpler than in [16℄ and the numbers are generallysmaller; the formulae there are obtained from those here by expanding the derivatives of� using the produ t rule. Noti e that the oeÆ ients depend only on the position of the�'s, and the oeÆ ients of the nonlinear terms in � are easily omputed as produ ts of the oeÆ ients of the linear terms, as is lear from the indu tive de�nition of ea h Dk.We are still free to hoose the normalization of the Killing form (� ; �): sin e �12 j�j2 isindependent of the long root �, we ould arrange that this is 1 and � = rD. This is thenormalization that gives the formulae stated in the introdu tion for the onformal ase.7. S ope of the onstru tionWe now show that the lass of operators onstru ted in the previous se tion in ludesmany standard invariant operators, at least for the `large' paraboli subgroups o uringin interesting examples. We shall also show that in onformal geometry, the operatorswe onstru t in lude (at least in the onformally at ase) those oming from AdS/CFT orresponden e for partially massless �elds in string theory.

20 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEK7.1. Lagrangian onta t stru ture. Let us onsider the ase of a Lagrangian onta tstru ture (see [10, 31, 33℄). This is the real split ase of the omplex paraboli algebra orresponding to the Dynkin diagram(7.1) . . .� � � ��1 �2 �n �n+1; n > 1:The Lie group is G = PSL(n+2;R) with Lie algebra g = sl(n+2;R), gr g being equippedwith the j2j-grading given by blo k matri es of size 1; n; 1. The g1 part de omposes furtherinto a dire t sum g1 = gL1 � gR1 of two irredu ibles; the g2 part is one-dimensional.In geometri terms, we have a onta t stru ture on a real manifold of dimension 2n + 1with a dire t sum de omposition of the onta t distribution into two Lagrangian subbundles.To be expli it, we onsider the ase n = 3, when the two irredu ible omponents of g�1are three dimensional. Let us denote roots orresponding to both by e1; e2; e3 and f1; f2; f3respe tively, ordered so that e1; f1 are the highest weights for g1, onsidered as a p0-module,and e3; f3 are the lowest ones. Let g be the root orresponding to g�2.� ......................g - �� e3 - �f1-� e2 - � ......................g --e3f1 - �- e3-f1 �f2-� e1 - � e2 -f1 - � e3-f2- �f3-� e1 -f1 - � ......................g --e2f2 - �- e2-f2 � e3-f3-� e1 -f2 - � e2-f3-� ......................g --e1f3 - �- e1-f3The (labelled) Hasse diagram for standard operators is shown above. The labels on thearrows indi ate the `dire tions' � for the orresponding operators. We have onstru ted alloperators indi ated by full arrows, hen e only the horizontal arrows are missing.A similar diagram applies in CR geometry, this being another real form of Lagrangian onta t geometry, ex ept that some representations and operators be ome onjugate.7.2. G2- ase. For a more exoti example, let us onsider the split real ase of the G2 omplex algebra. In this ase, the root system has 12 elements. If we denote simpleroots by �1 (the longer one) and �2 (the shorter one), then the set of positive roots isf�1; �2; �3; �4; �5; �6g with �3 = �1 + �2; �4 = �1 + 2�2; �5 = �1 + 3�2; �6 = 2�1 + 3�2.Let us onsider the ase that the paraboli is a Borel (maximal solvable) subgroup of G2.Then the asso iated graded algebra gr g is j5j-graded withg1 = R � f�1g �R � f�2g; g2 = R � f�3g; g3 = R � f�4g; g4 = R � f�5g; g5 = R � f�6g:

