ARCHIVUM MATHEMATICUM open parenthesis BRNO closing parenthesis \ centerlineTomus 47 open{ARCHIVUM parenthesis MATHEMATICUM 201 1 closing parenthesis ( BRNO ) comma} 309 endash 327 ELLIPTICITY .. OF .. THE .. SYMPLECTIC .. TWISTOR COMPLEX \ centerlineSvatopluk Kryacutesl{Tomus 47 ( 201 1 ) , 309 −− 327 } Abstract period .. For a Fedosov manifold open parenthesis symplectic manifold equipped with a \ centerline {ELLIPTICITY \quad OF \quad THE \quad SYMPLECTIC \quad TWISTOR COMPLEX } symplectic torsion hyphen free affine connectionARCHIVUM closing MATHEMATICUM parenthesis (admitting BRNO ) a metaplectic structure comma we shall investigate two sequences of first orderTomus differenti 47 ( 201 a-l 1 ) operators , 309 – 327 c-a t ing \ centerlineon sections of{ SvatoplukELLIPTICITY certain infinite Kr rank\ ’{ y vector} OFs l } bundles THE defined SYMPLECTIC over this manifold TWISTOR period COMPLEX The differential operators are symplectic analoguesSvatopluk of the twistor Krysl´ operators \ centerline { Abstract . \quad For a Fedosov manifold ( symplectic manifold equipped with a } known from RiemannianAbstract or Lorentzian . For spin a Fedosov geometry manifold period ( symplectic It is known manifold that equipped the with a mentioned sequences formsymplectic complexes torsion if the - free symplectic affine connection connection ) admitting is of a Ricci metaplectic structure , \ centerline { symplectic torsion − free affine connection ) admitting a metaplectic structure , } type period In this paperwe shall comma investigate we prove two sequences that certain of first part order s differenti of thesea complexes− l operators arec elliptic− a t ing period 1 period .. Introduction on sections of certain infinite rank vector bundles defined over this manifold . \ centerline {we shall investigate two sequences of first order differenti $ a−l $ operators $ c−a $ In this article comma we proveThe differential the ellipticity operators of are cert symplectic ain parts analogues of the so ofthe called twistor symplectic operators t ing } twistor complexes periodknown The symplectic from Riemannian twistor or Lorentziancomplexes spin are geometry two sequences . It is known of first that the order differential operatorsmentioned defined sequences over Ricci form type complexes Fedosov if the manifolds symplectic admitting connection is of Ricci \ centerline {on sections of certain infinite rank vector bundles defined over this manifold . } a metaplectic structuretype period . In thisThe paper mentioned , we prove parts that of certain these part complexes s of these will complexes be called are elliptic . truncated symplectic twistor complexes and will1 . be definedIntroduction later in this text period \ centerlineNow commaIn{The let this us article differential say a few, we words prove the about operators ellipticity the Fedosov of are cert manifolds ainsymplectic parts of period theanalogues so Formally called symplectic speaking of the twistor comma twistor complexes operators } a Fedosov. The manifold symplectic is a triple twistor open complexes parenthesis are two Mto sequences the power of first of 2 order l comma differential omega operators comma nabla defined closing over parenthesis where open\ centerline parenthesisRicci{known M type to theFedosov from power Riemannian manifolds of 2 l comma admitting or omega Lorentzian a closing metaplectic parenthesis spin structure geometry is a open . The parenthesis . mentioned It is for known parts definiteness of that these the } 2 l dimensionalcomplexes closing will beparenthesis called truncated symplectic symplectic manifold twistor and nabla complexes is a symplectic and will be torsion defined hyphen later in free this affine text \ centerlineconnection. period{ mentioned Connections sequences satisfying form these complexestwo properties if are the usually symplectic called Fedosov connection is of Ricci } connections in honor of BorisNow Fedosov , let us who say used a few them words to obtain about the a deformation Fedosov manifolds . Formally speaking , \ centerlinequantizationa Fedosov{ fortype symplectic manifold . In this manifolds is a triple paper period (M 2 ,l, ω,we open∇ prove) whereparenthesis that(M 2l, See ω certain) is Fedosov a ( for part open definiteness square s of bracket these 5 complexes closing square are bracket elliptic period . } closing parenthesis2l dimensional Let us also ) symplectic mention that manifold and ∇ is a symplectic torsion - free affine connection . Con- \ centerlinein contrarynections{ to1 torsion . satisfying\quad hyphenIntroduction these free two Levi properties hyphen} Civita are usually connections called commaFedosov the connections Fedosov onesin are honor not unique of Boris period We referFedosov an interested who used reader them to to Tondeur obtain open a deformation square bracket 1 8 closing square bracket and Gelfand comma Retakh comma ShubinIn this open articlequantization square bracket , wefor symplectic6 prove closing the square manifolds ellipticity bracket . ( for See Fedosov of cert [ 5 ] ain . ) Let parts us also of mention the so that called in contrary symplectic twistormore informationto complexes torsion period - free . Levi The - Civitasymplectic connections twistor , the Fedosov complexes ones are are not two unique sequences . of first orderTo formulate differentialWe refer the result an interested on operators the ellipticity reader defined to of Tondeur the truncated over [ 1 Ricci 8 symplectic ] and type Gelfand twistor Fedosov , Retakh manifolds , Shubin [ admitting6 ] for more acomplexes metaplecticinformation comma structure one . should know . The some mentioned basic facts onparts the structure of these of the complexes curvature will be called truncatedtensor fieldTo symplecticof a formulate Fedosov connection the twistor result on period complexes the Inellipticity Vaisman and of open will the truncatedsquare be defined bracket symplectic 1 later 9 closing twistor in square this complexes bracket text , commaone . one can find a proof of a should know some basic facts on the structure of the curvature tensor field of a Fedosov connection \ hspacetheorem∗{\. which Inf i Vaisman l l says}Now that [ , 1 such 9let ] , curvature one us can say find atensor few a proof field words of splits a theorem about into two whichthe parts Fedosov says if l greaterthat suchmanifolds equal curvature 2 comma . tensor Formally field speaking , namely intosplits the into symplectic two parts Ricci if l ≥ and2, namely symplectic into Weyl the symplectic curvature Riccitensor and fields symplectic period If Weyl curvature tensor \noindenthline fieldsa Fedosov . If manifold is a triple $ ( Mˆ{ 2 l } , \omega , \nabla ) $ where $ (20 10 M Mathematics ˆ{ 2 l Subject} , Classification\omega :) primary $ is 22 a E ( 46 for semicolon definiteness secondary 53 C 7 comma 53 C 80 comma 58 J 5 period Key words and phrases : Fedosov manifolds comma Segal hyphen Shale hyphen Weil represent at ion comma Kost ant quoteright s\noindent spinors comma$ 2 l $ dimensional ) symplectic manifold and $ \nabla $ is a symplectic torsion − f r e e a f f i n e connectionelliptic complexes . Connections period satisfying these two properties are usually called Fedosov connectionsThe author of thisin20 10 honor articleMathematics was of supported Boris Subject Classification Fedosov by the grant who: primary GA used Ccaron 22 them E 46 R ; secondary 201 to slash obtain 53 8 C slash 7 a , 53 deformation 397 C 80 of , the 58 J Grant 5 . Agency of Czech RepublicKey words period and phrases The work: Fedosov is a part manifolds of the , Segal research - Shale project - Weil representMSM 21 at 620839 ion , Kost financed ant ’ s spinors by , elliptic \noindentcomplexesquantization . for symplectic manifolds . ( See Fedosov [ 5 ] . ) Let us also mention that M Scaron MT Ccaron R period The author thanks to Ondrcaronejˇ Kalenda for a discussion period inReceived contrary JulyThe 8to author comma torsion of 201this article1− periodf r was e e Editor supported Levi J− period byCivita the Slovaacutekgrant connections GA C periodR 201 / 8 , / the397 of Fedosov the Grant Agency ones areof Czech not unique . Republic . The work is a part of the research project MSM 2 1 620839 financed by M Sˇ MT Cˇ R . The author \noindentthanksWe to refer OndˇrejKalenda an interested for a discussion reader . to Tondeur [ 1 8 ] and Gelfand , Retakh , Shubin [ 6 ] for more information . Received July 8 , 201 1 . Editor J . Slov´ak.

To formulate the result on the ellipticity of the truncated symplectic twistor complexes , one should know some basic facts on the structure of the curvature tensor field of a Fedosov connection . In Vaisman [ 1 9 ] , one can find a proof of a theorem which says that such curvature tensor field splits into two parts if $ l \geq 2 , $ namely into the symplectic Ricci and symplectic Weyl curvature tensor fields . If

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline {20 10 Mathematics Subject Classification : primary 22 E 46 ; secondary 53 C 7 , 53 C 80 , 58 J 5 . }

Key words and phrases : Fedosov manifolds , Segal − Shale − Weil represent at ion , Kost ant ’ s spinors , elliptic complexes .

The author of this article was supported by the grant GA $ \check{C} $ R 201 / 8 / 397 of the Grant Agency of Czech Republic . The work is a part of the research project MSM 2 1 620839 financed by M $ \check{S} $ MT $ \check{C} $ R . The author thanks to Ond\v{ r } ej Kalenda for a discussion .

\ centerline { Received July 8 , 201 1 . Editor J . Slov \ ’{ a}k . } 3 1 0 .. S period KR Yacute SL \noindentl = 1 comma3 1 only 0 \ thequad symplecticS . KR Ricci $ curvature\acute{Y tensor} $ field SL occurs period Fedosov manifolds with zero symplectic Weyl curvature are usually called of Ricci type period open parenthesis See also \noindentCahen comma$ lSchwachhodieresisfer = 1 , $ open only square the bracket symplectic 3 closing Ricci square bracketcurvature for another tensor but field related contextoccurs period . Fedosov closing manifolds parenthesiswith zero symplectic Weyl curvature are usually called of Ricci type . ( See also Cahen , Schwachh\”{o} fer [ 3 ] for another but related context . ) After introducing3 1 0 S . the KR underlyingY´ SL geometric structure comma let us st art describing the fields onl which= 1, only the the differential symplectic operators Ricci curvature from the tensor symplectic field occurs twistor . complexes Fedosov manifolds with zero symplectic Afteract period introducingWeyl These curvature fields the are are underlyingcertain usually exterior called geometric of differential Ricci type forms structure . ( See with also values Cahen , let in the , us Schwachh¨ofer[ so st called art describing 3 ] for another the fieldssymplectic onbut spinor which related bundle contextthe which differential . ) is an associated operators vector bundle from to the the symplecticmetaplectic twistor complexes bundle periodAfter We introducing shall introduce the underlying the metaplectic geometric bundle structure briefly now , let period us st art Because describing the first the fields on which \noindenthomotopytheact group differential . of These the operatorssymplectic fields from group are the Sp certain symplectic open parenthesis exterior twistor complexes2 l commadifferential R closing formsparenthesis with is isomorphic values in to Zthe comma so calledthere exists act . These fields are certain exterior differential forms with values in the so called \noindenta connectedsymplecticsymplectic two hyphen spinor fold spinor bundle covering which bundle of is this an which group associated period is vector an .. Theassociated bundle covering to the space vectormetaplectic is called bundle the bundle to. the We shall metaplectic bundlemetaplectic .introduce We group shall the comma metaplectic introduce and it is bundle usually the metaplectic briefly denoted now by . Mp Because bundle open theparenthesis briefly first homotopy 2 nowl comma group. Because R closing of the symplectic theparenthesis first period Let us fix homotopy group of the symplectic group Sp $ ( 2 l , R ) $ is isomorphic to $Z , $ an element ofgroup Sp (2l, R) is isomorphic to Z, there exists a connected two - fold covering of this group . The there exists the isomorphismcovering spaceclass of is all called connected the metaplectic 2 : 1 coverings group of , andSp open it is parenthesis usually denoted 2 l comma by Mp R(2 closingl, R). parenthesisLet us fix and denote it by a connected two − fold covering of this group . \quad The covering space is called the lambda periodan element of the isomorphism class of all connected 2 : 1 coverings of Sp (2l, R) and denote it by metaplectic group , and it is usually denoted byMp $ ( 2 l , R ) . $ Let us fix an element of In particularλ. In comma particular the mapping, the mapping lambdaλ :Mp open(2l, R parenthesis) → Sp (2 2l, lR comma) is a Lie R closing group parenthesis homomorphism right , arrow and in Sp open parenthesis 2the l comma isomorphism Rthis closing case itparenthesis is class also a ofLie is a group all Lie group connected representation homomorphism 2 . : A 1 metaplectic coverings comma structure of Sp on $ ( 2 l , R ) $ and denote it by $ \andlambda in thisa symplectic case. $ it is alsomanifold a Lie (M group2l, ω representation) is a notion parallel period to A that metaplectic of a spin structure knownon from Riemannian In particular , the mapping $ \lambda :$Mp$( 2 l , R ) \rightarrow $ Sp a symplecticgeometry manifold . In open particular parenthesis , one Mof toits the part power is a principalof 2 l commaMp omega(2l, R) closing− bundle parenthesisQ covering is a twice notion the parallel to that of a $( 2 l , R )$ is aLie grouphomomorphism , spin structurebundle of symplectic repe ` res P on (M, ω). This principal Mp (2l, R)− bundle is the mentioned andknown in from thismetaplectic Riemannian case bundle it geometry is and also will period abe Lie denoted In group particular by Q representationin comma this paper one . of its part. A is metaplectic a principal structure on Mp open parenthesis 2 l commaAs R we closing have already parenthesis written hyphen , the bundle fields we Q are covering interested twice in the are bundle certain of exterior symplectic rep egrave res P\noindent on open parenthesisdifferentiala symplectic Mforms comma on manifoldM omega2l with closing values $ (parenthesis in the Mˆsymplectic{ period2 l spinor} , bundle\omegawhich is) a $vector is bundle a notion over parallel to that of a spin structure known from Riemannian geometry . In particular , one of its part is a principal This principalM associated Mp open to parenthesis the chosen 2 principal l commaMp R closing(2l, R parenthesis)− bundle hyphenQ via an bundle ’ analytic is the derivate mentioned ’ of metaplectic the bundle and willMp be $ (Segal 2 - Sahle l - , Weil R representation ) − $ . The bundle Segal - $ Shale Q $ - Weil covering re - presentation twice the is a bundle faithful unitary of symplectic rep $ \grave{e} $ res $P$ on $( M , \omega ) . $ denotedrepresentation by Q in this paper of the period metaplectic group Mp (2l, R) ThisprincipalMp $(2 2 l , R ) − $ bundle is the mentioned metaplectic bundle and will be As we haveon the already vector written space commaL (L) the of complexfields we valuedare interested square in Lebesgue are certain integrable exterior functions defined on a denoted by $Q$ in this paper . differentialLagrangian forms on subspace M to theL powerof the of canonical 2 l with values symplectic in the vector symplectic space spinor bundle which is a 2l vector bundle(R , ω over0). For M associatedtechnical reasons to the chosen , we shall principal use the Mp so open called parenthesis Casselman 2 - l Wallach comma R glo closing - balization parenthesis of hyphen bundle Q\ hspace via an ∗{\thef iunderlying l l }As we Harish have - Chandraalready (g written, K˜ )− module , the of the fields Segal - we Shale are - Weil interested in are certain exterior quoterightrepresentation analytic derivate . Here quoteright, g is the Lie of the algebra Segal of hyphen the metaplectic Sahle hyphen group WeilG˜ and representationK˜ is a maximal period compact The Segal hyphen Shale \noindent differential forms on $ M ˆ{ 2 l }$ with values in the symplectic spinor bundle which is a hyphen Weilsubgroup re hyphen of the group G˜. The vector space carrying this globalization is the Schwartz space S vector bundle over $M$ associated to the chosen principal Mp $ ( 2 l , R ) − $ presentation:= S( isL) a of faithful smooth unitary functions representation on L rapidly of decreasing the metaplectic in infinity group with Mp its open usual parenthesis Fr´echet topology 2 l comma . This R closing parenthesis bundleon the vector $Q$Schwartz space via space L anto is the the power of 2 open parenthesis L closing parenthesis of complex valued square Lebesgue integrable ’ analytic derivate ’ of the Segal − Sahle − Weil representation . The Segal − Shale − Weil re − functions ’ analytic derivate ’ mentioned above . We shall denote the resulting representation of Mp (2n, R) presentation is a faithful unitary representation of the metaplectic group Mp $ ( 2 l , defined onon aS Lagrangianby L and call subspace it the Lmetaplectic of the canonical representation symplectic, i vector . e . , spacewe have L : Mp (2l, R) → Aut ( S ) R ) $ open parenthesis. Let us mention R to the that powerS ofdecomposes 2 l comma into omega two sub irreducible 0 closingMp parenthesis(2l, R)− periodsubmodules For technicalS+ and reasonsS−, i comma . we shall use the so callede Casselman. , S = hyphenS+ ⊕ WallachS−. The glo hyphen elements of S are usually called symplectic spinors . See Kostant \noindentbalization[ 1on of 1 the]the who underlying used vector them Harish space in the hyphen context $ L Chandra ˆ of{ geometric2 } open( quantization parenthesis L ) $ g . comma of complex K-tilde valued closing parenthesis square Lebesgue hyphen module integrable of functions thedefined Segal hyphen on a Shale Lagrangian hyphen Weil subspaceThe underlying $ L algebraic $ of the structure canonical of the symplectic symplectic spinor vector valued exterior space V• 2l ∗ L2l Vr 2l ∗ representationdifferential period forms Here is commathe vector g is space the LieE algebra:= of( theR ) metaplectic⊗ S = groupr=0 G-tilde(R ) and⊗ S tilde-K. Obviously is \noindenta maximal, this compact$ vector ( R subgroup space ˆ{ 2 is equipped of l the} group with, G-tilde the\omega following sub period{ tensor0 } .. The product) vector . $representation space For carrying technicalρ of this the reasonsmetaplectic , we shall use the so called Casselman − Wallach g l o − balization of the underlying Harish − Chandra $( g , \ tilde {K} ) − $ module of the Segal − Shale − Weil globalizationgroup isMp the(2 Schwartzl, R). For spacer = S0,: ..., =2 Sl, openg ∈ Mp parenthesis(2l, R) L closing parenthesis of smooth functions on L rapidly Vr 2l ∗ ∗∧r decreasingand inα infinity⊗ s ∈ with( R its) ⊗ usualS , Freacutechet we set ρ(g)(α topology⊗ s) := periodλ(g) Thisα ⊗ L Schwartz(g)s and space extend is this the prescription by \noindentquoterightlinearityrepresentation analytic . With derivate this quoteright notation . Here in mentioned mind$ , , the g above $ symplecticperiod is the We spinor Lie shall valued algebra denote exterior the of resultingthe differential metaplectic representation forms are group $ \ tilde {G} $ andof Mp $ \ opentildesections parenthesis{K of} the$ vector 2 i s n comma bundle RE closingassociated parenthesis on S by L and call it the metaplectic representation comma i period e perioda maximal comma we compact have subgroup of the group $ \ tilde {G} { . }$ \quad The vector space carrying this globalizationL : Mp open parenthesis is the 2 Schwartzl comma R closing space parenthesis S $ : right = arrow S Aut ( open L parenthesis ) $ of S smooth closing parenthesis functions period on Let $L$ us mentionr a p i d l y that S decomposes into two irreducible decreasingMp open parenthesis in infinity 2 l comma with R closing its usual parenthesis Fr \ ’{ hyphene} chet submodules topology S sub . This plus .. Schwartz and S sub space minus comma is the i period e period comma .. S .. = S sub plus oplus S sub minus period .. The elements of S are \noindentusually called’ analytic symplectic derivate spinors period ’ mentioned See Kostant above open square . We bracket shall 1 denote 1 closing the square resulting bracket who representation used them in the contextofMp $ ( 2 n , R ) $ onSby $L$ and call it the metaplectic representation , i . e . ,wehave $Lof geometric :$Mp$( quantization period 2 l , R ) \rightarrow $ Aut ( S ) . Let us mention that S decomposes into two irreducible MpThe $ underlying ( 2 algebraic l , structure R ) of the− $ symplectic submodules spinor valued $ S exterior{ + }$ \quad and $ S { − } , $ i . e . , \quad S \quad $ =differential S { + forms}\ isoplus the vectorS space{ −E .. } : =. bigwedge $ \quad toThe the power elements of bullet of open S are parenthesis R to the power of 2 l closing parenthesisusually to called the power symplectic of * oslash S spinors .. = bigoplus . See sub rKostant = 0 to the [ power 1 1 of ] 2who l bigwedge used them to the inpower the of r context open parenthesis R to theof power geometric of 2 l closing quantization parenthesis to . the power of * oslash S period .. Obviously comma this vector space is equipped with the following tensor product \ hspacerepresentation∗{\ f i l l rho}The of the underlying metaplectic algebraic group Mp open structure parenthesis of 2 thel comma symplectic R closing parenthesis spinor valued period For exterior r = 0 comma period period period comma 2 l comma g in Mp open parenthesis 2 l comma R closing parenthesis \noindentand alpha oslashdifferential s in bigwedge forms to the is power the of vector r sub open space parenthesis E \quad R to the$ : power = of 2\ lbigwedge closing parenthesisˆ{\ bullet to the power} ( of R* oslash ˆ{ 2 S comma l } we) set ˆ{ rho ∗ } open \otimes parenthesis$ g S closing\quad parenthesis$ = open\bigoplus parenthesisˆ{ alpha2 oslash l } { s closingr = parenthesis 0 }\ :bigwedge = lambda ˆ{ r } open( Rparenthesis ˆ{ 2 g l closing} ) parenthesis ˆ{ ∗ } \tootimes the power$ of * and r alpha oslash L open parenthesis g closing parenthesis s and extend S.this\ prescriptionquad Obviously by linearity , this period vector With this space notation is in equipped mind comma with the symplecticthe following spinor tensor product representationvalued exterior differential $ \rho forms$ are ofthemetaplecticgroupMp sections of the vector bundle E associated $( 2 l , R ) .$ For $r =0,...,2l,g \ in $Mp$( 2 l , R )$

\noindent and $ \alpha \otimes s \ in \bigwedge ˆ{ r } { ( } R ˆ{ 2 l } ) ˆ{ ∗ } \otimes $ S , we s e t $ \rho ( g ) ( \alpha \otimes s ) : = \lambda ( g ) ˆ{ ∗ \wedge r }\alpha \otimes L ( g ) s$ andextend this prescription by linearity . With this notation in mind , the symplectic spinor valued exterior differential forms are sections of the vector bundle $ E $ associated ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 3 1 1 \ hspaceto the∗{\ chosenf i l principal l }ELLIPTICITY Mp open parenthesis OF THE SYMPLECTIC 2 l comma R closingTWISTOR parenthesis COMPLEX hyphen\quad bundle3 1 Q 1 via rho comma i period e period comma E : = Q times sub rho E period Now comma we \noindentshall restricttothechosen our attention to principalMpthe mentioned specific $( symplectic 2 l spinor , valued R ) − $ bundle $Q$ via $ \rho ,$i.e$.exterior differential forms , period E : For each=Q r = 0 comma\times period{\ periodrho }$ period E .comma Now 2 , l we comma there exists a distinguished shall restrict our attention to the mentioned specific symplectic spinor valued irreducible submodule of bigwedge to the power ofELLIPTICITY r sub open OF parenthesis THE SYMPLECTIC R to the TWISTOR power of COMPLEX 2 l closing parenthesis 3 1 1 to the power of * oslash Sto sub the plusminux chosen principal which weMp denote(2l, R by)− Ebundle sub plusminuxQ via ρ, i to . thee ., E power:= Q of×ρ periodE . Now to the , we power shall of restrict r Actually comma the \noindentsubmodulesourexterior attentionE sub plusminux to differentialforms.Foreach the mentioned to the powerspecific of r aresymplectic the Cartan spinor components $r valued = of bigwedge 0 , to . the power . of . r open , parenthesis 2 l R ,$ to thethere power exists ofexterior 2 l closing a differential distinguished parenthesis forms to . the For power each r of= * 0 oslash, ..., 2l, Sthere sub plusminux exists a distinguished comma i period e period comma the highest Vr 2l ∗ r weight ofirreducible each of them submodule is the largest of R one) of⊗ theS± highestwhich we weights denote of by allE irreducible±. Actually , the \noindent irreducible submodule( of $ \bigwedge ˆ{ r } { ( } R ˆ{ 2 l } ) ˆ{ ∗ } \otimes constituentssubmodules of bigwedgeEr are to the the power Cartan of components r open parenthesis of Vr( R2l) to∗ ⊗ theS power, i . e of . ,2 the l closing highest parenthesis weight of to each the of power of * oslash S S {\pm }$ which± we denote by $ E ˆ{ r } {\Rpm ˆ{ ±. }}$ Actually , the sub plusminuxthem wrt is period the largest the st one andard of the choices highest period weights For of r all = irreducible 0 comma period period period comma 2 l comma we set E toconstituents the power of of rV :r =( E2l) sub∗ ⊗ plusS wrt to the . the power st andard of r oplus choices E sub . For minusr = 0 to, ..., the2l, powerwe set ofE rr and:= E Er to⊕ E ther power of r : = Q \noindent submodules $R E ˆ{ r± } {\pm }$ are the Cartan components of $ \bigwedge+ − ˆ{ r } ( times sub rho Row 1 r Row 2 periodr . comma let us denote the corresponding R ˆ{ 2and l }Er ):= ˆQ{× ∗E } \Fotimes urther , letS us denote{\pm the} corresponding, $ i . e . , the highest Mp open parenthesis 2ρ l comma. R closing parenthesis hyphen equivariant projection from bigwedge to the power of r open weight of each of them is the largest one of the highest weights of all irreducible parenthesis R to the power of 2 l closing parenthesis toVr the2 powerl ∗ of * oslashr S ontor E to the power of r by p to the power of r Mp (2l, R)− equivariant projection from (R ) ⊗ S onto E by p . We denote the lift of the period We denote the r \noindentprojectionconstituentsp to the of associated $ \bigwedge ( or ’ geometricˆ{ r ’} ) structures( R ˆ by{ the2 same l } ) ˆ{ ∗ } \otimes S {\pm }$ lift of the projection p to ther power of r toV ther 2 associatedl ∗ open parenthesisr or quoteright geometric quoteright closing parenthesis wrt.thestandardchoices.Forsymbol , i . e ., p : Γ(M,Q ×ρ ( (R $r) ⊗ S =)) → 0Γ(M,E ,). . . , 2 l ,$ we structures by theNow same , we are in a position to define the main subject of our investigation , namely the symplectic s esymbol t $ E comma ˆ{ r i period} : e period = E comma ˆ{ r p} to{ the+ power}\ ofoplus r : CapitalE Gammaˆ{ r } open{ − parenthesis }$ and M $ comma\ l e f t . Q E times ˆ{ subr } rho: open = Q \timestwistor{\ complexesrho } E .\ begin Let us{ considerarray }{ ac Fedosov} r \\ manifold. \end (M,{ array ω, ∇) } Further \ right . , $ let us denote the corresponding parenthesis bigwedge to the power of r open parenthesis R to the power of 2S l closing parenthesis to the power of * oslash S closing and suppose that (M, ω) admits a metaplectic structure . Let d∇ be the exterior covariant derivative parenthesis closing parenthesis right arrow Capital Gamma open parenthesis M comma E to the power of r closing parenthesis \noindentassociatedMp $ ( to ∇. 2For each lr ,= 0, R ..., 2l, )let us− restrict$ equivariant the projection from $ \bigwedge ˆ{ r } ( period S associated exterior covariant derivative d∇ to Γ(M,Er) and compose the restriction R ˆNow{ 2 comma l } we) are ˆ{ in ∗a position } \otimes to define$ the S main onto subject $ E of ˆ{ ourr investigation}$ by $ comma p ˆ{ namelyr } . $ We denote the with the projection pr+1. The resulting operator , denoted by T , will be called liftthe symplectic of the projection twistor complexes $ p period ˆ{ r Let}$ us consider to the a associated Fedosov manifoldr ( or open ’ geometricparenthesis M ’ comma ) structures omega comma by nablathe same symplectic twistor operator . In this way , we obtain two sequences of differential operators , namely closing parenthesis 0 → Γ(M,E0)T 0 Γ(M,E1)T 1 ···T l−1Γ(M,El) → 0 and 0 → Γ(M,El) T l Γ(M,El+1)T l+ \noindentand supposesymbol,i.e that open→ parenthesis $. M comma→ , pˆ omega→{ r closing} : parenthesis\Gamma admits(M,Q a metaplectic→ structure\times period→ {\ Letrho d} to the( 1 ···T 2l −1Γ(M,E2l) → 0. It is known , see Kry´− sl [ 1 4 ] , that these sequences form complexes power\bigwedge of nablaˆ{ tor the→} power( of R S ˆ be{ the2 exterior l } ) ˆ{ ∗ } \otimes $ S $ ) ) \rightarrow \Gamma ( provided the Fedosov manifold Mcovariant , E derivative ˆ{ r } associated) . $ to nabla period For each r = 0 comma period period period comma 2 l comma let us restrict the (M 2l, ω, ∇) is of Ricci type . These two complexes are the mentioned symplectic twistor complexes associated exterior covariant derivative d to the power of nabla to the power of S to Capital Gamma open parenthesis M comma . Let us notice , that we did not choose the full sequence of all NowE to the , we power are of inr closing a position parenthesis to and define compose the the main restriction subject of our investigation , namely symplectic spinor valued exterior differential forms together with the exterior thewith symplectic the projection twistor p to the power complexes of r plus . 1 Let period us The consider resulting operator a Fedosov comma manifold denoted by $ T ( sub Mr comma , will\omega be called , covariant derivative acting between them because for a general or even Ricci type Fedosov manifold \nablasymplectic) twistor $ operator period In this way comma we obtain two sequences of differential , this sequence would not form a complex in general . operators comma namely 0 right arrow Capital Gamma open parenthesis M comma E to the power of 0 closing parenthesis T As we have mentioned , we shall prove that some parts of these two complexes are elliptic . To right\noindent arrow 0 Capitaland suppose Gamma openthat parenthesis $ ( M M comma , \ Eomega to the power) $ of 1 admits closing parenthesis a metaplectic T right arrow structure 1 times times . Let times $ d ˆ{\nabla ˆ{ S }}$ obtain these parts , one should remove the last ( i . e . , the zero ) term and the second last term from Tbe right the arrow exterior l minus 1 Capital Gamma open parenthesis M comma E to the power of l closing parenthesis right arrow 0 and the first complex and the first term ( the zero space again ) from the second complex . The complexes covariant0 right arrow derivative Capital Gamma associated open parenthesis to M $ comma\nabla E to the.$Foreach$r power of l closing parenthesis = T 0 right , arrow . l Capital . .Gamma , obtained in this way will be called t runcated symplectic twistor complexes . Let us mention that by open2 parenthesis l ,$ M let comma us restrict E to the power the of l plus 1 closing parenthesis T right arrow l plus 1 times times times T right arrow 2 an elliptic complex , l minus 1 Capital Gamma open parenthesis M comma E to the power of 2 l closing parenthesis right arrow 0 period It is known we mean a complex of differential operators such that it s associated symbol sequence is an exact comma\noindent see associated exterior covariant derivative $ d ˆ{\nabla ˆ{ S }}$ to $ \Gamma (M sequence of the sheaves in question . ( See , e . g . , Wells [ 2 1 ] for details . ) ,Kryacute E ˆ{ r to} the) power $ and of minus compose sl open the square restriction bracket 1 4 closing square bracket comma that these sequences form complexes Let us make some remarks on the methods we have used to prove the ellipticity provided the Fedosov manifold of the truncated symplectic twistor complexes . We decided to use the so called Schur - Weyl - Howe \noindentopen parenthesiswith the M to projection the power of 2 $ l pcomma ˆ{ r omega + comma 1 } nabla. $ closing The resulting parenthesis is operator of Ricci type , denoted period These by two $ T { r } correspondence , which is referred to as the Howe correspondence complexes, $ will are be the called mentioned symplectic for simplicity in this text . The Howe correspondence in our case , i . e . , for the metaplectic twistor complexes period Let us notice comma that we did not choose the full sequence of all group Mp (2l, ) acting on the space E of symplectic spinor valued exterior forms , leads to the \noindentsymplecticsymplectic spinor valuedR exterior twistor differential operator forms . togetherIn this with way the , exterior we obtain two sequences of differential ortho - symplectic super Lie algebra osp(1 | 2) and a certain representation of this algebra on E . operatorscovariant derivative , namely acting $ between 0 \rightarrow them because for\ aGamma general or( even M Ricci , type E ˆ{ 0 } )T {\rightarrow } We decided to use the Howe type correspondence mainly because the spaces Er( defined above ) can 0 Fedosov\Gamma manifold( comma M ,this sequence E ˆ{ 1 would} )T not form{\ a complexrightarrow in general} period1 \cdot \cdot \cdot T {\rightarrow } be characterized via the mentioned representation of osp(1 | 2) easily and in a way described in this l As− we have1 mentioned\Gamma comma( M we shall , prove E ˆ that{ l some} ) parts\ ofrightarrow these two complexes0 $ and paper . See R . Howe [ 1 0 ] for more information on the Howe type correspondence in general . Let $are 0 elliptic\rightarrow period To obtain\Gamma these parts( comma M one , should E ˆ{ removel } the)T last open{\ parenthesisrightarrow i period} el period\Gamma comma the(M zero us also mention that besides this duality , the Cartan lemma on closing, E parenthesis ˆ{ l + term 1 } )T {\rightarrow } l + 1 \cdot \cdot \cdot T {\rightarrow } 2and l the− second1 last\Gamma term from the( first M complex , E and ˆ{ the2 first l term} ) open\ parenthesisrightarrow the zero0 space . $ It is known , see $again Kr closing\acute parenthesis{y} ˆ{ − from }$ the sl second [ 1 4 complex ] , that period these The complexes sequences obtained form in thiscomplexes way will be provided called the Fedosov manifold t runcated symplectic twistor complexes period Let us mention that by an elliptic complex comma \noindentwe mean a complex$ ( M of ˆdifferential{ 2 l operators} , such\omega that it s, associated\nabla symbol) sequence $ is of Ricci type . These two complexes are the mentioned symplectic twistoris an exact complexes sequence of . the Let sheaves us noticein question , period that openwe did parenthesis not choose See comma the e full period sequence g period comma of all Wells open square bracket 2 1 closing square bracket for details period closing parenthesis \noindentLet us makesymplectic some remarks spinor on the methods valued we exterior have used differential to prove the ellipticity forms together with the exterior of the truncated symplectic twistor complexes period We decided to use the so called \noindentSchur hyphencovariant Weyl hyphen derivative Howe correspondence acting between comma which them is becausereferred to for as the a Howe general correspondence or even Ricci type Fedosovfor simplicity manifold in this text , this period sequence .. The Howe would correspondence not form in a our complex case comma in i general period e period . comma for the metaplectic group Mp open parenthesis 2 l comma R closing parenthesis acting on the space E of symplectic spinor valued Asexterior we have forms mentioned comma leads , we to the shall ortho prove hyphen that symplectic some super parts Lie algebraof these osp open two parenthesis complexes 1 bar 2 closing parenthesis andare a elliptic . To obtain these parts , one should remove the last ( i . e . , the zero ) term andcertain the representation second last of this term algebra from on the E period first We complex decided to and use the the Howe first type term ( the zero space againcorrespondence ) from the mainly second because complex the spaces . TheE to complexesthe power of obtained r open parenthesis in this defined way willabove be closing called parenthesis can be characterizedt runcated symplectic twistor complexes . Let us mention that by an elliptic complex , via the mentioned representation of osp open parenthesis 1 bar 2 closing parenthesis easily and in a way described in this \noindentpaper periodwe Seemean R aperiod complex Howe ofopen differential square bracket operators1 0 closing square such bracket that it for s more associated information symbol on the Howesequence type correspondenceis an exact sequence of the sheaves in question . ( See , e . g . , Wells [ 2 1 ] for details . ) in general period Let us also mention that besides this duality comma the Cartan lemma on \ hspace ∗{\ f i l l } Let us make some remarks on the methods we have used to prove the ellipticity

