Higher algebraic K-theories related to the global program of Langlands
aa r X i v : . [ m a t h . G M ] S e p Higher algebraic K -theories related to the globalprogram of Langlands C. Pierre
Abstract
The paper revisits concretely the algebraic K -theory in the light of the globalprogram of Langlands by taking into account the new algebraic interpretation ofhomotopy viewed as deformation(s) of Galois representations given by compactifiedalgebraic groups.More concretely, we introduce higher algebraic bilinear K -theories referring tohomotopy and cohomotopy and related to the reducible bilinear global program ofLanglands as well as mixed higher bilinear KK -theories related to dynamical geo-metric bilinear global program of Langlands. ontents Introduction 11 Universal algebraic structures of the Langlands global program 92 Lower bilinear K -theory based on homotopy semigroups viewed as de-formations of Galois representations 233 Higher bilinear algebraic K -theories related to the reducible bilinearglobal program of Langlands 384 Mixed higher bilinear algebraic KK -theories related to the Langlandsdynamical bilinear global program 51References 62 ntroduction The higher algebraic K -theory of rings developed by D. Quillen [Quil] constitutes anabstract powerful tool [A-K-W] of algebraic topology which generalizes the lower K -groupsamong which the topological K -theory [Ati1], [Ati2] introduced by A. Grothendieck is verypopular among mathematicians and physicists.It is the aim of this paper to revisit concretely the algebraic K -theory [Kar1],[Kar2]in the light of the recent developments [Pie2], [Pie6] of the global program ofLanglands by taking into account the new algebraic interpretation of homotopy intro-duced here as a deformation of Galois representation given by compact(ified) algebraicgroups [Dol].More concretely, we introduce :a) lower (algebraic) bilinear K -theories (in the context of topological K -theory) referring to homotopy and cohomotopy and related to the irreduciblebilinear global program of Langlands ;b) higher algebraic bilinear K -theories referring to homotopy and cohomo-topy and related to the reducible bilinear global program of Langlands ;c) mixed lower and higher bilinear algebraic KK -theories related to thedynamical geometric bilinear global program of Langlands . Chapter 1 deals with the universal algebraic structures of the Langlands globalprogram. These algebraic structures are abstract bisemivarieties G (2 n ) ( F v × F v ) over products, right by left, F v × F v , of sets of increasing archimedean completions. Theseabstract bisemivarieties, in the heart of the Langlands global program, are universal in thesense that:a) they are (functional) representation spaces of the algebraic bilinear semigroupsGL n ( e F v × e F v ) being the 2 n -dimensional representations of the products, right byleft, W ab e F v × W ab e F v of global Weil semigroups;b) they have open coverings [Har] by affine bisemivarieties G (2 n ) ( e F v × e F v ) where e F v and e F v are symmetric increasing sets of Galois extensions;c) they generate cuspidal representations of GL n ( e F v × e F v ) after a suitable toroidalcompactification of these. 1 hapter 2 introduces bilinear versions [Pie3] of the “lower” (algebraic) K -theory related to the bilinear global program of Langlands .As these K -theories are contravariant functors whose objects are abstract bisemivarieties(or bisemifields) and as they are defined with respect to homotopy groups, it is logicalto want to give an algebraic interpretation of homotopy allowing it to resultfrom algebraic geometry . The fundamental group can then be expressed in terms of deformations ofGalois representations . Indeed, the equivalence classes of maps between the coeffi-cient semiring F v and the real linear (semi)variety G (2 n ) ( F v ) are the homotopy classes[Bott] corresponding to the classes of the quantum homomorphism Qh v + ℓ → v : F v + ℓ → F v sending F v into the deformed global coefficient semirings F v + ℓ obtained from F v by adding “ ℓ ” transcendental quanta covered by irreducibleclosed algebraic subsets according to section 2.6.Let then f h ℓ : F v + ℓ → G (2 n ) ( F v ) be a continuous map in such a say that: F H : F v × I −→ G (2 n ) ( F v ) , I = [0 , , be the homotopy map of f h = F H ( x,
0) with
F H ( x,
1) = f h ℓ . The homotopy classes , corresponding to the classes of the quantum homomorphism Qh v + ℓ → v , are characterized by integers “ ℓ ” which are in one-to-one corre-spondence with the values of the parameter t ∈ [0 ,
1] of the homotopy .Taking into account the existence of a cohomotopy of which classes are the inverseequivalence classes of the corresponding homotopy , we see that the set of homotopyclasses, being equivalence classes of maps between the set Ω( L v , G (2 n ) ( F v )) of orientedpaths and the semivariety G (2 n ) ( F v ) , forms a group, noted Π ( G (2 n ) ( F v ) , L v ) in thebig point L v , depending on deformations of Galois compact representationsof these paths corresponding to the increase of these by a(n) (in)finite numberof transcendental (or algebraic) quanta . The semigroup Π i ( G (2 n ) ( F v ) , L v ) of homotopy classes of maps s f h iℓ : s i ( ℓ ) −→ G (2 n ) ( F v ) , sending the base point of the 2 i -sphere S i to the base point L v i ( j ) of G (2 n ) ( F v ) ,or equivalently, of maps c f h iℓ : [0 , −→ G (2 n ) ( F v ) , from the i -cube [0 , i to the semivariety G (2 n ) ( F v ) , results from the deformationsof the Galois compact representation of the semigroup GL (2 i ) ( e F v ) given by the ernels G (2 i ) ( δF v + ℓ ) of the maps :GD iℓ : G i ( F v + ℓ ) −→ G (2 i ) ( F v ) , ∀ ℓ , ≤ ℓ ≤ ∞ ,t i → ℓ i , in such a way that the 2 i -th powers if the integers “ ℓ ” be in one-to-one correspondencewith the 2 i -th powers of the values of the parameter t ∈ [0 ,
1] .Similarly, the cohomotopy semigroup, noted Π i ( G (2 n ) ( F v ) , L v ) , is defined byclasses resulting from inverse deformations (GD (2 i ) ℓ ) − of the Galois represen-tations of GL (2 i ) ( e F v ) .If G (2 n ) ( F v ) denotes the semivariety dual of G (2 n ) ( F v ) , then the bilinear homotopy (resp.cohomotopy) semigroup will be given by Π i ( G (2 n ) ( F v × F v )) (resp. Π i ( G (2 n ) ( F v × F v )) )in such a way that its classes of (bi)maps: c f h iℓ × ( D ) c f h ih : [0 , iℓ × ( D ) [0 , iℓ −→ G (2 n ) ( F v × F v )result from the (resp. inverse) deformations of the Galois representations of the bisemiva-riety G (2 i ) ( e F v × e F v ) .It is then natural to associate to Π i ( G (2 n ) ( F v × F v )) (resp. to Π i ( G (2 n ) ( F v × F v )) ) the Π -cohomology (resp. the Π -homology) corresponding to the grouphomomorphism of Hurewicz : hH R × L : Π i ( G (2 n ) ( F v × F v )) −→ H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ )(resp. hcH R × L : Π i ( G (2 n ) ( F v × F v )) −→ H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) ) , where the entire bilinear cohomology H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) (resp. homology H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) ) refers to a bisemilattice deformed by the homotopy(resp. cohomotopy) classes of maps of Π i ( G (2 n ) ( F v × F v )) (resp. Π i ( G (2 n ) ( F v × F v )) ) .The topological (bilinear) K -theory K i ( G (2 n ) ( F v × F v )) of vector bibundles with base G (2 n ) ( F v × F v ) and (bi)fibre G (2 n − i +1) ( F v × F v ) , introduced in section 2.19, leads to setup the Chern character, restricted to the class c i , in the bilinear K -cohomology bythe homomorphism [Sus], [Wal]: c i ( G (2 n ) ( F v × F v )) : K i ( G (2 n ) ( F v × F v )) −→ H i ( G (2 n ) ( F v × F v ) , G (2 i ) ( F v × F v )) . Then, the lower bilinear (algebraic) K -theory referring to homotopy (resp.cohomotopy) will be given by the equality (resp. homomorphism) : K i ( G (2 n ) ( F v × F v )) = ( → ) Π i ( G (2 n ) ( F v × F v ))(resp. K i ( G (2 n ) ( F v × F v )) = ( → ) Π i ( G (2 n ) ( F v × F v )) )3 n such a way that the homotopy (resp. cohomotopy) classes of maps of Π i ( · )(resp. Π i ( · ) ) are (resp. correspond to) (inverse) liftings of quantum defor-mations of the Galois representation GL i ( e F v × e F v ) . Chapter 3 introduces bilinear versions of the higher algebraic K -theory [Mil]related to the reducible bilinear global program of Langlands . An infinite bilinear semigroup GL( F v × F v ) , depending on the geometric dimensions“ i ”, is given by GL( F v × F v ) = lim −→ i GL i ( F v × F v )and corresponds to the (partially) reducible (functional) representation spaceRED( F ) REPSP(GL n =2+ ··· +2 i + ··· +2 n s ( F v × F v )) of the bilinear semigroup ofmatrices GL n ( F v × F v ) with n → ∞ .An equivalent quantum infinite bilinear semigroup given by the set (cid:26) GL Q ( F iv × F iv ) = lim j =1 → r →∞ GL ( Q ) j ( F iv × F iv ) (cid:27) i , depending primarily on the algebraic dimension “ j ’ and based on the unitary representa-tion space of GL n ( F v × F v ) , is also introduced in sections 3.4 to 3.6. The classifying bisemispace BGL( F v × F v ) of GL( F v × F v ) , associated withthe partition 2 n = 2 + · · · + 2 i + . . . n s , n → ∞ , of the integer 2 n , is defined asthe base bisemispace of all equivalence classes of deformations of the Galois representationof GL( e F v × e F v ) given by the kernels GL( δF v + ℓ × δF v + ℓ ) of the maps:GD ℓ : GL( F v + ℓ × F v + ℓ ) −→ GL( F v × F v ) , ≤ ℓ ≤ ∞ . The “plus” constructed of Quillen , adapted to the bilinear case of the Langlandsglobal program, leads to consider the map:BG(1) : BGL( F v × F v ) −→ BGL( F v × F v ) + , in such a way that the classifying bisemispace BGL( F v × F v ) + is the base bisemispace of allequivalence classes of one-dimensional deformations of the Galois compact representationof GL( e F v × e F v ) given by the kernels n GL (1) ( δF v + ℓ × δF v + ℓ ) o ℓ of the maps GD(1) ℓ . The bilinear version of the algebraic K -theory [Blo], [Ger], [Gil2], of Quillenrelated to the Langlands global program is: K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + )where G (2 n )red ( F v × F v ) = RED( F ) REPSP(GL n ( F v × F v )) is the (partially) re-ducible (functional) representation space of the bilinear semigroup of matrices L n ( F v × F v ) in such a way that the partition 2 n = 2 + · · · + 2 i + · · · + 2 n s of thegeometric dimension 2 n , n ≤ ∞ , refers to the reducibility of GL n ( F v × F v ) .This higher version of the Langlands global program implies by the homotopy bisemi-group Π i ( · × · ) that the equivalence classes of 2 i -dimensional deformations of the Galoisrepresentations of the reducible bilinear semigroup GL n ( F v × F v ) result from quantumhomomorphisms of the global coefficient bisemiring F v × F v .This higher algebraic K -theory, referring to homotopy, implies the commutative dia-gram : K i ( G (2 n )red ( F v × F v )) Π i (BGL( F v × F v ) + ) H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) Chern higherrestricted characterin K -cohomology inverse restricted higherΠ-cohomology in such a way that:1. the classes of the entire bilinear cohomology H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ )refers to a bisemilattice deformed by the homotopy classes of maps ofΠ i (BGL( F v × F v ) + ) , corresponding to lifts of quantum deformations ofthe Galois representations of GL red2 n ( e F v × e F v ) .2. the restricted higher K -cohomology implies the restricted higher Π -cohomology. The total higher algebraic K -theory relative to homotopy is : K ∗ ( G (2 n )red ( F v × F v )) = Π ∗ (BGL( F v × F v ) + )where “ ∗ ” refers to the partition 2 n = 2 + · · · + 2 i + · · · + 2 n s of 2 n .Similarly, the higher bilinear algebraic K -theory relative to cohomotopy is given by theequality: K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + ) , where Π i (BGL( F v × F v ) + ) are the cohomotopy equivalence classes of 2 i -dimensionaldeformations of the Galois reducible representations of GL n ( e F v × e F v ) and its total versionis: K ∗ ( G (2 n )red ( F v × F v )) = Π ∗ (BGL( F v × F v ) + ) . Chapter 4 deals with mixed higher bilinear algebraic KK -theories [B-D-F],[Kas], [Jan] related to the Langlands dynamical bilinear global program andreferring to the existence of K ∗ K ∗ functors on the categories of elliptic biop-erators and (reducible) bisemisheaves F G (2 n )(red) ( F v × F v ) .5 prerequisite is the introduction of a bilinear contracting fibre F kR × L (TAN) inthe tangent bibundle TAN( F G (2 n ) ( F v × F v )) implying the homology: H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) ≃ Ad( F ) REPSP(GL k ( IR × IR ))in such a way that the homology [Nov], [Kas] of the 2 i -dimensional bisemisheaf F G (2 i [2 k ]) ( F v × F v ) shifted on 2 k dimensions, k ≤ i , be given by the adjoint functionalrepresentation space Ad( F ) REPSP(GL k ( IR × IR )) of GL k ( IR × IR ) .In order to introduce a mixed homotopy bisemigroup, we have to precise what mustbe the cohomotopy bisemigroup corresponding to the homology H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) .As a “Galois” cohomotopy bisemigroup refers to classes resulting from inverse deformationsof Galois representations, the searched cohomotopy bisemigroupΠ k ( F G (2 i [2 k ]) ( F v × F v )) must be described by classes resulting from inversedeformations of the differential Galois [Car] representations of GL k ( IR × IR )and depending on the