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 21The (labelled) Hasse diagram then has the form� ......................�3 - � ......................�6 - � ......................�4 - � ......................�5 - ���1 - ��2............-� ......................�5 --�1�2.............- � ......................�4 --..................................- �1�2 � ......................�6 --..................................- �1�2 � ......................�3 --..................................- �1�2 �..................................- �1-�2The operators onstru ted in se tion 6, indi ated by full arrows, are now not so numerous.There are two other paraboli subgroups of G2 up to isomorphism, one indu ing a j3j-grading, the other a j2j-grading of the Lie algebra of G2. For the j3j-grading, all roots ing1 are short, and so no operators are onstru ted. For the j2j-grading, we have:g1 = R � f�1g �R � f�3g �R � f�4g �R � f�5g; g2 = R � f�6g:The Hasse graph in this ase is� �1- � ............�3- � ............�6- � ............�4- � �5- �with approximately half of the operators onstru ted in se tion 6.7.3. The onformal ase. In the even dimensional ase, all operators in the BGG se-quen e are obtained by the onstru tion in se tion 6, although there are nonstandardoperators whi h are not. In the odd dimensional ase there is one arrow in the Hassediagram whi h has a spe ial hara ter, as was already noted in [25℄. In dimension 2n� 1,g = so(p; q;R) with p + q = 2n+ 1 and a j1j-grading. We denote positive roots with rootspa es in luded in g1 by ��1; : : :� �n; �n+1; where �n+1 is the short simple root. Supposethat the roots �1; : : :�n are ordered, i.e., that �1 is the highest among them and �n is thesmallest. Then the (labelled) Hasse diagram has the form� �1- � �2- � � � � � �n- � ............�n+1- � ��n- � � � � � ��2- � ��1- �The middle operator is labelled by the only short root, hen e the onstru tion of se tion 6does not apply; however, the other operators are all onstru ted.7.4. AdS/CFT for partially massless �elds. The operators onstru ted in the on-formal ase in lude some operators losely related to the AdS/CFT orresponden e forpartially massless �elds.We re all that, besides massive or stri tly massless �elds, partially massless �elds ofhigher spin were studied on va uum Einstein manifolds with a osmologi al onstant [8℄.In Dolan{Nappi{Witten [18℄, the following situation was onsidered. LetM be an Einsteinmanifold asymptoti to anti-de Sitter spa e of dimension 4 and let X be its boundarywith the indu ed onformal stru ture [27℄. Then the AdS/CFT orresponden e [1℄ yields a orresponden e between a partially massless �eld � on M and a �eld L on X , satisfying a ertain onformally invariant equation ( alled a `partial onservation law' in [18℄).In the ase that the �eld L is a symmetri tra eless tensor �eld Li1:::is , the equation is,to leading order, ri1 � � �ris�nLi1:::is + � � � = 0:

22 DAVID M. J. CALDERBANK, TAMMO DIEMER, AND VLADIM�IR SOU�CEKThe sour e and the target for the equation are easily identi�ed and the operator is thelast operator in the BGG sequen e (des ribed in x7.3 in the odd dimensional ase). Hen ese tion 6 provides an expli it formula for a onformally invariant operator of this form.A detailed study of the ase s = 2; n = 0 in [18℄ leads to the equation on RdrirjLij + 1d� 2RijLij = 0;whi h agrees with the formula of se tion 6 sin e the tra efree parts of 1n�2Rij and � agree.The higher order equations with urvature orre tions in se tion 6 are natural andidatesfor the orresponding higher order equations (the ases with s� n > 2) for the �eld L. Asan example, let us onsider the ase s = 3; n = 0, where we get(7.2) rirjrkLijk + 2d� 2[ri(RjkLijk) + RjkriLijk℄ = 0for a tra eless symmetri tensor �eld with three indi es on Rd. (Of ourse, after expandingthe ovariant derivatives using the Leibniz rule, we obtain the formulae already presentin [16, 25℄.) In the onformally at situation, this must be the equation arising from theAdS/CFT orresponden e, be ause representation theory shows that onformally invariantoperators are unique up to a multiple in this ase. In general, there may be onformallyinvariant urvature orre tions in the lower order terms.Appendix A. Weyl stru tures as redu tionsIn this appendix we relate our approa h to Weyl stru tures and the original approa hof �Cap and Slov�ak [13℄, who de�ne a Weyl stru ture to be a P0-equivariant se tion � of�0 : G ! G0. For this de�nition to make sense, an algebrai Weyl stru ture " must be �xedso that �0 : P ! P0 is split.By means of the equation yq(y) = �(�0(y)) su h a se tion � is equivalent to a P -invariant trivialization q : G ! exp p? of the prin ipal exp p?-bundle �0 : G ! G0, wherethe P -invarian e means that pq(yp)�0(p)�1 = q(y) for all p 2 P; y 2 G.On the other hand, using the algebrai Weyl stru ture ", any geometri Weyl stru tureon M is given by E = (Ad q)" for a unique P -invariant q : G ! exp p?. To summarize:A.1. Proposition. Let (G ! M; �) be a paraboli geometry and �x an algebrai Weylstru ture ". Then there is a natural bije tion between Weyl stru tures E on M and P0-equivariant se tions � of �0 : G ! G0.In our development, we used the Weyl stru ture E to give a P -invariant dire t sumde omposition E� : G �m� p0 �m�! G � g and hen e write(A.1) E�1� Æ � = �m + �p0 + �m� = �m + E�1� Æ �p = �m + E�1� Æ �E + �;where �p and �E are prin ipal P - onne tions on G and � is a P -invariant p?-valued horizontal1-form indu ing the normalized Ri i urvature rD. On the other hand, an algebrai Weylstru ture " gives a �xed dire t sum de omposition "� : m � p0 � m�! g, related to E� by onjugating with the a tion of q, where E = (Ad q)".Sin e the �xed de omposition is only P0-invariant, �Cap and Slov�ak de�ne the Weyl formto be the pull ba k � = ��� of � to G0, then de ompose � into a solder form, a prin ipalP0- onne tion and a P0-invariant p?-valued 1-form. In our approa h (A.1) is a P -invariantlift of this de omposition to G.