\noindent of the truncated symplectic twistor complexes . We decided to use the so called Schur − Weyl − Howe correspondence , which is referred to as the Howe correspondence

\noindent for simplicity in this text . \quad The Howe correspondence in our case , i . e . , for the metaplectic groupMp $ ( 2 l , R ) $ acting on the space E of symplectic spinor valued exterior forms , leads to the ortho − symplectic super Lie algebra $ osp ( 1 \mid 2 ) $ and a certain representation of this algebra on E . We decided to use the Howe type correspondence mainly because the spaces $ E ˆ{ r } ( $ defined above ) can be characterized via the mentioned representation of $ osp ( 1 \mid 2 ) $ easily and in a way described in this paper . See R . Howe [ 1 0 ] for more information on the Howe type correspondence in general . Let us also mention that besides this duality , the Cartan lemma on 3 1 2 .. S period KR Yacute SL \noindentexterior differential3 1 2 \quad forms wasS . used KR period $ \acute For other{Y} examples$ SL of elliptic complexes comma we refer an interested reader comma e period g period comma to Stein and Weiss open square bracket 1 7 closing square bracket comma\noindent Schmidexterior open square differential bracket 1 5 closing forms square was bracket used comma . For Hotta other open examples square bracket of elliptic 9 closing square complexes bracket comma , we andrefer an interested reader , e . g . , to Stein and Weiss [ 1 7 ] , Schmid [ 1 5 ] , Hotta [ 9 ] , and Branson3 open 1 2 square S . KR bracketY´ SL 2 closing square bracket period \noindentFor an applicationexteriorBranson differential of [symplectic 2 ] forms . spinors was used in mathematical . For other examples physics of comma elliptic see complexes comma e , period we refer g anperiod interested comma Shale open squarereader bracket , e . g 1 . 6 , closing to Stein square and Weiss bracket [ 1 .. 7 ]and , Schmid Green [ comma 1 5 ] , Hotta Hull open [ 9 ] square , and bracket 7 closing square bracket and the already\ hspace mentioned∗{\Bransonf i l l article} [For 2 ] . anof Kostant application .. open square of symplectic bracket 1 1 closing spinors square in bracket mathematical period In physics , see , e . g . , Shale the first reference commaFor one an application can find an of application symplectic of spinors these spinors in mathematical in quantizing physics of , see , e . g . , Shale \noindentKlein hyphen[ 1[ 6 1 ] Gordon 6 and ] \ Green fieldsquad and ,and Hull inGreen [the 7 ] second and , the Hull one already in [ the 7 mentioned 1 ] 0 anddimensional the article already super of Kostant hyphen mentioned string [ 1 1 ] theory . article In the period first of Kostant \quad [ 1 1 ] . In theThe first purposereference reference for taking , onesymplectic can , onefind an can spinor application find valued an of forms application these might spinors bej in ustified of quantizing these by the spinors of Klein - in Gordon quantizing fields and of Kleinintention− inGordon to the describe second fields higher one in andspin the boson 1 in 0 dimensional the fields second period super one - string in the theory 1 0 .dimensional The purpose for super taking− symplecticstring theory . TheIn the purpose secondspinor section for valued taking comma forms might wesymplectic recall be j some ustified knownspinor by the facts valued intention on symplectic forms to describe spinorsmight higher and be spin j ustified boson fields by . the intentionthe space ofIn to symplectic the describe second spinor section higher valued , we recall exteriorspin some boson forms known and fields facts its decompositionon . symplectic spinors into and the space of symplectic irreduciblespinor submodules valued exterior open parenthesis forms and Theorem its decomposition 1 closing parenthesis into irreducible period submodules In the third ( chapter Theorem comma 1 ) .In basic information on InFedosov the secondthe manifolds third section chapter and their , , basic curvature we informationrecall are mentioned some on Fedosov known and manifoldsthefacts symplectic on and symplectic theirtwistor curvature spinors are mentioned and and thecomplexes spacethe .. ofsymplectic are symplectic introduced twistor period complexes spinor .. In thevalued are fourth introduced exterior section comma. forms In the the fourth and symbol itssection sequence decomposition , the of symbol the sequence into of irreduciblesymplecticthe twistor symplectic submodules complexes twistor (is complexes computed Theorem is and 1 computed ) the . ellipticity In and the theof third ellipticity the truncated chapter of the truncated , basic information on Fedosovsymplecticsymplectic manifolds twistor complexes twistor and complexes their is proved curvature is open proved parenthesis ( Theorem are mentioned Theorem 7 ) . 7 closing and the parenthesis symplectic period twistor complexes2 period .. Symplectic\quad are spinor introduced valued2 forms . .Symplectic\quad Inspinor the fourth valued section forms , the symbol sequence of the symplecticIn this paperIn twistor the this Einstein paper complexes thesummation Einstein is convention summation computed is usedconvention and for the finite is ellipticity usedsums forcomma finite not of sums the , not truncated mentioning it mentioningexplicitly it explicitly unless unless otherwise otherwise is st ated is st . ated ( We period will not open use parenthesis this convention We will not use this convention \noindentin the proofinsymplectic the of proof the Lemma of the twistor 6Lemma and in 6 the complexes and it in em the 3 of it is emthe 3proved proof of the of proof the ( Theorem Theorem of the Theorem 7 7 only ) . period 7 only closing . ) The parenthesis category The categoryof representations of representations of Lie of groups Lie groups we shall we shall consider consider is that is that one one the the object of which are finite length \ centerlineobject ofadmissible which{2 are. \ representationsquad finite lengthSymplectic admissible of a fixed spinor representations reductive valued group of formsG aon fixed Fr´echet} reductive vector spaces and the morphisms group Gare on continuousFreacutechetG− vectorequivariant spaces and maps the between morphisms the objects are continuous . All manifolds G hyphen , vector equivariant bundles and their Inmaps this between papersections the the in objects this Einstein text period are supposed Allsummation manifolds to be smooth conventioncomma . vector The only bundles is manifolds used and for their which finite sections are allowed sums in to , be not of infinite mentioningthis textdimension are supposed it explicitly are theto be total smooth spaces unless period of vector otherwise Thebundles only manifolds is . If st this atedwhich is the are . case ( allowed We, the will bundles to not are use supposed this to convention be be of infiniteFr´echet dimension . The base are manifolds the total spaces are always of vector finite bundles dimensional period . The If this sheaves is the we case will comma consider are sheaves \noindentthe bundlesofin smooth are the supposed sections proof to of of be vector Freacutechet the Lemma period 6 and The in base the manifolds it em 3 are of always the finite proof of the Theorem 7 only . ) Thedimensional categorybundles period of . If The representationsE → sheavesM is a we Fr´echet will consider vector of Lie bundle are groups sheaves , we denote ofwe smooth shall thesections sheaf consider of ofsections vector is by that one the objectbundles ofΓ period, i which . e If., Γ( EU rightare) := arrowfinite Γ(U, E M) foris length a each Freacutechet open admissible set vectorU in M. bundle representationsFor m comma∈ M, we we denote denote of the a fixed stsheaf alk of of reductive sections Γ at m by by group $G$ on Fr \ ’{ e} chet vector spaces and the morphisms are continuous $ G − $ equivariant Capital GammaΓm. comma i period e period comma Capital Gamma open parenthesis U closing parenthesis : = Capital Gamma openmaps parenthesis between2 . 1 U . the commaSymplectic objects E closing linear . parenthesis All alge manifolds bra and for each basic ,open notation vector set U. in bundles M periodIn order and For to m their set in the M comma sections notation we , denote let in us the st alk this text are supposed to be smooth . The only manifolds which are allowed to of Capitalst Gammaart recalling at m some by Capital simple Gamma results from sub m symplectic period linear algebra . Let (V, ω0) be a real symplectic be2 period of infinitevector 1 period space .. dimension Symplectic of dimension linear are 2l, l the alge≥ 1. bra totalLet and us choose basicspaces notation two of Lagrangian vector period .. bundles In order to . set If the this notation is the comma case let , the bundles are supposed0 to be Fr \ ’{ e}0chet . The base manifolds are0 always finite us st artsubspaces recalling someL and simpleL , such results that fromV ' symplecticL ⊕ L 1. It linear is easy algebra to see period that dim LetL open= dim parenthesisL = l. V comma omega sub 0 closing dimensional . The sheaves we will consider are sheaves2l of smooth0 sections of vector parenthesisFurther be , let us choose an adapted symplectic basis {ei}i=1 of (V ' L ⊕ L , ω0), i . e . , 2l l 2l 0 a real symplectic{ei}i=1 is vector a symplectic space of basis dimension of (V, ω 20) lcomma and {ei} li greater=1 ⊆ L equaland {e 1i} periodi=l+1 ⊆ LetL . usThe choose two Lagrangian \noindent bundles . If $E 2l \rightarrow M $i 2l i s a Fr \ ’{ e} chet vector bundle , we denote the sheaf of sections by subspacesbasis L and dual L to to the the basis power{e ofi}i prime=1 will comma be denoted such that by { V} simeqi=1, i . L e oplus . , for Li, to j = the 1, power ..., 2l of prime 1 sub period It is easy to see that dim L = dimj L to the powerj of primej = l period we have  (ei) = ιei  = δi , where ιvα for an element v ∈ V and an exterior \noindent $ \GammaV• ∗ , $ i . e $ . , \Gamma ( U ) : = \Gamma (U,E Furtherform commaα ∈ let usV choose, denotes an adapted the contraction symplectic of the basis form openα braceby the e vector sub i closingv. Further brace sub i = 1 to the power of 2 l of open )$ for eachopenset $U$ in $M .$ Forij $m \ in M , $ we denote the st alk parenthesisfor V simeqi, j = L oplus 1, ..., 2 Ll, towe the set powerωij of:= primeω0(ei,comma ej) and omega define ω sub, i,0 j closing= 1, parenthesis ..., 2l, by comma i period e period comma o fopen $ brace\Gamma e sub$ i closing at $ brace m $ sub by i = $ 1 to\Gamma the power{ ofm 2} l is. a $ symplectic basis of open parenthesis V comma omega sub 0 closing parenthesis and open brace e sub i closing brace sub i = 1 to the power of l subset equal L and open brace e sub i closing brace\noindent sub i =2 l plus . 1 1 .to\ thequad powerSymplectic of 2 l subset linear equal L to alge the power bra and of prime basic period notation The . \quad In order to set the notation , let usbasis st dual art to recalling the basis open some brace simple e sub i results closing brace from sub symplectic i = 1 to the power linear of 2 lalgebra will be denoted . Let by $ open ( brace V epsilon , \ toomega { 0 } ) $ be the power of i closing brace sub1Let i =us 1 torecall the that power by Lagrangian of comma , towe the mean power maximal of 2 isotropic l i period wrt e.ω period0. comma for i comma j = 1 comma perioda real period symplectic period comma vector 2 l space of dimension $ 2 l , l \geq 1 . $ Let us choose two Lagrangian we have epsilon to the power of j open parenthesis e sub i closing parenthesis = iota sub e sub i epsilon to the power of j = delta sub\noindent i to the powersubspaces of j comma $L$ where iota and sub $L v alpha ˆ{\ forprime an element} , v $in V such and an that exterior $V \simeq L \oplus L ˆ{\prime } 1 form{ . alpha}$ inIt bigwedge is easy to to the see power that of bullet dim V $L to the power =$ of dim * comma $Lˆ denotes{\prime the contraction} = l of the . form $ alpha by the vector v period Further \noindentfor i commaFurther j = 1 comma , let period us period choose period anadapted comma 2 l symplectic comma we set omegabasis sub $ ij\{ : = omegae { subi }\} 0 open parenthesisˆ{ 2 l e} sub{ i =i comma 1 } e$ sub o j f closing $ ( parenthesis V \simeq and defineL omega\oplus to the powerL ˆ of{\ ijprime comma i} comma, j\ =omega 1 comma{ period0 } period) , period $ i comma . e . , 2 l comma by \noindenthline $ \{ e { i }\} ˆ{ 2 l } { i = 1 }$ is a symplectic basis of $ ( V , \omega1 sub Let{ us0 recall} ) that $ andby Lagrangian $ \{ commae { wei }\} mean maximalˆ{ l } isotropic{ i wrt = period 1 }\ omegasubseteq sub 0 periodL $ and $ \{ e { i }\} ˆ{ 2 l } { i = l + 1 }\subseteq L ˆ{\prime } . $ The

\noindent basis dual to the basis $ \{ e { i }\} ˆ{ 2 l } { i = 1 }$ will be denoted by $ \{\ epsilon ˆ{ i }\} ˆ{ 2 l } { i = 1 ˆ{ , }}$i.e.,for$i , j = 1 , . . . , 2 l $

\noindent we have $ \ epsilon ˆ{ j } ( e { i } ) = \ iota { e { i }}\ epsilon ˆ{ j } = \ delta ˆ{ j } { i } , $ where $ \ iota { v }\alpha $ for an element $ v \ in V $ and an exterior

\noindent form $ \alpha \ in \bigwedge ˆ{\ bullet } V ˆ{ ∗ } , $ denotes the contraction of the form $ \alpha $ by the vector $ v . $ Further

\noindent for$i , j = 1 , . . . , 2 l ,$weset$ \omega { i j } : = \omega { 0 } ( e { i } , e { j } ) $ and define $ \omega ˆ{ i j } , i , j=1,. . . ,2l ,$by

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ centerline { $ 1 { Let }$ us recall that by Lagrangian , we mean maximal isotropic wrt $ . \omega { 0 } . $ } ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 3 1 3 \ hspacethe equation∗{\ f iomega l l }ELLIPTICITY sub ij omega OFto the THE power SYMPLECTIC of kj = delta TWISTOR sub i to the COMPLEX power of\quad k for all3 i1 comma 3 k = 1 comma period period period comma 2 l period Let us remark that not only \noindentomega subthe ij = minusequation omega $sub\omega j i comma{ buti j also}\ omegaomega to theˆ{ powerkj } of= ij = minus\ delta omegaˆ{ tok the} { poweri }$ of j if o for r i a comma l l $ j i =, 1 comma k = period 1 period , period . . comma . 2 l , period 2 l .$ Letusremarkthatnotonly $ \omega { i j } = − \omega { j i } , $ but a l s o $ \omega ˆ{ i j } = − \omega ˆ{ j As in the Riemannian case comma we would likeELLIPTICITY to rise and OFlower THE indices SYMPLECTIC of tensor TWISTOR COMPLEX 3 1 3 i }$for$i , j=1kj k , . . . ,2 l .$ coordinatesthe periodequation Inω theijω symplectic= δi for case all i, comma k = 1one, ..., should2l. Let be us more remark careful that because not only ofω theij = −ωji, but also anti hyphenωij = symmetry−ωji for ofi, j omega= 1, ..., sub2l. 0 period For coordinates K sub a b period period period c period period period d to the power\ hspace of r∗{\ s periodf i l l } periodAs in period the t RiemannianAs period in the period Riemannian caseperiod , u case ofwe a , would tensor we would K like over like V to period rise rise and we and lower lower indices ofindices tensor of tensor denote thecoordinates expression . In omega the symplectic to the power case of , ione c K should sub a be b period more careful period because periodc of period the period period d to the power of r s \noindent coordinates . In the symplecticrs...t...u case , one should be more careful because of the period periodanti period - symmetry t by K of subω0 a. For b period coordinates periodK periodab...c...d toof the a tensorpowerK of iover subV period. we period period d to the power of r s period ic rs...t i rs...t rs...t...u rs...... u period perioddenote t and the K expressionsub a b periodω K periodab...c...d periodby Kab... c to...d theand powerKab...c of r sω periodti by K periodab...ci periodand similarly t period for period other period u omega sub t\noindent i by typesa n t of i tensors− symmetry and also of in the $ geometric\omega { 0 } . $ For coordinates $K { a b . . . cK . sub a .setting b period . when periodd } weˆ{ will periodr be sconsidering c tothe . power . tensor of . fieldsr s t period over . a period symplectic . period . manifold u sub}$ i to of(M the2 al, power ω tensor). of period $K$ period over period $V u and . $ we j j j similarly forLet other us remarktypes of that tensorsωi = and−ω alsoi = δ ini , the i, j = geometric 1, ..., 2l. Further , one can also define setting whenan isomorphism we will be] considering: V∗ → V, V tensor∗ 3 α 7→ fieldsα] ∈ overV, by a symplectic the formula manifold open parenthesis M to the power of 2 l comma \noindent denote the expression $ ]\omega ˆ{ i c } ∗ K { a b . . . c . . . d }ˆ{ r omega closing parenthesis period α(w) = ω0(α , w) for each α ∈ V and w ∈ V. s . . . t i }$ by $ K { a b . . . ] }i ˆ{ i } { .] i . . d }ˆ{ r s . . . Let us remarkFor α = thatαi omegaand j sub= 1 i, ...,to the2l, we power get α ofj j= =α minus(ej) = omegaω0((α to) e thei, ej) power = ωij of(α j) sub= i = delta sub i to the power of j comma t }$ and] $ K { a b] .] i . .i c }ˆ{ r s . . . t . . . u }\omega { t i comma j =(α 1) commaj which period implies periodα =( periodα ) ei comma= α ei. 2Thus l period , we Furthersee that comma the rising one of can indices also viadefine i }$ by an isomorphismthe form sharpω0 is : realized V to the by power the isomorphism of * right arrow]. V comma V to the power of * ni alpha arrowright-mapsto alpha to the power$ K of sharp{ aFinally in b V comma , . let us by . introduce the . formula c the}ˆ{ groupsr we s will . beusing . . .} Let{ usi denote}ˆ{ . the symplectic . . u group}$ of and similarly for other types of tensors and also in the geometric alpha open(V, ω parenthesis0) by G, i . w e closing., G := Sp parenthesis(V, ω0) ' =Sp omega(2l, R) sub. Because 0 open the parenthesis fundamental alpha group to the of G power= Sp of(V sharp, ω0) comma w closing \noindent setting when we will be considering tensor fields over a symplectic manifold $ ( M ˆ{ 2 parenthesisis ..Z for, there each .. exists alpha a in connected V to the 2power : 1 , of necessarily * .. and .. non w in - V universal period , covering of G by the so called l } , \omega ) . $ ˜ For alphametaplectic = alpha sub group i epsilonMp ( toV,the ω0) power denoted of by i andG in j = this 1 comma text . period period period comma 2 l comma we get alpha sub j = alpha openLet parenthesis us denote ethe sub mentioned j closing parenthesis two - fold covering = omega map sub by 0 openλ, in parenthesis particular λ open: G˜ → parenthesisG. ( See ,alpha e . g to . , the power of sharp closing\noindent parenthesisHabermannLet tous the remark , Habermann power of that i e [ sub 8 ]$ i. comma\ )omega e sub{ ji closing}ˆ{ j parenthesis} = − = omega \omega sub ijˆ open{ j parenthesis} { i } alpha= to\ delta the powerˆ{ j } { i } of, sharp i closing ,2 . 2 parenthesis j . Segal = - 1 Shaleto the , - power Weil . representation of .i = . , and 2 symplectic l .$ spinor Further,onecanalsodefine valued fo r − m s . The open parenthesisSegal - Shale alpha - Weil to the representation power of sharp is a distinguished closing parenthesis representation sub j which of the implies meta - alpha to the power of sharp = open \noindent an isomorphism˜ $ \sharp : V ˆ{ ∗ } \rightarrow V , V ˆ{ ∗ } \ ni \alpha parenthesisplectic alpha to group the powerG = Mp of sharp(V, ω closing0)2. This parenthesis representation to the is power unitary of i, efaithful sub i = and alpha does to not the descend power of to i ae sub i period Thus comma\mapsto we seerepresentation\ thatalpha theˆ rising{\ ofsharp the of indices symplectic}\ viain groupV . Its , underlying $ by the vector formula space the formis omega the vector sub 0 space is realized of complex by the valued isomorphism square Lebesgue sharp period integrable functions L2(L) \ centerline { $ \alpha ( w ) = \omega { 0 } ( ∞ \alpha2 ˆ{\sharp ∞} , w ) $ \quad f o r each \quad Finally commadefined let on us the introduce chosen Lagrangian the groups subspace we will beL. usingLet us period set S Let:= usV denote(HC( theL ( symplecticL))), where V is the $ \groupalpha ofCasselman open\ in parenthesis -V Wallach ˆ{ ∗ V }globalizationcomma$ \quad omega functorand sub\ 0quad and closingHC$ parenthesisdenotes w \ in the byV G comma . $ i period} e period comma G : = Sp open parenthesisforgetful V comma Harish omega - sub Chandra 0 closing functor parenthesis from the simeq category Sp open of G˜− parenthesismodules defined 2 l comma above R closing parenthesis period Because the\noindent fundamentalintoFor the category $ \alpha of Harish= - Chandra\alpha (g,{K˜ )i−modules}\ epsilon3. We shallˆ{ denotei }$and$j the resulting representation = 1 , . . . ,group 2 of lby GL = ,$wegetand Sp open call it parenthesis the metaplectic $ \ Valpha comma representation{ omegaj } sub= 0. Thusclosing\alpha , we parenthesis have( ise Z{ commaj } there) exists = a\omega connected{ 20 : 1} comma( ( \alpha ˆ{\sharp } ) ˆ{ i } e { i } , e { j } ) = \omega { i j } ( \alpha ˆ{\sharp } necessarily non hyphen universal comma L : Mp (V, ω0) → Aut ( S ). ) ˆcovering{ i } The of= G $elements by the so of calledS will metaplectic be called groupsymplectic Mp open spinors parenthesis. It is V well comma known omega that subS splits 0 closing into parenthesis two denoted by G-tilde in thisirreducible text periodMp (V, ω0)− submodules S+ and S−. Thus , we have S = S+ ⊕S−. See the foundational \noindentLet us denotepaper$ (the of A mentioned\ .alpha Weil [ 20ˆ two{\ ] for hyphensharp more fold} detailed covering) { informationj map}$ by which lambdaon the implies Segal comma - Shale in $ particular\ -alpha Weil representation lambdaˆ{\sharp : G-tilde} right= arrow ( \ Galpha ˆ{\sharp } period) ˆ{ i } ande Casselman{ i } = [ 4 ]\ onalpha this typeˆ{ ofi } globalizatione { i } . Let. $ us mention Thus , that wechoosing see that this the particular rising of indices via open parenthesisglobalization See seemscomma to e be period rather g technicalperiod comma Habermann comma Habermann open square bracket 8 closing square bracket\noindent periodfromthe closing the form point parenthesis of $ view\omega of the{ aim0 of}$ our articleis realized . by the isomorphism $ \sharp . $ 2 period 2 period .... Segal hyphenIn the Shale proof hyphen of the Weil ellipticity representation of the truncated .... and symplectic symplectic twistor .... spinor complexes valued , fo r-m s period .... TheFinally ,we let shall us need introduce some facts on the the groups underlying we vector will space be using of the metaplectic . Let us repre denote - sentation the symplectic . Let us groupofSegal hyphenmention $( Shale that V hyphen it is , known Weil\omega representation that S {is isomorphic0 } is a)$by$G distinguished to the Schwartz representation ,$i.e$. of the meta hyphen , G : =$Sp$( V,plectic group\omega G-tilde{ 0 = Mp} open) parenthesis\simeq VSp comma ( omega 2 sub l 0 closing , R parenthesis ) .$ 2 period Becausethefundamental This representation is unitary commagroupof faithful $G and does =$ not Sp $( V , \omega { 0 } ) $ is $ Z , $ there exists a connected 2 : 1 , necessarily non − u n i v e r s a l , coveringdescend to of a representation $G$ by of the the symplectic so called group metaplectic period Its underlying group Mp vector $ space ( V , \omega { 0 } ) $ denoted by $ \istilde the vector{G} $ space in of this complex text valued . square Lebesgue integrable functions L to the power of 2 open parenthesis L closing parenthesis 2T he names oscillator and metaplectic are also used in the literature . See , e . g . , Howe [ 10 ] . 3Here,g is the Lie ˜ ˜ ˜ \noindentdefined onLet the chosen us denote Lagrangianalgebra the mentionedsubspace of G and LK period twois the− maximal Letfold us set compact covering S : = Lie V subgroup to map the by of powerG. $ of\lambda infinity open, parenthesis $ in particular HC open parenthesis$ \lambda L to: the power\ tilde of 2{G open}\ parenthesisrightarrow L closingG parenthesis . $ closing parenthesis closing parenthesis comma (where See V, toe the. g power . ,of Habermann infinity is the , HabermannCasselman hyphen [ 8 Wallach] . ) globalization functor and HC denotes the forgetful Harish hyphen Chandra functor from the category of G-tilde sub hyphen modules defined above \noindentinto the category2 . 2 of . Harish\ h f i hyphenl l Segal Chandra− Shale open parenthesis− Weil representation g comma K-tilde closing\ h f i parenthesisl l and symplectic hyphen modules\ h f to i l the l spinor power valued fo of$ 3 r− periodm $ We s .shall\ h denote f i l l The the resulting representation by L and call it the metaplectic representation period Thus comma we have \noindentL : .. Mp openSegal parenthesis− Shale V− commaWeil omega representation sub 0 closing parenthesis is a distinguished right arrow Aut representation open parenthesis S of closing the parenthesismeta − period \noindentThe elementsplectic of S will group be called $ symplectic\ tilde {G spinors} = period $ Mp It is $ well ( known V that , S\ splitsomega { 0 } ) 2 { . }$ This representation is unitary , faithful and does not descendinto two irreducibleto a representation Mp open parenthesis of the V comma symplectic omega sub group 0 closing . Its parenthesis underlying hyphen vector submodules space S sub plus and S sub minus period Thus comma we have S = S sub plus oplus S sub minus period \noindentSee the foundationalis the vector paper of spaceA period of Weil complex open square valued bracket square 20 closing Lebesgue square bracket integrable for more functionsdetailed information $ L ˆ on{ the2 } (Segal L hyphen ) $ Shale hyphen Weil representation and Casselman open square bracket 4 closing square bracket on this type of globalization period Let \noindentus mentiondefined that choosing on the this particular chosen Lagrangian globalization seems subspace to be rather $ L technical . $ Let us set S $ : = Vˆ{\ infty } (from HC the ( point L of ˆ view{ 2 of} the( aim L of our ) article ) period ) , $ whereIn the proof $ V of ˆ{\ theinfty ellipticity}$ of the is truncated the Casselman symplectic− twistorWallach complexes globalization comma functor and $ HC $ denotes the we shall need some facts on the underlying vector space of the metaplectic repre hyphen \noindentsentation periodforgetful Let us mention Harish that− Chandra it is known functor that S is from isomorphic the to category the Schwartz of $ \ tilde {G} { − }$ modules defined above hline \noindent2 sub Theinto names the oscillator category and metaplectic of Harish are− alsoChandra used in the $( literature g period , See\ tilde comma{K} e period) g− periodmodules comma Howeˆ{ 3 } open. $ square We shall bracket denote 10 closing the square resulting bracket period representation3 sub Here comma by g is the $ L Lie $ algebra and ofcall G-tilde it and the tilde-K metaplectic is the maximal representation compact Lie subgroup . Thus of , G-tilde we have sub period \ centerline { $ L : $ \quad Mp $ ( V , \omega { 0 } ) \rightarrow $ Aut ( S ) . }

\noindent The elements of S will be called symplectic spinors . It is well known that S splits into two irreducible Mp $ ( V , \omega { 0 } ) − $ submodules $ S { + }$ and $ S { − } . $ Thus ,wehaveS $= S { + }\oplus S { − } . $ See the foundational paper of A . Weil [ 20 ] for more detailed information on the Segal − Shale − Weil representation and Casselman [ 4 ] on this type of globalization . Let us mention that choosing this particular globalization seems to be rather technical

\noindent from the point of view of the aim of our article .