classes of deformations of the Galois representations ofGL k ( e F v × e F v ) .Consequently, the mixed homotopy bisemigroup Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) ofthe shifted bisemisheaf F G (2 n [2 k ]) ( F v × F v ) under the action of differential bioperators will be given by the product :Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) = Π k ( F G (2 i [2 k ]) ( F v × F v )) × Π i ( F G (2 n [2 k ]) ( F v × F v ))where Π i ( F G (2 n [2 k ]) ( F v × F v )) is the homotopy bisemigroup of the bisemisheaf ( F G (2 n ) ( F v × F v )) shifted in 2 k dimensions.The mixed bilinear semigroup homomorphim of Hurewicz, introducing a restricted Π -homology- Π -cohomology, will be given by: mhH : Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) −→ H i − k ( F G (2 n [2 k ]) ( F v × F v ) , ZZ × ( D ) ZZ )where H i − k ( · ) is the entire mixed bilinear cohomology defined from: H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i [2 k ]) ( F v × F v ))= H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) × (cid:2) H k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 k ) ( F v × F v )) (cid:3) ⊕ H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i − k ) ( F v × F v ))as developed in sections 4.1 to 4.3.Similarly, the Chern mixed restricted character in the K -homology- K -cohomology, cor-responding to a bilinear version of the index theorem, is given by: c [ k ] · c i ( G (2 n [2 k ]) ( F v × F v )) : K i − k ( F G (2 n ) ( F v × F v )) −→ H i − k ( F G (2 n ) ( F v × F v ))6here K i − k ( F G (2 n ) ( F v × F v )) is the mixed topological (bilinear) K -theory ex-panded according to : K i − k ( F G (2 n ) ( F v × F v )) = K k ( F G (2 i [2 k ]) ( F v × F v )) × K i ( F G (2 n ) ( F v × F v ))with K k ( F G (2 i [2 k ]) ( F v × F v )) a topological bilinear contracting K -theory of contractingtangent bibundles. A mixed lower bilinear (algebraic) K -theory can then be defined by the equal-ity (resp. homomorphism) : K i − k ( F G (2 n ) ( F v × F v )) = ( → ) Π i [2 k ] ( F G (2 n ) ( F v × F v )) . The total Chern mixed character in the K -homology- K -cohomology is :ch ∗∗ ( F G (2 n [2 k ]) ( F v × F v )) : K ∗ K ∗ ( F G (2 n ) ( F v × F v )) −→ H ∗ H ∗ ( F G (2 n ) ( F v × F v ))leading to a simplified version of the (bilinear) version of the Riemann-Rochtheorem introduced in proposition 4.9.In order to develop a bilinear version of the higher algebraic mixed KK -theory, we have tointroduce a higher bilinear algebraic operator K -theory relative to cohomotopy .This implies the definition of the infinite bilinear classifying semisheaf BFGL( F v × F v )over the classifying bisemispace BGL( F v × F v ) as the base bisemispace of all equivalenceclasses of inverse deformations of the Galois differential representation ofGL( IR × IR ) = lim −→ k GL k ( IR × IR )acting on BGL( F v × F v ) .The mixed classifying bisemisheaf BFGL(( F v ⊗ IR ) × ( F v ⊗ IR )) then results from thebiaction of BFGL( IR × IR ) on BFGL( F v × F v ) . The bilinear version of the mixed higher algebraic KK -theory related to theLanglands dynamical bilinear global program is : K k ( F G (2 n )red ( IR × IR )) × K i ( F G (2 n )red ( F v × F v ))= Π k (BFGL( IR × IR ) + ) × Π i (BFGL( F v × F v ) + )written in condensed form according to: K i − k ( F G (2 n )red ( F v ⊗ IR ) × ( F v ⊗ IR )) = Π i [2 k ] (BFGL( F v ⊗ IR ) × ( F v ⊗ IR ) + )in such a way that the bilinear contracting K -theory K k ( F G (2 n )red ( IR × IR )) , responsiblefor a differential biaction, acts on the K -theory K i ( F G (2 n )red ( F v × F v )) of the reducible7unctional representation space F G (2 n )red ( F v × F v ) of the bilinear semigroup GL n ( F v × F v ) inone-to-one correspondence with the biaction of the cohomotopy bisemigroupΠ k (BFGL( IR × IR ) + ) of the “ + ” classifying bisemispace BFGL( IR × IR ) + .Finally, the bilinear version of the total mixed higher algebraic KK -theoryrelated to the dynamical reducible global program of Langlands is : K ∗ ( F G (2 n )red ( IR × IR )) × K ∗ ( F G (2 n )red ( F v × F v ))= Π ∗ (BFGL( IR × IR ) + ) × Π ∗ (BFGL( F v × F v ) + ) . Universal algebraic structures of the Langlandsglobal program a) Let e F denote a set of finite extensions of a number field k of characteristic 0 : e F is assumed to be a set of symmetric splitting fields composed of the right andleft algebraic extension semifields e F R and e F L being in one-to-one correspondence. e F L (resp. e F R ) is then composed of the set of complex (resp. conjugate complex)simple roots of the polynomial ring k [ x ] .If the algebraic extension fields are real, then the symmetric splitting fields e F + arecomposed of the left and right symmetric splitting semifields e F + L and e F + R beinggiven respectively by the set of positive and symmetric negative simple real roots.b) The left and right equivalence classes of infinite archimedean completions of e F L (resp. e F R ) are the left and right infinite complex places ω = { ω , . . . , ω j , . . . ,ω r } (resp. ω = { ω , . . . , ω j , . . . , ω r } ). In the real case, the infinite places aresimilarly v = { v , . . . , v j , . . . , v r } (resp. v = { v , . . . , v j , . . . , v r } ).c) All these (pseudoramified) completions, corresponding to transcendental extensions,proceed from the associated algebraic extensions by a suitable isomorphism of com-pactification [Pie7] and are built from irreducible subcompletions F v j (resp. F v j )characterized by a transcendence degreetr · d · F v j /k = tr · d · F v j /k = N equal to [ e F v j : k ] = [ e F v j : k ] = N which is the Galois extension degree of the associated algebraic closed subsets e F v j and e F v j . All these irreducible subcompletions (resp. (sub)extensions) are assumedto be transcendental (resp. algebraic) quanta [Pie1] .d)
The pseudoramified real extensions are characterized by degrees :[ e F v j : k ] = [ e F v j : k ] = ∗ + j N , ≤ j ≤ r ≤ ∞ , which are integers modulo N , ZZ /N ZZ ,where: 9 e F v j and e F v j are extensions corresponding to completions respectively of the v j -th and v j -th symmetric real infinite pseudoramified places; • ∗ denotes an integer inferior to N .Similarly, the pseudoramified complex extensions e F ω j and e F ω j correspond-ing to the completions F ω j and F ω j at the infinite places ω j and ω j arecharacterized by extension degrees[ e F ω j : k ] = [ e F ω j : k ] = ( ∗ + j N ) m ( j ) , where m ( j ) = sup( m j + 1) is the multiplicity of the j -th real extension covering its j -th complex equivalent.Let then e F v j,mj (resp. e F v j,mj ) denote a left (resp. right) pseudoramified real exten-sion equivalent to e F v j (resp. e F v j ).e) The corresponding pseudounramified real extensions e F nrv j,mj and e F nrv j,mj arecharacterized by their global class residue degrees: f v j = [ e F nrv j,mj : k ] = j and f v j = [ e F nrv j,mj : k ] = j . f) Let e F v j,mj (resp. e F v j,mj ) be the real pseudoramified extension corresponding to thecompletion F v j,mj and let e F nrv j,mj (resp. e F nrv j,mj ) be the corresponding pseudounram-ified extension.Let Gal( e F v j,mj /k ) (resp. Gal( e F v j,mj /k ) ) and Gal( e F nrv j,mj /k ) (resp. Gal( e F nrv j,mj /k ) )be the associated Galois subgroups.The corresponding global Weil subgroups W ( ˙ e F v j,mj /k ) = Gal( ˙ e F v j,mj /k ) (resp. W ( ˙ e F v j,mj /k ) = Gal( ˙ e F v j,mj /k ) ) are the Galois subgroups of the real pseudorami-fied extensions ˙ e F v j,mj /k (resp. ˙ e F v j,mj /k ) characterized by extension degrees d =0 mod N = jN .The global inertia subgroup I ˙ e F vj,mj (resp. I ˙ e F vj,mj ) of W ( ˙ e F v j,mj /k ) (resp. of W ( ˙ e F v j,mj /k ) ) is defined by I ˙ e F vj,mj = W ( ˙ e F v j,mj /k ) . W ( ˙ e F nrv j,mj /k )(resp. I ˙ e F vj,mj = W ( ˙ e F v j,mj /k ) . W ( ˙ e F nrv j,mj /k ) )10nd, having an order N , is considered as the subgroup of inner automorphisms ofWeil (or Galois). Remark that W ( e F nrv j,mj /k ) = Gal( ˙ e F nrv j,mj /k ) .Finally, the global Weil (semi)group W ab ˙ e F v = { W ( ˙ e F v j,mj /k ) } j,m j (resp. W ab ˙ e F v = { W ( ˙ e F v j,mj /k ) } j,m j ) , is the semigroup of all global Weil subsemigroups of real pseudoramified extensions˙ e F v j,mj (resp. ˙ e F v j,mj ) where˙ e F v = { ˙ e F v , . . . , ˙ e F v j,mj , . . . } (resp. ˙ e F v = { ˙ e F v , . . . , ˙ e F v j,mj , . . . } ) . The Galois sub(semi)groups Gal( e F v j,mj /k ) (resp. Gal( e F v j,mj /k ) ) of the exten-sions e F v j,mj (resp. e F v j,mj ) are in one-to-one correspondence with the sub(semi)-groups of automorphisms Aut k ( F v j,mj ) (resp. Aut k ( F v j,mj ) ) of the correspond-ing completions (or transcendental extensions) F v j,mj (resp. F v j,mj ) in such a way thata) the completions F v j,mj (resp. F v j,mj ) are characterized by a transcendence degree tr · d · F v j,mj /k (resp. tr · d · F v j,mj /k ) verifying tr · d · F v j,mj = [ e F v j,mj : k ] = ∗ + j · N (resp. tr · d · F v j,mj = [ e F v j,mj : k ] = ∗ + j · N ) which is the cardinal number of the transcendence base of F v j,mj (resp. F v j,mj ) over k .b) there is a one-to-one correspondence between the set of all transcendental extensionsubfields and the set of all sub(semi)groups of automorphisms of these. Proof : • The completions F v j,mj (resp. F v j,mj ) are transcendental extensions since they aregenerated from the corresponding algebraic extensions e F v j,mj (resp. e F v j,mj ) by iso-morphisms of compactifications c v j,mj : e F v j,mj −→ F v j,mj (resp. c v j,mj : e F v j,mj −→ F v j,mj )11ending e F v j,mj (resp. e F v j,mj ) by embedding into their compact isomorphic images F v j,mj (resp. F v j,mj ) which are closed compact subsets of IR + (resp. IR − ).As a result, certain points of these completions do not belong to algebraic extensionsand correspond to transcendental extensions. • If the degrees of the Galois sub(semi)groups correspond to the class zero of theintegers modulo N , i.e. if they are equal to d = 0 mod N , then, these Galoissub(semi)groups are global Weil sub(semi)groups of extensions ˙ e F v j,mj (resp. ˙ e F v j,mj )constructed from sets of “ j ” algebraic quanta (which are algebraic closed subsetscharacterized by an extension degree equal to N ).By the isomorphism of compactification c v j,mj (resp. c v j,mj ), the “ j ” (non compact)algebraic quanta of ˙ e F v j,mj (resp. ˙ e F v j,mj ) are sent into the corresponding com-pactified “ j ” transcendental (compact) quanta forming the completions˙ F v j,mj (resp. ˙ F v j,mj ) also noted simply F v j,mj (resp. F v j,mj ) . • The compact archimedean completion F v j,mj (resp. F v j,mj ) can also beviewed as resulting from the sub(semi)-group of automorphismsAut k ( F v j,mj ) (resp. Aut k ( F v j,mj ) ) in such a way that:a) Aut k ( F v j,mj ) (resp. Aut k ( F v j,mj ) ) is the compact sub(semi)group of the auto-morphisms of order “ j ” of a transcendental quantum F v j,mj (resp. F v j,mj );b) Aut k ( F v j,mj ) (resp. Aut k ( F v j,mj ) ) is a semigroup of reflections [Dol] of a tran-scendental quantum. • It is the evident that:a) Aut k ( F v j,mj ) ≃ Gal( e F v j,mj /k ) (resp. Aut k ( F v j,mj ) ≃ Gal( e F v j,mj /k ) );b) as in the Galois case, there is a one-to-one correspondence between all transcen-dental extension subfields F v ⊂ · · · ⊂ F v j,mj ⊂ · · · ⊂ F v r,mr (resp. F v ⊂ · · · ⊂ F v j,mj ⊂ · · · ⊂ F v r,mr )and the set of all normal sub(semi)groups of automorphisms of these:Aut k ( F v ) ⊂ · · · ⊂ Aut k ( F v j,mj ) ⊂ · · · ⊂ Aut k ( F v r,mr )(resp. Aut k ( F v ) ⊂ · · · ⊂ Aut k ( F v j,mj ) ⊂ · · · ⊂ Aut k ( F v r,mr ) ) . .3 Real algebraic bilinear semigroups a) Let B e F v (resp. B e F v ) be a left (resp. right) division semialgebra of real dimension2 n over the set e F v (resp. e F v ) of increasing real pseudoramified extensions e F v j,mj (resp. e F v j,mj ) of k .Then, B e F v (resp. B e F v ), which is a left (resp. right) vector semispace restricted tothe upper (resp. lower) half space, is isomorphic to the semialgebra of Borel upper(resp. lower) triangular matrices: B e F v ≃ T n ( e F v ) (resp. B e F v ≃ T t n ( e F v ) ) . This allows to define the algebraic bilinear semigroup of matricesGL n ( e F v × e F v ) by: B e F v ⊗ B e F v ≃ T t n ( e F v ) × T n ( e F v ) ≡ GL n ( e F v × e F v ) in such a way that its representation (bisemi)space is given by the tensor product f M v R ⊗ f M v L of a right T t n ( e F v ) -semimodule f M v R by a left T n ( e F v ) -semimodule f M v L .The GL n ( e F v × e F v ) -bisemimodule f M v R ⊗ f M v L is an algebraic bilinear (affine) semi-group noted G (2 n ) ( e F v × e F v ) verifying the commutative diagram:GL n ( e F v × e F v ) f M v R ⊗ f M v L ≡ G (2 n ) ( e F v × e F v )GL( f M v R ⊗ f M v L )where GL( f M v R ⊗ f M v L ) is the bilinear semigroup of automorphisms of f M v R ⊗ f M v L .Then, GL( f M v R ⊗ f M v L ) constitutes the 2 n -dimensional equivalent of theproduct W ab ˙ e F v × W ab ˙ e F v of the global Weil semigroups and the bilinear alge-braic semigroup G (2 n ) ( e F v × e F v ) becomes naturally the 2 n -dimensional (irreducible)representation space Irr Rep nW F + R × L ( W ab ˙ e F v × W ab ˙ e F v ) of ( W ab ˙ e F v × W ab ˙ e F v ) in such a way thatIrr Rep nW F + R × L ( W ab ˙ e F v × W ab ˙ e F v ) : GL( f M v R ⊗ f M v L ) −→ G (2 n ) ( e F v × e F v )implies the monomorphism: e σ v R × e σ v L : W ab ˙ e F v × W ab ˙ e F v −→ GL n ( e F v × e F v ) .