RICCI-CORRECTED DERIVATIVES AND INVARIANT OPERATORS 23Appendix B. Dependen e of D and rD on the Weyl stru tureIn equation (4.5), we obtained the (in�nitesimal) dependen e of the Ri i- orre ted Weyl onne tion D(1) on the Weyl stru ture. We now do the same for the Weyl onne tion Dand the normalized Ri i urvature rD (we do not need these results in the body of thepaper, but in lude them for general interest).B.1. Proposition. For 2 C1(M; p?M) and X 2 TM , � rD(X) = �DX + [ ;X ℄Ep?M,where X is lifted to gM and the Lie bra ket is proje ted onto p?M using E .Proof. rD is the p?M -valued 1-form on M indu ed by � = E��+ �m�, where E��y = �E (y)dEyand �m� is shorthand for the m� omponent (E�1� �)m�. Viewing as a P -invariant p?-valuedfun tion on G with � E = � , we easily ompute that � (E��) = �d � [ ; E��℄ and� �m� = (E�1� [ ; �℄)m�� [ ; �m�℄ = [ ; E��p0 ℄ + (E�1� [ ; E��m℄)m�. Hen e� � = �(d � [E��; ℄ + [E��p0 ; ℄) + [ ; �m℄Ep?as required, sin e �E = �E�� + E��p0 is the prin ipal onne tion indu ing D. �B.2. Proposition. Let V be a �ltered P -bundle and ' a se tion of V . Then for 2C1(M; p?M) and X 2 TM , � DX' = ([ ;X ℄EpM;0 +DX ) �'.Proof. D' = E�1V D(1)(EV') and so � DX' = E�1V ([ ;X℄EpM � (EV'))+DX( �')� �DX'.As gr V is semisimple, the result follows. �Sin e D(1)X ' = DX'+ rD(X) �', these omputations are not independent: we he k� (DX'+ rD(X) �') = [ ;X ℄EpM;0 �'+ [ ;X ℄Ep?M �' = [ ;X ℄EpM �'in a ordan e with equation (4.5). Unlike D(1), the Weyl onne tion D does not dependalgebrai ally on the Weyl stru ture. This, of ourse, was the whole reason for introdu ingRi i orre tions in the �rst pla e. Referen es[1℄ O. Aharony, S. Gubser, J. Malda ena, H. Ooguri and Y. Oz, Large N �eld theories, string theory andgravity, Phys. Rep. 323 (2000) 183{386.[2℄ T. N. Bailey, M. G. Eastwood and A. R. Gover, Thomas's stru ture bundle for onformal, proje tiveand related stru tures, Ro ky Mountain J. Math. 24 (1994) 1191{1217.[3℄ R. J. Baston, Almost hermitian symmetri manifolds, I Lo al twistor theory, II Di�erential invariants,Duke Math. J. 63 (1991) 81{138.[4℄ R. J. Baston, Verma modules and di�erential onformal invariants, J. Di�. Geom. 32 (1990) 851{898.[5℄ R. J. Baston and M. G. Eastwood, The Penrose Transform, Oxford University Press, Oxford (1989).[6℄ N. Bourbaki, Groupes et Algebres de Lie, Herman, Paris, 1975, Chapt. VIII, Exer ises, par. 9, no. 14.[7℄ T. P. Branson, Se ond order onformal ovariants, Pro . Amer. Math. So . 126 (1998) 1031{1042.[8℄ I. Bu hbinder, D. Gitman and B. Pershin, Causality of massive spin 2 �eld in external gravity, Phys.Lett. B 492 (2000) 161{170.[9℄ D. M. J. Calderbank and T. Diemer, Di�erential invariants and urved Bernstein{Gelfand{Gelfand se-quen es, J. reine angew. Math. 537 (2001) 67{103.[10℄ A. �Cap, Corresponden e spa es and twistor spa es for paraboli geometries, ESI preprint 989 (2001).[11℄ A. �Cap and A. R. Gover, Tra tor al uli for paraboli geometries, Trans. Amer. Math. So . 354 (2002)1511{1548.[12℄ A. �Cap and H. S hi hl, Paraboli geometries and anoni al Cartan onne tions, Hokkaido Math. J. 29(2000) 453{505.[13℄ A. �Cap and J. Slov�ak, Weyl stru tures for paraboli geometries, ESI Preprint 801 (1999).

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