\ hspace ∗{\ f i l l } In the proof of the ellipticity of the truncated symplectic twistor complexes ,

\noindent we shall need some facts on the underlying vector space of the metaplectic repre − sentation . Let us mention that it is known that S is isomorphic to the Schwartz

\ begin { a l i g n ∗} \ r u l e {3em}{0.4 pt } \end{ a l i g n ∗}

\ begin { c e n t e r } $ 2 { The }$ names oscillator and metaplectic are also used in the literature . See , e . g . , Howe [ 10 ] . $ 3 { Here , } g $ is the Lie algebra of $ \ tilde {G} $ and $ \ tilde {K} $ is the maximal compact Lie subgroup of $ \ tilde {G} { . }$ \end{ c e n t e r } 3 1 4 .. S period KR Yacute SL \noindentspace S open3 1 parenthesis 4 \quad LS closing . KR parenthesis $ \acute of{ smoothY} $ functions SL rapidly decreasing in the infinity equipped with the st andard open parenthesis lo cally convex closing parenthesis Freacutechet topology generated by the supremum semi hyphen norms\noindent period space $ S ( L ) $ of smooth functions rapidly decreasing in the infinity equipped with the stopen andard parenthesis ( lo See cally comma convex e period ) gFr period\ ’{ e} commachet topology Habermann generated comma Habermann by the open supremum square bracket semi − 8 closingnorms square . ( See , e . g . , Habermann , Habermann [ 8 ] or Borel , Wallach [ 1 ] . ) For the convenience of bracket or Borel3 1 4 comma S . KR WallachY´ SL open square bracket 1 closing square bracket period closing parenthesis For the convenience of the readerspace commaS(L) let of us smooth briefly functions recall the rapidly definition decreasing of the involved in the infinity semi hyphen equipped norms with period the st For andard each a ( comma lo \noindentb in N subcallythe 0 to convexthe reader power ) Fr´echet , of let comma topology us to briefly generatedthe power recall by of lthe the supremum semi the hyphen definition semi norm - norms q a of .comma ( the See ,b involved e is. g defined . , Habermann by semi the− formulanorms q a . comma For each b$ open a parenthesis , $, Habermann f closing [ 8 parenthesis ] or Borel , : Wallach = supremum [ 1 ] .sub ) For x inthe L convenience bar open parenthesis of x to the power of a partialdiff to the power of b f closingthe parenthesis reader , let open us briefly parenthesis recall x the closing definition parenthesis of the involved bar comma semi - norms . For each a, \noindent $l b \ in N ˆ{ l } { 0 ˆ{ , }}$ the semi − norm $qa b a , b$ is defined by the formula f in S openb ∈ parenthesisN0, the semi L - closing norm qa, parenthesis b is defined period by the Let formula us orderqa, the b(f set) := .... sup openx∈ | parenthesis(x ∂ f)(x) q|, a comma b closing parenthesis $ q a , b ( f ) : = \sup { x \ in L }\Lmid ( x ˆ{ a }\ partial ˆ{ b } sub a commaf b∈ inS the(L) st. Let andard us order quoteright the set lexicographical quoteright (qa, b)a, wayb in the st andard ’ lexicographical ’ way f ) ( x ) \mid , $ k and denoteand the denote resulting the resulting sequence sequence of semi hyphen of semi norms - norms by open by (q parenthesis)k ∈ N0. These q to the semi power - norms of k closing generate parenthesis a k in N sub 0 period Thesecomplete semi hyphen metric topology norms on S(L). Taking a = b = 0, one sees that \noindentgeneratethe a complete$ convergence f \ metricin with topologyS respect ( on toL S theopen ) semi parenthesis .$ - norms Letusordertheset L implies closing the parenthesis uniform period convergence\ h Taking f i l l immediately a$ = (b = 0 q comma . a one , sees b that ) { a , } b $ in the st andard ’ lexicographical ’ way the convergenceFurther ,with it is respect well known to the that semi the hyphen Schwartz norms space impliesS(L) the possesses uniform a convergence Schauder basis . For a complex immediatelymetric period ( e . Furtherg . , Fr´echet comma ) space it is wellF, an known ordered that countable the Schwartz set (fi space)i ∈ N S⊆ openF is parenthesis called a Schauder L closing basis parenthesis possesses a\noindentof andF if each denote element thef ∈ resultingF can be uni sequence - of semi − normsby $( qˆ{ k } ) k \ in N { 0 } . $ These semi − norms P∞ Schauderquely basis expressed period For as af complex= i=1 a metricifi for open some parenthesisai ∈ C. Notice e period that gfrom period the comma uniqueness Freacutechet closing parenthesis space generateacomplete metric topologyon $SP (∞ L ) . $ Taking $a = b = 0 ,$ F comma anof ordered the coefficients countableai immediately follows that 0 = aifi implies ai = 0 for all i ∈ N. From the one sees that i=1 set openbasic parenthesis mathematical f i closing analysis parenthesis courses i in , one N subset knows equal that in F theis called caseof a Schauder the Schwartz basis space of FS if( eachL), one element can f in F can be uni hyphen t ake , e . g . , the lexicographically ordered sequence \noindentquely expressedofthe Hermite convergence as ffunctions = sum sub in l with ivariables = 1 respect to the as the power Schauder to of the infinity basissemi a .− sub Wenorms idenote f i for implies this some basis a sub by the i in uniform C period convergence Notice that from the uniquenessimmediately . Further , it is well known that the Schwartz space $ S ( L ) $ possesses a Schauderof the coefficients basis a . sub For i immediately a complex follows metric that ( 0 =e sum . g sub . , i = Fr 1\ to’{ thee} chet power ) of space infinity a $ sub F i f i , implies $ an a sub ordered i = 0 for countable s e t $ ( f i ) i \ in N \subseteq F $ is called a Schauder basis of $ F $ if each element all (hi)i ∈ N. $ fi in N\ in periodF From $ the can basic be mathematical uni − analysis courses comma one knows that in the case of Now , we may define the so called symplectic Clifford multip lication · : V× S → S . For s ∈ S the Schwartz spacej S openj parenthesis L closing parenthesis comma one can t ake comma e period g period comma the lexicographically\noindent, xquely= x orderedej ∈ expressedL, sequence x ∈ R and asi, j $= f 1, ..., = l, let\ ussum set ˆ{\ infty } { i = 1 } a { i } f i $ f o r some $ a { i }\ in C . $ Notice that from the uniqueness of Hermite functions in l variables as the Schauder basis period We denote∂s this basis by e · s(x) := ıxis(x) and e · s(x) := (x). open parenthesis h sub i closing parenthesisi i in N period i+l ∂xi \noindentNow commaof we the may coefficients define the so called $ symplectic a { i } Clifford$ immediately multip lication follows times : V that times S$ right 0 arrow = \ Ssum periodˆ{\ infty } { i =For 1 s} inIn Sa comma physics{ i x} , = this xf to mapping the i$ power ( implies up of to j e a sub constant $a j in L multiple comma{ i } x )= tois usually the 0 power $ called f of o r j the in a lR canonicall and i comma quantization j = 1 comma . period period period$ i comma\Letin l us comma remarkN let . that $us set theFrom definition the basic is correct mathematical due to the preceding analysis paragraph courses . For ,each onev, w knows∈ V and that in the case of thee sub Schwartz i timess ∈ sS open space, one parenthesis can $ easily S x derive( closing L the parenthesis following ) , $ : commutation = onei x to can the power relation t ake of i , s opene . parenthesisg . , the x closing lexicographically parenthesis and e ordered sub sequence i plus l times s open parenthesis x closing parenthesis : = partialdiff s divided by partialdiff x to the power of i open parenthesis x closing\noindent parenthesisof Hermite period functions in $ l $ variables as the Schauder basis . We denote this basis by v · w · s − w · v · s = −ıω (v, w)s. (1) In physics comma this mapping open parenthesis up to a constant0 multiple closing parenthesis is usually called the canonical \ beginquantization{ a l i g n ∗} period Let us remark that the definition is correct due to the preceding ( h {( Seei } , e .) g . , i Habermann\ in N. , Habermann [ 8 ] . ) We shall use this relation repeatedly and without paragraphmentioning period For its each use .v Now comma , we w prove in V and that s the in .. symplectic S comma Clifford one can multipli easily derive - the following \endcommutation{ a l i g n ∗} relation cation by a fixed non - zero vector v ∈ V is inj ective as a mapping from S into S . Equation: open parenthesis˜ 1 closing parenthesis .. v times w times s minus w times v times s = minus i omega sub 0 open Now , weWe may shall define use the theG−equivariance so calledof the symplectic symplectic Clifford Clifford multiplication multip , lication i . e . , the fact $ L\(cdotg)(v ·s) =:V \times $ parenthesis v comma w closing parenthesis s period ˜ [λ(g)v] · L(g)s which holds for each g ∈ G,v ∈ V and s ∈ S ( see Habermann , Habermann [ 8 ] Sopen $ \rightarrow parenthesis See$ comma S . e period g period comma Habermann comma Habermann open square bracket 8 closing square ) . Thus , let us suppose that a fixed s ∈ S and a fixed 0 6= v ∈ V are given such that v · s = 0. bracketFor period $ s closing\ in $ parenthesis S $ , We xshall =use this x ˆrelation{ j } repeatedlye { j and}\ in L , x ˆ{ j }\ in R $ and Because the action of the symplectic group G on V − {0} is transitive and λ is a covering , there $iwithout , mentioning j = its 1 use , period . Now . comma . we , prove l that ,$letusset the symplectic Clifford multipli hyphen exists an element g ∈ G˜ such that λ(g)v = e1. Applying L(g) on the equation v · s = 0, we get cation byL( ag)( fixedv · s) non = 0 hyphen. Using zerothe above vector mentioned v in V is injequivariance ective as a of mapping the symplectic from S Clifford into S period multiplication , we \ [We e shall{ i use}\ thecdot G-tilde subs hyphen ( x equivariance ) : of =the symplectic\imath Cliffordx ˆ{ multiplicationi } s ( comma x i period ) and e period e comma{ i the + l }\cdotget 0 =sL(g)( (v · s x) = [λ )(g)v] · :(L(g =)s) =\ fe r1 a· c(L{\(g)partials). Denoting Ls (g}{\)s =:partialψ and using thex ˆ definition{ i }} of( x ) . \ ] fact L openthe parenthesissymplectic Clifford g closing multiplication parenthesis open , we parenthesis v times s closing parenthesis = open square bracket lambda open parenthesis g closing parenthesis v closing square bracket times L open parenthesis g closing parenthesis s which holds for each g in obtain ıx1ψ = 0, which implies ψ(x) = 0 for each x = (x1, ..., xl) ∈ L such that G-tilde sub comma1 v in V and s in S \noindentx In6= 0 physics. By continuity , this of ψ mapping∈ S , we get( upψ = to 0. aBecause constantL is a multiple group representation ) is usually , we get calleds = 0 the canonical open parenthesisfrom 0 = seeψ = HabermannL(g)s, i . ecomma . , the inj Habermann ectivity of open the symplectic square bracket Clifford 8 closing multiplication square bracket . closing parenthesis period Thusquantization comma let us . suppose Let us that remark a fixed s that in S and the a definition is correct due to the preceding paragraphfixed 0 negationslash-equal . For each v $ in v V are , given w such\ in that vV times $ ands = 0 period $ s Because\ in $ the\quad actionS of the , one symplectic can easily derive the following commutationgroup G on V minusrelation open brace 0 closing brace is transitive and lambda is a covering comma there exists an element g in G-tilde such that lambda open parenthesis g closing parenthesis v = e sub 1 period Applying L open parenthesis g closing parenthesis\ begin { a l on i g n the∗} equation v times s = 0 comma we get \ tagL open∗{$ parenthesis( 1 ) g $ closing} v parenthesis\cdot openw parenthesis\cdot s v times− s closingw \ parenthesiscdot v = 0\ periodcdot Usings the = above− mentioned \imath equivariance\omega { of0 the} symplectic( v Clifford , w ) s . \endmultiplication{ a l i g n ∗} comma we get 0 = L open parenthesis g closing parenthesis open parenthesis v times s closing parenthesis = open square bracket lambda open parenthesis g closing parenthesis v closing square bracket times open parenthesis L open parenthesis g\noindent closing parenthesis( See s , closing e . g parenthesis . , Habermann = e sub , 1 Habermanntimes open parenthesis [ 8 ] . L ) open We shall parenthesis use g this closing relation parenthesis repeatedly s closing and parenthesiswithout period mentioning Denoting its use . Now , we prove that the symplectic Clifford multipli − L open parenthesis g closing parenthesis s = : psi and using the definition of the symplectic Clifford multiplication comma we \noindentobtain i xcation to the power by a of fixed 1 psi = non 0 comma− zero which vector implies psi $ v open\ parenthesisin V $ x closing is inj parenthesis ective =as 0 a for mapping each x = from open S into S . parenthesis x to the power of 1 comma period period period comma x to the power of l closing parenthesis in L such that \noindentx to the powerWe shall of 1 equal-negationslash use the $ \ tilde 0 period{G} By{ continuity − equivariance of psi in S comma}$ we of get the psi =symplectic 0 period Because Clifford L is a group multiplication , i . e . , the representationfact$L comma ( g ) ( v \cdot s ) = [ \lambda ( g ) v ] \cdot L (we g get s ) = 0 s$from 0 whichholds = psi = L open for parenthesis each g $g closing\ parenthesisin \ tilde s comma{G} i{ period, } e periodv \ commain V the $ inj and ectivity $ of s the\ in $ symplecticS Clifford (multiplication see Habermann period , Habermann [ 8 ] ) . Thus , let us suppose that a fixed $ s \ in $ S and a f i x e d $ 0 \not= v \ in V$ are given such that $ v \cdot s = 0 . $ Because the action of the symplectic group $G$ on $V − \{ 0 \} $ is transitive and $ \lambda $ is a covering , there exists an element $ g \ in \ tilde {G} $ such that $ \lambda ( g ) v = e { 1 } . $ Applying $ L ( g )$ onthe equation $v \cdot s = 0 , $ we get $ L ( g ) ( v \cdot s ) = 0 . $ Using the above mentioned equivariance of the symplectic Clifford multiplication ,weget $0 = L ( g ) ( v \cdot s ) = [ \lambda ( g ) v ] \cdot ( L ( g ) s ) = e { 1 }\cdot ( L ( g ) s ) . $ Denoting $ L ( g ) s = : \ psi $ and using the definition of the symplectic Clifford multiplication , we

\noindent obtain $ \imath x ˆ{ 1 }\ psi = 0 ,$ which implies $ \ psi ( x ) = 0$ foreach $x = ( xˆ{ 1 } , . . . , x ˆ{ l } ) \ in L $ such that

\noindent $ x ˆ{ 1 }\ne 0 . $ By continuity of $ \ psi \ in $ S , we get $ \ psi = 0 . $ Because $ L $ is a group representation , weget $s = 0$ from $0 = \ psi = L ( g ) s ,$ i . e . , the inj ectivity of the symplectic Clifford multiplication . ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 3 1 5 \ hspaceHaving∗{\ definedf i l l the}ELLIPTICITY metaplectic representation OF THE SYMPLECTIC and the symplectic TWISTOR Clifford COMPLEX mul hyphen\quad 3 1 5 tiplication comma we shall introduce the underlying algebraic structure of the basic Havinggeometric defined object wethe are metaplectic interested in comma representation namely the space and E :the = bigwedge symplectic to the power Clifford of bullet mul V− to the power of * oslash Stiplication of sym hyphen , we shall introduce the underlying algebraic structure of the basic plectic spinor valued exterior forms period The vectorELLIPTICITY space EOF is THE considered SYMPLECTIC with TWISTOR its COMPLEX 3 1 5 \noindentcanonical opengeometricHaving parenthesis defined object Freacutechetthe metaplectic we are closing interested representation parenthesis in and direct , the namely sum symplectic topology the Clifford space induced mul E by - $the tiplication : metric = topology , we\bigwedge on the ˆ{\ bullet } V ˆopen{ ∗ parenthesis }shall \otimes introduce finite$ dimensional the S underlying o f sym closing− algebraic parenthesis structure space of the of exterior basic forms and the Freacutechet topology on S period The plectic spinor valued exterior forms . The vector spaceV E• is∗ considered with its metaplecticgeometric group G-tildeobject we acts are on interested E by the representationin , namely the space E := V ⊗ S of sym - plectic spinor rho : G-tildevalued right exterior arrow forms Aut open. The parenthesis vector space EE closingis considered parenthesis with .. definedits by the formula \noindentrho opencanonical parenthesiscanonical ( Fr´echet g closing ( Fr ) direct\ parenthesis’{ e} sumchet topology open ) direct parenthesis induced sum by alpha topology the oslash metric s topology inducedclosing parenthesis on by the the : metric = open parenthesis topology lambda on the open parenthesis( g finite closing dimensional parenthesis ) space to the of power exterior of forms * closing and parenthesis the Fr´echet totopology the power on S of.and Ther metaplectic alpha oslash group L open parenthesis g closing\noindent parenthesisG˜ (acts finite on s commaE by dimensional the representation ) space of exterior forms and the Fr \ ’{ e} chet topology on S . The metaplecticwhere alpha in group bigwedge $ to\ thetilde power{ρG:} ofG˜$ r→ VAut actsto the ( E onpower) E defined byof * thecomma by representation the s in formula S comma r = 0 comma period period period comma 2 l comma and it is extended by linearity also for \ centerlinenon hyphen{ homogeneous$ \rho elements: \ tilde period{ρG(g}\)(α ⊗ rightarrows) := (λ(g)∗)∧r$α ⊗ AutL(g)s, ( E ) \quad defined by the formula } For psi = alpha oslashr s in E comma v in V and beta in bigwedge to the power of bullet V to the power of * comma we set iota where α ∈ V ∗, s ∈ S , r = 0, ..., 2l, and it is extended by linearity also for non - homogeneous sub\ [ v\rho psi : = iota( sub g v alpha )V ( oslash\alpha s comma beta\otimes and psi :s = beta ) and : alpha = oslash ( s \lambda ( g ) ˆ{ ∗ } ) ˆ{\wedge r }\alphaelements\otimes . L ( g ) s , \ ] and v times psi : = alpha oslash v times s and extend• these definitions by linearity to non hyphen homogeneous For ψ = α ⊗ s ∈ E , v ∈ and β ∈ V ∗, we set ι ψ := ι α ⊗ s, β ∧ ψ := β ∧ α ⊗ s and elements period Obviously comma the contractionV commaV the exteriorv multiplicationv and the Clifford v · ψ := α ⊗ v · s and extend these definitions by linearity to non - homogeneous elements . Obviously multiplication by a fixed vector or co hyphen vector are continuous on E period , the contraction , the exterior multiplication and the Clifford multiplication by a fixed vector or co \noindentNow commawhere .. we shall $ \ describealpha the\ in decomposition\bigwedge of theˆ{ spacer } E intoV ˆ irreducible{ ∗ } , s \ in $ S $ , r = 0 - vector are continuous on E . ,G-tilde .sub . hyphen . , submodules 2 l period ,$ .. For andit i = 0 comma is extendedbylinearity period period period comma also l comma for let us set m sub i : = i comma Now , we shall describe the decomposition of the space E into irreducible G˜ submodules . ..non and− forhomogeneous i = l plus 1 comma elements period period . period 2 l comma − For i = 0, ..., l, let us set m := i, and for i = l + 1, ...2l, m := 2l − i, and define the m sub i : = 2 l minus i comma and definei the set Capital Xi of pairs of non hypheni negative integers set Ξ of pairs of non - negative integers ForCapital $ \ psi Xi : == braceleftbig\alpha open\ parenthesisotimes i commas \ in j closing$ E parenthesis $ , v in N\ subin 0 timesV $ N and sub 0 $bar\beta i = 0 comma\ in period\bigwedge ˆ{\ bullet } Vperiod ˆ{ ∗period } , comma $ we 2 l s comma e t $ j\ =iota 0 comma{ v period}\ periodpsi period: =comma\ iota m sub i{ bracerightbigv }\alpha period \otimes s , \beta Ξ := {(i, j) ∈ × | i = 0, ..., 2l, j = 0, ..., m }. \wedgeOne can say\ psi the set: Capital = Xi\beta has a shape\wedge of aN triangle0 \Nalpha0 if visualized\otimes in a 2 hyphensi $ plane period open parenthesis See andthe Figure $ vOne 1 canperiod\cdot say below the\ setpsi period Ξ has closing: a shape = parenthesis of\ aalpha triangle We if\ usevisualizedotimes the elements in av 2 - of plane\ Capitalcdot . ( See Xis forthe $ parameterizing Figure and 1 extend . below the these irreducible definitions by linearity to non − homogeneous elementssubmodules. ) . We of Obviously E use period the elements , the of contraction Ξ for parameterizing , the the exterior irreducible submodulesmultiplication of E . and the Clifford multiplication by a fixed vector or co − vector˜ are continuousij on E . In KryacuteslIn open Kr´ysl[ square 1 2 ] bracket for each 1 ( 2i, closingj) ∈ Ξ, squaretwo irreducible bracket forG− eachmodules openE parenthesis± were uniquely i comma defined j closing via parenthesis the in Capital Xi comma twohighest irreducible weights G-tilde of their sub underlying hyphen Harish modules - Chandra E sub plusminux modules and to the by powerthe fact of that ij were they uniquely are irreducible Now , \quad we shallVi describe∗ the decomposition of the space E intoij irreducible defined viasubmodules the highest of weightsV ⊗ ofS± their. For underlying convenience Harish for each hyphen (i, j) Chandra∈ Z × Z \ modulesΞ, we set andE± := 0, and for each $ \ tilde {G} { − }$ submodules . \quad For$i = 0 , . . . , l ,$letusset by the fact(i, j) that∈ Z they× Z, arewe define irreducible submodules of bigwedge to the power of i V to the power of * oslash S sub plusminux period$ m { Fori convenience} : = i , $ \quad andfor$i = l + 1 , . . . 2 l ,$ $for m each{ openi } parenthesis: = i 2 comma l j closing− i parenthesis , $ and in Z define times Z backslash the set Capital $ \Xi Xi$ comma ofpairs we set E of sub non plusminux− negative to integers ij ij ij the power of ij : = 0 comma and for each open parenthesisE := iE comma+ ⊕ E− j. closing parenthesis in Z times Z comma we define \ [ E\Xi to the power: = of ij\{ : = E sub( plus i to , the power j ) of ij oplus\ in EN sub{ minus0 }\ to thetimes power ofN ij period{ 0 }\mid i = 0 ,...,2l,j=0,...,m { i ˜}\} . \ ] In the following theoremIn comma the following the decomposition theorem , the of E decomposition into irreducible of G-tildeE into sub irreducible hyphenG submodules− submodules is describedis described period . Theorem 1 . For r = 0, ..., 2l, the fol lowing decomposition into Theoremi ..r 1− periodr educible .. For r = 0 comma period period period comma 2 l comma .. the fol lowing .. decomposition .. into .. i r-r\noindent educible One can say the set $ \Xi $ has a shape of a triangle if visualized in a 2 − plane . ( See the Figure 1 . below . ) We use the elements of $ \Xi $ for parameterizing the irreducible G-tilde sub hyphen modules r bigwedge V to the power of * oslash˜ S sub plusminux simeq bigoplus E sub plusminux to the powersubmodules of rj holds of period E . j open parenthesis r comma j closing parenthesisG−modules in Capital Xi Proof period .. See Kryacutesl open square bracket 1 2 closing square bracketr period square In Kr\ ’{ y} sl[12]foreach $( i , j ) \ in \Xi , $ two irreducible $ \ tilde {G} { − }$ The following remark on the multiplicity^ structure∗ of theM modulerj E is cru hyphen V ⊗ S± ' E± holds. modulescial period $ It E follows ˆ{ i j from} {\ thepm prescriptions}$ were for uniquely the highest weights of the underlying definedHarish hyphen via the Chandra highest modules weights of E sub of plusminux their to underlying the power of Harish ij openj parenthesis− Chandra see modules Kryacutesl and open square bracket 1 by the fact that they are irreducible submodules of $ \bigwedge ˆ{ i } V ˆ{ ∗ } \otimes S {\pm } 3 closing square bracket closing parenthesis period (r, j) ∈ Ξ . $Remark For period convenience .. 1 period .. For any open parenthesis r comma j closing parenthesis comma open parenthesis r comma k closing foreach $( i , j ) \ in Z \times Z \setminus \Xi , $ we s e t $ E ˆ{ i j } {\pm } parenthesisProof in Capital . XiSee such Kr´ysl[12] that j equal-negationslash.  The following k remark comma on we the have multiplicity structure of the module E :E = sub plusminux 0is cru ,$andforeach - to the power of rj negationslash-similarequal $( i , j E ) sub plusminux\ in Z to\ thetimes power ofZ rk ,$ wedefine cial . It follows from the prescriptions for the highest weights of the underlying \ begin { a l i g n ∗} ij Harish - Chandra modules of E±( see Kr´ysl[ 1 3 ] ) . E ˆ{ i j Remark} : .=1 E . ˆ{ Fori j any} ({r,+ j),}\(r, k)oplus∈ Ξ such thatE ˆj{ 6=ik, j we} { have − } . \end{ a l i g n ∗} Erj 6' Erk \ hspace ∗{\ f i l l } In the following theorem , the± decomposition± of E into irreducible $ \ tilde {G} { − }$ submodules

\noindent is described . Theorem \quad 1 . \quad For$r=0 , . . . , 2 l ,$ \quad the fol lowing \quad decomposition \quad i n t o \quad i $ r−r $ e d u c i b l e

\ begin { a l i g n ∗} \ tilde {G} { − } modules \\ r \\\bigwedge V ˆ{ ∗ } \otimes S {\pm }\simeq \bigoplus E ˆ{ r j } {\pm } holds . \\ j \\ ( r , j ) \ in \Xi \end{ a l i g n ∗}

\noindent Proof . \quad See Kr\ ’{ y} s l $ [ 1 2 ] . \ square $ The following remark on the multiplicity structure of the module E is cru −

\noindent cial . It follows from the prescriptions for the highest weights of the underlying

\noindent Harish − Chandra modules of $ E ˆ{ i j } {\pm } ( $ see Kr\ ’{ y} s l [ 1 3 ] ) .