13) The isomorphisms Aut k ( F v j,mj ) ≃ Gal( e F v j,mj /k )(resp. Aut k ( F v j,mj ) ≃ Gal( e F v j,mj /k ) ) , ∀ j, m j , between the subgroups of automorphisms of the completions F v j,mj (resp. F v j,mj )and the corresponding Galois subgroups of the extensions e F v j,mj (resp. e F v j,mj ), asdeveloped in proposition 1.1.2, naturally leads to the commutative diagram: W ab ˙ e F v × W ab ˙ e F v G (2 n ) ( e F v × e F v )Aut k ( F v ) × Aut k ( F v ) G (2 n ) ( F v × F v ) e σ vR × e σ vL σ vR × σ vL ∼ ∼ where σ v R × σ v L is the monomorphism between the product, right by left, of the semi-groups of automorphisms of the set of completions F v and F v and the completelocally compact (algebraic) bilinear semigroup G (2 n ) ( F v × F v ) defining anabstract bisemivariety.Thus, the isomorphism W ab ˙ e F v × W ab ˙ e F v ∼ → Aut k ( F v ) × Aut k ( F v ) implies the homomor-phism G (2 n ) ( e F v × e F v ) ≃ G (2 n ) ( F v × F v ) between bilinear semigroups suchthat the abstract bisemivariety G (2 n ) ( F v × F v ) be covered by the algebraic(affine) semigroup G (2 n ) ( e F v × e F v ) .c) Let G (2 n ) ( e F nrv × e F nrv ) be the algebraic bilinear semigroup over the product of thesets of increasing pseudounramified extensions with e F nrv = { e F nrv , . . . , e F nrv j,mj , . . . } and e F nrv = { e F nrv , . . . , e F nrv j,mj , . . . } .Then, the kernel Ker( G (2 n ) e F → e F nr ) of the map: G (2 n ) e F → e F nr : G (2 n ) ( e F v × e F v ) −→ G (2 n ) ( e F nrv × e F nrv )is the smallest bilinear normal pseudoramified subgroup of G (2 n ) ( e F v × e F v ) :Ker( G (2 n ) e F → e F nr ) = P (2 n ) ( e F v × e F v ) , i.e. the minimal bilinear parabolic subsemigroup P (2 n ) ( e F v × e F v ) over theproduct ( e F v × e F v ) of the sets e F v = { e F v , . . . , e F v j,mj , . . . , e F v r,mr } and e F v = { e F v , . . . , e F v j,mj , . . . , e F v r,mr } of unitary archimedean pseudoramified extensions e F v j,mj and e F v j,mj in e F v j,mj and e F v j,mj respectively. 14) At every infinite biplace v j × v j of F v × F v corresponds a conjugacy class g (2 n ) v R × L [ j ]of the algebraic bilinear semigroup G (2 n ) ( e F v × e F v ) . The number of represen-tatives of g (2 n ) v R × L [ j ] corresponds to the number of equivalent extensions of e F v j × e F v j .So, we have the injective morphism: I F − G v : { e F v j,mj × e F v j,mj } m j −→ { G (2 n ) ( e F v j,mj × e F v j,mj ) } m j leading to the homeomorphism:Π n ( e F v j,mj × e F v j,mj ) ≃ G (2 n ) ( e F v j,mj × e F v j,mj )where G (2 n ) ( e F v j,mj × e F v j,mj ) is the ( j, m j ) -th conjugacy class representative of G (2 n ) ( e F v j,mj × e F v j,mj )e) G (2 n ) ( e F v × e F v ) acts on the bilinear parabolic subsemigroup P (2 n ) ( e F v × e F v )by conjugation in such a way that the number of conjugates of P (2 n ) ( e F v j × e F v j ) in G (2 n ) ( e F v j,mj × e F v j,mj ) is the index of the normalizer P (2 n ) ( e F v j × e F v j ) in G (2 n ) ( e F v j,mj × e F v j,mj ) : (cid:12)(cid:12)(cid:12) G (2 n ) ( e F v j,mj × e F v j,mj ) : P (2 n ) ( e F v j × e F v j ) (cid:12)(cid:12)(cid:12) = j . f) Let Out( G (2 n ) ( e F v × e F v )) = Aut( G (2 n ) ( e F v × e F v )) (cid:14) Int( G (2 n ) ( e F v j × e F v j )) be the(bisemi)group of (Galois) automorphisms of the algebraic bilinear semigroup G (2 n ) ( e F v × e F v ) where Int( G (2 n ) ( e F v × e F v )) is the (bisemi)group of (Galois) innerautomorphisms.As we have that Int( G (2 n ) ( e F v × e F v )) = Aut( P (2 n ) ( e F v × e F v )) , the bilinear parabolicsemigroup P (2 n ) ( e F v × e F v ) can be considered as the unitary irreducible representa-tion space of the algebraic bilinear semigroup GL n ( e F v × e F v ) of matrices [Pie2]. Similarly as it was done for the real case in section 1.3, let us consider the complex caseand, especially:a) Let B e F ω (resp. B e F ω ) be a left (resp. right) division semialgebra of complex dimen-sion n over the set e F ω = { e F ω , . . . , F ω j,mj , . . . } (resp. e F ω = { e F ω , . . . , F ω j,mj , . . . } )of increasing complex pseudoramified extensions of k .15his allows to define the algebraic bilinear semigroup GL n ( e F ω × e F ω ) by: B e F ω ⊗ B e F ω ≃ GL n ( e F ω × e F ω ) ≡ T tn ( e F ω ) × T n ( e F ω )in such a way that its representation space be given by the GL n ( e F ω × e F ω ) -bisemi-module f M ω R ⊗ f M ω L which is also a complex affine algebraic bilinear semigroup G (2 n ) ( e F ω × e F ω ) homeomorphic to the complete (algebraic) bilinear semigroup G (2 n ) ( F ω × F ω ) over the sets of completions F ω and F ω .Let GL( f M ω R ⊗ f M ω L ) denote the bilinear semigroup of automorphisms of ( f M ω R ⊗ f M ω L )verifying: GL n ( e F ω × e F ω ) f M ω R ⊗ f M ω L ≡ G (2 n ) ( e F ω × e F ω )GL( f M ω R ⊗ f M ω L ) ∼ because GL( f M ω R ⊗ f M ω L ) constitutes the n -dimensional complex equivalent of theproduct W ab ˙ e F ω × W ab ˙ e F ω of the corresponding global Weil semigroups. So, G (2 n ) ( e F ω × e F ω ) is the n -dimensional (or 2 n -dimensional real) complex(irreducible) representation space of W ab ˙ e F ω × W ab ˙ e F ω given by:Irr Rep nW FR × L ( W ab ˙ e F ω × W ab ˙ e F ω ) : GL( f M ω R ⊗ f M ω L ) ∼ −→ G (2 n ) ( e F ω × e F ω )implying the morphism: σ ω R × σ ω L : W ab ˙ e F ω × W ab ˙ e F ω −→ GL n ( e F ω × e F ω ) . b) At every biplace ( ω j × ω j ) of ( F ω × F ω ) corresponds a conjugacy class g (2 n ) e ω R × L [ j ]of G (2 n ) ( e F ω × e F ω ) leading to the injective morphism: I F − G ω : { e F ω j,mj × e F ω j,mj } m j −→ { G (2 n ) ( e F ω j,mj × e F ω j,mj ) } m j where G (2 n ) ( e F ω j,mj × e F ω j,mj ) is the ( j, m j ) -th conjugacy class representative of G (2 n ) ( e F ω × e F ω ) .c) G (2 n ) ( e F ω × e F ω ) acts by conjugation on the bilinear parabolic semigroup P (2 n ) ( e F ω × e F ω ) which can be considered as the unitary irreducible repre-sentation space of the complex algebraic bilinear semigroup GL n ( e F ω × e F ω ) of matrices because the (bisemi)group of (Galois) inner automorphisms of G (2 n ) ( e F ω × e F ω ) verifies: Int( G (2 n ) ( e F ω × e F ω )) = Aut( P (2 n ) ( e F ω × e F ω )) . .5 Inclusion of real (algebraic) bilinear semigroups into theircomplex equivalents The complex GL n ( e F ω × e F ω ) -bisemimodule f M ω R ⊗ f M ω L is the representation space of thealgebraic bilinear semigroup of matrices GL n ( e F ω × e F ω ) .Assume that each conjugacy class representative G (2 n ) ( e F ω j × e F ω j ) of G (2 n ) ( e F ω × e F ω ) ≡ f M ω R ⊗ f M ω L is unique in the j -th class.Then, the set { G (2 n ) ( e F ω j × e F ω j ) } rj =1 of conjugacy class representatives of G (2 n ) ( e F ω × e F ω ) is the representation (bisemi)space of the restricted complex algebraic bilinearsemigroup GL (res) n ( e F ω × e F ω ) .As a result, each complex conjugate class representative G (2 n ) ( e F ω j × e F ω j ) of G (2 n )(res) ( e F ω × e F ω ) is covered by the m j real conjugacy class representatives G ( n ) ( e F v j,mj × e F v j,mj ) of G ( n ) ( e F v × e F v ) [Pie2].So, the complex bipoints of G (2 n )(res) ( e F ω × e F ω ) are in one-to-one correspondence with the realbipoints of G (2 n ) ( e F v × e F v ) and we have the inclusion: G (2 n ) ( e F v × e F v (cid:14) G ( n ) ( e F v × e F v ) ≃ f M v R ⊗ f M v L G (2 n )(res) ( e F ω × e F ω ) ≡ f M (res) ω R ⊗ f M (res) ω L where f M (res) ω L (resp. f M (res) ω R ) is the left (resp. right) restricted T (res) n ( e F ω ) -semimodule(resp. T t (res) n ( e F ω ) -semimodule). G (2 n )(res) ( e F ω × e F ω ) is then said to be covered by G (2 n ) ( e F v × e F v ) . a) Providing a cuspidal representation of the complex bilinear algebraic semigroup G (2 n )(res) ( e F ω × e F ω ) consists in finding a cuspidal form on G (2 n )(res) ( e F ω × e F ω ) by sum-ming the cuspidal subrepresentations of its conjugacy class representatives G (2 n ) ( e F ω j × e F ω j ) .Let then γ TF ωj : e F ω j −→ F Tω j (resp. γ TF ωj : e F ω j −→ F Tω j )be the toroidal isomorphism mapping each left (resp. right) extension e F ω j (resp. e F ω j ) into its toroidal compact equivalent F Tω j (resp. F Tω j ) which is a complex one-dimensional semitorus localized in the upper (resp. lower) half space.17hen, the morphism: T n ( F Tω j ) : F Tω j −→ T (2 n ) ( F Tω j ) = T nL [ j ](resp. T n ( F Tω j ) : F Tω j −→ T (2 n ) ( F Tω j ) = T nR [ j ] )of the respective fibre bundle sends F Tω j (resp. F Tω j ) into the n -dimensional complexsemitorus T nL [ j ] (resp. T nR [ j ] ) corresponding to the upper (resp. lower) conjugacyclass representative T (2 n ) ( F Tω j ) = G (2 n ) ( F Tω j ) (resp. T (2 n ) ( F Tω j ) = G (2 n ) ( F Tω j ) ).So, we have a homeomorphism G (2 n ) ( e F ω j × e F ω j ) ≃ G (2 n ) ( F Tω j × F Tω j ) between theconjugacy class representative G (2 n ) ( e F ω j × e F ω j ) of G (2 n ) ( e F ω × e F ω ) and the conjugacyclass representative G (2 n ) ( F Tω j × F Tω j ) of G (2 n ) ( F Tω × F Tω ) where F Tω = { F Tω , . . . , F Tω j , . . . } (resp. F Tω = { F Tω , . . . , F Tω j , . . . } ) . b) Every left (resp. right) function on the conjugacy class representative G (2 n ) ( F Tω j ) (resp. G (2 n ) ( F Tω j ) ) is a function (resp. cofunction) φ L ( T nL [ j ])(resp. φ R ( T nR [ j ]) ) on the complex semitorus T nL [ j ] (resp. T nR [ j ] ) havingthe analytic development : φ L ( T nL [ j ]) = λ (2 n, j ) e πijz (resp. φ R ( T nR [ j ]) = λ (2 n, j ) e − πijz )where: • ~z = n Σ d =1 z d ~e d is a complex point of G (2 n ) ( F Tω j ) ; • λ (2 n, j ) is a product of the eigenvalues of the j -th coset representative of theproduct, right by left, of Hecke operators [Pie2].c) This left (resp. right) function φ L ( T nL [ j ]) (resp. φ R ( T nR [ j ]) ) constitutes the cusp-idal representation Π ( j ) ( G (2 n ) ( e F ω j )) (resp. Π ( j ) ( G (2 n ) ( e F ω j )) ) of the j -th conjugacyclass representative of G (2 n )(res) ( e F ω ) (resp. G (2 n )(res) ( e F ω ) ) in such a way that the cuspidalbiform of GL (res) n ( F ω × F ω ) is given by the Fourier biseries :Π(GL (res) n ( e F ω ⊕ × D e F ω ⊕ )) = r ⊕ j =1 Π ( j ) (GL (res) n ( e F ω j × D e F ω j )) , ≤ r ≤ ∞ , where: • e F ω ⊕ = Σ j e F ω j ; • GL (res) n ( e F ω × D e F ω ) is a bilinear “diagonal” algebraic semigroup.18 .7 Proposition (Langlands global correspondence on GL n ( e F ω × D e F ω ) ) The Langlands global correspondence on the complex (diagonal) bilinear algebraic semigroup GL (res) n ( e F ω × D e F ω ) is given by the isomorphism: LGC IC : σ (res) e ω R × L ( W ab ˙ e F ω × D W ab ˙ e F ω ) −→ Π(GL (res) n ( e F ω × D e F ω )) between the set σ (res) e ω R × L ( W ab ˙ e F ω × D W ab ˙ e F ω ) of the n -dimensional complex conjugacy class rep-resentatives of the diagonal products, right by left, of global Weil subgroups given by thediagonal algebraic bilinear semigroup G (2 n )(res) ( e F ω × D e F ω ) and its cuspidal representation givenby Π(GL (res) n ( e F ω × D e F ω )) in such a way that Π(GL (res) n ( e F ω ⊕ × D e F ω ⊕ )) = Σ j (cid:16) λ (2 n, j ) e − πijz × D λ (2 n, j ) e πijz (cid:17) be the cuspidal biform of GL (res) n ( e F ω × D e F ω ) . Proof : From section 1.1.4, it results that: σ (res) e ω R × L ( W ab ˙ e F ω × D W ab ˙ e F ω ) = GL (res) n ( e F ω × D e F ω ) , where σ (res) e ω R × L = σ (res) ω R × D σ (res) ω L , for the restricted case introduced in section 1.1.5.According to section 1.1.6, the j -th cuspidal representation of the j -th conjugacy classof G (2 n )(res) ( e F ω × D e F ω ) is given by Π j ( G (2 n )(res) ( e F ω j × D e F ω j )) (or by Π j (GL (res) n ( e F ω j × D e F ω j )) ).So we get the commutative diagram: σ (res) e ω R × L ( W ab ˙ e F ω × D W ab ˙ e F ω ) Π(GL (res) n ( e F ω × D e F ω )) { G (2 n )(res) ( e F ω j × D e F ω j ) } rj =1 ∼ toroidal isomorphisms { γ TFωj × γ TFωj } j where Π(GL (res) n ( e F ω × D e F ω )) = { Π j ( G (2 n ) ( e F ω j × D e F ω j ) } j = { φ R ( T nL [ j ]) ⊗ D φ L ( T nL [ j ]) } j asresulting from section 1.6. a) A real cuspidal representation, covering the complex cuspidal representa-tion Π(GL (res) n ( e F ω × D e F ω )) , can be obtained for the real diagonal bilinear algebraicsemigroup G (2 n ) ( e F v × D e F v j ) by summing the cuspical subrepresentations of ts conjugacy class representatives taking into account the inclusion (whichis also a covering) G (2 n ) ( e F v × D e F v ) G (2 n )(res) ( e F ω × D e F ω )mentioned in section 1.1.5.Let then: γ TF vj,mj : e F v j,mj −→ F Tv j,mj (resp. γ TF vj,mj : e F v j,mj −→ F Tv j,mj )be the toroidal isomorphism mapping each left (resp. right) extension e F v j,mj (resp. e F v j,mj ) into its toroidal compact equivalent F Tv j,mj (resp. F Tv j,mj ) which is a semicirclelocalized in the upper (resp. lower) half space.The fibre bundle morphism: T n ( F Tv j,mj ) : F Tv j,mj −→ T (2 n ) ( F Tv j,mj )(resp. T n ( F Tv j,mj ) : F Tv j,mj −→ T (2 n ) ( F Tv j,mj ) )sends F Tv j,mj (resp. F Tv j,mj ) into the 2 n -dimensional real semitorus T (2 n ) ( F Tv j,mj )(resp. T (2 n ) ( F Tv j,mj ) ).b) Every left (resp. right) function on the conjugacy class representative G (2 n ) ( F Tv j,mj )(resp. G (2 n ) ( F Tv j,mj ) ) is a function (resp. cofunction) φ L ( T nL [ j, mj ]) (resp. φ R ( T nR [ j, m j ]) ) on the real semitorus T nL [ j, m j ] (resp. T nR [ j, m j ] ) having the ana-lytic development: φ L ( T nL [ j, m j ]) = λ (2 n, j, m j ) e πijz , x ∈ IR n , (resp. φ R ( T nR [ j, m j ]) = λ (2 n, j, m j ) e − πijz )where λ (2 n, j, m j ) is the product of the eigenvalues of the ( j, m j ) -th coset repre-sentative of the product, right by left, of Hecke operators.This function (resp. cofunction) φ L ( T nL [ j, m j ]) (resp. φ R ( T nR [ j, m j ]) ) is the cuspi-dal representation Π ( j,m j ) ( G (2 n ) ( e F v j,mj )) (resp. Π ( j,m j ) ( G (2 n ) ( e F v j,mj )) ) of the ( j, m j ) -th conjugacy class representative of G (2 n ) ( e F v ) (resp. G (2 n ) ( e F v ) ) becauseΠ(GL n ( e F v ⊗ × D e F v ⊗ )) is given by the Fourier biseriesΠ(GL n ( e F v ⊗ × D e F v ⊗ )) = ⊕ j,m j Π ( j,m j ) (GL n ( e F v j,mj × D e F v j,mj )) , where e F v ⊕ = Σ j,m j e F v j,mj , and corresponds to a cuspidal biform.20 .9 Proposition (Langlands global correspondence on GL n ( e F v × D e F v ) ) The Langlands global correspondence on the real diagonal bilinear algebraic semigroup GL n ( e F v × D e F v ) is given by the isomorphism: LGC IR : σ e v R × L ( W ab ˙ e F v × D W ab ˙ e F v ) −→ Π(GL n ( e F v × D e F v )) between the set σ e v R × L ( W ab ˙ e F v × D W ab ˙ e F v ) of the n -dimensional real conjugacy class represen-tatives of the diagonal products, right by left, of global Weil subgroups given by the algebraicbilinear semigroup G (2 n ) ( e F v × D e F v ) and its cuspidal representation Π(GL n ( e F v × D e F v )) in such a way that Π(GL n ( e F v ⊕ × D e F v ⊕ )) be a “cuspidal biform” on GL n ( e F v × D e F v ) . Proof : As in proposition 1.7, the proposition results from the commutative diagram: σ e v R × L ( W ab ˙ e F v × D W ab ˙ e F v ) Π(GL n ( e F v × D e F v )) { G (2 n ) ( e F v j,mj × D e F v j,mj ) } j,m j LGC IR ∼ toroidal isomorphisms { γ TFvj,mj × γ TFvj,mj } j,mj The commutative diagram: σ e ω R × L ( W ab ˙ e F ω × D W ab ˙ e F ω ) Π(GL (res) n ( e F ω × D e F ω )) σ e v R × L ( W ab ˙ e F v × D W ab ˙ e F v ) Π(GL n ( e F v × D e F v )) LGC IC LGC IR implies that the cuspidal biform Π(GL (res) n ( e F ω ⊕ × D e F ω ⊕ )) is covered by the product, rightby left, Π(GL n ( e F v ⊕ × D e F v ⊕ )) of Fourier series over real archimedean completions. Proof : This is a consequence of the propositions 1.7 and 1.9.
