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Centerline{ELLIPTICITY \Quad OF ARCHIVUM MATHEMATICUM open parenthesis BRNO closing parenthesis n centerlineTomus 47 openfARCHIVUM parenthesis MATHEMATICUM 201 1 closing parenthesis ( BRNO ) commag 309 endash 327 ELLIPTICITY .. OF .. THE .. SYMPLECTIC .. TWISTOR COMPLEX n centerlineSvatopluk KryacuteslfTomus 47 ( 201 1 ) , 309 −− 327 g Abstract period .. For a Fedosov manifold open parenthesis symplectic manifold equipped with a n centerline fELLIPTICITY nquad OF nquad THE nquad SYMPLECTIC nquad TWISTOR COMPLEX g 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 n centerlineon sections off SvatoplukELLIPTICITY certain infinite Kr rankn 'f y vectorg OFs l g bundles THE defined SYMPLECTIC over this manifold TWISTOR period COMPLEX The differential operators are symplectic analoguesSvatopluk of the twistor Krysl´ operators n centerline f Abstract . nquad For a Fedosov manifold ( symplectic manifold equipped with a g 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 , n centerline f symplectic torsion − free affine connection ) admitting a metaplectic structure , g 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 . n centerline fwe 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 g 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 n centerline fon sections of certain infinite rank vector bundles defined over this manifold . g a metaplectic structuretype period . 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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 n hspacetheorem∗fn. which Inf i Vaisman l l saysgNow 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 nnoindenthline fieldsa Fedosov . If manifold is a triple $ ( M^f 2 l g , nomega , nnabla ) $ where $ (20 10 M Mathematics ^f 2 l Subjectg , Classificationnomega :) 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 snnoindent spinors comma$ 2 l $ dimensional ) symplectic manifold and $ nnabla $ 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 nnoindentcomplexesquantization . 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 nnoindentthanksWe 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 ngeq 2 , $ namely into the symplectic Ricci and symplectic Weyl curvature tensor fields . If n begin f a l i g n ∗g n r u l e f3emgf0.4 pt g nendf a l i g n ∗g n centerline f20 10 Mathematics Subject Classification : primary 22 E 46 ; secondary 53 C 7 , 53 C 80 , 58 J 5 . g 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 $ ncheckfCg $ 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 $ ncheckfSg $ MT $ ncheckfCg $ R . The author thanks to Ondnvf r g ej Kalenda for a discussion . n centerline f Received July 8 , 201 1 . Editor J . Slov n 'f agk
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