\noindent Remark . \quad 1 . \quad Forany$( r , j ) , ( r , k ) \ in \Xi $ such that $ j \ne k , $ we have

\ [ E ˆ{ r j } {\pm }\not\simeq E ˆ{ rk } {\pm }\ ] 3 1 6 .. S period KR Yacute SL \noindentLine 1 E sub3 1 plusminux 6 \quad toS the . KR power $ of\ 0acute E sub{Y plusminux} $ SL to the power of 1 0 E sub plusminux to the power of 2 0 E sub plusminux to the power of 3 0 E sub plusminux to the power of 40 E sub plusminux to the power of 5 0 E sub plusminux to the power\ [ \ begin of 60{ Linea l i g n2 e E d sub} E plusminux ˆ{ 0 } {\ to thepm power} E of ˆ 1{ 11 E sub 0 plusminux} {\pm to} theE power ˆ{ 2 of 2 0 1} E{\ subpm plusminux} E ˆ to{ the3 power 0 } of{\ 3 pm } E1 E ˆ{ sub40 plusminux} {\pm to} theE power ˆ{ 5 of 4 10 E} sub{\ plusminuxpm } E to ˆ{ the60 power} {\ ofpm 5 1}\\ Line 3 E sub plusminux to the power of 2 2 E sub E ˆ{ 1 1 } {\pm } E ˆ{ 2 1 } {\pm } E ˆ{ 3 1 } {\pm } E ˆ{ 4 1 } {\pm } E ˆ{ 5 plusminux to3 1 the 6 power S . KR ofY´ 3SL 2 E sub plusminux to the power of 42 Line 4 E sub plusminux to the power of 3 3 1 }Fig{\ periodpm }\\ .. 1 : .. Decomposition of bigwedge to the power of bullet V to the power of * oslash S sub plusminux for 2 l = 6 E ˆ{ 2 2 } {\pm } E ˆ{ 30 2 } 10{\pm20 } 30E ˆ{40 42 50} {\60pm }\\ period E± E± E± E± E± E± E± E ˆ{ 3 3 } {\pm }\end{ a l i g n e d }\ ] 11 21 31 41 51 open parenthesis any combination of plusminux atE both± E sides± ofE± theE preceding± E± relation is allowed closing parenthesis period Thus in particular comma bigwedge to the power of r V to the power of * oslash S is multiplicity hyphen free for each r = 0 E22 E32 E42 comma period period period comma 2 l period ± ± ± \ centerline { Fig . \quad 1 : \quad Decomposition of $ \bigwedge33 ˆ{\ bullet } V ˆ{ ∗ } \otimes 2 period .. Moreover comma it is known that E sub plusminux to the powerE of± rj simeq E sub minusplus to the power of sj .. for S {\pm }$for$2 l = 6 .$ } each open parenthesis r comma j closing parenthesis comma open parenthesisV• ∗ s comma j closing parenthesis in Capital Xi period .. Fig . 1 : Decomposition of V ⊗ S± for 2l = 6. One \ hspace ∗{\ f i l l }( any combination( any combination of $ \pm of ±$at bothat both sides of sides the preceding of the relation preceding is allowed relation ) . is allowed ) . cannot change the order of plus and minusV atr precisely∗ one side of the preceding Thus in particular , V ⊗ S is multiplicity - free for each r = 0, ..., 2l. isomorphism without2 changing . Moreover its trueness , it is known period that Erj ' Esj for each (r, j), (s, j) ∈ Ξ. One \ centerline3 period .. From{Thus the in preceding particular two items $ comma , \ onebigwedge gets± immediatelyˆ{ r∓ } thatV ˆ there{ ∗ are } no \otimes $ S is multiplicity − free for each $r=0,...,2l.$cannot change the order of + and − at precisely} one side of the preceding isomorphism without submoduleschanging of bigwedge its trueness to the . power of i V to the power of * oslash S isomorphic to E sub plusminux to the power of i plus 1 comma i plus 1 for each i = 0 comma period period period comma l minus 1 period \ hspace ∗{\ f i l l }2 . \quad Moreover3 . From the, it preceding is known two items that , one $ E gets ˆ{ immediatelyr j } {\pm that}\ theresimeq are no E ˆ{ s j } {\mp }$ In the Figure 1 comma one can see the decompositionVi structure∗ of bigwedge toi+1 the,i+1 power of bullet V to the power of * oslash S \quad foreach$( r ,submodules j ) of ,V (⊗ S sisomorphic , j to )E± \ infor each\Xi i = 0., ..., $ l −\quad1. One sub plusminux in the V• ∗ In the Figure 1 , one can see the decomposition structure of V ⊗ S± in the case of lcase = 3 ofperiodl = 3 For. For i =i = 0 0 comma, ..., 6, the periodith column period period constitutes comma of the 6 comma irreducible the i modules to the power in which of t hthe columnS − constitutes of the irreduciblecannot change modules the order of $ + $ and $ − $ at precisely one side of the preceding± isomorphismvalued without exterior forms changing of form its- degree truenessi decompose . . in which the S sub plusminux hyphen valued exterior forms∗ of formij hyphen degree i decompose˜ period In the next theorem , the decomposition of V ⊗ E , (i, j) ∈ Ξ, into irreducible G− submodules In the nextis described theorem . comma Let us remindthe decomposition the reader that of V due to the to our power convention of * oslash E to the power of ij comma open parenthesis i comma\ hspace j closing∗{\ f i lparenthesis l }3 . \quad in CapitalFrom Xi the comma preceding into irreducible two items , one gets immediately that there are no Eij = 0 for (i, j) ∈ Z × Z \ Ξ. We will use this theorem in the proofs of Lemma 6 and Theorem 7 on G-tilde sub hyphen submodules is described period Let us remind the reader that due to our convention \ hspace ∗{\thef iellipticity l l } submodules of the truncated of $ symplectic\bigwedge twistorˆ{ complexesi } V ˆ .{ ∗ } \otimes $ S isomorphic to $ E ˆ{ i E to theTheorem power of ij 2 = . 0 forFor open(i, parenthesisj) ∈ Ξ, we have i comma j closing parenthesis in Z times Z backslash Capital Xi period We will +use this 1 theorem , i in the + proofs 1 } of{\ Lemmapm } 6$foreach$i = 0 , . . . , l − 1 . $ and Theorem 7 on the ellipticity of the truncated symplectic twistor complexes periodi + 1 \ hspace ∗{\ f i l l } In the Figure 1 , one can see the decomposition structure of $ \bigwedge ˆ{\ bullet } Theorem 2 period .. For open parenthesis∗ ij i comma^ ∗ j closingi+1 parenthesis,j−1 i+1 in,j Capitali+1,j Xi+1 comma we have V ˆLine{ ∗ 1 } i plus \otimes 1 Line 2 openS parenthesis{\(Vpm⊗ E}$ V) ∩ toin( the theV power⊗ S) of' E * oslash E⊕ toE the power⊕ E of ij. closing parenthesis cap open parenthesis bigwedge V to the power of * oslash S closing parenthesis simeq E to the power of i plus 1 comma j minus 1 oplus E to the power Proof . See Kr´ysl[13].  of\noindent i plus 1 commacaseof$l j oplus E to the = power 3 of i .$For$iplus 1 comma j plus 1 = period 0 , . . . , 6 ,$the$iˆ{ t h }$ columnRemark constitutes . ofRoughly the irreducible speaking , the modules theorem says that the wedge multiplication sends Proof period .... See Kryacutesl openij square bracket 1 3 closing square bracket period squarest in whicheach the irreducible $ S {\ modulepm }E −into$ at valued most three exterior “ neighbor forms ” modules of form in the− (i +degree 1) $ i $ decompose . Remarkcolumn period .... . ( Roughly See the Figure speaking 1 . comma ) the theorem says that the wedge multiplication sends each irreducible2 . 3 . moduleOperators E to related the power to a Howe of ij type into correspondence at most three quotedblleft. In this neighbor section , quotedblright we will introduce modules in the open parenthesisIn the next i plus theorem 1 closing parenthesis , the decomposition to the power of of s t $ V ˆ{ ∗ } \otimes E ˆ{ i j } , ( i , j ) \ in five\Xi continuous, $ linear into operators irreducible acting on the space E of symplectic spinor valued exterior forms . column periodLet us open mention parenthesis that these See operators the Figure are 1 periodrelated closing to the parenthesisso called Howe type correspondence for the $2 period\ tilde 3{ periodG} { .. − Operators }$ submodules related to a Howeis described type correspondence . Let us period remind .. In this the section reader comma that we due will to our convention metaplectic group Mp (V, ω0) acting on introduce five continuous linear operators acting on the space E of symplectic \noindentspinor valued$ exterior E ˆ{ i forms j } period= 0$for$( Let us mention that these i operators , j are ) related\ in to theZ \times Z \setminus \Xi . $so called We will Howe use type this correspondence theorem for in the the metaplectic proofs group of Lemma Mp open 6 parenthesis V comma omega sub 0 closing parenthesis actingand onTheorem 7 on the ellipticity of the truncated symplectic twistor complexes . \noindent Theorem 2 . \quad For $ ( i , j ) \ in \Xi , $ we have

\ [ \ begin { a l i g n e d } i + 1 \\ ( V ˆ{ ∗ } \otimes E ˆ{ i j } ) \cap ( \bigwedge V ˆ{ ∗ } \otimes S) \simeq E ˆ{ i + 1 , j − 1 }\oplus E ˆ{ i + 1 , j }\oplus E ˆ{ i + 1 , j + 1 } . \end{ a l i g n e d }\ ]

\noindent Proof . \ h f i l l See Kr\ ’{ y} s l $ [ 1 3 ] . \ square $

\noindent Remark . \ h f i l l Roughly speaking , the theorem says that the wedge multiplication sends

\noindent each irreducible module $ E ˆ{ i j }$ into at most three ‘‘ neighbor ’’ modules in the $ ( i + 1 ) ˆ{ s t }$

\noindent column . ( See the Figure 1 . )

\noindent 2 . 3 . \quad Operators related to a Howe type correspondence . \quad In this section , we will introduce five continuous linear operators acting on the space E of symplectic spinor valued exterior forms . Let us mention that these operators are related to the so called Howe type correspondence for the metaplectic group Mp $ ( V , \omega { 0 } ) $ a c t i n g on ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 3 1 7 ELLIPTICITYE via the representation OF THE SYMPLECTIC rho period For TWISTOR r = 0 comma COMPLEX period\quad period3 period 1 7 comma 2 l and alpha oslash s in bigwedge to the powerE via of r the V to representation the power of * oslash $ S\ commarho we.$For$r set = 0 , . . . , 2 l$and$ \alpha \otimesLine 1 r r pluss 1\ 2in l Line\ 2bigwedge F to the powerˆ{ r of} plusV : bigwedgeˆ{ ∗ } V \ tootimes the power$ of S * , oslash we s S e t right arrow bigwedge V to the power of * oslash S comma F to the power of plus open parenthesis alpha oslash s closing parenthesis : = i divided by 2 sum i = 1 epsilon \ [ \ begin { a l i g n e d } r r + 1 2 l \\ to the power ofELLIPTICITY i and alphaOF oslash THE e SYMPLECTIC sub i times s TWISTOR COMPLEX 3 1 7 E via the representation ρ. For F ˆ{ + } : \bigwedge VrV∗ ˆ{ ∗ } \otimes S \rightarrow \bigwedge V ˆ{ ∗ } \otimes and r = 0, ..., 2l and α ⊗ s ∈ V ⊗ S , we set SF to , the F power ˆ{ + of} minus( : r\ bigwedgealpha V\ tootimes the powers of * oslash ) : S right = arrow\ f r ar c {\ minusimath 1 bigwedge}{ 2 V}\ to thesum power{ ofi * oslash = 1S } comma\ epsilon F toˆ the{ poweri }\ ofwedge minus open\alpha parenthesis\otimes alpha oslashe s closing{ i }\ parenthesiscdot r : = rs 1+\ divided 1end 2{l a by l i g 2 n sum e d }\ from] i = 1 to 2 l omega ^ ^ ı X to the power of ij iota sub eF sub+ : i alpha oslash∗ ⊗ S → e sub j∗ times⊗ S,F s +(α ⊗ s) := i ∧ α ⊗ e · s V V 2 i and extend them linearly period Further comma we shall introduce the operatorsi=1 H comma E to the power of plus and \noindentE to the powerand of minus acting also continuously on the space E = bigwedge to the power of bullet V to the power of * oslash and S period We define \ [ F ˆ{ − } : r {\bigwedge } V ˆ{ ∗ } \otimes S \rightarrow r − 1 {\bigwedge } V ˆ{ ∗ } H : = 2 open brace F to the power of plus comma F to the power of minus closingi=1 brace and E to the power of plusminux : = \otimes S , F ˆ{ −^ } ( \alpha^ \otimes s )1 X : = \ f r a c { 1 }{ 2 }\sum ˆ{ i = plusminux 2 open brace F− to: ther power∗ ⊗ ofS plusminux→ r − 1 comma∗ ⊗ S,F F to− the(α ⊗ powers) := of plusminuxωijι α closing⊗ e · s brace comma 1 } { 2 l }\omega ˆ{ i jV }\ iota { Ve { i }}\alpha 2 \otimesei j e { j }\cdot s \ ] where open brace comma closing brace denotes the anti hyphen commutator in2l the associative algebra End open parenthesis E closing parenthesis period By a and extend them linearly . Further , we shall introduce the operators H,E+ and E− acting also direct computation comma we get • continuously on the space E = V ∗⊗ S . We define \noindentEquation:and open extend parenthesis them 2 closing linearly parenthesisV . Further .. E to the , power we shall of minus introduce open parenthesis the operators alpha oslash s $ closing H parenthesis , E ˆ{ + }$ =and i divided by 2 omega to the power of ij iota sub e sub i iota sub e sub j alpha oslash s H := 2{F +,F −} and E± := ±2{F ±,F ±}, $for E any ˆ{ alpha − }$ oslash acting s in bigwedge also continuously to the power of on bullet the V space to the powerE $ =of * oslash\bigwedge S periodˆ{\ Thusbullet comma} weV see ˆ that{ ∗ the } operator\otimes E$ towhere the S .power{ We, } denotes ofdefine minus the acts anti on the - commutator form hyphen in part the associative algebra End ( E ) . By a direct of a symplecticcomputation spinor , valued we get exterior form only period Because of that we will write \ [E H to the : power = of 2 minus\{ alphaF oslash ˆ{ + s} instead, of F E ˆ{ to − the } power \} of minusand open E ˆ parenthesis{\pm } alpha: = oslash\pm s closing2 parenthesis\{ F ˆ{\pm } , F ˆ{\pm }\} , \ ] simply period ı E−(α ⊗ s) = ωijι ι α ⊗ s (2) In the next lemma comma we sum hyphen up some known2 factsei andej derive some new information on the operators F to the power of plusminux comma E to the power of plusminux and H which we shall need in the proof of \noindent where \{ ,V•\} ∗ denotes the anti − commutator− in the associative algebra End ( E ) . By a the ellipticityfor any α⊗s ∈ V ⊗ S . Thus , we see that the operator E acts on the form - part of a symplectic directof the truncated computationspinor valued symplectic exterior , we twistor form get onlycomplexes . Because period of that we will write E−α ⊗ s instead of E−(α ⊗ s) simply Lemma. 3 period .. 1 period .. The operators F to the power of plusminux comma E to the power of plusminux and H are G-tilde\ begin sub{ a l hyphen i g n ∗} equivariantIn the period next lemma , we sum - up some known facts and derive some new information \ tag2 period∗{$ (on .. Forthe 2 ioperators = ) 0 comma $} EF ± ˆ,E period{± −and } periodH( which period\alpha we comma shall\ need lotimes comma in the the proofs operator of ) the F = ellipticity to the\ f r power a c of{\ the ofimath minustruncated}{ bar2 E to}\ theomega power ofˆ{ i j } im\ iota i = 0 and{ symplectice for{ i =i l}}\ comma twistoriota period complexes{ periode . { periodj }}\ commaalpha 2 l comma\otimes the operators \end{ a l i g n ∗} ± ± F to theLemma power of 3 plus . bar1 . subThe E im operators sub i = 0F period,E and H are G˜−equivariant. 3 period .. The associative2 . algeFor bra i = 0, ..., l, the operator F −|Eimi = 0 and for i = l, ..., 2l, the operator \noindentEnd sub G-tildef o r any open $parenthesis\alpha E closing\otimes parenthesiss :\ in = open\bigwedge brace A : Eˆ right{\ bullet arrow E} .. continuousV ˆ{ ∗ } .. bar \otimes A rho open$ S . Thus , we see that the operator $ E ˆ{ − }$ acts on the form − part + parenthesis g closing parenthesis = rho open parenthesisF g|E closingimi = parenthesis 0. A for al l g in tilde-G closing brace ofis acomma symplectic .. as an associative spinor algebravalued comma exterior finitely form generated only by . F Because to the power of of that plus .. we and will F to writethe power of minus .. and 3 . The associative alge bra the$ E ˆ{ − } \alpha \otimes s $ instead of $Eˆ{ − } ( \alpha \otimes s ) $ simply . End ( E ) := {A : E → E continuous | Aρ(g) = ρ(g)A for al l g ∈ G˜} G-tilde sub hyphen equivariantG˜ projections p plusminux : S right arrow S sub plusminux period is , as an associative algebra , finitely generated by F + and F − and the \ hspace4 period∗{\ ..f For i l l alpha} In theoslash next s in bigwedge lemma , to we the sum power− ofup r V some to the known power of facts * oslash and S comma derive the some fol lowing new re information la tions hold on E G˜ projectionsp± : S → S . \noindentEquation:on open the parenthesis operators 3 closing $ parenthesis F ˆ−{\equivariantpm ..} open, square E ˆ{\ bracketpm } E$ to± the and power $ H of $ plus which comma we E to shall the power need of minus in the proof of the ellipticity of the truncated symplectic twistorVr ∗ complexes . closing square bracket = H4 comma. For openα ⊗ s square∈ V bracket⊗ S , E the to fol the lowing power re of la minus tions comma hold on FE to the power of plus closing square bracket = minus F to the power of minus comma Equation: open parenthesis 4 closing parenthesis .. H open parenthesis alpha oslash\noindent s closingLemma parenthesis 3 . \ =quad 1 divided1 . by\quad 2 openThe parenthesis operators r minus $ l Fclosing ˆ{\ parenthesispm } , alpha E ˆ oslash{\pm s comma}$ and Equation: $ H $ open are + − − + − parenthesis$ \ tilde { 5G closing} { − parenthesisequivariant .. open brace . F} to$ the power[E of,E plus] = commaH, [E iota,F sub] v= closing−F , brace open parenthesis(3) alpha oslash s closing parenthesis = i divided by 2 alpha oslash v times s and open square1 bracket F to the power of minus comma v times \ hspace ∗{\ f i l l }2 . \quad For $i = 0 , .H(α .⊗ s) .= (r ,− l)α l⊗ s, ,$ theoperator(4) $Fˆ{ − }{ \mid } closing square bracket open parenthesis alpha oslash s closing parenthesis = i divided2 by 2 iota sub v alpha oslash s period E ˆ{ im } i = 0$andfor $i = l , . . . , 2 l ,$ theoperator Proof period .. See Kryacutesl+ open square bracketı 1 3 closing square− bracket forı the proof of the items 1 and 2 comma and {F , ιv}(α ⊗ s) = α ⊗ v · s and [F , v·](α ⊗ s) = ιvα ⊗ s. (5) Kryacutesl open square bracket 1 2 closing square2 bracket for a 2 \ [proof F ˆ{ of+ the}{\ itemmid 3 and} { ofE the} relationsim { i in} the= rows 0 open . parenthesis\ ] 3 closing parenthesis and open parenthesis 4 closing Proof . See Kr´ysl[ 1 3 ] for the proof of the items 1 and 2 , and Kr´ysl[ 1 2 ] for a proof of the parenthesis period Now comma suppose item 3 and of the relations in the rows ( 3 ) and ( 4 ) . Now , suppose we are given an element v = v to the power of i e sub i in V comma v to the power of i in R comma i = 1 comma period period we are given an element v = vie ∈ , vi ∈ , i = 1, ..., 2l, and a homogeneous element α⊗s ∈ Vj ∗⊗ period\ centerline comma 2{3 l comma . \quad andThe a homogeneous associativei V algeR bra } V S , j = 0, ..., 2l. First , let us prove the first relation element alpha oslash s in bigwedge to the power of j V to the power of * oslash S comma j = 0 comma period period period in the row ( 5 ) . Using the definition of F +, we may write {F +, ι }(α ⊗ s) = comma\ hspace 2∗{\ l periodf i l l First} $ comma End {\ let ustilde prove{G the}} first( relation $ E $ ) : = \{ A : $v E $ \rightarrow $ E \quad continuous \quad $ \inmid the rowA ....\ openrho parenthesis( g 5 closing ) = parenthesis\rho period( ....g Using ) the A$ definition forall of F to $g the power\ in of plus\ tilde comma{G we}\} may $ write open brace F to the power of plus comma iota sub v closing brace open parenthesis alpha oslash s closing parenthesis = \ hspaceF to the∗{\ powerf i l l } ofi s plus , \ openquad parenthesisas an associative iota sub v alpha algebra oslash s , closing finitely parenthesis generated plus i dividedby $ Fby ˆ 2{ iota+ } sub$ v\quad open and parenthesis$ F ˆ{ − epsilon }$ \quad to theand power the of i and alpha oslash e sub i times s closing parenthesis = i divided by 2 open square bracket epsilon to the power of i and iota sub v alpha oslash e sub i times s plus v to the power of i alpha oslash e sub i times s minus epsilon\ [ \ tilde to the{G power} { of − i andequivariant iota sub v alpha} oslashprojections e sub i times s p closing\pm square:S bracket =\rightarrow i divided by 2 alphaS oslash{\pm v times} . s\ ] period Thus comma the first relation of open parenthesis 5 closing parenthesis follows now by linearity period Now comma let us prove the\ centerline second {4 . \quad For $ \alpha \otimes s \ in \bigwedge ˆ{ r } V ˆ{ ∗ } \otimes $ S , the fol lowing re la tions hold on E }

\ begin { a l i g n ∗} \ tag ∗{$ ( 3 ) $} [ E ˆ{ + } , E ˆ{ − } ] = H , [ E ˆ{ − } , F ˆ{ + } ] = − F ˆ{ − } , \\\ tag ∗{$ ( 4 ) $} H( \alpha \otimes s ) = \ f r a c { 1 }{ 2 } ( r − l ) \alpha \otimes s , \\\ tag ∗{$ ( 5 ) $}\{ F ˆ{ + } , \ iota { v } \} ( \alpha \otimes s ) = \ f r a c {\imath }{ 2 }\alpha \otimes v \cdot s and [ F ˆ{ − } , v \cdot ]( \alpha \otimes s ) = \ f r a c {\imath }{ 2 } \ iota { v }\alpha \otimes s . \end{ a l i g n ∗}

\noindent Proof . \quad See Kr\ ’{ y} sl [ 1 3 ] for the proof of the items 1 and 2 , and Kr\ ’{ y} s l [ 1 2 ] f o r a proof of the item 3 and of the relations in the rows ( 3 ) and ( 4 ) . Now , suppose

\noindent we are given an element $ v = v ˆ{ i } e { i }\ in V , v ˆ{ i }\ in R , i = 1 , . . . , 2 l ,$ andahomogeneous element $ \alpha \otimes s \ in \bigwedge ˆ{ j } V ˆ{ ∗ } \otimes $ S $ , j = 0 , . . . , 2 l .$ First ,letusprovethefirst relation

\noindent in the row \ h f i l l ( 5 ) . \ h f i l l Using the definition of $ F ˆ{ + } , $ we may write $ \{ F ˆ{ + } , \ iota { v }\} ( \alpha \otimes s ) = $

\ [ F ˆ{ + } ( \ iota { v }\alpha \otimes s ) + \ f r a c {\imath }{ 2 }\ iota { v } ( \ epsilon ˆ{ i }\wedge \alpha \otimes e { i }\cdot s ) = \ f r a c {\imath }{ 2 } [ \ epsilon ˆ{ i }\wedge \ iota { v }\alpha \otimes e { i }\cdot s + v ˆ{ i } \alpha \otimes e { i }\cdot s − \ epsilon ˆ{ i }\wedge \ iota { v }\alpha \otimes e { i }\cdot s ] = \ f r a c {\imath }{ 2 }\alpha \otimes v \cdot s . \ ]