The functoriality conjecture introduced by R. Langlands deals with the product of cuspi-dal representations of algebraic linear groups over adele rings. Transposed in this bilinear21ontext, this problem is easily solvable by taking into account the cross binary operationof bilinear (algebraic) semigroups introduced in [Pie3]. Indeed, the Langlands functorial-ity conjecture then results from the reducibility of representations of bilinear (algebraic)semigroups [Pie4], covering their linear equivalents.This new bilinear approach useful in the decomposition of the bilinear cohomology of thebilinear (algebraic) semigroups GL n ( e F v × D e F v ) can be stated as follows: The cuspidal (and holomorphic) representation Π(GL n ( e F v × D e F v )) of the (bi-linear (algebraic) semigroup GL n ( e F v × D e F v ) is (non orthogonally) completelyreducible if it decomposes: a) diagonally according to the direct sum n ⊕ ℓ =1 Π (2 ℓ ) (GL ℓ ( e F v × D e F v )) of irreducible cuspidal (and holomorphic) representations of the (algebraic) bilinearsemigroups GL ℓ ( e F v × D e F v ) ;b) and off-diagonally according to the direct sum r ⊕ k = ℓ =1 (cid:16) Π (2 k ) (GL k ( e F v )) ⊗ Π (2 ℓ ) (GL ℓ ( e F v )) (cid:17) of the products of irreducible cuspidal (and holomorphic) representations of crosslinear (algebraic) semigroups GL k ( e F v ) × GL ℓ ( e F v ) ≡ T t k ( e F v ) × T ℓ ( e F v ) . Proof : The thesis directly results from the definition of a bilinear semigroup introducedin [Pie3] and was developed in [Pie3]. 22
Lower bilinear K -theory based on homotopy semi-groups viewed as deformations of Galois represen-tations It results from chapter 1 that the Langlands global program refers mainly to the (func-tional) representation space (F)REPSP(GL n ( F ω × F ω )) ≡ G (2 n ) ( F ω × F ω ) of the complexcomplete (algebraic) bilinear semigroup GL n ( F ω × F ω ) , covered by its real equivalent(F)REPSP(GL n ( F v × F v )) ≡ G (2 n ) ( F v × F v ) , because these (bisemi)spaces are represen-tations of the products, right by left, of global Weil semigroups. Related to the reducibility of representations of bilinear (algebraic) semigroups (which areabstract bisemivarieties), recalled in proposition 1.12, a general bilinear cohomologytheory was defined in section 3.2 of [Pie2] as a contravariant bifunctor :H ∗ : { smooth abstract (algebraic) bisemivarieties G (2 n ) ( F ω × F ω ) = (F)REPSP(GL n ( F ω × F ω )) (cid:9) −→ { graded (functional) representation spaces of thecomplete (algebraic) bilinear semigroups GL ∗ ( F ω × F ω ) } written in the conventional form: H ∗ ( G (2 n ) ( F ω × F ω ) , (F)REPSP(GL ∗ ( F ω × F ω )))= ⊕ i H i ( G (2 n ) ( F ω × F ω ) , (F)REPSP(GL i ( F ω × F ω ))) . Taking into account the inclusion G (2 n ) ( F v × F v ) G (2 n )(res) ( F ω × F ω )of the real (algebraic) bilinear semigroup G (2 n ) ( F v × F v ) , which is an abstract real bisemi-variety, into the corresponding complex (algebraic) abstract bisemivariety G (2 n )(res) ( F ω × F ω )as developed in section 1.5, the general bilinear cohomology can be rewrittenin function of rational (bi)coefficients (algebraic case) or in function of real bi)coefficients (abstract (complete) general case) : H ∗ ( G (2 n ) ( F ω × F ω ) , FREPSP(GL ∗ ( F v × F v )))= ⊕ ii ≤ n H i ( G (2 n ) ( F ω × F ω ) , FREPSP(GL i ( F v × F v ))) . This corresponds to Hodge bisemicycles [D-M-O-S], [Riv], sending the abstract (and, thus,also the algebraic) complex bisemivariety G (2 n ) ( F ω × F ω ) into the abstract (and, thus, alsoalgebraic) real bisemivarieties G (2 i ) ( F v × F v ) ≡ FREPSP(GL i ( F v × F v )) [Pie2] in sucha way that there is a bifiltration F pR × L on the right and left cohomology semigroups of H i ( G (2 n ) ( · × · ) , − ) given by F pR × L H i ( G (2 n ) ( F ω × F ω ) , G (2 i ) ( F v × F v ))= ⊕ i = p + q H p + q ) ( G (2 n ) ( F ω × F ω ) , G p + q ) ( F v × F v )) . In addition to the bifiltration on Hodge bisemicycles, the general bilinear cohomology ischaracterized by the following properties.a)
A bisemicycle map [Mur], [Mor]: γ iG (2 n ) ω × ω : Z i ( G (2 n ) ( F ω × F ω )) −→ H i ( G (2 n ) ( F ω × F ω ) , G (2 i ) ( F v × F v )) • from the bilinear semigroup Z i ( G (2 n ) ( F ω × F ω )) of compactified (resp. non-compactified) bisemicycles of codimension i , in the abstract (resp. algebraic)case, on the bilinear complete (resp. algebraic) semigroup G (2 n ) ( F ω × F ω ) (resp. G (2 n ) ( e F ω × e F ω ) ) • into the bilinear cohomology H i ( G (2 n ) ( F ω × F ω ) , G (2 i ) ( F v × F v )) in such a waythe the embedding G (2 i ) ( F v × F v ) G (2 i ) ( F ω × F ω )of the real bisemivariety G (2 i ) ( F v × F v ) into its complex equivalent G (2 i ) ( F ω × F ω )is directly related to the Hodge bisemicycles according to section 2.2.b) A K¨unneth isomorphism : H i ( G (2 n ) ( F ω ) , G (2 i ) ( F v )) ⊗ F v × F v H i ( G (2 n ) ( F ω ) , G (2 i ) ( F v )) −→ H i ( G (2 n ) ( F ω × F ω ) , G (2 i ) ( F v × F v )) , n ≥ i , G (2 i ) ( F v × F v ) ofreal dimension 2 i , covering its complex equivalent G (2 i ) ( F ω × F ω ) , in the (tensor)product of a right complex abstract semivariety G (2 n ) ( F ω ) of complex dimension n by its left equivalent G (2 n ) ( F ω ) . The next step consists in finding the K -theory associated with the general bilinear coho-mology and in defining the corresponding Chern classes. But, as K -theories are relatedto homotopy and as the proposed bilinear cohomology is essentially a motivic (bilinear)cohomology theory or a Weil (bilinear) cohomology theory [Pie2], the homotopy mustbe proved to result from algebraic geometry in order that this general contextbe coherent .It will then be proved that the concept of homotopy in topology corresponds to a defor-mation of Galois representation as introduced in [Pie7] and briefly recalled now. Two kinds of deformations of (2) n -dimensional representations of global Weil(or Galois) (semi)groups given by bilinear (algebraic) semigroups over completeglobal Noetherian bisemirings were envisaged [Pie7], [Pie1].a) global bilinear quantum deformations leaving invariant the orders of inertiasubgroups;b) global bilinear deformations inducing the invariance of their bilinear residue (i.e.pseudounramified) semifields.Case a) will be only taken into account in this paper because the inertia subgroups, beingthe subgroups of automorphisms of algebraic space quanta, are supposed to be stable. Then, a global quantum deformation results from a global coefficient semiring quantumhomomorphism. 25 left (resp. right) global (compactified) coefficient semiring F v (resp. F v ) is given bythe set of infinite pseudoramified archimedean embedded completions: F v ⊂ · · · ⊂ F v j,mj ⊂ · · · ⊂ F v r,mr (resp. F v ⊂ · · · ⊂ F v j,mj ⊂ · · · ⊂ F v r,mr ) , as developed in section 1.1, where two neighbouring completions F v j and F v j +1 differ bya transcendental quantum F v j characterized by a transcendence degree tr · d · F v j (cid:14) k = N .Let F v + ℓ (resp. F v + ℓ ) denote another left (resp. right) global coefficient semiring of whichcompletions are those of F v (resp. F v ) increased by “ ℓ ” transcendental quanta: F v ℓ ⊂ · · · ⊂ F v j + ℓ ⊂ · · · ⊂ F v r + ℓ (resp. F v ℓ ⊂ · · · ⊂ F v j + ℓ ⊂ · · · ⊂ F v r + ℓ ) . A uniform quantum homomorphism between global coefficient semirings isgiven by : Qh F v + ℓ → F v : F v + ℓ −→ F v (resp. Qh F v + ℓ → F v : F v + ℓ −→ F v )in such a way that:1) the kernel K ( Qh F v + ℓ → F v ) (resp. K ( Qh F v + ℓ → F v ) ) of the quantum homomorphism Qh F v + ℓ → F v (resp. Qh F v + ℓ → F v ), inducing an isomorphism on their global inertia sub-groups, is characterized by a transcendence degreetr · d · F v + ℓ (cid:14) k − tr · d · F v (cid:14) k = N × ℓ × Σ j m j (if m j + ℓ = m j ).2) this quantum homomorphism corresponds to a base change from F v (resp. F v ) into F v + ℓ (resp. F v + ℓ ) of which transcendence extensions degree istr · d · F v + ℓ (cid:14) k − tr · d · F v (cid:14) k which means an increment of ℓ quanta on each completion of the coefficient semiring F v (resp. F v ); A global bilinear quantum deformation representative, resulting from a globalbilinear coefficient semiring quantum homomorphism Qh F v + ℓ × F v + ℓ → F v × F v : F v + ℓ × F v + ℓ −→ F v × F v ,
26s an equivalence class representative ρ F ℓ of liftingGal( ˙ e F v + ℓ /k ) × Gal( ˙ e F v + ℓ /k ) Gal( ˙ e F v /k ) × Gal( ˙ e F v /k )GL n ( F v + ℓ × F v + ℓ ) GL n ( F v × F v ) Qh Fℓ → F Qh Gℓ → G ρ Fℓ ρ F with the notations of sections 1.1 and 1.3. A n -dimensional global bilinear quantum deformation of ρ F is an equivalenceclass of liftings { ρ F ℓ } ℓ , 1 ≤ ℓ ≤ ∞ , described by the following diagram:1 Gal( δ ˙ e F v + ℓ /k ) Gal( ˙ e F v + ℓ /k ) Gal( ˙ e F v /k ) 1 × Gal( δ ˙ e F v + ℓ /k ) × Gal( ˙ e F v + ℓ /k ) × Gal( ˙ e F v /k )1 GL n ( δF v + ℓ × δF v + ℓ ) GL n ( F v + ℓ × F v + ℓ ) GL n ( F v × F v ) 1 δρ Fℓ ρ Fℓ ρ F of which “Weil kernel” is Gal( δ ˙ e F v + ℓ /k ) × Gal( δ ˙ e F v + ℓ /k ) and “ GL n ( · × · ) ” kernel isGL n ( δF v + ℓ × δF v + ℓ ) .This equivalence class of liftings { ρ F ℓ } ℓ is then given by ρ F ℓ = ρ F + δρ F ℓ , ∀ ℓ , ≤ ℓ ≤ ∞ , in such a way that two liftings ρ F ℓ and ρ F ℓ are strictly equivalent if they can be trans-formed one into another by conjugation by bielements of GL n ( F v + ℓ × F v + ℓ ) in the kernelof Qh G ℓ → G . The transformation of kernels GL n ( δF v + ℓ × δF v + ℓ ) −→ GL n ( δF v + ℓ × δF v + ℓ ) corresponds to a base change from GL n ( F v + ℓ × F v + ℓ ) into GL n ( F v + ℓ × F v + ℓ ) of whichdimension is given by the difference of ranks δr G n ( ℓ − ℓ ) = N n ( f n v + ℓ − f n v + ℓ ) where f v + ℓ is the sum of all global residue degrees corresponding to the conjugacy classrepresentatives of GL n ( F v + ℓ × F v + ℓ ) . roof : Referring to section 2.7, it is clear that the liftings ρ F ℓ and ρ F ℓ are respectivelycharacterized by the kernels GL n ( δF v + ℓ × δF v + ℓ ) and GL n ( δF v + ℓ × δF v + ℓ ) .The kernel GL n ( δF v + ℓ × δF v + ℓ ) is characterized by a rank r δ Gnℓ = f n ℓ × N n and thekernel GL n ( δF v + ℓ × δF v + ℓ ) is characterized by a rank r δ Gnℓ = f n ℓ × N n .These ranks r δ Gnℓ and r δ Gnℓ describe the increase of the algebraic dimensions respectivelyof all the conjugacy class representatives of GL n ( F v + ℓ × F v + ℓ ) and GL n ( F v + ℓ × F v + ℓ ) .So, the difference of ranks ( r δ Gnℓ − r δ Gnℓ ) characterizes the difference of liftings ( ρ F ℓ − ρ F ℓ ) and describes the base change from GL n ( F v + ℓ × F v + ℓ ) to GL n ( F v + ℓ × F v + ℓ ) . Let Qh v + ℓ → v : F v + ℓ → F v be a uniform quantum homomorphism sending the globalcoefficient semiring F v into the deformed global coefficient semiring F v + ℓ obtained from F v by adding “ ℓ ” transcendental quanta according to section 2.6.Let f h ℓ : F v + ℓ → G (2 n ) ( F v ) be a continuous map from F v + ℓ into the real abstract linear(semi)variety G (2 n ) ( F v ) over the set F v of archimedean completions.Then, there exists a continuous map F H : F v × I −→ G (2 n ) ( F v ) , I = [0 , , such that F H ( x,
0) = f h and
F H ( x,
1) = f h ℓ , ∀ x ∈ F v , x being a point or a big point(i.e. a quantum), where f h is the continuous map: f h : F v → G (2 n ) ( F v ) . This continuous map
F H is thus the homotopy of f h , and will be called theGalois homotopy of f h . The Galois homotopy of the continuous map f h : F v → G (2 n ) ( F v ) results froma quantum homomorphism between global coefficient semirings . Proof : It is sufficient to prove that the homotopy classes for all the functions f h ℓ corre-spond to the classes of the quantum homomorphism Qh v + ℓ → v .Let cor F H : F v × I −→ F v + ℓ t → ℓ , t ∈ [0 , , denote the one-to-one correspondence between the product of the basic coefficient semiring F v by the unit interval [0 ,
1] and the deformed global coefficient semiring F v + ℓ in such a28ay that to any t ∈ [0 ,
1] corresponds an integer ℓ labelling the number of quanta addedto F v .Then, the homotopy F H : F v × I → G (2 n ) ( F v ) , interpreted as a family of continuous maps f h ( t ) : F v → G (2 n ) ( F v ) by the relation f h ( t ) ( x ) = F H ( x, t ) , 0 ≤ t ≤ t ” with the places v + ℓ of F v + ℓ which are the classes of the deformed global coefficient semiring F v + ℓ .This homotopy, resulting from a quantum deformation of the global coefficient semiring,will be called a Galois homotopy because the deformed coefficient semiring F v + ℓ is home-omorphic to the algebraic coefficient semiring˙ e F v + ℓ = n ˙ e F v ℓ , ˙ e F v j + ℓ , ˙ e F v r + ℓ o given by this set of real pseudoramified extensions according to sections 1.1 and 2.6. The Galois cohomotopy is the inverse Galois homotopy defined by the relation
CF H ( x, t ) = F H ( x, − t )and corresponding to the homotopy between f h ℓ and f h in such a way that it resultsfrom the inverse quantum homomorphism Qh − v + ℓ → v between global coefficient semir-ings. Let F v = { F v , . . . , F v r } denote the unit subset of F v composed of one quantum in eachcompletion of F v . The global coefficient semiring F v + ℓ is said to be retract if the Galois homotopy CF H ( x,
1) =
F H ( x,
0) corresponds to the constant homotopy, i.e. if F v + ℓ is sent to itssubsemiring F v . The global coefficient semiring F v + ℓ is said to be strongly retract if F v + ℓ is sentto the unit subset F v of F v . The equivalence classes of maps between a fixed basic coefficient semiring F v and thereal linear (semi)variety G (2 n ) ( F v ) are the homotopy classes corresponding to the classes29f the quantum homomorphism Q v + ℓ → v characterized by the integers “ ℓ ” which are inone-to-one correspondence with the values of the parameter t ∈ [0 ,
1] of the homotopy.Let f h ℓ : F v + ℓ → G (2 n ) ( F v ) and f h ℓ,d : F ( v + ℓ )+ d → G (2 n ) ( F v ) be two maps relativerespectively to f h : F v → G (2 n ) ( F v ) and f h ℓ .They belong to two difference equivalence classes of maps characterized respectively bythe integers ℓ and d .Being homotopic is then an equivalence relation compatible with the product of equivalenceclasses.That is to say, if { f h ℓ } denotes the set of continuous maps from F v + ℓ into G (2 n ) ( F v )with respect to F v , and { f h ℓ,d } denotes the set of continuous maps from F ( v + ℓ )+ d into G (2 n ) ( F v ) with respect to F v + ℓ , { f h ℓ } × { f h ℓ,d } will correspond to the product of theequivalence classes { f h ℓ } × { f h ℓ,d } .Taking into account the existence ofa) the Galois cohomotopy of which classes are the inverse equivalence classes of thecorresponding Galois homotopy,b) the null homotopy associated with identity homotopy maps,we see that the set of equivalence classes of Galois homotopy forms a group notedΠ( F v , G (2 n ) ( F v )) .If L v is the image in G (2 n ) ( F v ) of F v , we get the fundamental groupΠ ( G (2 n ) ( F v ) , L v ) in the big point L v , which is one quantum or the center of blowup ofthis one [Pie5].Remark that the equivalence classes of maps between the coefficient semiring F v and the real linear abstract (semi)variety G (2 n ) ( F v ) are the equivalence classesof maps between the set Ω( L v , G (2 n ) ( F v )) of oriented paths, which are the set F v of archimedean completions , and G (2 n ) ( F v ) .These equivalence classes thus depend on the deformations of the Galois compact repre-sentations of these paths corresponding to the increase of these by a(n) (in)finite numberof transcendental or algebraic quanta. The definition of the fundamental (Galois) homotopy [D-N-F] group Π ( G (2 n ) ( F v ) , L v ) interms of deformations of Galois representations of paths (or loops) can be easily generalized30o the i -th homotopy group Π i ( G (2 n ) ( F v ) , L v j ) ) of the given (semi)variety G (2 n ) ( F v )with base point L v j ) .The set of homotopy classes of maps S f h iℓ : S i −→ G (2 n ) ( F v ) , sending the base point b of the i -sphere S i to the base point L v i ( j ) of G (2 n ) ( F v ) ,are equivalently described by maps C f h iℓ : [0 , i −→ G (2 n ) ( F v )from the i -cube [0 , i to G (2 n ) ( F v ) by taking its boundary δ [0 , i to L v i ( j ) . The (semi)group Π i ( G (2 n ) ( F v ) , L v j ) ) of homotopy classes of maps S f h iℓ : S i ( ℓ ) −→ G (2 n ) ( F v )or C f h iℓ : [0 , iℓ −→ G (2 n ) ( F v )results from the deformations of the Galois compact representation of the semi-group GL i ( e F v ) of real dimension i given by the kernels G (2 i ) ( δF v + ℓ ) of the maps: GD iℓ : G (2 i ) ( F v + ℓ ) −→ G (2 i ) ( F v ) ,t i → ℓ i ∀ ℓ , ≤ ℓ ≤ ∞ , in such a way that the i -th powers of the integers “ ℓ ” be in one-to-one correspondencewith the i -th powers of the values of the parameter t ∈ [0 , . Proof : The structure of (semi)group of Π i ( G (2 n ) ( F v ) , L v j ) ) results from the compositionof its homotopy classes.Let C f h iℓ : [0 , iℓ → G (2 n ) ( F v ) be the homotopy class of maps C f h iℓ characterized bythe value(s) t iℓ ∈ [0 , i of the parameter t i of the 2 i -cube [0 , i .Let C f h id : [0 , id → G (2 n ) ( F v ) be another homotopy class of maps C f h id characterizedby the value t id ∈ [0 , i of the parameter t i .Then, the composition (i.e. sum) of these two homotopy classes is given by the homotopyclass of maps: C f h iℓ + d : [0 , iℓ + d −→ G (2 n ) ( F v )31haracterized by the value t iℓ + d ∈ [0 , i of the parameter t i .Referring to lemma 2.10 and section 2.7, it appears that the homotopy class of maps C f h iℓ : [0 , iℓ → G (2 n ) ( F v ) is a deformation of the semivariety G (2 i ) ( F v ) ⊂ G (2 n ) ( F v )and corresponds to the deformation of the Galois compact representation of the linearsemigroup GL i ( e F v ) given by the kernel G (2 i ) ( δF v + ℓ ) of the map:GD iℓ : G (2 i ) ( F v + ℓ ) −→ G (2 i ) ( F v ) . This kernel ( G (2 i ) ( F v + ℓ )) is then characterized by the integer ℓ i whicha) denotes the number of quanta added to G (2 i ) ( F v ) by the envisaged deformation;b) is in one-to-one correspondence with the parameter t iℓ .It is then clear that the composition C f h iℓ + d of the two homotopy classes of maps C f h iℓ and C f h id results from the kernel G (2 i ) ( δF v + ℓ + d ) of the composition GD id ◦ GD iℓ of themaps GD iℓ and GD id :GD id ◦ GD iℓ : G (2 i ) ( F v + ℓ + d ) −→ G (2 i ) ( F v ) . If the homotopy group Π i ( G (2 n ) ( F v ) , L v j ) ) lacks for inverse homotopy classes (and null-homotopy class), it becomes a homotopy semigroup whose dual is the cohomotopy semi-group noted Π i ( G (2 n ) ( F v ) , L v j ) ) .Thus, the cohomotopy semigroup Π i ( G (2 n ) ( F v ) , L v j ) ) is defined by classes re-sulting from inverse deformations (GD iℓ ) − : G (2 i ) ( F v ) → G (2 i ) ( F v + ℓ ) of theGalois representations of GL i ( e F v ) [Pie7]. Let G (2 n ) ( F v ) denote the semivariety dual of G (2 n ) ( F v ) and let L v j ) be the base point of G (2 n ) ( F v ) .Then, the (Galois) bilinear homotopy (semi)group will be given byΠ i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) in such a way that its classes of (bi)maps: C f h iℓ × ( D ) C f h iℓ : [0 , iℓ × ( D ) [0 , iℓ −→ G (2 n ) ( F v × F v )result from the deformations of the Galois (compact) representations of the bisemivariety G (2 i ) ( e F v × e F v ) given by the (bi)kernels G (2 i ) ( δF v + ℓ × δF v + ℓ ) of the (bi)mapsGD iℓ R × L : G (2 i ) ( F v + ℓ × F v + ℓ ) −→ G (2 i ) ( F v × F v ) , ∀ ℓ . i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) in such a way that its classes of (bi)maps are inverse of those of thebilinear (Galois) homotopy (semi)group Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) . Taking into account the group homomorphism of Hurewicz: hH : Π i ( G (2 n ) ( F v ) , L v j ) ) −→ H i ( G (2 n ) ( F v ) , Z ) , we can specialize it to the Galois bilinear homotopy and cohomotopy semigroups accordingto: hH R × L : Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) −→ H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) hCH R × L : Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) −→ H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) where:1) hH R × L is the bilinear semigroup homomorphism from the bilinear homotopy Π i ( · ) into the bilinear cohomology H i ( · ) ;2) hCH R × L is the bilinear semigroup homomorphism from the bilinear cohomotopy Π i ( · ) into the bilinear homology H i ( · ) . Proof : As Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) (resp. Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) ) is a bi-linear homotopy (resp. cohomotopy) semigroup resulting from deformations (resp. inversedeformations) of the Galois (compact) representations of the bilinear semigroup GL i ( e F v × e F v ) ⊂ GL n ( e F v × e F v ) , it is natural to associate to it by the homomorphism hH R × L (resp. hCH R × L ) the entire bilinear cohomology (resp. homology) H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ )(resp. H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) ) where ZZ × ( D ) ZZ refers to a bisemilattice deformedby the classes of deformations (resp. inverse deformations) of Galois representations ofGL i ( e F v × ( D ) e F v ) in one-to-one correspondence with the classes of homotopy (resp. coho-motopy).Indeed, the cohomology (resp. the homology) is defined with respect to a coboundary(resp. boundary) homomorphism increasing (resp. decreasing) the dimension of one unit.33 .19 Topological bilinear K -theory As the universal cohomology theory is bilinear [Pie2] referring to the Tannakian category[Riv] of representations of affine group schemes, the (topological) K -theory of the compactreal (resp. complex) bisemivariety G (2 i ) ( F v × F v ) (resp. G (2 i ) ( F ω × F ω ) ) can be naturallyintroduced and is proved to correspond to the classical definition of the K -theory.Indeed, classically, if M denotes the abelian semigroup (or monoid) of classes of isomor-phism of k -vector bundles over a compact space X , the topological K -theory K ( X )is the symmetrized group of M , i.e. the quotient of M × M by the equivalence re-lation identifying ( x, y ) to ( x ′ , y ′ ) , or is the set of cosets of ∆( M ) in M × M where∆ : M → M × M is a diagonal homomorphism of semigroups [Ati1].The locally constant function r : X → IN given by r ( x ) = dim E x (where E is thevector bundle over X ) defines the group homomorphism K ( X ) → H ( X ; ZZ ) , where H ( X ; ZZ ) is the first ˇCech cohomology group of X given by locally constant functionsover X with values in ZZ [Kar1].In this context, let K L ( G (2 i ) ( F v )) denote the abelian semigroup (or monoid) of classesof isomorphism of k -vector bundles over the compact left semivariety G (2 i ) ( F v ) .Then, the topological (bilinear) K -theory, noted K R × L ( G (2 i ) ( F v × F v )) orsimply K ( G (2 i ) ( F v × F v )) , is the set of cosets of K R ( K L ( G (2 i ) ( F v )))where K R : K L ( G (2 i ) ( F v )) → K R × L ( G (2 i ) ( F v × F v ))) is the diagonal ho-momorphism sending the left abelian semigroup K L ( G (2 i ) ( F v )) on the left semivariety G (2 i ) ( F v ) into the diagonal bilinear semigroup [Pie3] K R × L ( G (2 i ) ( F v × F v )) on the prod-uct, right by left, of the semivarieties G (2 i ) ( F v ) and G (2 i ) ( F v ) .Remark that, in the diagonal bilinear semigroup, the cross products are not considered.Taking into account the derived functors of the K -theory of the variety X , K − n ( X ) = K ( X × IR n )introduced by Atiyah and Hirzebruch, and the periodicity of the Clifford algebra C n , thegroup K n ( X ) can be introduced from the category of k -fibre bundles in graded moduleson C n as a general cohomology theory n → K n ( X ) on the category of pointed compactspaces (or locally compact spaces) [Kar2].In this context, let G (2 i ) ( F v × F v ) → ( F v × F v ) be a vector (bi)bundle with (bi)fibreGL i − ( F v × F v ) (or IR i − ).Let K i ( G (2 n ) ( F v × F v )) denote the topological (bilinear) K -theory of vector (bi)bundleswith base G (2 n ) ( F v × F v ) and bifibre G (2 n − i +1) ( F v × F v ) . K i ( G (2 n ) ( F v × F v )) is then equivalent to K i ( F v × F v ) since the total space of these twovector (bi)bundles is the bisemivariety G (2 i ) ( F v × F v ) .34he cohomology class of obstruction of these fibre bundles is α i ∈ H n − i +1 ( F v × F v , Π n − i +1 ( G (2 i ) ( F v × F v )))leading to the Stiefel-Whitney class W i = α n − i +1 ∈ H i ( F v × F v , ZZ × ( D ) ZZ ) of whichpolynomial is W ( x ) = 1 + W x + · · · + W i x i + . . . Similarly, in the complex case, the Chern classes C i = β n − i +1 ∈ H i ( F ω × F ω , ZZ × ( D ) ZZ )of the fibre bundles G (2 i ) ( F ω × F ω ) → ( F ω × F ω ) with base ( F ω × F ω ) are included intothe Chern polynomial: C ( X ) = 1 + C x + · · · + C i x i + . . . . K -cohomology The