\noindent Thus , the first relation of ( 5 ) follows now by linearity . Now , let us prove the second ı ı ı F +(ι α ⊗ s) + ι (i ∧ α ⊗ e · s) = [i ∧ ι α ⊗ e · s + viα ⊗ e · s − i ∧ ι α ⊗ e · s] = α ⊗ v · s. v 2 v i 2 v i i v i 2 Thus , the first relation of ( 5 ) follows now by linearity . Now , let us prove the second 3 1 8 .. S period KR Yacute SL \noindentrelation at3 the 1 8row\quad open parenthesisS . KR $ 5\ closingacute parenthesis{Y} $ SL period Using the definition of F to the power of minus and the commutationrelation relationat the row ( 5 ) . Using the definition of $ F ˆ{ − }$ and the commutation relation open parenthesis 1 closing parenthesis comma we get F to the power of minus open parenthesis alpha oslash v times s closing parenthesis\ [( 1 = 1 ) divided , by we 2 open get parenthesis Fˆ{ omega − } to( the power\alpha of ij iota\otimes sub e sub iv alpha\cdot oslash e subs j times ) = v times\ f r as cclosing{ 1 }{ 2 } ( \omega ˆ{ i j }\ iota { e { i }}\alpha \otimes e { −j }\cdot v \cdot s ) parenthesis3 = 1 1 8 divided S . KR byY 2´ omegaSL relation to the at power the row of ij ( iota 5 ) sub . Using e sub the i alpha definition oslash of openF parenthesisand the commutation v times e sub j times s minus =i omega\ f r a sub c {relation 01 open}{ parenthesis2 }\omega e subˆ j{ commai j }\ v closingiota parenthesis{ e { si closing}}\ parenthesisalpha \ =otimes ( v \cdot e { j } \cdotv timess F to− the power \imath of minus\omega open parenthesis{ 0 } alpha( oslash e { sj closing} , parenthesis v ) plus s i divided ) by= 2\ ] omega to the power of ij iota sub e sub i alpha oslash v sub j s = v times1 F to the power of minus1 open parenthesis alpha oslash s closing parenthesis plus i divided by 2 iota sub(1), v wegetF alpha− oslash(α ⊗ v s· periods) = Thus(ωijι commaα ⊗ e the· v · seconds) = ω relationijι α ⊗ (v · e · s − ıω (e , v)s) = 2 ei j 2 ei j 0 j \noindentat the row open$ v parenthesis\cdot 5 closingF ˆ{ − parenthesis } ( is\alpha proved period\otimes square s ) + \ f r a c {\imath }{ 2 }\omega ˆ{ i j } \ iota { e { i }}\alpha \otimes v { j } s = v \cdot F ˆ{ − } ( \alpha \otimes Remarkv period· F −(α The⊗ s operators) + ı ωijι Fα to⊗ thev s power= v · F of−( plusminuxα ⊗ s) + ı commaι α ⊗ s. EThus to the , the power second of plusminux relation and H satisfy the commutation s ) + \ f r a c {\imath2 ei}{ 2 j }\ iota { v }\2 v alpha \otimes s . $ Thus , the second relation and anti hyphenat the commu row ( 5 hyphen ) is proved .  t ation relationsRemark identical . The operatorsto that onesF ± which,E± and areH satisfiedsatisfy by the the commutation usual generators and anti of - commu - \noindent at the row ( 5 ) is proved $ . \ square $ the orthot hyphenation relations symplectic identical super toLie that algebra ones osp which open are parenthesis satisfied 1by bar the 2 usualclosing generators parenthesis of period the ortho - 3 periodsymplectic .. Symplectic super twistor Lie algebra complexesosp(1 and| 2) their. elliptic parts \noindent Remark . The operators $ F ˆ{\pm } , E ˆ{\pm }$ and $ H $ satisfy the commutation and anti − commu − In this section comma3 we . defineSymplectic the notion twistor of a Fedosov complexes manifold and comma their recall elliptic some informa parts hyphen tion on its curvatureIn this section comma , we introduce define the a symplectic notion of analogue a Fedosov of manifold the spin structure , recall some open informa parenthesis - tion the on its \noindent t ation relations identical to that ones which are satisfied by the usual generators of metaplecticcurvature structure , introduce closing parenthesis a symplectic and analogue define the of the symplectic spin structure twistor ( complexes the metaplectic period structure ) and the ortho − symplectic super Lie algebra $ osp ( 1 \mid 2 ) . $ Let opendefine parenthesis the symplectic M comma twistor omega complexes closing parenthesis . be a symplectic manifold period Let us consider an affine torsion hyphen free sym hyphen Let (M, ω) be a symplectic manifold . Let us consider an affine torsion - free sym - \ centerline {3 . \quad Symplectic twistor complexes and their elliptic parts } plectic connectionplectic connection nabla on∇ openon ( parenthesisM, ω) and denote M comma the omega induced closing connection parenthesis on Γ(M, andV denote2 T ∗M) the by induced∇ as well connection . on Capital Gamma open parenthesis M comma bigwedge to the power of 2 T to the power of * M closing parenthesis In this sectionLet us recall , wethat define by torsion the - free notion and symplectic of a Fedosov , we mean manifoldT (X,Y ) := ,∇ recallX Y − ∇Y someX − [X,Y informa] = 0 − by nablafor as allwellX,Y period∈ X Let(M us) and recall∇ thatω = by 0. Such torsion connections hyphen free are and usually symplectic called comma Fedosov we connections mean T open , and parenthesis X comma Ytion closing on parenthesis its curvature : = , introduce a symplectic analogue of the spin structure ( the metaplecticthe triple structure (M, ω, ∇) ) a Fedosovand define manifold the . See symplectic the Introduction twistor and the complexes references . therein for more nabla subinformation X Y minus on nabla these sub Y X minus open square bracket X comma Y closing square bracket = 0 for all X comma Y in X open parenthesis M closing parenthesis and nabla∇ omega = 0 period Such connections are \ hspace ∗{\connectionsf i l l } Let . The $ curvature( M tensor , \Romegaof a Fedosov) $ connection be a symplectic is defined in manifold the . Let us consider an affine torsion − f r e e sym − usually calledclassical Fedosov way , connectionsi . e . , formally comma by andthe same the triple formula open as parenthesis in the Riemannian M comma geometry omega comma . It nabla closing parenthesis a Fedosov manifold period ∇ \noindentis knownplectic , see connection Vaisman [ 1 9 ] $ , that\nablaR splits$ on into $two ( parts M , namely , \ intoomega the extended) $ and denote the induced connection on See the Introduction and the references therein for more information on these ∇ ∇ symplectic Ricci and Weyl curvature tensor fields , here denoted by σe and W respectively . Let $ \connectionsGamma (M, period The curvature\bigwedge tensor Rˆ{ to2 the} powerT ˆ{ of nabla∗ } ofM a Fedosov ) $ connection is defined in the by $ \nablaus display$ the as definitions well . Letof these us two recall curvature that parts by although torsion − free and symplectic , we mean $ T classical way comma i period e period comma formally by the same formula2l as in the Riemannian geometry period It ( X ,we shall Y not ) use : them = explicitly $ . For a symplectic frame (U, {ei}i=1),U ⊆ M, we have the following is knownlo comma cal formulas see Vaisman open square bracket 1 9 closing square bracket comma that R to the power of nabla splits into two$ parts\nabla comma{ namelyX } Y into the− extended \nabla { Y } X − [ X , Y ] = 0$forall$X , Y \ in X ( M ) $ and $ \nabla \omega = 0 . $ Such connections are symplectic Ricci and Weyl curvature tensor fields comma here denoted by sigma-tildewideσ := Rk to, the power of nabla and W to the usually called Fedosov connections , and the triple $ ( Mij , ikj\omega , \nabla ) $ a Fedosov manifold . power of nabla ∇ See the Introduction2(l and+ 1)ij theσ kn references:= ωinσjk − ωik thereinσjn + ωjnσ forik − moreωjkσin information+ 2σijωkn and on these respectively period Let us displaye the definitions of these two curvature parts although ∇ ∇ ∇ we shall not use them explicitly period For a symplectic frame open parenthesisW := UR comma− σe open, brace e sub i closing brace sub \noindent connections . The curvature tensor $ R ˆ{\nabla }$ of a Fedosov connection is defined in the i = 1 to thewhere poweri, of j, 2 k, l n closing= 1, ..., parenthesis2l. Let us call comma a Fedosov U subset manifold equal M (M, comma ω, ∇) ofweRicci type if have the following lo cal formulas \noindentLine 1 sigmaclassical sub ij : = R way to the , power i . ofe k . sub , iformally kj comma Line by 2 the 2 open same parenthesis formula l plus as 1 in closing the parenthesis Riemannian ij sigma-tildewide geometry . It to the power of nabla sub kn : = omega sub in sigma subW jk∇ = minus 0. omega sub i k sigma sub jn plus omega sub jn sigma sub i k minus\noindent omega subis known j k sigma , sub see in Vaisman plus 2 sigma [ 1 sub 9 ij ] omega , that sub kn $Rˆ and{\ Linenabla 3 W to}$ the powersplits of nabla into : two = R parts to the power , namely of into the extended nabla minusRemark sigma-tildewide . to the powerBecause of nabla the comma Ricci curvature tensor field σij is symmetric ( see Vaisman \noindent symplectic Ricci and Weyl curvature tensor fieldsij , here denoted by $ \ widetilde {\sigma} ˆ{\nabla }$ where i comma[ 1 9 ] ) j ,comma a possible k comma candidate n = for 1 thecomma scalar period curvature period , namely period commaσ ωij, is 2 l zero period . Let us call a Fedosov manifold open parenthesisand $ WExample M ˆ{\ commanabla omega. }$It comma is easy nabla to see closing that each parenthesis Riemann of surfaceRicci type equipped if with its volume form as the respectivelyW to thesymplectic power . of Let nabla form us and = 0 display with period the Riemann the definitions connection is of a Fedosov these manifold two curvature parts although l Remarkof period Ricci .... type Because . Further the for Ricci any curvaturel ≥ 1, the tensor Fedosov field manifold sigma sub (CP ij is, ω symmetricFS, ∇) is also open a Fedosov parenthesis manifold see Vaisman \noindent we shall not use them explicitly . For a symplectic frame $ ( U , \{ e { i } open squareof Ricci bracket type 1 . 9 Hereclosing, ω squareFS is the bracket K¨ahlerform closing parenthesis associated comma to the Fubini a possible - Study candidate metric for and the to scalar the curvature comma namely\} ˆ{ sigma2complex l to} the{ structure poweri = of onij omega 1 the} complex sub),U ij comma projective is zero\ spacesubseteq period M , $ we have the followingl lo cal formulas ExampleCP period, and ..∇ Itis is theeasy Riemannian to see that each connection Riemann associated surface equipped to the Fubini with its- Study volume metric . Now , let us form as theintroduce symplectic the metaplectic form and with structure the Riemann the definition connection of which is a we Fedosov have manifold \ [ \ofbegin Ricci{ typesketa l i g periodched n e d briefly}\ Furthersigma in for the any{ Introductioni l j greater} : equal . =For 1 comma a R symplectic ˆ{ thek } Fedosov{ manifoldi manifold kj (M}2l,open ω,) of\\ parenthesis dimensionCP 2l, let to the us power of l comma omega2 sub ( FSdenote l comma +the nabla bundle 1 closing )of symplectic i parenthesis j {\ repwidetildee `isres also in aTM{\sigmaby P and}} theˆ{\ footnabla - point} { kn } : = \omega { in } \sigmaFedosov{ manifoldjk } of− Ricci \omega type period{ i Here k comma}\sigma omega sub{ FSjn is} the+ Kadieresishler\omega form{ jn associated}\sigma to the{ i k } − \omegaFubini hyphen{ j Study k }\ metricsigma and to{ thein complex} + structure 2 \sigma on the complex{ i j }\ projectiveomega space{ kn } and \\ WCP ˆ{\ to thenabla power} of l: comma = and R nablaˆ{\nabla is the Riemannian} − \ widetilde connection associated{\sigma} toˆ the{\ Fubininabla hyphen} , Study\end{ metrica l i g n period e d }\ ] Now comma let us introduce the metaplectic structure the definition of which we have sket ched briefly in the Introduction period For a symplectic manifold open parenthesis M to the power of 2 l comma omega closing\noindent parenthesiswhere of dimension $i , j , k , n = 1 , . . . , 2 l .$ LetuscallaFedosovmanifold $ (2 l comma M , let us\omega denote the, bundle\nabla of symplectic) $ rep of egrave Ricci res in type TM by if P and the foot hyphen point \ begin { a l i g n ∗} W ˆ{\nabla } = 0 . \end{ a l i g n ∗}

\noindent Remark . \ h f i l l Because the Ricci curvature tensor field $ \sigma { i j }$ is symmetric ( see Vaisman

\noindent [ 1 9 ] ) , a possible candidate for the scalar curvature , namely $ \sigma ˆ{ i j } \omega { i j } , $ i s zero .

\noindent Example . \quad It is easy to see that each Riemann surface equipped with its volume form as the symplectic form and with the Riemann connection is a Fedosov manifold

\noindent of Ricci type . Further for any $ l \geq 1 , $ the Fedosov manifold $ ( CPˆ{ l } , \omega { FS } , \nabla ) $ i s a l s o a Fedosov manifold of Ricci type . Here $ , \omega { FS }$ i s the K\”{a} hler form associated to the Fubini − Study metric and to the complex structure on the complex projective space

\noindent $ CP ˆ{ l } , $ and $ \nabla $ is the Riemannian connection associated to the Fubini − Study metric . Now , let us introduce the metaplectic structure the definition of which we have

\noindent sket ched briefly in the Introduction . For a symplectic manifold $ ( M ˆ{ 2 l } , \omega ) $ of dimension $ 2 l , $ let us denote the bundle of symplectic rep $ \grave{e} $ res in $TM$ by $ P $ and the foot − point ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 3 1 9 \ hspaceprojection∗{\ f from i l l } PELLIPTICITY onto M by p period OF THE Thus SYMPLECTIC open parenthesis TWISTOR p : P right COMPLEX arrow\ Mquad comma3 G1 closing 9 parenthesis comma where G simeq Sp open parenthesis 2 l comma R closing parenthesis comma is a \noindentprincipal Gprojectionfrom hyphen bundle over M$P$ period onto As in the $M$ subsection by 2 $p period .$1 comma Thus let lambda $( : p G-tilde : right P arrow\rightarrow G be a Mmember , G ) ,$ where $G \simeq Sp ( 2 l , R ) ,$isa principal $ G − $ bundle over $M . $ As in the subsection 2 . 1 , let $ \lambda : of the isomorphism class of the non hyphen trivialELLIPTICITY two hyphen OF fold THE coverings SYMPLECTIC of the TWISTOR symplectic COMPLEX 3 1 9 \ tilde {G}\rightarrow G $ be a member group Gprojection period In from particularP onto commaM by G-tildep. Thus simeq (p : P Mp→ M,G open), parenthesiswhere G ' Sp 2 l(2 commal, R), is R a closing principal parenthesisG− bundle period Now comma let us considerover aM. principalAs in the tilde-G subsection sub hyphen 2 . 1 bundle , let λ : G˜ → G be a member \noindentopen parenthesisofof the the isomorphism q : isomorphism Q right arrowclass of M class the comma non of G-tilde - trivial the closing non two− - parenthesis foldt r i v coverings i a l overtwo of the− thefold chosen symplectic coverings symplectic group manifold ofG. In the open symplectic parenthesis group $G . $˜ In particular $ , \ tilde {G}\simeq˜ $Mp$( 2˜ l , R ) .$ M comma omegaparticular closing, G parenthesis' Mp (2l, periodR). Now We , letcall us the consider pair open a principal parenthesisG− Qbundle comma (q Capital: Q → M, LambdaG) over closing the parenthesis Nowmetaplectic , letchosen us structure consider symplectic if Capital amanifold principal Lambda (M, ω :) Q. We $ right\ calltilde arrow the{G pair P} is ( a{Q, surj −Λ) } ective metaplectic$ bundle bundle structure morphism if compatibleΛ : Q → P withis a $the ( actions qsurj of ective : G on Q bundle P and\rightarrow morphism that of G-tildecompatible onM, Q andwith with the\ tilde the actions covering{G} of G lambdaon) $P and over in that the the sameof G˜ chosenon Q and symplectic with the manifold $ (way Mas incovering the , Riemannian\omegaλ in the same spin)geometry way .$ as in Wecallthepair period the Riemannian open parenthesis spin geometry $( For a more Q . ( For elaborate , a more\Lambda definition elaborate see) definition $ comma e period g period commametaplecticsee , e structure . g . , Habermann if $ , Habermann\Lambda [ 8:Q ] . ) Let us\rightarrow remark , that typicalP $ examples is a surj of symplectic ective bundle morphism compatible with theHabermann actionsmanifolds comma of admitting $G$ Habermann aon metaplectic open $P$ square structure and bracket that 8are closing of cotangent $ square\ tilde bundles bracket{G} of period$ orientable on closing $Q$ manifolds parenthesis and ( phase with Let us the remark covering comma that$ \lambda typicalspaces examples$ in ) , Calabi theof symplectic same - Yau manifolds and the complex projective spaces waymanifolds as in admitting the Riemannian a metaplectic spin structure geometry are cotangent . ( For bundles a more of orientable elaborate definition see , e . g . , Habermannmanifolds open , Habermann parenthesis phase [ 8 ] spaces . ) closing Let us parenthesis remark comma , that Calabi typical hyphen examples Yau manifolds of symplectic and the complex projective manifolds admitting a metaplectic structure2k+1 are cotangent bundles of orientable spaces CP , k ∈ N0. manifoldsCP to the power ( phase of 2 k spaces plus 1 comma ) , Calabi k in N sub− 0Yau period manifolds and the complex projective spaces ˜ Now commaNow let us , let denote us denote the Freacutechet the Fr´echet vector vector bundle bundle associated associated to to the the introduced introduced principal G− bundle \ begin { a l i g n ∗} ˜ principal(q G-tilde: Q → subM, hyphenG) via bundle the metaplectic open parenthesis representation q : Q rightL on arrowS by M commaS. Thus tilde-G , we have closingS = parenthesisQ×L S . via the metaplectic representationCP ˆ{ 2We Lk shall on S+ call by this 1 } associated, k vector\ in bundleN {S →0 }M the. symplectic spinor bundle . The sections \end{ a l i g n ∗} S periodφ Thus∈ Γ(M,S comma) will we be have called S =symplectic Q times sub spinor L S period fields We. Let shall us putcallE this:= associatedQ×ρ E . vector For r = bundle 0, ..., 2 Sl, rightwe arrow M r r the symplecticdefine E spinor:= Q bundle×ρ E period, The sections phi in Capital Gamma open parenthesis M comma S closing parenthesis will Now , let us denote the Fr \rm’{ e} chet vector bundle associated to the introduced be called symplecticwhere Er abbreviates E r T he smooth sections Γ(M,E) will be called symplectic p rspinor i n c i p fields a l period $ \ tilde Let{ usG} put{ E − : =. }$ Q times bundle sub rho $( E period q For : r = Q 0 comma\rightarrow period periodM, period comma\ tilde 2 l{ commaG} we) $ via the metaplectic representation $ L $ on S by ± ± define E tospinor the power valued of r exterior : = Q times differential sub rho forms E to . the Because power ofthe r operatorscomma E ,F and H are G˜−equivariant( $Swhere E .$see to the Thus,wehave Lemma power of 3 r it abbreviates em 1 ) , $S they Row lift = 1 to rm operators Q sub r\ Rowtimes acting 2 period on{ sectionsL .} smooth$ of S the . sections We corresponding shall Capital call associatedGamma this open associated parenthesis vector M bundle comma$ S E\rightarrow closingvector parenthesis bundles .M The will $ same be called is t rue symplectic about the projections the symplecticij spinor bundle . The sections $ \phi \ in \Gamma ( M , S ) $ will be called symplectic spinor valuedp , (i, exterior j) ∈ Z × differentialZ. We shall forms use the period same Because symbols the as foroperators the mentioned E to the operators power of as plusminux for their “comma lifts ” F to the power of plusminuxspinor and fieldsto the H are associated . Let us vector put bundle $E structure : =. Q \times {\rho }$E.For$r = 0 , . .G-tilde . sub, hyphen 2 l equivariant ,$Now wedefine open , we parenthesis shall make $Eˆ asee use{ ther of} Lemmathe Fedosov: 3 = it connectionem Q 1 closing\times . The parenthesis Fedosov{\rho comma connection} E they ˆ{ r lift} to operators, $ acting on sections∇ determines of the induced principal G− bundle connection on the principal bundle (p : P → M,G). \noindent where $ E ˆ{ r }$ abbreviates $\ l e f t .E\ begin { array }{ c} rm { r }\\ . \end{ array }The\ right . $ the correspondingThis connection associated lift s vector to a principal bundles periodG˜− bundle The sameconnection is t rue on about the principal the projections bundle (q : Q → M, G˜) smoothp to the sectionsand power defines of ij $thecomma\Gamma associated open parenthesis covariant( M derivative i , comma E j closing ) $ parenthesis will be called in Z times symplectic Z period We shall use the same symbols as for the mentionedon the symplectic operators bundle S, which we shall denote by ∇S, and call it the sym - plectic spinor covariant \noindentas for theirderivativespinor quotedblleft . valuedSee ,lifts e . quotedblright g exterior . , Habermann differential to the , Habermann associated forms [ vector 8 ] for bundle . this Because structure the period operators $ E ˆ{\pm } , F ˆ{\pm }$ andNow $ comma Hclassical $ weare shall construction make a use . The of the symplectic Fedosov spinor connection covariant period derivative The Fedosov∇S induces connection the S $nabla\ tilde determinesexterior{G} covariant{ the − inducedequivariant derivative principald∇ G} hyphen( $ bundle see connection the Lemmaacting on on 3 the Γ( it principalM,E em). 1For bundle)r ,= they 0, ..., 2l, liftwe have to operators acting on sections of theopen corresponding parenthesis∇S p : P associated rightVr arrow∗ M vector comma bundles G closingV parenthesis.r+1 The∗ same period is t This rue connection about the lift s projections to a principal G-tilde sub d : Γ(M,Q ×ρ ( V ⊗ S )) → Γ(M,Q ×ρ ( V ⊗ S ) ) . Now , we are able to define the hyphen bundlesymplectic connection twistor on the operators . For r = 0, ..., 2l, we set \noindentprincipal bundle$ p openˆ{ iparenthesis j } , q ( : Q iright arrow, j M comma ) \ G-tildein Z closing\times parenthesisZ and . defines $ We the shall associated use covariant the same symbols as for the mentioned operators as for their ‘‘ lifts ’’ to the associated vector bundle structure . derivative r r+1 r+1,mr+1 S Tr : Γ(M,E ) → Γ(M,E ),Tr := p d∇|Γ(M,Er) on the symplectic bundle S comma which we shall denote by nabla to the power of S comma and call it the sym hyphen \ hspaceplectic∗{\ spinorandf i call l covariantl } theseNow operators, derivative we shallsymplectic period make See twistor a comma use operators of e period the . Fedosov gInformally period comma connection , one Habermann can say . that The comma the Fedosov operators Habermann connection open square bracket 8 closingare going square on bracketthe lower for edges this of the triangle at the Figure 1 . Let − S \noindentclassicalus construction notice$ \nabla that periodF$(∇ determines The− T symplectic0) is , up the spinorto a induced non covariant - zero principal scalar derivative multiple nabla $ , G the to the so− powercalled$ bundle symplecticof S induces connection Dirac the on the principal bundle $exterior ( poperator covariant : introduced derivative P \rightarrow d by to K the . Habermann powerM of nabla . , See to G, the e . powerg ) . , Habermann of. S$ .... This acting , Habermann connection on Capital [ Gamma 8 ] lift . open s to parenthesis a principal M comma $ \ tilde {G} { − }$ Ebundle closing parenthesisconnectionIn theperiod next on theorem Forthe r = ,0 we comma st ateperiod that the period sequences period consisting comma 2 of lcomma the symplectic we have twistor operators principald to theform power bundle complexes of nabla $( . to These the q powersequences : of Q S will : Capital\ berightarrow called Gamma symplectic openM, twistor parenthesis sequences\ tilde M comma{ orG} complexes Q) times $ . and sub rho defines open parenthesis the associated covariant derivative bigwedge to the power of r V to the power of * oslash S closing parenthesis closing parenthesis right arrow Capital Gamma open parenthesis\noindent Mon comma the Q symplectic times sub rho bundle open parenthesis $ S ,bigwedge $ which to the we power shall of r denote plus 1 V by to the $ \ powernabla of *ˆ{ oslashS } S closing, $ and call it the sym − parenthesisplectic closing spinor parenthesis covariant period derivative Now comma . we See are able , e to . g . , Habermann , Habermann [ 8 ] for this define the symplectic twistor operators period For r = 0 comma period period period comma 2 l comma we set \noindentT sub r : Capitalclassical Gamma construction open parenthesis . M The comma symplectic E to the power spinor of r closing covariant parenthesis derivative right arrow $ Capital\nabla Gammaˆ{ S open}$ parenthesisinduces the M comma E to the power of r plus 1 closing parenthesis comma T sub r : = p to the power of r plus 1 comma m sub r plus 1 d nabla sub bar Capital Gamma open parenthesis M comma E r closing parenthesis to the power of S \noindentand call theseexterior operators covariant symplectic twistor derivative operators $ period d ˆ{\ Informallynabla commaˆ{ S }} one$ can\ h say f i l that l a c t i n g on $ \Gamma ( Mthe , operators E ) are going .$For$r on the lower edges = of the 0 triangle , . at the . Figure . 1 ,period 2 Let l ,$wehave us notice that F to the power of minus open parenthesis nabla to the power of S minus T sub 0 closing parenthesis is comma up\noindent to a non hyphen$ d zero ˆ{\ scalarnabla multipleˆ{ S comma}} : the\ soGamma called (M,Q \times {\rho } ( \bigwedge ˆ{ r } V ˆsymplectic{ ∗ } \ Diracotimes operator$ S introduced $ ) by) K\ periodrightarrow Habermann\ periodGamma See comma(M,Q e period g period\times comma Habermann{\rho } comma( \bigwedge ˆ{ r +Habermann 1 } V ˆ open{ ∗ square } \otimes bracket 8$ closing S ) square ) . Now bracket , we period are able to defineIn the next thesymplectic theorem comma twistor we st ate that operators the sequences .For consisting $r of = the symplectic 0 , . . . , 2 l ,$ weset twistor operators form complexes period These sequences will be called symplectic twistor \ [Tsequences{ r or} complexes: \Gamma period ( M , E ˆ{ r } ) \rightarrow \Gamma ( M , E ˆ{ r + 1 } ),T { r } : = p ˆ{ r + 1 , m { r + 1 }} d{\nabla }ˆ{ S } {\mid \Gamma ( M , E r ) }\ ]

\noindent and call these operators symplectic twistor operators . Informally , one can say that the operators are going on the lower edges of the triangle at the Figure 1 . Let

\noindent us notice that $ F ˆ{ − } ( \nabla ˆ{ S } − T { 0 } ) $ is ,upto anon − zero scalar multiple , the so called symplectic Dirac operator introduced by K . Habermann . See , e . g . , Habermann , Habermann [ 8 ] .

In the next theorem , we st ate that the sequences consisting of the symplectic twistor operators form complexes . These sequences will be called symplectic twistor sequences or complexes . 320 .. S period KR Yacute SL \noindentTheorem ..320 4 period\quad .. LetS . l greater KR $ equal\acute 2 ..{ andY} open$ SL parenthesis M to the power of 2 l comma omega comma nabla closing parenthesisTheorem ..\quad be a Fedosov4 . \quad manifoldLet of Ricci $ l type\geq 2 $ \quad and $ ( M ˆ{ 2 l } , \omega , \nablaadmitting) a $ metaplectic\quad be s tru a c Fedosov ture period manifold .. Then of Ricci type 0 right arrow Capital Gamma open parenthesis M comma E to the power of 0 closing parenthesis T right arrow 0 Capital \noindent admitting a metaplectic s tru c ture . \quad Then2l Gamma open320 parenthesis S . KR Y´ MSL commaTheorem E to the 4 power . Let of 1l 1 closing≥ 2 parenthesisand (M T, ω, right∇) arrowbe a 1 Fedosov times times manifold times T right arrow l minus 1 Capitalof Ricci Gamma type open parenthesis M comma E to the power of l l closing parenthesis right arrow 0 \ [and 0 \rightarrowadmitting a metaplectic\Gamma s tru c( ture M . , Then E ˆ{ 0 } )T {\rightarrow } 0 \Gamma (M ,0 Eright ˆ{ arrow1 Capital1 } )T Gamma{\ openrightarrow parenthesis M} comma1 \ Ecdot to the power\cdot of l l\cdot closing parenthesisT {\rightarrow T right arrow} l Capitall − 1 \Gamma ( M , E ˆ{ l0 l } ) 11\rightarrow 0 \ ] ll Gamma open parenthesis M comma0 → Γ( EM,E to the)T power→0 Γ( ofM,E l plus) 1T comma→1 ··· l plusT→l 1− closing1Γ(M,E parenthesis) → 0 T right arrow l plus 1 times times times T right arrow 2 l minus 1 Capital Gamma open parenthesis M comma E to the power of 2 l comma 2 l closing parenthesis and right arrow 0 \noindentare complexesand period 0 → Γ(M,Ell)T l Γ(M,El+1,l+1)T l + 1 ··· T 2l − 1Γ(M,E2l,2l) → 0 Proof period .... See Kryacutesl open→ square bracket 1 4 closing→ square→ bracket period square \ [4 0 period\rightarroware .. Ellipticity complexes of . the\Gamma symplectic( twistor M complex , E ˆ{ l l } )T {\rightarrow } l \Gamma ( M , E ˆ{ l + 1 , l + 1 } )T {\rightarrow } l + 1 \cdot \cdot \cdot After theProof preceding . summarizing parts comma we now tend to the proof the ellipticity See Kr´ysl[14].  T of{\ therightarrow truncated symplectic} 24 twistor . l Ellipticity− complexes1 \ of periodGamma the Let symplectic us( recall M that twistor , by E an ˆ ellipticcomplex{ 2 l , 2 l } ) \rightarrow 0 \complex] of differentialAfter the preceding operators summarizing we mean a complex parts , we of now differential tend to operators the proof acting the ellipticity of the truncated on the sectionssymplectic of Freacutechet twistor complexes bundles . such Let that us recall the associated that by an complex elliptic of complex symbols of of differential operators the consideredwe mean differential a complex operators of differential forms operatorsan exact sequence acting on of the sheaves sections period of Fr´echet Let us bundles such that the \noindentrecall thatassociatedare a sequence complexes complex open parenthesis of . symbols Capital of the considered Gamma open differential parenthesis operators F to the forms power an of exact bullet sequence closing parenthesis of comma pi to the powersheaves of bullet . Let closing us recall parenthesis that a sequence in the category (Γ(F •), of π• complexes) in the category of sheaves of complexes of sections of sheaves of sections \noindent Proof . \ h f i l l• See Kr\ ’{ y} s l $ [ 1 4i ] . i−1 \ square $ of Freacutechetof Fr´echet bundles bundles F toF theis powercalled exactof bullet if the is called stalks exact [ Ker if (π the)]m stalks, [ Im open (π square)]m satisfy bracket Ker open parenthesis pi to the i i−1 power of i closingthe equality parenthesis [ Ker (π closing)]m = square [ Im (π bracket)]m for sub each m commai ∈ Z and open each squarem ∈ M, bracketwhere Im always open when parenthesis arriving pi to the power of i minus\ centerline 1 closingat a{ parenthesis preshaef4 . \quad and closing notEllipticity at square a sheaf bracket , we of consider subthe m symplectic satisfy it s sheafification twistor not distinguishing complex } it at the notation the equalitylevel open . Let square us notice bracket that Ker in openthe case parenthesis of symbols pi to , we the may power speak of i closingabout fibers parenthesis and not closing necessarily square bracket sub m = openAfter square theabout bracket preceding st Imalks open because summarizing parenthesis the symbols pi toparts the power , we of now i minus tend 1 closing to the parenthesis proof closing the ellipticity square bracket sub m for each i in Zof and the each truncatedare m in bundle M comma and symplectic not where only sheaf twistor morphisms complexes . See the classical . Let textus recall - book of that Wells by[ 2 1 an ] for elliptic more on complexalways whenellipticity of arrivingdifferential of complexes at a preshaef operators of differential and not at we operators a meansheaf comma a . complex we consider of differential it s sheafification operators acting onnot the distinguishing sectionsAfter this it of at introductory the Fr \ notation’{ e} chet paragraph level bundles period , we Let start such us with notice that a simple that the in lemma the associated case in ofwhich symbols thecomplex symbol comma of of the symbols exte- of thewe may consideredrior speak covariant about differential fibers symplectic and not spinor necessarily operators derivative about forms associated st alks an because to exact a Fedosov the sequence symbols manifold of admitting sheaves a metaplectic . Let us recallare bundle thatstructure and a not sequence is onlycomputed sheaf .$ morphisms ( \Gamma period See( the F classical ˆ{\ bullet text hyphen} ), book of Wells\ pi openˆ{\ squarebullet bracket} ) 2 $ 1 closing in the category of complexes of sheaves of sections square bracketLemma 5 . Let (M, ω, ∇) be a Fedosov manifold admitting a metaplectic s tructure , \noindent o f Fr \ ’{ e} chet bundles $ F ˆ{\ bullet }$ isS called exact if the stalks [ Ker $ ( for moreS on→ ellipticityM be the of corresponding complexes of symplectic differential sp operators inor b undle period and d∇ denotes the exterior \ pi ˆ{ i } )] { m } , [ $ Im $ ( \ pi ˆ{ i − 1 } )] { m }$ s a t i s f y After thiscovariant introductory derivative paragraph . comma Then we for start each withξ ∈ aΓ( simpleM,T ∗M lemma) inand whichα the⊗ φ ∈ Γ(M,E), the S symbol ofsymbol the exteriorσξ of covariantd∇ is symplectic given by spinor derivative associated to a Fedosov \noindentmanifold admittingthe equality a metaplectic [ Ker structure $ ( is\ computedpi ˆ{ i period} )] { m } = [ $ Im $ ( \ pi ˆ{ i − 1 } )] { m }$ f o r each $ i \ in Z$ and each $m \ in M , $ where Lemma 5 period .... Let open parenthesis M commaσξ(α ⊗ omegaφ) = ξ comma∧ α ⊗ φ. nabla closing parenthesis be a Fedosov manifold admitting aalways metaplectic when s tructure arriving comma at a preshaef and not at a sheaf , we consider it s sheafification ∞ ∗ notS right distinguishing arrowProof M . be the itcorresponding at theFor notation symplecticf ∈ C ( levelM sp) inor, ξ ∈ .bΓ( undleM,T Let us andM) notice and d toα the⊗ s that power∈ Γ(M,E in of nabla the), let to caseus the compute powerof symbols of S denotes , the exteriorwe may speak about fibers and not necessarily about st alks because the symbols covariant derivativeS period .... ThenS for each xi in Capital GammaS open parenthesisS M comma T to the power of * M closing d∇ (fα ⊗ s) − fd∇ (α ⊗ s) = df ∧ α ⊗ s + fd∇ (α ⊗ s) − fd∇ (α ⊗ s) = df ∧ α ⊗ s. parenthesis\noindent ....are and bundle alpha oslash and phi not in Capital only sheaf Gamma morphisms open parenthesis . See M comma the classical E closing parenthesis text − book comma of .... Wells the [ 2 1 ] for more on ellipticity of complexes of differential operators . symbol sigmaUsing to this the computation power of xi , of we d get to the the power statement of nabla of the to lemma the power. of S .. is given by sigma to the power of xi open parenthesis alpha oslash phi closing parenthesis = xi and alpha oslash phi period From now on , we shall denote the projections pimi onto Ei by pi simply , i = 0, ..., 2l.( In order AfterProof this period introductory .... For f in C to paragraph the power of , infinity we start open parenthesis with a simple M closing lemma parenthesis in which comma the xi in Capital Gamma open not to cause a possible confusion , we will make no use of the projections from E onto Vi ∗⊗ S or parenthesissymbol of M comma the exterior T to the power covariant of * M closing symplectic parenthesis spinor and alpha derivative oslash s in Capital associated GammaV to open a Fedosovparenthesis M comma of their lift s to the associated geometric structures . ) Due to the previous lemma and the definition Emanifold closing parenthesis admitting comma a let metaplectic us compute structure is computed . of the symplectic twistor operators , we get easily that for each i = 0, ..., 2l and ξ ∈ Γ(M,T ∗M), the d to the power of nabla to the power of S open parenthesis f alpha oslash s closing parenthesis minus fd to the power of nabla symbol to\noindent the power ofLemma S open 5 parenthesis . \ h f i l l alphaLet oslash $ ( s closing M parenthesis, \omega = df and, alpha\nabla oslash s) plus $ fd be to a the Fedosov power of nablamanifold to the admitting a metaplectic s tructure , iξ of the symplectic twistor operator T is given by the formula power of S openσ parenthesis alpha oslash s closingi parenthesis minus fd to the power of nabla to the power of S open parenthesis alpha\noindent oslash s closing$ S parenthesis\rightarrow = df andM alpha $ be oslash the s period corresponding symplectic sp inor b undle and $ d ˆ{\nabla ˆ{ S }}$ iξ (α ⊗ s) := pi+1(ξ ∧ α ⊗ s) denotesUsing this the computation exterior comma we get the statementσ of the lemma period square From now on comma we shall denote the projections p to the power of im sub i onto E to the power of i by p to the power of i simply\noindent commacovariant i = derivative . \ h f i l l Then for each $ \ xi \ in \Gamma ( M , T ˆ{ ∗ } M0 comma ) $ \ periodh f i l l periodand period $ \alpha comma 2\ lotimes period open\phi parenthesis\ in In order\Gamma not to( cause M a possible , E confusion ) comma , $ \ weh f willi l l the make no use of the \noindentprojectionssymbol from E onto $ \ bigwedgesigma ˆ to{\ thexi power}$ of o fi V $to dthe ˆ power{\nabla of * oslashˆ{ S S}} or$ of their\quad lift si s to given the associated by geometric structures period closing parenthesis Due to the previous lemma and the definition of the symplectic twistor \ [ operators\sigma commaˆ{\ xi we} get( easily\alpha that for each\otimes i = 0 comma\phi period) period = period\ xi comma\wedge 2 l and\alpha xi in Capital\otimes Gamma open\phi parenthesis. \ ] M comma T to the power of * M closing parenthesis comma the symbol i sigma to the power of xi of the symplectic twistor operator T sub i is given by the formula i sigma to the power of xi open parenthesis alpha oslash s closing parenthesis : = p to the power of i plus 1 open parenthesis xi and\noindent alpha oslashProof s closing . \ h parenthesis f i l l For $ f \ in C ˆ{\ infty } (M), \ xi \ in \Gamma ( M , T ˆ{ ∗ } M ) $ and $ \alpha \otimes s \ in \Gamma ( M , E ) , $ let us compute