Stiefel-Whitney character restricted to the class W i in the bilinear K -cohomology of the abstract real bisemivariety G (2 n ) ( F v × F v ) given by the homomor-phism: SW i ( G (2 n ) ( F v × F v )) : K i ( G (2 n ) ( F v × F v )) −→ H i ( G (2 n ) ( F v × F v ) , G (2 i ) ( F v × F v )) , ∀ i ≤ n , and corresponds to the Chern character restricted to the class C i in the bilin-ear K -cohomology of the abstract complex bisemivariety G (2 n ) ( F ω × F ω ) given by thehomomorphism C i ( G (2 n ) ( F ω × F ω )) : K i ( G (2 n ) ( F ω × F ω )) −→ H i ( G (2 n ) ( F ω × F ω ) , G (2 i ) ( F ω × F ω )) . Let hH R × L : Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) −→ H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ )35 e the bilinear semigroup homomorphism of Hurewicz from the “Galois” bilinear homotopy Π i ( · ) into the entire bilinear cohomology H i ( · ) : it can be called (restricted) Π -cohomology with reference to K -cohomology .Let C i ( G (2 n ) ( F v × F v )) : K i ( G (2 n ) ( F v × F v )) −→ H i ( G (2 n ) ( F v × F v ) , G (2 i ) ( F v × F v )) be the restricted Chern character in the bilinear K -cohomology of the abstract real bisemi-variety G (2 n ) ( F v × F v ) .Then, the lower bilinear (algebraic) K -theory will be given by the equality (resp.homomorphism) K i ( G (2 n ) ( F v × F v )) = ( → ) Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) in such a way that the homotopy classes of maps of Π i ( · ) are (resp. correspondto) liftings of quantum deformations of the Galois representation GL i ( e F v × e F v ) . Proof : Referring to proposition 1.9, the functional representation space of the prod-uct, right by left, of global Weil (semi)groups is given by the real abstract bisemivariety G (2 n ) ( F v × F v ) in the frame of the Langlands global program.So, the relations between the bilinear cohomology, homotopy and topological K -theoryof G (2 n ) ( F v × F v ) are given according to sections 2.2, 2.15 and 2.19 by the commutativediagram K i ( G (2 n ) ( F v × F v )) Π i ( G (2 n ) ( F v × F v ) , L v ( j ) × L v j ) ) H i ( G (2 n ) ( F v × F v ) , G (2 i ) ( F v × F v )) lower (algebraic) K -theoryinverse restrictedChern character inverse restricted K -cohomology Hurewicz homomorphism in such a way that the classes of the entire bilinear cohomology H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) as well as those of the bilinear K -theory K i ( G (2 n ) ( F v × F v )) are thehomotopy classes of maps of Π i ( G (2 n ) ( F v × F v )) corresponding to the lifts ofquantum deformations of the Galois compact representations of GL i ( e F v × e F v ) according to proposition 2.15.It then results that: K i ( G (2 n ) ( F v × F v )) = Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) defining a lower bilinear (algebraic) K -theory with reference of the higher (bilinear)algebraic K -theory (of Quillen) reexamined afterwards according to the Langlands globalprogram. 36 .22 Corollary Let hCH R × L : Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) −→ H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) be the bilinear semigroup homomorphism of Hurewicz from the bilinear “Galois” cohomo-topy Π i ( · ) into the entire bilinear homology H i ( · ) .It can be called restricted Π -homology with reference to the K -homology.Let C i ( G (2 n ) ( F v × F v )) : K i ( G (2 n ) ( F v × F v )) −→ H i ( G (2 n ) ( F v × F v ) , G (2 i ) ( F v × F v )) be the restricted Chern character in the bilinear K -homology.Then, the lower bilinear (algebraic) K -theory, referring to the cohomotopy ,will be given by the equality (resp. homomorphism): K i ( G (2 n ) ( F v × F v )) = ( → ) Π i ( G (2 n ) ( F v × F v ) , L v j ) × L v j ) ) in such a way that the cohomotopy classes of maps Π i ( · ) are (resp. correspond to) inverseliftings of inverse quantum deformations of the Galois (compact) representation GL i ( e F v × e F v ) . Proof : The proof of proposition 2.21, transposed to the lower bilinear algebraic K -theoryreferring to the cohomotopy, is evident here if section 2.16 is taken into account as well asthe commutative diagram: K i ( G (2 n ) ( F v × F v )) Π i ( G (2 n ) ( F v × F v ) , L v ( j ) × L v j ) ) H i ( G (2 n ) ( F v × F v ) , G (2 i ) ( F v × F v )) inverse restricted K -homology restricted Π-homology Higher bilinear algebraic K -theories related to thereducible bilinear global program of Langlands It was noticed in section 2.1 that the main tool of the Langlands global program is the (func-tional) representation space (F)REPSP(GL n ( F v × F v )) (resp. (F)REPSP(GL n ( F ω × F ω )) ), of the real (resp. complex) (algebraic) bilinear semigroup GL n ( F v × F v ) (resp.GL n ( F ω × F ω ) ), that is to say a real (resp. complex) abstract bisemivariety G (2 n ) ( F v × F v )(resp. G (2 n ) ( F ω × F ω ) ).This led us to define a “lower” bilinear (algebraic) K -theory on the basis of this abstractbisemivariety G (2 n ) ( F v × F v ) (resp. G (2 n ) ( F ω × F ω ) ).In order to introduce a “higher” bilinear algebraic K -theory referring to the Langlandsglobal program, we have also to take into account the unitary (functional) representationspace of the bilinear (algebraic) semigroup GL n ( F v × F v ) . Let P (2 n ) ( F v × F v ) (resp. P (2 n ) ( F ω × F ω ) ) be the real (resp. complex) parabolicbilinear semigroup viewed as the smallest bilinear normal pseudoramified subgroup of G (2 n ) ( F v × F v ) (resp. G (2 n ) ( F ω × F ω ) ) according to section 1.3 (resp. 1.4),where F v = { F v , . . . , F v j , . . . , F v r } is the set of classes of unitary pseudoramified realarchimedean completions.Referring to [Pie2], the (bisemi)group Int( G (2 n ) ( e F v × e F v )) of Galois inner automorphismsof G (2 n ) ( e F v × e F v ) corresponds to the (bisemi)group Aut( P (2 n ) ( e F v × e F v )) of Galois au-tomorphisms of the bilinear parabolic subsemigroup P (2 n ) ( e F v × e F v ) .It then results that P (2 n ) ( e F v × e F v ) can be considered as the unitary represen-tation space of the algebraic bilinear semigroup GL n ( F v × F v ) because it is theisotropy subgroup of G (2 n ) ( F v × F v ) fixing its bielements.On the other hand, as G (2 n ) ( e F v × e F v ) is a smooth reductive bilinear affine semigroup, wehave that P (2 n ) ( e F v × e F v ) ≈ ( e F v ) n × ( D ) ( e F v ) n . .3 Lemma The unitary (functional) representation space
U(F)REPSP(GL n ( F v × F v )) of the (alge-braic) bilinear semigroup GL n ( F v × F v ) is given by U(F)REPSP(GL n ( F v × F v )) = (F)REPSP( P n ( F v × F v ))= ( F v ) n × ( D ) ( F v ) n . Sketch of proof : As G (2 n ) ( e F v × e F v ) is a reductive bilinear semigroup, P (2 n ) ( e F v × e F v ) , being its isotropy subgroup, is (isomorphic to) the product, right by left, of unitaryalgebraic semitori e F nv and e F nv . K -theories Two types of equivalent “higher” bilinear algebraic K -theories on the basis of the globalprogram of Langlands will now be introduced.1) The first “classical” depends on the geometric dimensions of the classify-ing bisemispace BGL( F v × F v ) of GL( F v × F v ) whereGL( F v × F v ) = lim −→ GL m ( F v × F v )in such a way that GL m ( F v × F v ) embeds in GL m +1 ( F v × F v ) and GL m ( F v × F v ) ≃ ( F v ) m × ( D ) ( F v ) m .2) The second, called “quantum”, refers at first sight to the algebraic dimen-sions “ j , i.e. Galois extension degrees corresponding to global residue degrees (seesection 1.1), of the classifying bisemispace BGL ( Q ) ( F iv × F iv ) of GL ( Q ) ( F iv × F iv )where GL ( Q ) ( F iv × F iv ) = lim j =1 → r GL ( Q ) j ( F iv × F iv ) , ∀ i , ≤ i ≤ n , in such a way that:a) GL ( Q )1 ( F iv × F iv ) = P i ( F v × F v ) is the unitary, i.e. parabolic, bilinear semi-group of GL i ( F v × F v ) ;b) GL ( Q ) j ( F iv × F iv ) = GL i ( F v j × F v j ≃ ( F iv j × F iv j ) where the integer “ j ” denotesa global residue degree and the integer “ 2 i ” denotes a geometric dimension.c) GL ( Q ) j ( F iv × F iv ) ⊂ GL ( Q ) j +1 ( F iv × F iv ) ;d) GL ( Q ) j ( F iv × F iv ) ⊂ GL ( Q ) j ( F i +1 v × F i +1 v ) : geometric inclusion.39 .5 Lemma The set of “quantum” infinite general bilinear semigroups n GL ( Q ) ( F iv × F iv ) o i = (cid:26) lim j =1 → r ≤∞ GL ( Q ) j ( F iv × F iv ) (cid:27) i corresponds to the “classical” infinite general bilinear semigroups GL( F v × F v ) = lim −→ GL m ( F v × F v ) where F v = { F v , . . . , F v j , . . . , F v r } (resp. F v = { F v , . . . , F v j , . . . , F v r } ) is the set of r classes of archimedean pseudoramified real completions and F v = { F v , . . . , F v j , . . . , F v r } is the corresponding set of unitary completions. Proof : The “quantum” infinite bilinear semigroup GL ( Q ) ( F iv × F iv ) generates the set: G (1) , ( Q ) ( F iv × F iv ) ⊂ · · · ⊂ G ( j ) , ( Q ) ( F iv × F iv ) ⊂ · · · ⊂ G ( r ) , ( Q ) ( F iv × F iv ) , ≤ i ≤ n , of embedded (abstract) bisemispaces which are respectively (isomorphic to) the classes ofproducts, right by left, of embedded algebraic semitori (increasing algebraic filtration): F iv × ( D ) F iv ⊂ · · · ⊂ F iv j × ( D ) F iv j ⊂ · · · ⊂ F iv r × ( D ) F iv r since F iv j = j × F iv .As the geometric dimension, given by the integer “ i ”, varies, we have n = n + · · · + i + n s such increasing filtrations with n s → ∞ .On the other hand, the “classical” infinite general bilinear semigroup GL( F v × F v ) gener-ates the set: G (1) ( F v × F v ) ⊂ · · · ⊂ G ( m ) ( F v × F v ) ⊂ · · · ⊂ G (2 n s ) ( F v × F v )of embedded (abstract) bisemispaces which are 1 . . . m . . . n s -dimensional products, rightby left, of symmetric towers of increasing algebraic semitori. G (2 i ) ( F v × F v ) is then the 2 i -th algebraic filtration: F iv × ( D ) F iv ⊂ · · · ⊂ F iv j × ( D ) F iv j ⊂ · · · ⊂ F iv r × ( D ) F iv r , i.e. GL ( Q ) ( F iv × F iv ) which corresponds to the (functional) representation space(F)REPSP(GL i ( F v × F v )) of the bilinear semigroup GL i ( F v × F v ) .So, the quantum infinite general bilinear semigroupGL ( Q ) ( F iv × F iv ) = lim j =1 → r →∞ (GL ( Q ) j ( F iv × F iv ))40s GL i ( F v × F v ) .And, the set of “quantum” infinite bilinear semigroups { GL ( Q ) ( F v × F v ) , . . . , GL ( Q ) ( F mv × F mv ) , . . . , GL ( Q ) ( F nv × F nv ) } corresponds to the “classical” infinite bilinear semigroupGL( F v × F v ) = lim −→ GL m ( F v × F v ) . The classical (and quantum) infinite bilinear semigroupGL( F v × F v ) = lim −→ i GL i ( F v × F v )corresponds to the (partially) reducible (functional) representation space RED (F)REPSP(GL n =2+ ··· +2 i + ··· +2 n s ( F v × F v )) of the bilinear semigroup GL n ( F v × F v ) with n → ∞ . Proof : The infinite bilinear semigroup GL( F v × F v ) is the disjoint union of the GL i ( F v × F v ) modulo an equivalence relation together with morphisms mgℓ i : GL i ( F v × F v ) −→ GL n ( F v × F v )of GL i ( F v × F v ) into GL n ( F v × F v ) .So, we have that:GL( F v × F v ) = GL ( F v × F v ) ∪ · · · ∪ GL i ( F v × F v ) ∪ · · · ∪ GL n s ( F v × F v )= lim −→ GL i ( F v × F v )with GL i +2 ( F v × F v ) ⊂ GL i +2 ( F v × F v ) .And, thus, GL( F v × F v ) generates to the (partially) reducible (functional) representationspace RED (F)REPSP(GL n ( F v × F v )) which decomposes according to the partition 2 n =2 + · · · + 2 i + · · · + 2 n s :RED (F)REPSP(GL n ( F v × F v )) = (F)REPSP(GL ( F v × F v )) ⊞ · · · ⊞ (F)REPSP(GL i ( F v × F v )) ⊞ · · · ⊞ (F)REPSP(GL n s ( F v × F v ))as introduced in [Pie2]. Summarizing, we have :GL( F v × F v ) = lim −→ GL i ( F v × F v ) ≃ RED (F)REPSP(GL n ( F v × F v ))= { GL ( Q ) ( F iv × F iv ) } i = lim −→ GL ( Q ) ( F iv × F iv ) , ∀ i ∈ partition 2 n = 2 + · · · + 2 i + · · · + 2 n s , n → ∞ .41 .7 The classifying bisemispace BGL( F v × F v ) The classifying bisemispace BGL( F v × F v ) of GL( F v × F v ) is the quotient of a weaklycontractible bisemispace EGL( F v × F v ) by a free action of GL( F v × F v ) ; that is to say,generalizing the homotopy linear definition of a classifying space, the contractible bisemis-pace EGL( F v × F v ) is the total bisemispace of a universal principal GL( F v × F v ) -bibundleover the classifying bisemispace BGL( F v × F v ) given by the continuous mappingGD ℓ : EGL( F v × F v ) −→ BGL( F v × F v ) . This approach is more basic than the condition implying classically that the higher homo-topy groups are trivial (or vanish).
The (continuous) mapping
GD : EGL( F v × F v ) −→ BGL( F v × F v ) of the principal GL( F v × F v ) -bibundle over the classifying bisemispace BGL( F v × F v ) isa homotopy map corresponding to the deformations of the Galois compact representationof BGL( e F v × e F v ) given by the (bi)fibres of GD ℓ , ∀ ℓ , ≤ ℓ ≤ ∞ . Proof : Referring to proposition 2.15 introducing homotopy maps as deformations ofGalois representations of linear semigroups, we see that the map GD ℓ of the principalGL( F v × F v ) -bibundle corresponds to a deformation of the Galois compact representationof GL( e F v × e F v ) : GD ℓ : GL( F v + ℓ × F v + ℓ ) −→ GL( F v × F v )in such a way that the kernel GL( δF v + ℓ × δF v + ℓ ) of GD ℓ is responsible for the increaseof sets of powers of “ ℓ ” biquanta to GL( F v × F v ) .GD ℓ then belongs to an equivalence class of homotopy maps, given by deformations ofthe Galois compact representation of GL( e F v × e F v ) .And, the set { GD ℓ } ℓ of all equivalence classes of homotopy maps is the continuous mappingGD : EGL( F v × F v ) −→ BGL( F v × F v ) . .9 Corollary The classifying bisemispace BGL( F v × F v ) is the base bisemispace of all equiva-lence classes of deformations of the Galois representation of GL( e F v × e F v ) givenby the kernels GL( δF v + ℓ × δF v + ℓ ) of the maps GD ℓ : GL( F v + ℓ × F v + ℓ ) −→ GL( F v × F v ) , ≤ ℓ ≤ ∞ . The “plus” construction, adapted to the bilinear case of the Langlands global program,leads to consider a mapBG(1) : BGL( F v × F v ) −→ BGL( F v × F v ) + , unique up to homotopy, such that:1) the kernel of Π (BG(1)) be one-dimensional deformations of the Galois compactrepresentation of GL( e F v × e F v ) ;2) the homotopy fibre of BG(1) has the same integral homology as a point (orBGL( F v × F v ) and BGL( F v × F v ) + have the same integral homology). Let { GL (1) ( δF v + ℓ × δF v + ℓ ) } ℓ denote the set of kernels of the maps: GD(1) ℓ : GL (1) ( F v + ℓ × F v + ℓ ) −→ GL (1) ( F v × F v ) , ≤ ℓ ≤ ∞ , where GL (1) ( F v × F v ) denote the set of one-dimensional irreducible components of thebisemispace GL( F v × F v ) .Then, the classifying bisemispace BGL( F v × F v ) + is the base bisemispace of all equivalenceclasses of one-dimensional deformations of the Galois compact representation of GL( e F v × e F v ) given by the kernels { GL (1) ( δF v + ℓ × δF v + ℓ ) } ℓ of the maps GD(1) ℓ . Proof : The classifying bisemispace BGL( F v × F v ) is the base bisemispace of the principalGL( F v × F v ) -bibundle whose map is:GD : EGL( F v × F v ) −→ BGL( F v × F v ) . F v × F v ) + must be the base bisemispaceof the principal GL (1) ( F v × F v ) -bibundle whose map is:GD(1) : EGL( F v × F v ) + −→ BGL( F v × F v ) + , where EGL( F v × F v ) + is the total bisemispace verifying the equivalent conditions:a) EGL( F v × F v ) + = Π (BGL( F v × F v ) + ) ;b) EGL( F v × F v ) + corresponds to all equivalence classes of one-dimensional deforma-tions GL (1) ( F v + ℓ × F v + ℓ ) of the Galois compact representations of GL( e F v × e F v ) :EGL( F v × F v ) + = { GL (1) ( F v + ℓ × F v + ℓ ) } ℓ and Π (BGL( F v × F v ) + ) = GL( F v × F v ) (cid:14) GL (1) ( F v × F v ) . The “ + ” construction leads to the following commutative diagram : EGL( F v × F v ) EGL( F v × F v ) + BGL( F v × F v ) BGL( F v × F v ) +EG(1)BG(1)GD GD(1) where EG(1) is the map:
EG(1) : { GL( F v + ℓ × F v + ℓ ) } ℓ −→ { GL (1) ( F v + ℓ × F v + ℓ ) } ℓ , from deformations { GL( F v + ℓ × F v + ℓ ) } ℓ of the Galois compact representations GL( e F v × e F v ) to one-dimensional deformations { GL ( F v + ℓ × F v + ℓ ) } ℓ of the Galois compact representa-tions of GL( e F v × e F v ) . The bilinear version of the algebraic K -theory of Quillen adapted to the Lang-lands global program is : K (2 i ) ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + ) , where G (2 n )red ( F v × F v ) = RED (F)REPSP(GL n ( F v × F v )) is the (partially) reducible (func-tional) representation space of the bilinear semigroup of matrices GL n ( F v × F v ) , in sucha way that: ) the partition n = 2 + · · · + 2 i + · · · + 2 n s of the geometric dimension n , n ≤ ∞ ,refers to the reducibility of GL n ( F v × F v ) ;b) the dimension i of the bisemigroup of homotopy Π i (BGL( F v × F v ) + ) must beinferior or equal to each term of the partition of n in order that this homotopybisemigroup be non trivial. Proof : Referring to proposition 3.6, the infinite bisemigroup GL( F v × F v ) isGL( F v × F v ) = lim −→ GL i ( F v × F v ) ≃ RED (F)REPSP(GL n ( F v × F v )) , i.e. the decomposition of the partially reducible bisemivariety G (2 n )red ( F v × F v ) into G (2 n )red ( F v × F v ) = G (2) ( F v × F v ) ⊕ · · · ⊕ G (2 i ) ( F v × F v ) ⊕ · · · ⊕ G (2 n s ) ( F v × F v ) . It is then evident that the homotopy bisemigroup Π i (BGL( F v × F v ) + ) is null for thebisemivarieties G (2 h )red ( F v × F v ) whose geometric dimension h < i . The bilinear version of the algebraic K -theory K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + ) , relative to the Langlands global program and corresponding to a higher version of thisglobal program, is in one-to-one correspondence with the “quantum” bilinear version of thealgebraic K -theory: K i ( G (2 n )red ( F v × F v )) = Π i (BGL ( Q ) ( F iv × F iv ) + ) . Proof : Indeed, according to lemma 3.5 and proposition 3.6, we have that the “classical”infinite bisemigroup GL( F v × F v ) = lim −→ i GL i ( F v × F v )is equal to its “quantum” version given by:GL( F v × F v ) = lim −→ i GL ( Q ) ( F iv × F iv )45or every i belonging to the partition of 2 n associated with the dimensions of the re-ducibility of the bisemivariety G (2 n )red ( F v × F v ) = RED (F)REPSP(GL n ( F v × F v )) . The “quantum” version works explicitly with the algebraic dimensions “ j ” by the map-ping: GL ( Q ) j −−−→ : F iv × F iv −→ lim −→ j GL ( Q ) j ( F iv × F iv ) = GL ( Q ) ( F iv × F iv )= GL i ( F v × F v )while the classical version is based on the fibre bundle:GL i : F v × F v −→ GL i ( F v × F v ) , with “geometric” bifibre GL i − ( F v × F v ) . The higher version of the Langlands global program K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + )implies that the equivalence classes of 2 i -dimensional deformations of the Ga-lois compact representations of the reducible bilinear semigroup GL n ( F v × F v ) result from quantum homomorphisms of the global coefficient bisemiring F v × F v . Proof : Referring to proposition 3.11, the “plus” classifying bisemispace BGL( F v × F v ) + is the base bisemispace of the principal GL (1) ( F v × F v ) -bibundle in such a way that thetotal bisemispace EGL( F v × F v ) + verifies:EGL( F v × F v ) = Π (BGL( F v × F v ) + )and corresponds to quantum homomorphisms of the global coefficient bisemiring F v × F v : Qh F v + ℓ × F v + ℓ → F v × F v : F v + ℓ × F v + ℓ −→ F v × F v according to section 2.7.Then, the 2 i -th homotopy bisemigroup Π i (BGL( F v × F v ) + ) , describing the equivalenceclasses of 2 i -dimensional deformations of the Galois representations of GL n ( F v × F v ) ,implies the monomorphism:Π (BGL( F v × F v ) + ) −→ Π i (BGL( F v × F v ) + ) . .16 Restricted Chern character The Chern character restricted to the class C i in the higher bilinear K -cohomology isgiven by the homomorphism: C i ( G (2 n )red ( F v × F v )) : K i ( G (2 n )red ( F v × F v )) −→ H i ( G (2 n )red ( F v × F v ) , G (2 i ) ( F v × F v ))where G (2 n )red ( F v × F v ) is the reducible representation of GL n ( F v × F v ) , i.e. a compactbisemivariety decomposing into: G (2 n )red ( F v × F v ) = G (2) ( F v × F v ) ⊕ · · · ⊕ G (2 i ) ( F v × F v ) ⊕ · · · ⊕ G (2 n s ) ( F v × F v ) . The higher bilinear K -cohomology restricted to the class “ i ” implies the “higher” bilinearsemigroup homomorphisms of Hurewicz, i.e. a higher restricted Π -cohomology : hhH R × L : Π i (BGL( F v × F v ) + ) −→ H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) from the “Galois” higher bilinear homotopy Π i ( · ) into the entire “higher” bilinear coho-mology H i ( · ) .This leads to the commutative diagram: K i ( G (2 n )red ( F v × F v )) Π i (BGL( F v × F v ) + ) H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) higher algebraic K -theory −− “Chern” higherrestricted character inverse restrictedhigher Π -cohomology in such a way that the classes of the entire bilinear cohomology H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) refer to a bisemilattice deformed by the homotopy classes of maps of Π i (BGL( F v × F v ) + ) , corresponding to lift of quantum deformations of the Galois representations of GL (red)2 n ( e F v × e F v ) . Proof : The higher algebraic K -theory K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + ) , relative to the Langlands global program,together with the restricted “higher” K -cohomology C i ( G (2 n )red ( F v × F v )) : K i ( G (2 n )red ( F v × F v )) −→ H i ( G (2 n )red ( F v × F v ) , G (2 i ) ( F v × F v ))implies the restricted higher Π -cohomology:Π i (BGL( F v × F v ) + ) −→ H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) . .18 Proposition The total Chern character in the bilinear K -cohomology of the reducible rep-resentation G (2 n )red ( F v × F v ) of GL n ( F v × F v ) : ch ∗ ( G (2 n )red ( F v × F v )) : K ∗ ( G (2 n )red ( F v × F v )) −→ H ∗ ( G (2 n )red ( F v × F v ) , G ∗ ( F v × F v )) , where ∗ is the partition n = 2 + · · · + 2 i + ˙+2 n s of n ,implies the total higher algebraic K -theory: K ∗ ( G (2 n )red ( F v × F v )) = Π ∗ (BGL( F v × F v ) + ) . Proof : This results from the preceding sections.
The total higher algebraic K -theory associated with the reducible global program of Lang-lands is based on the commutative diagram: K ∗ ( G (2 n )red ( F v × F v )) Π ∗ (BGL( F v × F v ) + ) H ∗ ( G (2 n )red ( F v × F v ) , G ∗ ( F v × F v )) total higher algebraic K -theory −− “Chern” totalhigher character ch ∗ higher inverse Π -cohomology K -theory referring to cohomology As a lower bilinear (algebraic) K -theory referring to cohomotopy was introduced in corol-lary 2.22, a higher bilinear algebraic K -theory relative to cohomotopy can beintroduced by the equality : K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + )where Π i (BGL( F v × F v ) + ) are the cohomotopy equivalence classes of 2 i -dimensionaldeformations of the Galois reducible representations of GL n ( e F v × e F v ) .48 .21 Proposition The higher bilinear algebraic K -theory relative to cohomotopy implies thecommutative diagram : K i ( G (2 n )red ( F v × F v )) Π i (BGL( F v × F v ) + ) H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) higher algebraic K -theory −− referring tocohomotopy“Chern” higherrestricted characterrelative to homology higher inverserestricted Π -homology where:a) C i ( G (2 n )red ( F v × F v )) : K i ( G (2 n )red ( F v × F v )) → H i ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) is theChern higher restricted character relative to the K -homology where H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) is the entire bilinear homology of the reducible bisemivariety G (2 n )red ( F v × F v ) in the real bisemilattice deformed by the cohomotopy classes of maps of Π i (BGL( F v × F v ) + ) , corresponding to lifts of inverse quantum deformations of theGalois representations of GL (red)2 n ( e F v × e F v ) ;b) hhCH (2 i ) R × L : Π i (BGL( F v × F v ) + ) → H i ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) is the Hurewiczhigher homomorphism relative to cohomotopy. Sketch of proof : The higher bilinear algebraic K -theory referring to cohomotopygiven by the equality K i ( G (2 n )red ( F v × F v )) = Π i (BGL( F v × F v ) + )together with the Chern higher restricted character relative to homology implies theHurewicz higher homomorphism hhCH R × L . The total higher bilinear algebraic K -theory relative to cohomotopy is givenby the equality : K ∗ ( G (2 n )red ( F v × F v )) = Π ∗ (BGL( F v × F v ) + )49 nd implies the commutative diagram: K ∗ ( G (2 n )red ( F v × F v )) Π ∗ (BGL( F v × F v ) + ) H ∗ ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ )) total higheralgebraic K -theory −− relative tocohomotopy“Chern” total highercharacter relative to homology ch ∗ higher inverse Π -homology where:a) Ch ∗ ( G (2 n )red ( F v × F v )) : K ∗ ( G (2 n )red ( F v × F v )) → H ∗ ( G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) is theChern character of the reducible bisemivariety G (2 n )red ( F v × F v ) in the higher bilinear K -homology;b) hhCH ( ∗ ) R × L : H ∗ ( G (2 n )red ( F v × F v ) , ZZ × ( D ) ZZ ) → Π ∗ (BGL( F v × F v ) + ) is the correspondingHurewicz total higher inverse homomorphism relative to cohomotopy. Sketch of proof : The framework of the total higher bilinear algebraic K -theoryrelative to cohomotopy is similar to that of homotopy handled in proposition 3.18 andcorollary 3.20. 50 Mixed higher bilinear algebraic
K K -theories re-lated to the Langlands dynamical bilinear globalprogram
The lower and higher versions of the Langlands dynamical global program refer respectivelyto dynamical lower and higher bilinear (algebraic) K -theories related to the existenceof K ∗ K ∗ functors on the categories of elliptic bioperators and (reducible)bisemisheaves F G (2 n )red ( F v × F v ) , being (reducible) functional representationspaces of the (algebraic) general bilinear semigroups GL n ( F v × F v ) . Let then