\ [ d ˆ{\nabla ˆ{ S }} ( f \alpha \otimes s ) − fd ˆ{\nabla ˆ{ S }} ( \alpha \otimes s ) = df \wedge \alpha \otimes s + fd ˆ{\nabla ˆ{ S }} ( \alpha \otimes s ) − fd ˆ{\nabla ˆ{ S }} ( \alpha \otimes s ) = df \wedge \alpha \otimes s . \ ]

\noindent Using this computation , we get the statement of the lemma $ . \ square $

From now on , we shall denote the projections $ p ˆ{ im { i }}$ onto $ E ˆ{ i }$ by $ p ˆ{ i }$ simply $, i =$ $0 , . . . , 2 l . ($ In order not to causea possible confusion ,wewill makenouse of the projections from E onto $ \bigwedge ˆ{ i } V ˆ{ ∗ } \otimes $ S or of their lift s to the associated geometric structures . ) Due to the previous lemma and the definition of the symplectic twistor operators ,wegeteasilythatforeach $i = 0 , . . . , 2 l$ and $ \ xi \ in \Gamma ( M , T ˆ{ ∗ } M ) ,$ thesymbol

\noindent $ i {\sigma }ˆ{\ xi }$ of the symplectic twistor operator $ T { i }$ is given by the formula

\ [ i {\sigma }ˆ{\ xi } ( \alpha \otimes s ) : = p ˆ{ i + 1 } ( \ xi \wedge \alpha \otimes s ) \ ] ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 32 1 \ hspacefor each∗{\ alphaf i l l oslash}ELLIPTICITY s in Capital OF Gamma THE open SYMPLECTIC parenthesis TWISTOR M comma COMPLEX E to the power\quad of32 i closing 1 parenthesis period In order to prove the ellipticity of the appropriate parts of the symplectic twistor \ begincomplexes{ a l i g comma n ∗} we need to compare the kernels and the images of the symbols maps f oi r sigma each to the power\alpha of xi for\otimes any xi in Capitals \ Gammain \Gamma open parenthesis( M M comma , E T ˆto{ thei } power). of * M closing parenthesis \end{ a l i g n ∗} backslash open brace 0 closing brace period ThereforeELLIPTICITY comma we OF prove THE the SYMPLECTIC following TWISTOR st atement COMPLEX in 32 1 which the projections p to the power of i are more specified period In order to prove the ellipticity of the appropriate parts of the symplectic twistor Lemma 6 period .. For i = 0 comma period period period comma l minusi 1 comma xi in V to the power of * and alpha oslash scomplexes in E to the power , we of need i comma to we compare have theforeachα kernels⊗ ands ∈ Γ( theM,E images). of the symbols maps Equation: openIn order parenthesis to prove 6 the closing ellipticity parenthesis of the appropriate .. p to thepower parts of of the i plus symplectic 1 open parenthesistwistor complexes xi and , alpha we oslash s closing parenthesis\noindent = xi$ and i alpha{\sigma oslash s}ˆ plus{\ betaxi } F$ to the f o r power any of $ plus\ xi open parenthesis\ in \Gamma alpha oslash( xi M to the , power T ˆ of{ sharp ∗ } timesM) s \setminusneed\{ to compare0 the\} kernels. $ and Therefore the images of , thewe symbols prove themaps following st atement in closing parenthesisξ plus gamma open∗ parenthesis E to the power of plus iota sub xi sharp alpha oslash s closing parenthesis where iσ for any ξ ∈ Γ(M,T M) \{0}. Therefore , we prove the following st atement in beta = 2 divided by i minus l and gammai = i divided by i minus l sub period \noindentwhichwhich the projectionsthe projectionsp are more $ specified p ˆ{ i . }$ are more specified . For i = l plus 1 comma period period period comma∗ 2 l and psi ini E to the power of i minus 1 comma m i minus 1 oplus E to Lemma 6 . For i = 0, ..., l − 1, ξ ∈ V and α ⊗ s ∈ E , we have the power of i minus 1 comma m i minus 1 minus 1 oplus E to the power of i minus 1 comma m i minus 1 minus 2 sub comma we have\noindent Lemma 6 . \quad For$i=0 , . . . , l − 1 , \ xi \ in V ˆ{ ∗ }$ and $ \alpha \otimes i+1 s \ in E ˆ{ i } +, $ we] have + Equation: open parenthesisp 7 closing(ξ ∧ α ⊗ parenthesiss) = ξ ∧ α ..⊗ ps to+ βF the power(α ⊗ ξ of· s i) minus + γ(E 1ι psiξ]α =⊗ psis) plus 4 divided (6)by l minus i F to the power of minus F to the power of plus psi plus 1 divided by l minus i E to the2 power of minusı E to the power of plus psi period \ beginProof{ perioda l i g n ∗} .. We prove the first relation only period The secondwhereβ formula= canandγ be= derived following i − l i − l . \ tagthe∗{ same$ ( l ines 6 of reasoning ) $} p ˆused{ i for proving + 1 the} first( one\ xi period\wedge We split the\alpha proof of open\otimes parenthesiss 6 closing ) = parenthesis\ xi \wedge \alpha \otimes s + \beta F ˆ{ +i−1},m ( \alphai−1,m \otimesi−1,m \ xi ˆ{\sharp }\cdot s into four parts periodFor i = l + 1, ..., 2l and ψ ∈ E i − 1 ⊕ E i − 1 − 1 ⊕ E i − 1 − 2, we have )1 +period\gamma .. In this item( comma E ˆ{ we+ }\ proveiota that for{\ a fixedxi }\ i in opensharp brace 0\alpha comma period\otimes period periods comma ) \\ where l closing brace\beta = \ f r a c { 2 }{ i − l } and \gamma = \ f r a c {\imath }{ i − l } { . } and any k = 0 comma period period period comma i comma4 1 \end{ a l i g n ∗} pi−1ψ = ψ + F −F +ψ + E−E+ψ. (7) there exists alpha i sub k in C such that l − i l − i Line 1 i Line 2 p to the power of i = sum i alpha sub k open parenthesis F to the power of plus closing parenthesis to the power of\ hspace k open∗{\ parenthesisProoff i l l .}For$i FWe to the prove power the = of first minus l relation + closing only 1 parenthesis . The , second . to .the formula power . can of , bek Line 2 derived 3 l$and$ k =following 0 the same\ psi \ in E ˆ{ i − with1 alpha ,l ines i m sub of} reasoning 0 =i 1 for− used each1 for i = proving\ 0oplus comma the period firstE ˆ one{ periodi . We− period split1 the comma proof , lm of period (} 6 )i into .. Because− four parts1 for .− each i1 = 0\ commaoplus periodE ˆ{ i −period1 period , comma1 m .} In l commathisi item− the ,1 we prove− that2 for{ a, fixed}$i we∈ {0 have, ..., l} and any k = 0, ..., i, there exists αik ∈ C projectionssuch p that to the power of i are G-tilde sub hyphen equivariant comma they can be expressed as open parenthesis finite closing\ begin parenthesis{ a l i g n ∗} linear i \ tagcombinations∗{$ ( 7 of the ) elements $} p ˆ{ ofi the− finite1 dimensional}\ psi vector= space\ psi End+ sub\ G-tildef r a c { open4 }{ parenthesisl − Ei closing} F parenthesis ˆ{ − } F ˆ{ + } \ psi + \ f r a c { 1 }{ l − i } E ˆi { −X } E+ ˆ{k + −}\k psi . period p = iαk (F ) (F ) \endDue{ a to l i g the n ∗} Lemma 3 item 3 open parenthesis cf period also Kryacutesl open square bracket 1 2 closing square bracket closing k = 0 parenthesis comma we know that the complex \noindent Proof . \quad We prove the first relation only . The second formula cani be derived following associative algebrawith αi0 End= sub 1 G-tilde for each openi parenthesis= 0, ..., l. E closingBecause parenthesis for each i is generated= 0, ..., l, bythe F to projections the powerp of plus .. and F to the same l ines of reasoning used for proving the first one . We split the proof of ( 6 ) the power ofare minusG˜−equivariant, .. and by thethey can be expressed as ( finite ) linear combinations of the elements of the finite into four parts . projectionsdimensional p sub plusminux vector space periodEnd ItG is˜ ( easyE ) to . Duesee that to the the Lemma projections 3 item p sub 3 ( plusminux cf . also Kr´ysl[ can be 1 omitted 2 ] ) , we + − from anyknow expression that the for complex p to the associative power of i algebraand thusEnd comma˜ ( E each) is projection generated pby toF the powerand F of i canand be by expressed the 1 . \quad In this item , we prove that for aG fixed $ i \ in \{ 0 , . . . , l j ust usingprojections F to thep power±. It is of easy plus to and see F that to the the power projections of minusp± periodcan be Due omitted to the defining relation H = 2 open brace F to the power\} $andany$k of plus commafrom any F to expression the = power 0 for ofp minus,i and . closing thus . , braceeach . projection and , ipi ,$can be expressed j ust using F + and therethe relation existsF −. openDue $ parenthesis to\alpha the defining{ 4i closing} relation{ k parenthesis}\H in= 2{ onFC+ the,F $− values} suchand of the that H on relation homogeneous ( 4 ) on elements the values comma of H oneon can order thehomogeneous operators F toelements the power , one of can plus order and F the to operators the powerF of+ minusand F in− in an an expression expression for for p topi thein the power way of i in the way that \ [ \thebegin operators{thata l i g the nF e to doperators} thei power\\ F + ofappear plus appear on the on left the - hand left hyphen and the hand operators and theF − operatorson the F to the power of minus on the pright ˆ{ hypheni } right= hand -\ hand sidesum period sidei . In{\ thisalpha way comma, we{ expressk we}} expresspi(as a p F linear to ˆ{ the+ combination power} ) of ˆ i{ as ofk a the} linear expressions( combination F ˆ{ of − type } of the) ˆ{ k }\\ k = 0+ \aend−{ba l i g n e d }\ ] expressions(F of) ( typeF ) openfor some parenthesisa, b ∈ N F0. toSince the the power projection of plus closing parenthesis to the power of a open parenthesis F to the power of minusp closingi does not parenthesis change the to formthe power degree of of b a for symplectic some a comma spinor b valued in N sub exterior 0 period form Since and F the− projectionand F + p to thedecreases power of iand does increases not change the formthe form degree degree by one of a , symplectic spinor valued exterior withform $ and\alpha F to{ thei power} { 0 of minus} respectively= .. and 1$foreach$i F to , the the relation power ofa plus= b follows decreases = . 0 Because and , increases the . operator the . form .F − degreedecreases , by l one .$ comma\quad Because for each $i=0respectively comma , the . relation . .athe = b ,form follows ldegree period ,$the by Because one , the the summands operator ( F+ to)k the(F − power)k for ofk > minus i actually decreases do projectionsthe form degree $ by p one ˆnot{ commai o ccur}$ the in are the summands expression $ \ tilde open for{G parenthesis the} projection{ − Fequivariant topi thewritten power above of plus . , Thus} closing$ , they parenthesis can be to expressed the power of as k ( finite ) linear opencombinations parenthesis F ofto the the power elements of minus of closing the parenthesis finite dimensional to the power of vector k for k greater space i actually $ End do{\ tilde {G}} ( $ E).not o ccur in the expression for the projection p to the power of i written above period Thus comma Duei Equation: to the open Lemma parenthesis 3 item 8 3 closing ( cf parenthesis . also Kr ..\ p’{ toy the} sl power [ 1 of 2 i ] = ) sum , we i alpha know sub that k open the parenthesis complex F to the power ofassociative plus closing parenthesis algebra to the $ End power{\ of ktilde open parenthesis{Gi}}X ( F $ to+ E) thek power− isk generated of minus closing by parenthesis $Fˆ{ + to} the$ power\quad ofand k k = $0 F ˆ{ − }$ p = iαk (F ) (F ) (8) for\quad someand alpha by i sub the k in C comma k = 0 comma period period period comma i period projections $ p {\pm } . $ It is easy to see thatk = 0 the projections $ p {\pm }$ can be omitted

forsomeαik ∈ C, k = 0, ..., i. from any expression for $ p ˆ{ i }$ and thus , each projection $ p ˆ{ i }$ can be expressed j ust using $Fˆ{ + }$ and $ F ˆ{ − } . $ Due to the defining relation $H = 2 \{ F ˆ{ + } , F ˆ{ − } \} $ and the relation ( 4 ) on the values of $ H $ on homogeneous elements , one can order the operators $ F ˆ{ + }$ and $ F ˆ{ − }$ in an expression for $ p ˆ{ i }$ in the way that the operators $ F ˆ{ + }$ appear on the left − hand and the operators $ F ˆ{ − }$ on the

r i g h t − hand side . In this way , we express $ p ˆ{ i }$ as a linear combination of the expressions of type $ ( F ˆ{ + } ) ˆ{ a } ( F ˆ{ − } ) ˆ{ b }$ forsome $a , b \ in N { 0 } . $ Since the projection

$ p ˆ{ i }$ does not change the form degree of a symplectic spinor valued exterior form and $ F ˆ{ − }$ \quad and $ F ˆ{ + }$ decreases and increases the form degree by one ,

\ hspace ∗{\ f i l l } respectively , the relation $ a = b $ follows . Because the operator $ F ˆ{ − }$ d e c r e a s e s

\ hspace ∗{\ f i l l } the form degree by one , the summands $ ( F ˆ{ + } ) ˆ{ k } ( F ˆ{ − } ) ˆ{ k }$ f o r $ k > i $ actually do

\ centerline { not o ccur in the expression for the projection $ p ˆ{ i }$ written above . Thus , }

\ begin { a l i g n ∗} i \\\ tag ∗{$ ( 8 ) $} p ˆ{ i } = \sum i {\alpha { k }} ( F ˆ{ + } ) ˆ{ k } ( F ˆ{ − } ) ˆ{ k }\\ k = 0 \\ f o r some \alpha{ i } { k }\ in C , k = 0 , . . . , i . \end{ a l i g n ∗} 322 .. S period KR Yacute SL \noindentNow comma322 we\ shallquad proveS . theKR equation $ \acute alpha{Y} i sub$ 0 SL = 1 comma i = 0 comma period period period comma l period By evaluating Nowthe , left we hyphen shall hand prove side the of open equation parenthesis $\ 8alpha closing{ parenthesisi } { 0 on} an=1,i=0,..., element phi in E to the power of i we get phi comma whereasl . $ at the By evaluating the l e f t − hand side of ( 8 ) on an element $ \phi \ in E ˆ{ i }$ we get $ \phi , $ whereas at the right hyphen322 hand S . KR sideY´ theSL only summand which remains is the one indexed by zero periodNow open , parenthesis we shall prove The the other equation summandsαi0 = vanish 1, i = because0, ..., l. By F toevaluating the power the of left minus - hand is G-tilde side of sub ( 8 )hyphen equivariant comma\ hspace decreases∗{\onf an i l lelement} r i g h tφ ∈−Ehandi we get sideφ, whereas the only at the summand which remains is the one indexed by the form degree by one and there isright no summand - hand side in bigwedge the only summandto the power which of i remains minus 1 is V the to the one power indexed of * by oslash S isomorphic \ hspace ∗{\ f i l l } zero . ( The other summands vanish because $ F− ˆ{ − }$ i s $ \ tilde {G} { − equivariant to E sub plus to the power of i orzero to . E ( sub The minus other tosummands the power vanish of period because to theF poweris G˜−equivariant, of i See thedecreases Remark item 3 below the , }$ d e c r e a s e s Vi−1 ∗ i i Theorem 1 periodthe closing form degree parenthesis by one and there is no summand in V ⊗ S isomorphic to E+ or to E−. See 2 periodthe .. Now Remark comma item suppose 3 below xi the in V Theorem to the power 1 . ) of * and alpha oslash s in E to the power of i comma i = 0 comma period the form degree by one and there is no summand in $ \bigwedge ˆ{ i − 1 } V ˆ{ ∗ } \otimes $ period period comma2 . Now l minus , suppose 1 periodξ ∈ Due∗ and to theα ⊗ Theorems ∈ Ei, i 2= comma 0, ..., l − 1. Due to the Theorem 2 , we know that S isomorphic V we knowφ that:= ξ phi∧ α :⊗ =s ∈ xiE andi+1,i alpha−1 ⊕ E oslashi+1,i ⊕ sE ini+1 E,i+1 to. theApplying powerp ofi+1 i plusto 1 comma i minus 1 oplus E to the power of i plus 1 to $ E ˆ{ i } { + }$ or to $ E ˆ{ i } { − ˆ{ . }}$ See the Remark item 3 below the Theorem 1 . ) comma i oplus E to the power of ithe plus element 1 commaφ, only i plus the 1 period zeroth Applying , first , and p to second the power summand of i plus in the 1 to expression the elementp phii+1φ comma= Pi only+1 ki the+1(F zeroth+)k(F comma−)kφ remains first comma . ( andFor secondk > 2, summandthe kth summand in the expression vanishes in the 2 . \quad Now , supposek=0 α $ \ xi \ in V ˆ{ ∗ }$ and $ \alpha \otimes s \ in E ˆ{ i } p to theexpression power of i for pluspi 1+1 phiφ because = sumF sub− is k =G˜ 0 to the powerdecreases of i plus 1 k alpha to the power of i plus 1 open parenthesis F to , i = 0 , . . . ,− l equivariant,− 1 . $ Due to the Theorem 2 , the power of plus closing parenthesis to the power of k open parenthesis F to the powerV ofi−2 minus∗ closing parenthesis to the power we know that $ \phi :the = form\ degreexi by\wedge one and there\alpha is no summand\otimes in s V ⊗\ inS isomorphicE ˆ{ i + 1 , of k phi remainsto periodEi+1,i− open1 or parenthesisEi+1,i or Ei For+1,i+1 k. greaterSee the 2 item comma 3 of the the k Remark to the power below of the t h Theorem summand 1 . ) i vanishes− 1 in}\ the expressionoplus± forE± ˆ p{ toi the± power+ 1 of i plus , 1 i phi}\ becauseoplus F to theE powerˆ{ i of minus+ 1 is G-tilde , sub i hyphen + 1 equivariant} . $ Applying $ p ˆ{ i3 + . Due 1 } to$ the to previous item , we already know that for the element φ = ξ ∧ α ⊗ s comma decreases chosen above , we get the form degree by one and there is no summand in bigwedge2 to the power of i minus 2 V to the power of * oslash S isomorphic \ hspaceto E sub∗{\ plusminuxf i l l } the to element the power of $ i\ plusphi 1 comma, $ i only minus the 1 or E zeroth sub plusminux , first to the , and power second of i plus summand 1 comma in i or the E sub expression plusminux to the power of i plus 1 comma i plusi+1 1 periodX See+1 the+ itemk − 3 ofk the Remark below the p φ = iα (F ) (F ) φ. $Theorem p ˆ{ i 1 period + 1closing}\ parenthesisphi = \sum ˆ{ i +k 1 } { k = 0 } k {\alpha }ˆ{ i + 1 } (3 periodF ˆ{ + .. Due} ) to ˆthe{ k previous} ( item F comma ˆ{ − } we already) ˆ{ k know}\ thatphi fork$= the 0 remains element phi . =( xi For and alpha$ k oslash> s2 , $ the $ k ˆ{ t h }$ summand chosen above comma we get Using the relations ( 4 ) and ( 2 ) , we may write vanishes2 in the expression for $ p ˆ{ i + 1 }\phi $ because $Fˆ{ − }$ i s $ \ tilde {G} { − equivariant , }$ d e c r e a s e s Line 1 p to the power of i plus 1 phi =pi sum+1(ξ i∧ alphaα ⊗ s sub) = kξ ∧ toα the⊗ s power+ i+1F of+ plusF −( 1ξ ∧ openα ⊗ parenthesiss) F to the power of plus closing α1 parenthesis to the power of k open parenthesis F to the power of minus closing parenthesis to the power of k phi period Line 2 k = +i+1(F +)2(F −)2(ξ ∧ α ⊗ s) 0\ hspace ∗{\ f i l l } the form degree by one and thereα is2 no summand in $ \bigwedge ˆ{ i − 2 } V ˆ{ ∗ } \otimes $ S isomorphic 1 Using the relations open parenthesis= ξ ∧ α ⊗ 4 closings + i+1 parenthesisF +ωij[(ι andξ)α ⊗ opene · parenthesiss − ξ ∧ ι α 2⊗ closinge · s] parenthesis comma we may write α1 ei j ei j Line 1 p to the power of i plus 1 open parenthesis2 xi and alpha oslash s closing parenthesis = xi and alpha oslash s plus i to $ E ˆ{ i + 1 , i − 1 } {\pm }$+1 or+ ı $ij E ˆ{ i + 1 , i } {\pm }$ or $ E ˆ{ i alpha sub 1 to the power of plus 1 F to the power of plus F to−i theE powerω ofιe minusιe (ξ ∧ openα ⊗ s parenthesis) xi and alpha oslash s closing α2 32 i j +parenthesis 1 , Line i2 plus + i alpha 1 } sub{\ 2pm to the} power. $ of Seeplus 1 the open item parenthesis 3 of F the to the Remark power of below plus closing the parenthesis to the power 1 ofTheorem 2 open parenthesis 1 . ) F to the power of minus= ξ ∧ α closing⊗ s − parenthesisi+1 F +[α ⊗ toξ the] · s power+ 2ξ ∧ ofF − 2( openα ⊗ s parenthesis)] xi and alpha oslash s closing α1 2 parenthesis Line 3 = xi and alpha oslash s plus i alpha sub 1 to the power of plus 1 1 divided by 2 F to the power of plus omega \ hspace ∗{\ f i l l }3 . \quad Due to the previous+1 + ı itemij , we already know that for the element $ \phi to the power of ij open square bracket open parenthesis−iα E iota subω eιe subi (ξjα i xi⊗ s closing− ξ ∧ parenthesisιej α ⊗ s). alpha oslash e sub j times s minus xi 2 32 =and iota\ xi sub e\ subwedge i alpha\ oslashalpha e sub\ jotimes times s closings $ square bracket Line 4 minus i alpha sub 2 to the power of plus 1 E to the i − power of plus iBecause divided byα ⊗ 32s omega∈ E , we to the get powerF (α of⊗ s ij) iota = 0 sub by Lemma e sub i iota 3 item sub 2 e . sub Using j open the parenthesis last written xi equation and alpha oslash s closing parenthesis\ centerline, Line we{ maychosen 5 = write xi and above alpha ,oslash we get s minus} i alpha sub 1 to the power of plus 1 1 divided by 2 F to the power of plus open square bracket alpha oslash xi to the power of sharp times s plus 2 xi+1 and F to the power of minus open parenthesis alpha oslash \ centerline {2 } iα s closing parenthesis closing square bracketpi+1(ξ Line∧ α ⊗ 6s minus) = ξ ∧ i alphaα ⊗ s sub− 21 F to+ the(α ⊗ powerξ] · s) of plus 1 E to the power of plus i divided by 32 omega to the power of ij iota sub e sub i open parenthesis xi sub2 j alpha oslash s minus xi and iota sub e sub j alpha oslash s \ [ \ begin { a l i g n e d } p ˆ{ i + 1 ıi}\+1 phi = \sum2i+1 i {\alpha { k }}ˆ{ + 1 } ( F ˆ{ + } closing parenthesis period α2 + i α2 − − E (2ξ ιei α ⊗ s + ξ ∧ E α ⊗ s). ) ˆBecause{ k } alpha( oslash F ˆ{ s − in } E to) the ˆ{ powerk }\32 ofphi i comma. we\\ get F toı the power of minus open parenthesis alpha oslash s closing parenthesisk = = 0 0 by\end Lemma{ a l i g 3 n item e dThe}\ 2] period last summand Using the in this expression vanishes due to the Lemma 3 it em 2 be - last writtencause equation first commaE− = − we4F may−F − write( Eqn . ( 2 ) ) and second α ⊗ s ∈ Ei. Summing - up , we have Line 1 p to the power of i plus 1 open parenthesis xi and alpha oslash s closing parenthesis = xi and alpha oslash s minus i \ centerline { Using the relations ( 4 ) and1 ( 2 ) , we mayı write } alpha sub 1 to the power of plusp 1i+1 dividedφ = ξ ∧ byα 2⊗ Fs to− i the+1 powerF +(α of⊗ ξ plus] · s) open− i+1 parenthesisE+ι ]α alpha⊗ s, oslash xi to the power of sharp times α1 α2 ξ s closing parenthesis Line 2 minus i i alpha sub 2 to the2 power of plus 1 divided16 by 32 E to the power of plus open parenthesis 2 xi to\ [ \ thebegin power{ a of l ig in iota e d } subpwhich e ˆ sub{ iis i alpha aformula + oslash 1 of} s the plus( form 2\ ixiwritten alpha\ subwedge in the2 to statement the\ poweralpha of of the plus\ lemmaotimes 1 divided . bys i xi ) and = E to the\ xi power\wedge of minus\alpha alpha\ oslashotimes s closings parenthesis + i period{\alpha { 1 }}ˆ{ + 1 } F ˆ{ + } F ˆ{ − } ( \ xi \wedge \alphaThe last\ summandotimes in thiss expression ) \\ vanishes due to the Lemma 3 it em 2 be hyphen +cause i first{\ Ealpha to the power{ 2 of}} minusˆ{ + = minus 1 } 4( F to Fthe ˆ power{ + } of minus) ˆ{ F2 to} the( power F ˆof{ minus − } open) ˆ{ parenthesis2 } ( Eqn\ xi period\wedge open\alpha parenthesis\otimes 2 closings parenthesis ) \\ closing parenthesis and second alpha oslash s in E to the power of i period Summing hyphen up= comma\ xi \wedge \alpha \otimes s + i {\alpha { 1 }}ˆ{ + 1 }\ f r a c { 1 }{ 2 } F ˆ{ + } \omegawe haveˆ{ i j } [( \ iota { e { i }}\ xi ) \alpha \otimes e { j }\cdot s − \ xip to the\wedge power of\ iiota plus 1 phi{ e = xi{ andi }}\ alpha oslashalpha s minus\otimes i alpha sube 1 to{ thej }\ powercdot of plus 1s 1 divided ] \\ by 2 F to the power of− plus openi {\ parenthesisalpha alpha{ 2 oslash}}ˆ{ xi+ to the 1 } powerE ˆof{ sharp+ }\ timesf r a cs{\ closingimath parenthesis}{ 32 minus}\omega i alpha subˆ{ 2i j to}\ the poweriota of{ pluse { i }} 1\ iota i divided{ bye 1{ 6 Ej to}} the power( \ ofxi plus\ iotawedge sub xi sharp\alpha alpha oslash\otimes s commas ) \\ =which\ xi is a formula\wedge of the\ formalpha written\ inotimes the statements of− thei lemma{\ periodalpha { 1 }}ˆ{ + 1 }\ f r a c { 1 }{ 2 } F ˆ{ + } [ \alpha \otimes \ xi ˆ{\sharp }\cdot s + 2 \ xi \wedge F ˆ{ − } ( \alpha \otimes s ) ] \\ − i {\alpha { 2 }}ˆ{ + 1 } E ˆ{ + }\ f r a c {\imath }{ 32 }\omega ˆ{ i j }\ iota { e { i }} ( \ xi { j }\alpha \otimes s − \ xi \wedge \ iota { e { j }}\alpha \otimes s ) . \end{ a l i g n e d }\ ]

Because $ \alpha \otimes s \ in E ˆ{ i } , $ we get $ F ˆ{ − } ( \alpha \otimes s ) = 0$ byLemma3 item 2 . Using the last written equation , we may write

\ [ \ begin { a l i g n e d } p ˆ{ i + 1 } ( \ xi \wedge \alpha \otimes s ) = \ xi \wedge \alpha \otimes s − \ f r a c { i {\alpha { 1 }}ˆ{ + 1 }}{ 2 } F ˆ{ + } ( \alpha \otimes \ xi ˆ{\sharp }\cdot s ) \\ − \ f r a c {\imath i {\alpha { 2 }}ˆ{ + 1 }}{ 32 } E ˆ{ + } ( 2 \ xi ˆ{ i }\ iota { e { i }} \alpha \otimes s + \ f r a c { 2 i {\alpha { 2 }}ˆ{ + 1 }}{\imath }\ xi \wedge E ˆ{ − } \alpha \otimes s ) . \end{ a l i g n e d }\ ]

\ hspace ∗{\ f i l l }The last summand in this expression vanishes due to the Lemma 3 it em 2 be −

cause first $ E ˆ{ − } = − 4 F ˆ{ − } F ˆ{ − } ( $ Eqn . ( 2 ) ) and second $ \alpha \otimes s \ in E ˆ{ i } . $ Summing − up , we have

\ [ p ˆ{ i + 1 }\phi = \ xi \wedge \alpha \otimes s − i {\alpha { 1 }}ˆ{ + 1 }\ f r a c { 1 }{ 2 } F ˆ{ + } ( \alpha \otimes \ xi ˆ{\sharp }\cdot s ) − i {\alpha { 2 }}ˆ{ + 1 }\ f r a c {\imath }{ 1 6 } E ˆ{ + }\ iota {\ xi }\sharp \alpha \otimes s , \ ]

\ centerline {which is a formula of the form written in the statement of the lemma . } ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 323 \ hspace4 period∗{\ ..f In i l this l }ELLIPTICITY item comma we OF shall THE determine SYMPLECTIC the numbers TWISTOR beta COMPLEX comma gamma\quad in C323 period Using the fact that p to the power of i plus 1 is an idempotent open parenthesis open parenthesis p to the power of i plus 1 closing parenthesis to\ hspace the power∗{\ off i 2 l = l } p4 to . the\quad powerIn of i this plus 1 item closing , parenthesis we shall comma determine we get i thealpha numbers sub 1 to the $ power\beta of plus, 1 =\ 4gamma slash open\ in Cparenthesis . $ l Using minus i the closing fact parenthesis and i alpha sub 2 to the power of plus 1 = 1 6 slash openELLIPTICITY parenthesis OF l THEminus SYMPLECTIC i closing parenthesis TWISTOR COMPLEX after a tedious 323 but straightforward \ hspace ∗{\ f i l l } that $ p ˆ{ i + 1 }$ is anidempotent $( ( pˆ{ i + 1 } ) ˆ{ 2 } calculation period 4 . In this item , we shall determine the numbers β, γ ∈ C. Using the fact = p ˆ{ i + 1 } ) , $that wepi get+1 is an $ idempotent i {\alpha ((pi+1{)2 =1 p}}i+1ˆ){, we+ get 1i+1} == 4/(l − 4i) and / ( l − Thus comma comparing the last written formula of the preceding it em and the α1 i ) $ and i+1 = 16/(l − i) after a tedious but straightforward calculation . Eqn period open parenthesisα2 6 closing parenthesis comma we get beta = 2 slash open parenthesis i minus l closing parenthesis and gamma = iThus slash , open comparing parenthesis the last i minus written l closing formula parenthesis of the preceding period it em and the Eqn . ( 6 ) , we get \ centerlinesquare β ={ 2/$(i − i l){\ andalphaγ = ı/(i −{ l)2. }}ˆ{ + 1 } = 1 6 / ( l − i ) $ after a tedious but straightforward calculation . } Remark period .. For i = l comma period period period comma 2 l comma xi in V to the power of * and alpha oslash s in E to theThus power , comparing of i comma the the formula last for written p to theformula power of i plusof the 1 reads preceding it em and the Eqnsimply . (6) ,weget $ \beta = 2 / ( i − l ) $ and $ \gamma = \imath / ( i − l ) . $ p to the power of i plus 1 open parenthesis xi and∗ alpha oslashi s closing parenthesisi+1 = xi and alpha oslash s becauseRemark of the Theorem . For 2 andi = thel, ..., items2l, 1ξ ∈ andV 2and of theα ⊗ Remarks ∈ E , belowthe formula the Theorem for p 1reads period simply \ begin { a l i g n ∗} open parenthesis Notice that one may also usei+1 the relation open parenthesis 7 closing parenthesis period closing parenthesis \ squareNow comma we are prepared to prove the ellipticityp (ξ ∧ ofα the⊗ s) truncated = ξ ∧ α ⊗ symplectics twistor \end{ a l i g n ∗} complexesbecause period of the Theorem 2 and the items 1 and 2 of the Remark below the Theorem 1 . ( Notice that Theoremone .... may 7 period also use .... the Let relation open parenthesis ( 7 ) . ) M to the power of 2 l comma omega comma nabla closing parenthesis .... be a \noindent Remark . \quad For$i=l , . . . ,2 l , \ xi \ in V ˆ{ ∗ }$ Fedosov manifold of Ricci typeNow admitting , we are a prepared to prove the ellipticity of the truncated symplectic twistor and $ \alpha \otimes s \ in E ˆ{ i } , $ the formula for $ p ˆ{ i + 1 }$ reads metaplecticcomplexes s tructure . comma l greater equal 2 period .. Then the truncated symplectic twistor complexes simply 0 right arrowTheorem Capital Gamma 7 . openLet parenthesis(M 2l, ω, M∇) commabe E a to Fedosov the power manifold of 0 closing of Ricci parenthesis type admitting T right a arrow 0 Capital Gamma openmetaplectic parenthesis s tructure M comma, l E≥ to2. theThen power the of 1truncated closing parenthesis symplectic Ttwistor right arrowcomplexes 1 times times times T right arrow l minus 2\ [ Capital p ˆ{ Gammai + open 1 parenthesis} ( \ xi M comma\wedge E to the\ poweralpha of l minus\otimes 1 closings parenthesis ) = \ xi \wedge \alpha \otimes s \ ] 0 1 l−1 and 0 → Γ(M,E )T→0 Γ(M,E )T→1 ··· T→l − 2Γ(M,E ) Capital Gamma open parenthesis M comma E to the power of l closing parenthesis T right arrow l Capital Gamma open parenthesisand M comma E to the power of l plus 1 closing parenthesis T right arrow l plus 1 times times times T right arrow 2 l minus \noindent because of the Theorem 2 and the items 1 and 2 of the Remark below the Theorem 1 . 1 Capital Gamma open parenthesis Ml comma E to thel+1 power of 2 l closing parenthesis2l right arrow 0 (are Notice e l l iptic that period one mayΓ( alsoM,E use)T→l theΓ(M,E relation)T→l + ( 1 7··· )T .→2 )l − 1Γ(M,E ) → 0 Proof periodare e .. l l We iptic should . prove the equations Ker open parenthesis i sigma to the power of xi closing parenthesis sub m = Im \ hspace ∗{\ f i l l }Now , we are prepared to prove theξ ellipticityξ of the truncated symplectic twistor open parenthesisProof i pi. subWe minus should 1 to prove the power the equations of xi closing Ker parenthesis (iσ)m = Im sub (iπ m−1 for)m thefor appropriate the appropriate indices i indices iand and for for each each point pointm m∈ inM. MHere period the Here constituents the constituents of the previous of the previous equation equation \noindent complexes . are fibersare of fibers the corresponding of the corresponding shaeves periodshaeves . 1 period .. First1 . comma First we, we prove prove that that the the sequences sequences mentioned mentioned in the in formulation the formulation of the of theorem the theorem are \noindent Theorem \ h f i l l 7 . \ h f i l l Let $ ( M ˆ{ 2 l } , \omega , \nabla ) $ \ h f i l l be a Fedosov manifold of Ricci type admitting a are complexescomplexes period . For Fori i= = 0 0, ..., comma l − 2, period l, ..., 2l period− 1, ψ ∈ periodΓ(M,E commai) and l minusa differential 2 comma 1 - l form commaξ ∈ periodΓ(M,T period∗M), period comma 2 l minus 1 commawe may psi writein Capital 0 = p Gammai+2(0) = openpi+2(( parenthesisξ ∧ ξ) ∧ ψ) M = comma E to the power of i closing parenthesis and a differential \noindent metaplectic s tructure $ , l \geq 2 . $ \quad Then the truncated symplectic twistor complexes 1 hyphen form xi in Capital Gammai+2 open parenthesisi+2 M commaPmi+1 T toi+1 the,j power of * M closing parenthesis comma we may write p (ξ∧ Id (ξ ∧ ψ)) = p (ξ ∧ j=0 p (ξ ∧ ψ)). Due to the Theorem 2 , we 0 = p to the power of i plus 2 open parenthesis 0 closing parenthesis = p to the poweri+2 of ii plus+1 2 open parenthesisξ ξ open parenthesis \ [ 0 \rightarrow \Gammaknow that( the M last , written E ˆ expression{ 0 } )T equals p{\(ξrightarrow∧ p (ξ ∧ ψ))} = iσ0+1σi\ψGamma (M xi and xi closing parenthesis and psi closing parenthesis = ξ ξ , E ˆ{ 1 } )T {\rightarrow } and1 thus\cdotiσ+1iσ =\ 0cdot. \cdot T {\rightarrow } l − 2 p to the power of i plus 2 open parenthesis xi and Id openξ parenthesis xiξ and psi closing parenthesis∗ closing parenthesis = p to 2 . Second , we prove the relation Ker (i )m ⊆ Im (i −1)m for each 0 6= ξ ∈ T M and the\Gamma power of( i plus M 2 open , parenthesis E ˆ{ l xi− and sum1 } sub) j =\ ] 0 toσ the power ofσ m sub i plus 1 p to the powerm of i plus 1 comma j open i = 0, ..., l − 2. Here σξ = 0 is to be understood . Suppose a homogeneous element α ⊗ s ∈ Ei is parenthesis xi and psi closing parenthesis−1 closing parenthesis period Due to the Theorem 2 comma we m given such that iξ (α ⊗ s) = 0.( In the next item , we will treat the general non - homogeneous case . know that the last writtenσ expression equals p to the power of i plus 2 open parenthesis xi and p to the power of i plus 1 open ) Due to the paragraph below parenthesis\noindent xiand and psi closing parenthesis closing parenthesis = i sigma sub plus 1 to the power of xi sigma sub i to the power of xi the Lemma 5 , we know that 0 = iξ (α ⊗ s) = pi+1(ξ ∧ α ⊗ s). We shall find psi σ an element ψ ∈ Ei−1 such that pi(ξ ∧ ψ) = α ⊗ s. \ [ and\Gamma thus i sigma( sub M plus , 1 to E the ˆ{ powerl } of)T xi i sigmam {\ torightarrow the power of xi} = 0l period\Gamma ( M , E ˆ{ l + Using formula ( 6 ) for the projection ( Lemma 6 ) , we may rewrite the 1 }2 period)T .. Second{\rightarrow comma we prove} l the + relation 1 Ker\cdot open parenthesis\cdot i sigma\cdot to theT power{\rightarrow of xi closing parenthesis} 2 l sub− m subset1 \Gamma equal Im open( parenthesis M , iE sigma ˆ{ 2 sub minus l } 1) to the\rightarrow power of xi closing0 \ parenthesis] sub m for each 0 negationslash-equal xi in T sub m to the power of * M equationpi+1(ξ ∧ α ⊗ s) = 0into and i = 0 comma period period period comma l minus+ 2 period] Here sigma+ sub minus 1 to the power of xi = 0 is to be understood period\noindent Supposeare a homogeneous e l l iptic . ξ ∧ α ⊗ s + βF (α ⊗ ξ · s) + γE ιξ]α ⊗ s = 0. (9) element alpha oslash s in E sub m to the power of i is given such that i sigma to the power of xi open parenthesis alpha oslash s\noindent closing parenthesisProof = . 0\ periodquad openWe should parenthesis prove In the the next equations item comma Ker we $ ( i {\sigma }ˆ{\ xi } ) { m } = $will Im treat $ the ( general i {\ nonpi hyphen}ˆ{\ homogeneousxi } { − case1 period} ) closing{ m parenthesis}$ for the Due to appropriate the paragraph below indicesthe Lemma $ 5 i comma $ and weknow for each that 0 point = i sigma $m to the\ powerin ofM xi open . $ parenthesis Here the alpha constituents oslash s closing of parenthesis the previous = p to equation the power of i plus 1 open parenthesis xi and alpha oslash s closing parenthesis period We shall find \noindentan elementare psi in fibers E sub m of to the the power corresponding of i minus 1 such shaeves that p to . the power of i open parenthesis xi and psi closing parenthesis = alpha oslash s period 1 .Using\quad formulaFirst .. open , we parenthesis prove that 6 closing the parenthesis sequences .. for mentioned the projection in the.. open formulation parenthesis Lemma of the 6 closing theorem parenthesis commaarecomplexes.For we may rewrite the $i = 0 , . . . , l − 2 , l , . . . , 2equation l − p to1 the power , of\ psi i plus 1\ openin parenthesis\Gamma xi and( alpha M oslash , Es closing ˆ{ i parenthesis} ) $ =and 0 into a Equation:differential open parenthesis 91 closing− form parenthesis $ \ xi .. xi\ andin alpha\Gamma oslash s plus( Mbeta F , to the T ˆ power{ ∗ } of plusM open ) parenthesis ,$ wemaywrite alpha oslash xi $0 to the power= pˆ of { i +sharp 2 times} s( closing 0 parenthesis ) = plus p ˆ gamma{ iE + to the 2 } power(( of plus iota\ xi sub xi\wedge sharp alpha\ xi oslash) s = 0\wedge period \ psi ) = $

\ hspace ∗{\ f i l l } $ p ˆ{ i + 2 } ( \ xi \wedge $ Id $ ( \ xi \wedge \ psi )) = p ˆ{ i + 2 } ( \ xi \wedge \sum ˆ{ m { i + 1 }} { j = 0 } p ˆ{ i + 1 , j } ( \ xi \wedge \ psi ) ) . $ Dueto the Theorem2 ,we

\ hspace ∗{\ f i l l }know that the last written expression equals $ p ˆ{ i + 2 } ( \ xi \wedge p ˆ{ i + 1 } ( \ xi \wedge \ psi ) ) = i {\sigma }ˆ{\ xi } { + 1 }\sigma ˆ{\ xi } { i } \ psi $

\ centerline {and thus $ i {\sigma }ˆ{\ xi } { + 1 } i {\sigma }ˆ{\ xi } = 0 . $ }

2 . \quad Second , we prove the relation Ker $ ( i {\sigma }ˆ{\ xi } ) { m }\subseteq $ Im $ ( i {\sigma }ˆ{\ xi } { − 1 } ) { m }$ f o r each $ 0 \not= \ xi \ in T ˆ{ ∗ } { m } M $ and$i=0 , . . . , l − 2 . $ Here $ \sigma ˆ{\ xi } { − 1 } = 0 $ is to be understood . Suppose a homogeneous element $ \alpha \otimes s \ in E ˆ{ i } { m }$ is given such that $ i {\sigma }ˆ{\ xi } ( \alpha \otimes s ) = 0 . ($ Inthenextitem,we will treat the general non − homogeneous case . ) Due to the paragraph below

\ hspace ∗{\ f i l l } the Lemma5 , weknow that $ 0 = i {\sigma }ˆ{\ xi } ( \alpha \otimes s ) = p ˆ{ i + 1 } ( \ xi \wedge \alpha \otimes s ) .$ Weshall find

\ centerline {an element $ \ psi \ in E ˆ{ i − 1 } { m }$ such that $pˆ{ i } ( \ xi \wedge \ psi ) = \alpha \otimes s . $ }

\ hspace ∗{\ f i l l } Using formula \quad ( 6 ) \quad for the projection \quad ( Lemma 6 ) , we may rewrite the

\ begin { a l i g n ∗} equation p ˆ{ i + 1 } ( \ xi \wedge \alpha \otimes s ) = 0 i n t o \\\ tag ∗{$ ( 9 ) $}\ xi \wedge \alpha \otimes s + \beta F ˆ{ + } ( \alpha \otimes \ xi ˆ{\sharp } \cdot s ) + \gamma E ˆ{ + }\ iota {\ xi }\sharp \alpha \otimes s = 0 . \end{ a l i g n ∗} 324 .. S period KR Yacute SL \noindentApplying the324 operator\quad ES to the. KR power $ of\acute minus open{Y} parenthesis$ SL formula open parenthesis 2 closing parenthesis closing parenthesis on the both sides of the previous \ hspaceequation∗{\ andf i l using l } Applying the first commutation the operator relation $ in E the ˆ{ row − } open( parenthesis $ formula 3 closing ( 2 parenthesis) ) on the from both Lemma sides 3 comma of the previous we get equation and using the first commutation relation in the row ( 3 ) from Lemma 3 , i divided324 by 2 omega S . KR toY´ SL the power of ij iota sub e sub i iota sub e sub j open parenthesis xi and alpha closing parenthesis oslash swe plus get beta E to the power of minusApplying F to the the power operator of plusE open−( formula parenthesis ( 2 ) ) alpha on the oslash both xi sides to the of the power previous of sharp times s closing parenthesis plusequation gamma andopen using parenthesis the first E commutation to the power relationof plus E in to the the row power ( 3 )of from minus Lemma minus 3 2 , H we closing get parenthesis iota sub xi \ [ \ f r a c {\imath }{ 2 }\omega ˆ{ i j }\ iota { e { i }}\ iota { e { j }} ( \ xi \wedge sharp alpha oslash s = 0 periodı \alphaUsing the) graded\otimes Leibnizωij propertyι sι (ξ +∧ ofα iota) ⊗\betas sub+ βE xi to−FE the+ ˆ(α{ power⊗ −ξ] }· ofs) sharpF + γ ˆ({E comma+E}− − the(2H relation)ι\]αalpha⊗ s open= 0. parenthesis\otimes 4 closing\ xi ˆ parenthesis{\sharp } 2 ei ej ξ for\cdot the valuess of ) + \gamma ( E ˆ{ + } E ˆ{ − } − 2 H ) \ iota {\ xi }\sharp \alpha \otimes s = 0 . \ ] H on form hyphen homogeneousUsing elements the and graded the secondLeibniz relation property in of theιξ] row, the open relation parenthesis ( 4 ) for 3 the closing values parenthesis of from Lemma 3 commaH on formwe obtain - homogeneous elements and the second relation in the row ( 3 ) from Lemma 3 , we Line 1 iobtain divided by 2 open parenthesis minus 2 iota sub xi sharp minus 2 i xi and E to the power of minus closing parenthesis open\ hspace parenthesis∗{\ f i l alpha l } Using oslash the s closing graded parenthesis Leibniz plus property beta F to the of power $ \ ofiota plus E{\ to thexi powerˆ{\ ofsharp minus}} open, parenthesis $ the alpha relation ( 4 ) for the values of ı − + − ] − ] oslash xi to the power of sharp(− times2ιξ] − s2 closingıξ ∧ E parenthesis)(α ⊗ s) + minusβF E beta(α ⊗ Fξ to· thes) − powerβF (α of⊗ minusξ · s) open parenthesis alpha oslash xi to $ H $ on form − homogeneous2 elements and the second relation in the row ( 3 ) from the power of sharp times s closing parenthesis Line 2 plus+ gamma− E to the power of plus E to the power of minus iota sub xi to the +γE E ι ] α ⊗ s + γ(l − i + 1)ι ] α ⊗ s = 0. powerLemma of sharp 3 , we alpha obtain oslash s plus gamma open parenthesis lξ minus i plus 1 closingξ parenthesis iota sub xi to the power of sharp alpha oslash s =The 0 period operator E− commutes with the operator of the symplectic Clifford multiplication ( by the \ [ \ begin { a l i g n e d }\]f r a c {\imath }{ 2 } ( − 2 \−iotaı ij{\ xi }\sharp − 2 \imath \ xi The operatorvector E field to theξ ) power and also of minus with the commutes contraction withιξ the] because operatorE of= the2 ω symplecticιei ιej ( Cliffordformula ( 2 ) ) . Using \wedgemultiplicationtheseE ˆ{ two open − facts } parenthesis)( , we get by\alpha the vector\ fieldotimes xi to thes power ) of + sharp\ closingbeta parenthesisF ˆ{ + } andE also ˆ{ with − } the( contraction\alpha iota\otimes sub xi sharp\ xi becauseˆ{\sharp }\cdot s ) − \beta F ˆ{ − } ( \alpha \otimes \ xi ˆ{\sharp } \cdot s ) \\ ı − + ] − − ] E to the power of minus =(− i2 dividedιξ] − 2ıξ by∧ 2E omega)(α ⊗ tos) the + βF powerξ · ofE ij(α iota⊗ s sub) − βF e sub(α i⊗ iotaξ .s sub) e sub j open parenthesis formula 2 open+ parenthesis\gamma 2 closingE ˆ{ parenthesis+ } E ˆ{ closing − } parenthesis \ iota {\ periodxi Usingˆ{\sharp these two}}\ factsalpha comma we\ getotimes s + \gamma +γE+ι ]E−α ⊗ s + γ(l − i + 1)ι ]α ⊗ s = 0. (Line l 1 i− dividedi by + 2 open 1 parenthesis ) \ iota minus{\ 2 iotaxi subˆ{\ξ xisharp sharp minus}}\ 2alpha i xi andξ E\ tootimes the powers of minus = closing 0 . parenthesis\end{ a l i g n e d }\ ] open parenthesisBecause alpha oslashF −(α ⊗ s closings) = 0( parenthesis Lemma 3 item plus 2 beta ) , we F to have theE power−α ⊗ ofs = plus 4F − xiF to−( theα⊗ powers) = 0 of. Thus sharp , timeswe E to the power of minus openobtain parenthesis the identity alpha oslash s closing parenthesis minus beta F to the power of minus open parenthesis alpha oslash xi toThe the operator power of sharp $ E period ˆ{ − s closing}$ commutes parenthesis with Line 2 the plus operator gamma E to of the the power symplectic of plus iotaClifford sub xi sharp E to the power of multiplication ( by the vector field− $ \ xi] ˆ{\sharp } ) $ and also with the contraction $ \ iota {\ xi } minus alpha oslash s plus gamma− openıιξ]α parenthesis⊗ s − βF l(α minus⊗ ξ · is plus) + γ 1(l closing− i + 1) parenthesisιξ]α ⊗ s = iota 0. sub xi sharp alpha oslash s = 0 period \sharpBecause$ F because to the power of minus open parenthesis alpha oslash s closing parenthesis = 0 open parenthesis Lemma 3 item 2 Substituting the second relation in the row ( 5 ) into the previous equation and using the fact closing$ E parenthesis ˆ{ − } = comma\ f r awe c {\ haveimath E to the}{ power2 }\ of minusomega alphaˆ{ i oslash j }\ s =iota 4 F to{ thee power{ i of}}\ minusiota F to the{ powere { ofj minus}} ( $ F −(α ⊗ s) = 0 again , we get openformula parenthesis ( 2 ) alpha ) . oslash Using these two facts , we get s closing parenthesis = 0 period Thus comma we obtain the identity ı \ [ \ begin { a l i g n e d }\ f r a c {\imath−ıι}{]α2⊗}s −(βξ] ·−F −(α2⊗ s) −\ iotaβ ι ]α{\⊗ s xi }\sharp − 2 \imath \ xi minus i iota sub xi sharp alpha oslash sξ minus beta F to the power of2 minusξ open parenthesis alpha oslash xi to the power of \wedge E ˆ{ − } )( \alpha \otimes s ) + \beta F ˆ{ + }\ xi ˆ{\sharp }\cdot sharp times s closing parenthesis plus gamma open parenthesis+γ(l − li minus+ 1)ιξ i]α plus⊗ s 1= closing 0. parenthesis iota sub xi sharp alpha oslash s E= 0 ˆ{ period − } ( \alpha \otimes s ) − \beta F ˆ{ − } ( \alpha \otimes \ xi ˆ{\sharp } .Substituting s ) \\Using the second the prescription relation in for the the row numbers open parenthesisβ and γ( 5Lemma closing 6 parenthesis ) and the already into the twice previous used equation relation + \gamma− E ˆ{ + }\ iota {\ xi }\ısharp E ˆ{ − } \alpha \otimes s + \gamma and usingF the(α fact⊗ s) F = to 0, thewe get power (−ı of+ minusγ(l − i open+ 1) parenthesis− β 2 )ιξ]α ⊗ alphas = − oslash2ıιξ]α s⊗ closings = 0 fromparenthesis which =the 0 equation again comma we get (Line l 1 minus− ii iota + sub xi 1 sharp ) alpha\ iota oslash s{\ minusxi beta}\ xisharp to the power\alpha of sharp times\otimes F to the powers = of minus 0 open . \ parenthesisend{ a l i g n e d }\ ] alpha oslash s closing parenthesis minus beta i divided by 2 iota sub xi sharp alpha oslash s Line 2 plus gamma open parenthesis l ι ] α ⊗ s = 0 (10) minus i plus 1 closing parenthesis iota sub xi sharp alphaξ oslash s = 0 period BecauseUsing the $ prescription F ˆ{ − } for( the numbers\alpha beta\otimes and gammas open )parenthesis = 0 Lemma ($ 6 closing Lemma3item2) parenthesis and the ,wehave already $Eˆ{ − } \alpha \otimes s = 4 F ˆ{ − } Ffollows ˆ{ − . } ( \alpha \otimes $ twice used relation F to the power of minus open parenthesis alpha oslash s closing parenthesisi = 0 comma we get open $s ) = 0 . $ ThusSubstituting ,weobtain this relation the identityinto the prescription for the projection p ( Eqn . parenthesis minus i plus gamma open parenthesis( 9 ) ) , we lget minus for i i= plus 0, ..., 1closing l − 2 the parenthesis equation minus beta i divided by 2 closing parenthesis iota sub xi sharp alpha oslash s = \ [ minus− \ 2imath i iota sub\ xiiota sharp alpha{\ xi oslash}\ ssharp = 0 from which\alpha the equation\otimes s − \beta F ˆ{ − } ( \alpha \otimes \ xi ˆ{\sharp }\cdot s ) + \gamma ( l − i + 1 ) \ iota {\ xi } Equation: open parenthesis 1 0 closing0 = pi+1 parenthesis(ξ ∧ α ⊗ s ..) = iotaξ ∧ subα ⊗ xis + toβF the+ power(α ⊗ ξ] of· s sharp). alpha oslash s =(11) 0 \sharpfollows period\alpha \otimes s = 0 . \ ] SubstitutingApplying this relation the contraction into the prescription operator ι forξ] to the the projection previous p equation to the power and using of i open the first parenthesis formula Eqn in the period open parenthesisrow ( 5 ) 9 from closing Lemma parenthesis 3 , we obtain closing parenthesis comma we get for i = 0 comma period period period comma l minus 2Substituting the equation the second relation in the row ( 5 ) into the previous equation ı andEquation: using open the parenthesis fact $ F 10 1=ˆ{ closing− −ξ ∧ }ι ]α parenthesis( ⊗ s −\alphaβF ..+ι 0]( =α \⊗ potimesξ to] · thes) + powerβ sα ⊗ ofξ] i).( plusξ] · =s 1). open 0$ parenthesis again,weget xi and alpha oslash s ξ ξ 2 closing parenthesis = xi and alpha oslash s plus beta F to the power of plus open parenthesis alpha oslash xi to the power of sharp times\ [ \ begin s closing{ a liUsing parenthesis g n e d the} − fact period that \imath the contraction\ iota and{\ symplecticxi }\sharp Clifford multiplication\alpha \ commuteotimes , wes have− \beta \ xi ˆ{\sharp } \cdot F ˆ{ − } ( \alpha \otimes s ) − \beta \ f r a c {\imath }{ 2 }\ iota {\ xi } Applying the contraction operator iota sub xi sharp to the previous equationı and using \sharp \alpha \otimes0 = −ξs∧\\ι ]α ⊗ s − βF +ξ] · (ι ]α ⊗ s) + β α ⊗ ξ] · (ξ] · s). the first formula in the row open parenthesisξ 5 closing parenthesisξ from2 Lemma 3 comma we obtain +0 = minus\gamma xi and( iota subl xi− sharpi alpha + oslash 1 s ) minus\ iota beta F{\ to thexi power}\ ofsharp plus iota\ subalpha xi sharp\otimes open parenthesiss = alpha 0 oslash. \end xi{ toa l ithe g n e power d }\ ] of sharp times s closing parenthesis plus beta i divided by 2 alpha oslash xi to the power of sharp period open parenthesis xi to the power of sharp times s closing parenthesis period Using the fact that the contraction and symplectic Clifford multiplication Usingcommute the comma prescription we have for the numbers $ \beta $ and $ \gamma ( $ Lemma 6 ) and the already twice0 = minus used xi relation and iota sub $ xi F sharp ˆ{ − alpha } ( oslash\alpha s minus beta\otimes F to the powers ) of plus = xi 0 to the ,$weget power of sharp $( times open− parenthesis\imath iota+ sub\gamma xi sharp alpha( oslash l − s closingi parenthesis + 1plus ) beta− i \ dividedbeta by\ f 2 ralpha a c {\ oslashimath xi to}{ the2 power} ) of sharp\ iota times{\ xi } open\sharp parenthesis\alpha xi to the\otimes power of sharps times = $ s closing parenthesis period $ − 2 \imath \ iota {\ xi }\sharp \alpha \otimes s = 0 $ from which the equation

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 0 ) $}\ iota {\ xi ˆ{\sharp }}\alpha \otimes s = 0 \end{ a l i g n ∗}

\ centerline { f o l l o w s . }

\ hspace ∗{\ f i l l } Substituting this relation into the prescription for the projection $ p ˆ{ i } ( $ Eqn .

\ centerline {(9)),wegetfor$i = 0 , . . . , l − 2 $ the equation }

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 1 ) $} 0 = p ˆ{ i + 1 } ( \ xi \wedge \alpha \otimes s ) = \ xi \wedge \alpha \otimes s + \beta F ˆ{ + } ( \alpha \otimes \ xi ˆ{\sharp } \cdot s ) . \end{ a l i g n ∗}

Applying the contraction operator $ \ iota {\ xi }\sharp $ to the previous equation and using the first formula in the row ( 5 ) from Lemma 3 , we obtain

\ [ 0 = − \ xi \wedge \ iota {\ xi }\sharp \alpha \otimes s − \beta F ˆ{ + } \ iota {\ xi }\sharp ( \alpha \otimes \ xi ˆ{\sharp }\cdot s ) + \beta \ f r a c {\imath }{ 2 } \alpha \otimes \ xi ˆ{\sharp } .( \ xi ˆ{\sharp }\cdot s ) . \ ]

Using the fact that the contraction and symplectic Clifford multiplication commute , we have

\ [ 0 = − \ xi \wedge \ iota {\ xi }\sharp \alpha \otimes s − \beta F ˆ{ + } \ xi ˆ{\sharp }\cdot ( \ iota {\ xi }\sharp \alpha \otimes s ) + \beta \ f r a c {\imath }{ 2 } \alpha \otimes \ xi ˆ{\sharp }\cdot ( \ xi ˆ{\sharp }\cdot s ) . \ ] ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 325 ELLIPTICITYSubstituting OF the THEEqn period SYMPLECTIC open parenthesis TWISTOR 1 0COMPLEX closing parenthesis\quad 325 into the previous equation comma we obtain Substitutingalpha oslash xi to the the Eqn power . of ( sharp 1 0 times) into open the parenthesis previous xi to equation the power of , sharp we obtain times s closing parenthesis = 0 period Substituting the definition of F to the power of plus into the equation open parenthesis 1 1 closing parenthesis multiplying it by \ [ xi\alpha to the power\otimes of sharp and\ usingxi ˆ{\ the equationsharp }\ iota subcdot xi sharp( alpha\ xi oslashˆ{\ ssharp = 0 open}\ parenthesiscdot Eqns period ) open = parenthesis 0 . \ ] 1 0 closing parenthesisELLIPTICITY closing OF parenthesisTHE SYMPLECTIC again TWISTOR comma we COMPLEX get 325 Substituting the Eqn . ( 1 0 ) into the Line 1 0previous = xi and equation alpha oslash , we obtain xi to the power of sharp times s plus beta i divided by 2 epsilon to the power of i and alpha oslashSubstituting xi to the power the of definition sharp times e of sub i $ times F ˆ xi{ to+ the}$ power into of the sharp equation times s comma ( 1 Line 1 ) 2 0multiplying = xi and alpha it oslash by xi to the power$ \ ofxi sharpˆ{\ timessharp s plus}$ beta and i divided using by the 2 epsilon equationα ⊗ toξ] the· (ξ] power $· s)\ =iota of 0. i and{\ alphaxi }\ oslashsharp open parenthesis\alpha e sub\otimes i times xi tos the =power 0 of sharp ($ times Eqn. xi to (10))again the power of sharp times ,weget minus i omega sub 0 open parenthesis xi to the power of sharp comma e sub i Substituting the definition of F + into the equation ( 1 1 ) multiplying it by ξ] and using the closing parenthesis xi to the power of sharp times closing parenthesis s period equation ι ]α ⊗ s = 0( Eqn . ( 1 0 ) ) again , we get \ [ \Substitutingbegin { a l i g the n e d identityξ} 0 alpha = oslash\ xi xi\wedge to the power\alpha of sharp times\otimes xi to the\ xi powerˆ{\ ofsharp sharp times}\ scdot = 0 intos the +previous\beta equation\ f r a c {\ commaimath we}{ 2 }\ epsilon ˆ{ i }\wedge \alphaı \otimes \ xi ˆ{\sharp }\cdot e { i } 0 = ξ ∧ α ⊗ ξ] · s + β i ∧ α ⊗ ξ] · e · ξ] · s, \cdotobtain \ xi ˆ{\sharp }\cdot s , \\ 2 i 0 = \ xi \wedge \alpha \otimes \ xi ˆ{\sharp }\cdot s + \beta \ f r a c {\imath }{ 2 } 0 = open parenthesis 1 plus 1 divided by 2] beta closingı i parenthesis] xi] and alpha] oslash] xi to the power of sharp times s period 0 = ξ ∧ α ⊗ ξ · s + β  ∧ α ⊗ (ei · ξ · ξ · −ıω0(ξ , ei)ξ ·)s. \ epsilonIf i = 0 commaˆ{ i }\ periodwedge period period\alpha comma\ lotimes minus2 2 comma( the e coefficient{ i }\ 1cdot plus beta\ xi slashˆ{\ 2 equal-negationslashsharp }\cdot 0 comma\ xi ˆ{\sharp } \cdot − \imath \omega { 0 } ( \ xi ˆ{\sharp } , e { i } ) \ xi ˆ{\sharp }\cdot and thus by dividingSubstituting comma the we identity get α ⊗ ξ] · ξ] · s = 0 into the previous equation , we obtain )xi s and alpha . \end oslash{ a l ixi g nto e dthe}\ power] of sharp times s = 0 period Because the symplectic Clifford multiplication by a non hyphen 1 zero 0 = (1 + β)ξ ∧ α ⊗ ξ] · s. vector is inj ective open parenthesis see the subsection2 2 period 2 closing parenthesis comma we have Substituting the identity $ \alpha \otimes \ xi ˆ{\sharp }\cdot \ xi ˆ{\sharp }\cdot Equation: open parenthesis 1 2 closingIf i = parenthesis 0, ..., l − 2, the .. 0 coefficient = xi and alpha 1 + β/ oslash2 6= 0, sand period thus by dividing , we get s = 0 $ into the previous equation , we 3 period .. Inξ ∧ thisα ⊗ itemξ] · s comma= 0. Because we will the st symplectic ill suppose Clifford i = 0 comma multiplication period period by a nonperiod - zero comma vector l minus is inj ective2 period Let us consider obtain a general ( see the subsection 2 . 2 ) , we have element phi in Ker open parenthesis i sigma to the power of xi closing parenthesis sub m subset equal E sub m to the power of\ [ i 0and denote = ( the basis 1 + of bigwedge\ f r a c { to1 the}{ power2 }\ ofbeta i T sub m) to the\ xi power\wedge of * M by\ openalpha parenthesis\otimes alpha sub\ xi i kˆ{\ closingsharp } parenthesis\cdot s sub k . =\ 1] to the power of comma to the power0 = ξ of∧ nα sub⊗ s. i (12) n sub i in N period Due to the finite dimensionality of bigwedge to the power of i T sub m to the power of * M comma there exist complex 3 . In this item , we will st ill suppose i = 0, ..., l − 2. Let us consider a general element φ ∈ Ker \ hspace ∗{\ξf i l l } If$i=0,.i Vi ∗ . .ni ,l − 2 ,$ the coefficient $1 + numbers(i aσ) subm ⊆ jkE commam and denote j in N commathe basis k of = 1 commaTmM by period (αik) periodk=1, period comma n sub i comma such that phi = sum sub k \beta / 2 \ne 0 , $ and thus byV dividingi ∗ , we get = 1 to the powerni of∈ nN sub. Due i sum to the sub finite j = 1 dimensionality to the power of of infinityTmM, a subthere jk exist alpha complex to the numberspower ofa ijk k, oslash j ∈ N,h k sub= j ) Pni P∞ ik (h j j∈ where sub1, ..., of Sni open, such parenthesis that φ = L sub closingajk parenthesisα ⊗hj where to the powerNS of. openistheSchauderBecause parenthesis h sub simeqtheoperators to the power of j to the $ \ xi \wedge \alpha k\=1otimesj=1 \ xi ˆ{\sharpofS(L)}\' cdotm s = 0 . $ Because the symplectic Clifford multiplication by a non − zero power of closing parenthesismcorresponding j sub S sub m period to the powerbasis of in N is the Schauder BecausePn subi P the∞ operators basis of S F to vector isbasis inj of S ective± ( see, the± subsectionto theξ∧Schauder 2 . 2 ) , weon haveEm, we get 0 = ajkξ∧ the power of plusminuxF sub commaH,E to the,ιξand power of m correspondingarecontinuous sub H comma E to the powerk=1 of plusminuxj=1 comma iota sub ik xi and to theα xi⊗ andhj precisely Schauder in are the continuous same way to as the we power obtained of basis the formula ( 1 2 ) in the homogeneous situation \ beginon E{ suba l( i mitg n em comma∗} 2 of thiswe get proof 0 = ) sum . sub k = 1 to the power of n sub i sum sub j = 1 to the power of infinity a sub j k xi and \ tag ∗{$ ( 1 2 ) $} 0 = \ xi \wedge \alpha \otimes s . Pni alpha to the powerUsing of the i k definition oslash h sub of the j precisely Schauder in basis the same again way , we as have we for each j ∈ N the equation k=1 ajkξ ∧ ik \endobtained{ a l i gα n the∗}= formula 0. Using open the parenthesisCartan lemma 1 2 on closing exterior parenthesis differential in systemsthe homogeneous , we get the situation existence open of parenthesisa family it em 2 of this Pni ik proof closing(β parenthesisj)j ∈ N of (i period− 1) forms such that ξ ∧ βj = k=1 ajkα . It is possible to see ( e . g . by t aking 3 .Using\quad thethe definitionInthisitem,wewillst st andard of Hodge the Schauder - type metric basis again on illsuppose the comma space we of forms have $i for ) that each = one j in can 0 N the choose , the . family . ( .βj)j ∈ ,N l − 2 . $ Let us consider a general P∞ equationin sum such sub a way k = that 1 toψ the:= powerj=1 ofβj n⊗ subhj converges i a sub j k . xi Thus and , alpha we may to the write power of i k = 0 period Using the Cartan lemma onelement exterior differential $ \phi \ in $ Ker $ ( i {\sigma }ˆ{\ xi } ) { m }\subseteq E ˆ{ i } { m }$ and denote the basis of $ \bigwedge ˆ{ i } T ˆ{ ∗ } { m } M $ by $ ( \alpha { i k } ) ˆ{ n { i }} { k systems comma we get the existence of a family open parenthesis beta sub j closingni parenthesis sub j in N of open parenthesis i = 1 ˆ{ , }}$ ξ P∞ i P∞ i P∞ X ik minus 1 closing parenthesisiσ− forms1( j=1 suchβj ⊗ thathj) = p ( j=1ξ ∧ βj ⊗ hj) = p ( j=1 ajkα ⊗ hj) = xi and beta sub j = sum sub k = 1 to the power of n sub i a sub jk alpha tok=1 the power of i k period It is possible to see open parenthesis$ n { i e period}\ in g periodN by . t $ aking Due the to st andard the finite dimensionality of $ \bigwedge ˆ{ i } T ˆ{ ∗ } { m } M , $ there existi complex P∞ Hodge hyphen type metricp (φ) on = the spaceφ. Summing of forms closing - up , we parenthesis have that thatψ one= canj=1 chooseβj ⊗ thehj familyis the desired numbers $ a { jk } , j ξ \ in N,k=1,...,nξ ξ { i } , $ such that open parenthesispreimage beta . Thus sub, j φ closing∈ Im parenthesis (iσ−1)m. 4 sub . j Now in N in, we such prove a way that that Ker psi (i :σ =)m sum⊆ Im sub (i jσ =−1 1)m tofor thei power= of infinity beta $ \phi = \sum ˆ{ n { i }}∗ { k = 1 }\sumξ ˆ{\ inftyi+1} { j = 1 } a { jk }\alpha ˆ{ i sub j oslashl h+ sub 1, ..., j converges2l, 0 6= ξ ∈ periodΓ(M,T ThusM). commaIf φ = weα ⊗ mays ∈ Ker write (iσ)m, then 0 = p (ξ ∧ φ) = ξ ∧ α ⊗ s. Due k }\otimes h { j }$ Vi−1 ∗ i sigma sub minus 1 to the powerto the of xi Cartan open parenthesis lemma , we to know the powerthat there of sum is a sub form j =β 1∈ to theT powermM such of infinity that beta sub j oslash h sub$ where j closing{ parenthesiso f S = pξ(∧ toβ the L⊗ s} power=ˆ{α (⊗ ofs. iDefine h open} { parenthesisψ):=}pˆi{−1(jβ to⊗ ˆ{s the).)Using power} j the of} sumformula{\ subsimeq ( j 7 = )} 1 ,ˆ theto{\ the equationin powerN of} infinity{ S { xim and} beta. } subis j oslash the h sub Schauder j closing parenthesis{ξ ∧Becauseβ = α =and p} to the{ the assumptionthe power operators of iF open+(α ⊗ parenthesiss)} =$ 0( b implied a tos i s the o by powerfα $⊗ Ssof∈ sumE{ imF subi), ˆ{\one j =pm 1 to}} theˆ{ powerm of corresponding } { , } { H infinity, E sum ˆ{\ subpm k} = 1, to the\ iota powercan of{\ n provexi sub} i that a subandξ ∧ jkψ} alpha$= α to⊗ tos thein $ the an power{\ analogous ofxi i k}\ oslash waywedge as h we sub proceeded j closingSchauder parenthesis the item{ are = continuous }ˆ{ b a s i s }$ onp to $ the E power{ m of} i open, $ parenthesis we get2 of thisphi $ proof0 closing = . The parenthesis\sum dehomogenizationˆ{ =n phi{ periodi goes}} ..{ in Summingk the steps = hyphen similar 1 }\ uptosum that commaˆ ones{\ weinfty have that} { psij = =sum sub 1 } j =a 1 to{ thej power k }\ of infinityxi beta\wedge subwritten j oslash\alpha in hthe sub precedingˆ j{ ..i is the item k desired}\ . otimes h { j }$ precisely in the same way as we obtainedpreimage period the formula Thus comma ( 1 phi 2 )in in Im openthe homogeneousparenthesis i sigma situation sub minus ( 1 it to emthe power2 of of this xi closing proof parenthesis ) . sub m period Using4 period the .. definition Now comma we of prove the that Schauder Ker open basis parenthesis again i sigma , we to have the power for of each xi closing $ j parenthesis\ in N sub $ m subsetthe equal Imequation open parenthesis $ \sum i sigmaˆ{ subn { minusi }} 1 to{ thek power = of 1 xi} closinga { parenthesisj k }\ sub mxi for i\ =wedge l plus 1 comma\alpha periodˆ{ periodi k period} = comma0 . 2 $ l comma Using 0 theequal-negationslash Cartan lemma xi in on exterior differential systemsCapital Gamma , we get open the parenthesis existence M comma of a T family to the power $ ( of * M\beta closing{ parenthesisj } ) period{ j }\ If phiin = alphaN $ oslash o f s $ in ( Ker openi − parenthesis1 ) i sigma $ forms to the such power ofthat xi closing parenthesis sub m comma then 0 = p to the power of i plus 1 open parenthesis xi and$ \ phixi closing\wedge parenthesis\beta = xi and{ j alpha} = oslash\sum s periodˆ{ Duen { i }} { k = 1 } a { jk }\alpha ˆ{ i k } . $to the It Cartan is possible lemma comma to see we know ( e that . g there . by is t a form aking beta the in bigwedge st andard to the power of i minus 1 T sub m to the power of *Hodge M such− thattype metric on the space of forms ) that one can choose the family $xi ( and beta\beta oslash{ sj =} alpha) oslash{ j }\ s periodin DefineN$ psi in : = such p to the a waypower that of i minus $ \ 1psi open parenthesis: = \ betasum oslashˆ{\ sinfty closing} { j =parenthesis 1 }\ periodbeta Using{ j the}\ formulaotimes open parenthesish { j } 7$ closing converges parenthesis . comma Thus , the we equation may write xi and beta = alpha and the assumption F to the power of plus open parenthesis alpha oslash s closing parenthesis = 0 open parenthesis\ begin { a l implied i g n ∗} by alpha oslash s in E to the power of im i closing parenthesis comma one i can{\ provesigma that} xiˆ{\ andxi psi} ={ alpha − oslash1 } s in( an ˆ{\ analogoussum }ˆ way{\ infty as we proceeded} { j the = item 1 }\beta { j }\otimes h 2{ ofj this} proof) period = p The ˆ{ dehomogenizationi } ( ˆ{\sum goes}ˆ in{\ theinfty steps similar} { j to that = ones 1 }\ xi \wedge \beta { j }\otimes h written{ j } in) the preceding = p ˆ item{ i period} ( ˆ{\sum }ˆ{\ infty } { j = 1 }\sum ˆ{ n { i }} { k = 1 } a square{ jk }\alpha ˆ{ i k }\otimes h { j } ) = \end{ a l i g n ∗}

\ hspace ∗{\ f i l l } $ p ˆ{ i } ( \phi ) = \phi . $ \quad Summing − up , we have that $ \ psi = \sum ˆ{\ infty } { j = 1 }\beta { j }\otimes h { j }$ \quad is the desired

\noindent preimage . Thus $ , \phi \ in $ Im $ ( i {\sigma }ˆ{\ xi } { − 1 } ) { m } . $ 4 . \quad Now , we prove that Ker $ ( i {\sigma }ˆ{\ xi } ) { m }\subseteq $ Im $ ( i {\sigma }ˆ{\ xi } { − 1 } ) { m }$for$i=l+1 , . . . ,2 l , 0 \ne \ xi \ in $ $ \Gamma ( M , T ˆ{ ∗ } M ) . $ I f $ \phi = \alpha \otimes s \ in $ Ker $ ( i {\sigma }ˆ{\ xi } ) { m } ,$ then $0 = pˆ{ i + 1 } ( \ xi \wedge \phi ) = \ xi \wedge \alpha \otimes s . $ Due

\ hspace ∗{\ f i l l } to the Cartan lemma , we know that there is a form $ \beta \ in \bigwedge ˆ{ i − 1 } T ˆ{ ∗ } { m } M $ such that

\ hspace ∗{\ f i l l } $ \ xi \wedge \beta \otimes s = \alpha \otimes s . $ Define $ \ psi : = p ˆ{ i − 1 } ( \beta \otimes s ) . $ Using the formula ( 7 ) , the equation

\ hspace ∗{\ f i l l } $ \ xi \wedge \beta = \alpha $ and the assumption $ F ˆ{ + } ( \alpha \otimes s ) = 0 ($ impliedby $ \alpha \otimes s \ in E ˆ{ im } i ) , $ one

\ hspace ∗{\ f i l l }can prove that $ \ xi \wedge \ psi = \alpha \otimes s $ in an analogous way as we proceeded the item

\ hspace ∗{\ f i l l }2 of this proof . The dehomogenization goes in the steps similar to that ones

\ centerline { written in the preceding item . }

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KR Y´ SL complexes areIn notthe elliptic future by, we finding would an like example to interpret of a suitable the appropriate Ricci type Fedosov( reduced ) cohomology groups of \noindentmanifoldthe admittingOne truncated can a also metaplectic symplectic try to structuretwistor prove complexes period that .the Eventually full ( , onei . can e. search , not for truncatedan application ) of symplectic the twistor complexesReferencessymplectic are not twistor elliptic complexes by in finding representation an example theory . of a suitable Ricci type Fedosov manifoldopen squareOne admitting canbracket also 1 try closing a to metaplectic prove square that bracket the structure full Borel ( i . ecomma . ., not A truncated period comma ) symplectic Wallach twistor comma complexes N period are comma Continuous cohomologynot comma elliptic dis by crete finding subgroups an example comma of and a suitable representations Ricci type of Fedosov manifold admitting a metaplectic \ centerlinereductivestructure groups{ References period . Second} edition comma Math period Surveys Monogr period 67 open parenthesis 2000 closing parenthesis comma xviii plus 260 pp period References \ hspace ∗{\ f i l l } [ 1 ] Borel , A . , Wallach , N . , Continuous cohomology , dis crete subgroups , and representations of open square bracket[ 12 ] closing Borel , A square . , Wallach bracket , N . Branson , Continuous comma cohomology T period , dis commacrete subgroups Stein ,hyphen and representations Weiss operators of and e l lipticity comma J period Functreductive period groups Anal . periodSecond 1 edition 5 1 open , Math parenthesis . Surveys Monogr 2 closing . 67 parenthesis( 2000 ) , xviii open+260 parenthesispp . 1997 closing parenthesis \ centerline { reductive groups . Second edition , Math . Surveys Monogr . 67 ( 2000 ) , xviii $ + comma 334 endash[ 2 ] Branson 383 period , T . , Stein - Weiss operators and e l lipticity , J . Funct . 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Math . 7 ( 1 970 ) , 539 endasheds 570 . period ) , vol . 3 1 , American Mathematical Society , Providence , Rhode - Island , Mathematical Surveys and 1 1 7 −− 1 60 . open squareMonographs bracket , 1 1 989 1 closing . square bracket Kostant comma B period comma Symplectic Spinors comma Symposia Mathe- matica comma[ 1 6 vol ] Shale period , D . XIV , Linear commasymmet Cambridger−i es of Univ free boson period fields Press, Trans comma . Amer . Math . Soc . 1 3 ( 1 962 ) , 149 – \noindent [ 10 ] Howe , R . , Remarks on classical invariant theory , Trans . Amer . Math . Soc . 3 1 3 ( 2 ) ( 1 989 ) , 1 974 comma1 67 . pp period 1 39 endash 1 52 period 539 −− 570 . open square[ 17 ] bracket Stein , E 1 . , 2 Weiss closing , G square. , Generalization bracket Kryacuteslof the Cauchy comma – Riemann S period equations comma and representations Howe duality for metaplectic group acting on symplectic spinor valued foof r-m the s rotation comma group , Amer . J . Math . 90 ( 1 968 ) , 1 63 – 1 96 . \noindent [ 1 1 ] Kostant , B . , Symplectic Spinors , Symposia Mathematica , vol . XIV , Cambridge Univ . Press , accepted[ in18 ]J Tondeur period ,Lie P . Theory , Affine Zus period ammenh¨angeauf Mannigfaltigkeiten mit fast - s ymplektis cher St r − u ktur , 1 974 , pp . 1 39 −− 1 52 . open square bracket 1 3 closing squareComment bracket . Math Kryacutesl . Helv . 36 comma( 1 96 1 S ) period , 234 – 244 comma . Symplectic spinor forms and the invariant operators acting[ 1 9 ]between Vaisman , them I . , Symplectic comma ArchCu r period− v ature Tens ors , Monatshefte f¨urMath . , vol . 100 , Springer - Verlag , \noindent [ 1 2 ] Kr\ ’{ y} sl , S . , Howe duality for metaplectic group acting on symplectic spinor valued fo Math periodWien ,open 1 985 parenthesis , pp . 299 – 327 Brno . closing parenthesis 42 open parenthesis Supplement closing parenthesis open parenthesis 2006$ r−m closing $ s parenthesis , comma 279 endash 290 period acceptedopen square in bracket J . Lie 14 closing Theory square . bracket Kryacutesl comma S period comma A complex of symplectic twistor operators in symplectic spin geometry comma Monatsh period \noindentMath period[ 1 1 613 ]open Kr\ parenthesis’{ y} sl , 4 S closing . , Symplectic parenthesis open spinor parenthesis forms 20 and 10 closing the invariant parenthesis comma operators 381 endash acting 398 between them , Arch . period \ centerlineopen square{ bracketMath . 1 (5 closing Brno )square 42 ( bracket Supplement Schmid comma ) ( 2006 W period ) , comma 279 −− Homogeneous290 . } complex manifolds and repre- sentations of semisimple Lie group comma \noindentRepresent[ at 14 ion ] theory Kr\ ’ and{ y} sl harmonic , S . analysis , A complex on semisimple of symplectic Lie groups period twistor .. open operators parenthesis in Sally symplectic comma P period spin geometry , Monatsh . comma Vogan comma D period comma \ centerlineeds period closing{Math parenthesis . 1 61 ( comma 4 ) (vol 20 period 10 3) 1 , comma 381 −− American398 . Mathematical} Society comma Providence comma Rhode hyphen Island comma Mathematical \noindentSurveys and[ Monographs 1 5 ] Schmid comma , W 1 989 . , period Homogeneous complex manifolds and representations of semisimple Lie group , open square bracket 1 6 closing square bracket Shale comma D period comma Linear symmet to the power of r-i es of free boson fields\ hspace comma∗{\ Transf i l l } periodRepresent Amer period at ion Math theory period and Soc period harmonic 1 3 open analysis parenthesis on 1 semisimple 962 closing parenthesis Lie groups comma . \quad ( Sally ,P . , Vogan ,D. , 149 endash 1 67 period edsopen . ) square , vol bracket . 3 17 1 closing , American square bracket Mathematical Stein comma Society E period , comma Providence Weiss comma , Rhode G period− Island comma , Generalization Mathematical of theSurveys Cauchy endash and Monographs Riemann equations , 1 989 and representations. of the rotation group comma Amer period J period Math period 90 open parenthesis 1 968 closing parenthesis comma 1 63 endash\noindent 1 96 period[ 1 6 ] Shale , D . , Linear $ symmet ˆ{ r−i }$ es of free boson fields , Trans . Amer . Math . Soc . 1 3 ( 1 962 ) , 149open−− square1 67 bracket . 18 closing square bracket Tondeur comma P period comma Affine Zus ammenhadieresisnge auf Mannig- faltigkeiten mit fast hyphen s ymplektis cher St r-u ktur comma \noindentComment period[ 17 ]Math Stein period , EHelv . period , Weiss 36 open , G parenthesis . , Generalization 1 96 1 closing parenthesis of the Cauchy comma−− 234 endashRiemann 244 equations period and representations open square bracket 1 9 closing square bracket Vaisman comma I period comma Symplectic Cu r-v ature Tens ors comma Monatshefte\ centerline fudieresisr{ of the Math rotation period comma group vol , period Amer 100 . J comma . Math Springer . 90 hyphen ( 1 968 Verlag ) comma, 1 63 −− 1 96 . } Wien comma 1 985 comma pp period 299 endash 327 period \noindent [ 18 ] Tondeur , P . , Affine Zus ammenh\”{a}nge auf Mannigfaltigkeiten mit fast − s ymplektis cher St $ r−u $ ktur ,

\ centerline {Comment . Math . Helv . 36 ( 1 96 1 ) , 234 −− 244 . }

\noindent [ 1 9 ] Vaisman , I . , Symplectic Cu $ r−v $ ature Tens ors , Monatshefte f \”{u} r Math . , vol . 100 , Springer − Verlag , Wien , 1 985 , pp . 299 −− 327 . ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX .. 327 \ hspaceopen square∗{\ f i bracket l l }ELLIPTICITY 20 closing square OF THE bracket SYMPLECTIC Weil comma TWISTOR A period COMPLEX comma Sur\ certainsquad 327 groups d quoteright op eacuterateurs unitaires comma Acta Math period 1 1 1 open parenthesis 1 964 closing parenthesis comma 143 endash 2 1 1 period \noindentopen square[ bracket 20 ] Weil 2 1 closing , A square . , Sur bracket certains Wells comma groups R period d ’ comma op \ ’ Differential{ e} rateurs analysis unitaires on complex , Acta manifolds Math comma . 1 1 1 ( 1 964 ) , 143 −− 2 1 1 . Grad[ 2 period 1 ] Wells Text s in , Math R . period , Differential comma vol period analysis 65 comma on Springer complex comma manifolds , Grad . Text s in Math . , vol . 65 , Springer , New York comma 2008 period ELLIPTICITY OF THE SYMPLECTIC TWISTOR COMPLEX 327 \ centerlineCharles[ University 20{ ]New Weil ,York A of . Prague , Sur , certains 2008 comma groups . } d ’ op ´erateurs unitaires , Acta Math . 1 1 1 ( 1 964 ) , 143 – 2 1 1 . [ 2 1 ] SokolovskaacuteWells , R .83 , Differential comma Praha analysis 8 comma on complex Czech manifolds Republic, Grad . Text s in Math . , vol . 65 , Springer , \noindentE hyphen mailCharles : krys University l at karl in period of mfPrague f periodNew , c Yorku-n i , period 2008 . cz Charles University of Prague , \noindentSokolovskSokolovska´ 83 ,\ Praha’{ a} 883 , Czech , Praha Republic 8 , Czech Republic E - mail : krys l @ karl in . mf f . c u − n i . cz \noindent E − mail : krys l $@$ karl in .mff . c $u−n $ i . cz