F G (2 i ) ( F v × F v ) = FREPSP(GL i ( F v × F v )) denote the functional representationspace of GL i ( F v × F v ) , i ≤ n ≤ ∞ , which splits into:FREPSP(GL i ( F v × F v ))= FREPSP(GL k ( F v × F v )) ⊕ FREPSP(GL i − k ( F v × F v )) , k ≤ i , in such a way that FREPSP(GL k ( F v × F v )) is the functional representation space ofgeometric dimension 2 k of the bilinear semigroup GL k ( F v × F v ) on which acts the ellipticbioperator D kR ⊗ D kL .Let then D kR ⊗ D kL be the product of a right linear differential (elliptic) operator D kR acting on 2 k variables by its left equivalent D kL [Sat], [Kash].This bioperator D kR ⊗ D kL is defined by its biaction: D kR ⊗ D kL : F G (2 i ) ( F v × F v ) −→ F G (2 i [2 k ]) ( F v × F v )where F G (2 i [2 k ]) ( F v × F v ) is the functional representation space of GL i ( F v × F v ) shifted in(2 k × ( D ) k ) dimensions, i.e. bisections of a 2 i -dimensional bisemisheaf shifted in (2 k × ( D ) k ) dimensions, of differentiable bifunctions on the abstract bisemivariety G i [2 k ] ( F v × F v ) .Referring to chapter 3 of [Pie6], the shifted bisemisheaf F G (2 i [2 k ]) ( F v × F v ) decomposesinto: F G (2 i [2 k ]) ( F v × F v ) = (∆ kR × ∆ kL ) ⊕ F G (2 i − k ) ( F v × F v )where:a) ∆ kR × ∆ kL ≃ Ad (F)REPSP(GL k ( IR × IR )) × (F)REPSP(GL k ( F v × F v )) ≃ (F)REPSP(GL k ( F v × IR )( F v × IR ))51ith Ad (F)REPSP(GL k ( IR × IR )) being the adjoint functional representation spaceof GL k ( IR × IR ) corresponding to the biaction of the bioperator ( D kR ⊗ D kL ) onthe bisemisheaf F G (2 k ) ( F v × F v ) which is the functional representation space ofGL (2 k ) ( F v × F v ) ;b) F G (2 i − k ) ( F v × F v ) = FREPSP(GL i − k ( F v × F v )) is the (2 i − k ) (geometric)dimensional bisemisheaf being the functional representation space of GL i − k ( F v × F v ) .In fact, (∆ kR × ∆ kL ) is the total bisemispace of the tangent (bi)bundleTAN( F G (2 k ) ( F v × F v )) to the bisemisheaf F G (2 k ) ( F v × F v ) of which bilinear fibre F kR × L (TAN) = (AdF) REPSP(GL k ( IR × IR ))is isomorphic to the adjoint functional representation space of GL k ( IR × IR ) .And, Aut(TAN e ( F G (2 k ) ( F v × F v ))) is an open subset of the bilinear vector semispace ofendomorphisms of TAN e ( F G (2 k ) ( F v × F v )) at the identity element “ e ” in order to definedifferentials on it.If the bisemisheaf F G (2 k ) ( F v × F v ) = FREPSP(GL k ( F v × F v )) is the functional representa-tion space over the abstract bisemivariety G (2 k ) ( F v × F v ) , then ∆ kR × ∆ kL is in one-to-onecorrespondence with a Galois (bisemi)group(oid) of which isomorphism is generated by thebilinear fibre (AdF) REPSP(GL k ( IR × IR )) .Indeed, the Galois (bisemi)group(oid) associated with the shifted bisemisheaf (∆ kR × ∆ kL )refers essentially to the bilinear semigroup GL(∆ kR × ∆ kL ) of “Galois” automorphismsof (∆ kR × ∆ kL ) , i.e. to the bilinear semigroup of shifted “Galois” automorphisms of thebase bisemisheaf G (2 k ) ( F v × F v ) = (F)REPSP(GL k ( F v × F v )) endowed with a nontrivialfundamental bisemigroup Π ( G (2 k ) ( F v × F v )) . The existence of a bilinear contracting fibre F kR × L (TAN) in the tangent bi-bundle TAN( F G (2 k ) ( F v × F v )) implies the homology : H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) ≃ Ad (F)REPSP(GL k ( IR × IR )) in such a way that the cohomoloby of the shifted bisemisheaf F G (2 i [2 k ]) ( F v × F v ) be givenby: H k ( F G (2 i [2 k ]) ( F v × F v ) , ∆ kR × L )= H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) × H k ( F G (2 i [2 k ]) ( F v × F v ) , F G (2 k ) ( F v × F v ))= FREPSP(GL k ( F v × IR ) × ( F v × IR )) 52 here ∆ kR × L = ∆ kR × ∆ kL . Proof : The cohomology H k ( F G (2 i [2 k ]) ( F v × F v ) , ∆ kR × L ) of the bisemisheaf F G (2 i [2 k ]) ( F v × F v ) shifted under the action of the bioperator D kR ⊗ D kL must be expressed by meansof the homology H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) with value in the bilinear fibre F kR × L (TAN) as developed in chapter 2 of [Pie6].As H k ( F G (2 i [2 k ]) ( F v × F v ) , F G (2 k ) ( F v × F v )) = FREPSP(GL k ( F v × F v ))and as H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) = FREPSP(GL k ( IR × IR )) ≃ Ad FREPSP(GL k ( IR × IR )) , we get the thesis. The bilinear cohomolgy of the shifted bisemisheaf (also called bilinear mixed cohomology)
F G (2 n [2 k ]) ( F v × F v ) is given by the functional representation space of the bilinear generalsemigroup GL i ( F v × F v ) shifted in k real geometric dimensions according to: H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i [2 k ]) ( F v × F v )) = FREPSP(GL i [2 k ] ( F v ⊗ IR ) × ( F v ⊗ IR )) where FREPSP(GL i [2 k ] ( F v ⊗ IR ) × ( F v ⊗ IR )) is a condensed notation for FREPSP(GL k ( IR × IR )) × FREPSP(GL i ( F v × F v )) . Proof : Referring to section 4.1 giving the decomposition of the shifted bisemisheaf
F G (2 i [2 k ]) ( F v × F v ) into: F G (2 i [2 k ]) ( F v × F v ) = (∆ kR × ∆ kL ) ⊕ F G (2 i − k ) ( F v × F v ) , we see that the cohomology H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i [2 k ]) ( F v × F v )) must similarlydecompose into: H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i [2 k ]) ( F v × F v ))= H k ( F G (2 n [2 k ]) ( F v × F v ) , F kR × L (TAN)) × (cid:2) H k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 k ) ( F v × F v )) (cid:3) ⊕ (cid:2) H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i − k ) ( F v × F v )) (cid:3) . H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) × H k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 k ) ( F v × F v ))= FREPSP(GL k ( F v × IR ) × ( F v × IR ))and that H i − k ( F G (2 n [2 k ]) ( F v × F v ) , F G (2 i − k ) ( F v × F v )) = FREPSP(GL i − k ( F v × F v )) , we get the thesis. In order to introduce a mixed homotopy bisemigroup in relation to a mixed Hurewicz ho-momorphism to be defined, we have to precise what must be the cohomotopy bisemigroupcorresponding to the homology H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) with coefficients inthe bifibre F kR × L (TAN) .As H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) = (F)REPSP(GL k ( IR × IR )) and as a cohomo-topy bisemigroup refers, according to section 2.16, to classes resulting from inverse deforma-tions of Galois representations, under the circumstances of the shifted general bilinear semi-group GL i [2 k ] ( e F v × e F v ) , the searched cohomotopy bisemigroup Π k ( F G (2 i [2 k ]) ( F v × F v )) must be described by classes :a) resulting from inverse deformations( GD (2 k ) ) − : F G (2 k ) ( IR × IR ) −→ F G (2 k ) ( IR × IR )of the differential Galois representation [Car] of GL k ( IR × IR ) ;b) depending on the classes of deformations of the Galois representation of GL k ( e F v × e F v ) , i.e. the homotopy semigroup Π k ( F G (2 i [2 k ]) ( F v × F v )) .Consequently, the mixed homotopy bisemigroup Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) ofthe shifted bisemisheaf F G (2 n [2 k ]) ( F v × F v ) , will be defined by the product :Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) = Π k ( F G (2 i [2 k ]) ( F v × F v )) × Π i ( F G (2 n [2 k ]) ( F v × F v ))of the cohomotopy Π k ( F G (2 i [2 k ]) ( F v × F v )) resulting from the action of a differential biop-erator ( D kR ⊗ D kL ) on the bisemisheaf F G (2 n ) ( F v × F v ) by the homotopyΠ i ( F G (2 n [2 k ]) ( F v × F v )) of the bisemisheaf F G (2 n ) ( F v × F v ) shifted in 2 k geometric di-mensions. 54 .5 Proposition The mixed bilinear semigroup homomorphism of Hurewicz will be given by: mhH : Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) −→ H i − k ( F G (2 n [2 k ]) ( F v × F v ) , ZZ × ( D ) ZZ ) , i.e. a restricted Π -homology- Π -cohomology . Proof : Indeed, the classes of the entire mixed bilinear cohomology H i − k ( F G (2 n [2 k ]) ( F v × F v ) , ZZ × ( D ) ZZ ) are the classes of the mixed homotopy bisemigroup Π i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) and correspond to the bisemilattice Λ (2 i ) v ⊗ Λ (2 i ) v ⊂ G (2 i ) ( F v × F v ) deformed by themixed deformations of the considered Galois representation associated withΠ i [2 k ] ( F G (2 n [2 k ]) ( F v × F v )) . Let C [ k ] ( F G (2 i [2 k ]) ( F v × F v )) : K k ( F G (2 i [2 k ]) ( F v × F v )) −→ H k ( F G (2 i [2 k ]) ( F v × F v ) , F kR × L (TAN)) denote the restricted Chern character in the operator bilinear K -homology in such a waythat C [ k ] corresponds to the Todd class J ( F G (2 i [2 k ]) ( F v × F v )) ≡ J (TAN( F G (2 k ) ( F v × F v ))) according to chapter 3 of [Pie6]and let C i ( F G (2 n ) ( F v × F v )) : K i ( F G (2 n ) ( F v × F v )) −→ H i ( F G (2 n ) ( F v × F v ) , F G (2 i ) ( F v × F v )) be the Chern restricted character in the bilinear K -cohomology.Then, the Chern mixed restricted character in the K -homology- K -cohomologycorresponds to a bilinear version of the index theory [A-S] and is given by : C [ k ] · C i ( F G (2 n [2 k ]) ( F v × F v )) : K i − k ( F G (2 n ) ( F v × F v )) −→ H i − k ( F G (2 n ) ( F v × F v )) where the mixed topological (bilinear) K -theory K i − k ( F G (2 n ) ( F v × F v )) is given by: K i − k ( F G (2 n ) ( F v × F v )) = K k ( F G (2 i [2 k ]) ( F v × F v )) × K i ( F G (2 n ) ( F v × F v )) where K k ( F G (2 i [2 k ]) ( F v × F v )) is the topological bilinear contracting K -theory of con-tracting tangent-bibundles (with contracting bifibres). ketch of proof : The differential bioperator ( D kR ⊗ D kL ) defined by its biaction D kR ⊗ D kL : F G (2 i ) ( F v × F v ) −→ F G (2 i [2 k ]) ( F v × F v )leads to the Chern mixed restricted character C [ k ] · C i ( F G (2 n [2 k ]) ( F v × F v )) = J ( F G (2 i [2 k ]) ( F v × F v )) × C i ( F G (2 n ) ( F v × F v ))corresponding to a bilinear version of the index theorem according to chapter 3 of [Pie6],in such a way that the restricted Chern character C [ k ] ( F G (2 i [2 k ]) ( F v × F v )) in the operatorbilinear K -homology corresponds to the Todd class J ( F G (2 i [2 k ]) ( F v × F v )) = J (TAN( F G (2 k ) ( F v × F v )))= C [ k ] ( D kR ⊗ D kL ) . If the classes of mixed bilinear cohomology H i − k ( F G (2 n ) ( F v × F v ) , ZZ × ( D ) ZZ ) are classesdeformed by the mixed deformations of the Galois representation associated with Π i [2 k ] ( F G (2 n ) ( F v × F v )) , then we can define a mixed lower bilinear (algebraic) K -theory by the equality (resp. homomorphism): K i − k ( F G (2 n ) ( F v × F v )) = ( → ) Π i [2 k ] ( F G (2 n ) ( F v × F v )) implying that the mixed topological bilinear K -theory K i − k ( F G (2 n ) ( F v × F v )) is (resp.corresponds to) a bisemigroup of deformed vector bibundles. Proof : The thesis results from the commutative diagram: K i − k ( F G (2 n ) ( F v × F v )) Π i [2 k ] ( F G (2 n ) ( F v × F v )) H i − k ( F G (2 n ) ( F v × F v )) mixed lower bilinear K -theory −− inverse Chern mixedrestricted character mixed homomorphismof Hurewicz implying that K i − k ( F G (2 n ) ( F v × F v )) is a bisemigroup of vector bibundles deformed bymixed deformations of the Galois representations associated with Π i [2 k ] ( F G (2 n ) ( F v × F v )) .56 .8 Proposition The total Chern mixed character in the K -homology- K -cohomology ch ∗∗ ( F G (2 n [2 k ]) ( F v × F v )) : K ∗ K ∗ ( F G (2 n ) ( F v × F v )) −→ H ∗ H ∗ ( F G (2 n ) ( F v × F v )) where:a) ch ∗∗ ( F G (2 n [2 k ]) ( F v × F v )) = Σ k Σ i C [ k ] C i ( F G (2 n [2 k ]) ( F v × F v )) ,b) K ∗ K ∗ ( F G (2 n ) ( F v × F v )) = Σ k Σ i K i − k ( F G (2 n ) ( F v × F v )) ,c) H ∗ H ∗ ( F G (2 n ) ( F v × F v )) = Σ k Σ i H i − k ( F G (2 n ) ( F v × F v )) ,as well as the total mixed bilinear semigroup homomorphism of Hurewicz: mhH ∗ : Π ∗ [2 ∗ ] ( F G (2 n ) ( F v × F v )) −→ H ∗ H ∗ ( F G (2 n ) ( F v × F v )) where Π ∗ [2 ∗ ] ( F G (2 n ) ( F v × F v )) = Σ k Σ i Π i [2 k ] ( F G (2 n ) ( F v × F v )) implies the total mixed lowerbilinear (algebraic) K ∗ K ∗ -theory given by the equality: K ∗ K ∗ ( F G (2 n ) ( F v × F v )) = Π ∗ [2 ∗ ] ( F G (2 n ) ( F v × F v )) . Proof : This results evidently from the preceding sections.
Let f : F G (2 n ) Y ( F v × F v ) −→ F G (2 n ) X ( F v × F v ) be a morphism between two compact bisemisheaves and let f !! K ∗ K ∗ ( F G (2 n ) Y ( F v × F v )) −→ K ∗ K ∗ ( F G (2 n ) X ( F v × F v )) be the homomorphism between the corresponding total mixed K ∗ K ∗ -theories.Then, the bilinear version of the Riemann-Roch theorem [B-S], [A-H], [Gil1] assertsthat the diagram: K ∗ K ∗ ( F G (2 n ) Y ( F v × F v )) K ∗ K ∗ ( F G (2 n ) X ( F v × F v )) H ∗ H ∗ ( F G (2 n ) Y ( F v × F v )) H ∗ H ∗ ( F G (2 n ) X ( F v × F v )) f !! f ∗∗ ch ∗∗ Y ch ∗∗ X s commutative,or that: f ∗∗ ◦ ch ∗∗ Y = ch ∗∗ X ◦ f !! . Proof : • The linear classical version [B-S], [Gil1], [A-H] of the Riemann-Roch theorem is f ∗ ( ch ( y ) · T ( Y )) = ch ( f ! ( y )) · T ( X )for any proper morphism f : Y → X between nonsingular, irreducible quasiprojec-tive varieties where: – f ! : K ( Y ) → K ( X ) , y ∈ K ( Y ) , – f ∗ : H ∗ ( Y, IQ ) → H ∗ ( X, IQ ) , – T ( Y ) is the Todd class of the tangent bundle to Y . • The mixed bilinear version of the Riemann-Roch theorem f ∗∗ ◦ ch ∗∗ Y = ch ∗∗ X ◦ f !! then corresponds to the linear classical version if the total Todd class J ( F G (2 ∗ [2 k ]) ( F v × F v )) = Σ k Σ i c [ k ] ( F G (2 i [2 k ]) ( F v × F v )) is the total Chern character ch ∗ in the operatorbilinear K -homology: ch ∗ : K ∗ ( F G (2 n ) ( F v × F v )) −→ H ∗ ( F G (2 n ) ( F v × F v )) . Remark that this way of envisaging the Riemann-Roch theorem by operator K -homology- K -cohomology is much more natural than the classical one working onlyin the form of K -cohomology. In order to develop a bilinear version of the algebraic mixed KK -theory relative to thedynamical global program of Langlands, we have to introduce a higher bilinear algebraicoperator K -theory relative to cohomotopy.58n this respect, we have to introduce the classical (and quantum) infinite bilinear semi-group acting by the biactions of the differential bioperator { ( D k ) R ⊗ D kL ) } k on the in-finite bilinear semisheaf FGL( F v × F v ) over the infinite bilinear semigroup GL( F v × F v ) = lim −→ i GL i ( F v × F v ) corresponding to the reducible functional representation space(F)REPSP(GL n =2+ ··· +2 i + ··· +2 n s ( F v × F v )) of the reducible bilinear semigroup GL n ( F v × F v )with n → ∞ .Referring to the preceding sections, it appears that the searched operator infinitebilinear semisheaf must be FGL( IR × IR ) = lim −→ k FGL k ( IR × IR ) acting onFGL( F v × F v ) by the biaction FGL IR × IR : FGL( F v × F v ) −→ FGL( IR × IR ) × FGL( F v × F v ) = FGL(( F v ⊗ IR ) × ( F v ⊗ IR ))in such a way that:a) each factor of FGL( IR × IR ) acts on a factor of FGL( F v × F v ) ;b) the factor “ 2 k ” FGL k ( IR × IR ) ⊂ FGL( IR × IR ) acts on the factor “ 2 i ” FGL i ( F v × F v ) ⊂ FGL( F v × F v ) in such a way that the (adjoint) functional representation spaceof GL i [2 k ] (( F v ⊗ IR ) × ( F v ⊗ IR )) is given by the shifted bisemisheaf F G (2 i [2 k ]) ( F v × F v ) = FREPSP(GL i [2 k ] ( F v ⊗ IR ) × ( F v ⊗ IR )) with bilinear fibre FREPSP(GL k ( IR × IR )) in the sense of proposition 4.3. BFGL( IR × IR ) The classifying bisemisheaf BFGL( IR × IR ) of FGL( IR × IR ) is the quotient of a weaklycontractible bisemisheaf EFGL( IR × IR ) by a free action of FGL( IR × IR ) in such a waythat the continuous mapping:GD IR : EFGL( IR × IR ) −→ BFGL( IR × IR )of the principal FGL( IR × IR ) -bibundle over BFGL( IR × IR ) is a cohomotopy mapcorresponding to inverse Galois deformations of the Galois differential repre-sentation of BFGL( IR × IR ) .The classifying bisemisheaf BFGL( IR × IR ) is then the base bisemisheaf of all equivalenceclasses of inverse deformations of the Galois differential representation of FGL( IR × IR ) inthe one-to-one correspondence with the kernels FGL( δF v + ℓ × δF v + ℓ ) of the mapsGD ℓ : FGL( F v + ℓ × F v + ℓ ) −→ FGL( F v × F v ) , ≤ ℓ ≤ ∞ , F v ⊗ IR ) × ( F v ⊗ IR )) then results from the(bi)action of BFGL( IR × IR ) on BFGL( F v × F v ) . BFGL( IR × IR ) The “plus” construction of the mixed bilinear case of the Langlands dynamical globalprogram is based on the map:BFG IR (1) : BFGL( IR × IR ) −→ BFGL( IR × IR ) + , unique up to cohomotopy, in such a way that:1) the kernel of the fundamental cohomotopy bisemigroup Π (BFG IR (1)) be one-dimensional inverse deformations of the Galois differential representation ofFGL( IR × IR ) ;2) the cohomotopy fibre of BFG IR (1) has the same integral homology as a (bi)point.The classifying bisemisheaf BFGL( IR × IR ) + is the base bisemisheaf of all equivalenceclasses of one-dimensional inverse deformations of the Galois differential representation ofFGL( IR × IR ) in one-to-one correspondence with the one-dimensional deformations of theGalois representation of GL( e F v × e F v ) given by the kernel { GL (1) ( δF v + ℓ × δF v + ℓ ) } ℓ of themaps GD(1) ℓ according to proposition 3.11. The bilinear version of the mixed higher algebraic KK -theory of Quillenadapted to the Langlands dynamical bilinear global program is: K k ( F G (2 n )red ( IR × IR )) × K i ( F G (2 n )red ( F v × F v )) = Π k (BFGL( IR × IR ) + ) × Π i (BFGL( F v × F v ) + ) written in condensed form according to: K i − k ( F G (2 n )red (( F v × IR ) × ( F v × IR ))) = Π i [2 k ] (BFGL(( F v ⊗ IR ) × ( F v ⊗ IR )) + ) in such a way that the bilinear contracting K -theory K k ( F G (2 n )red ( IR × IR )) , responsiblefor a differential biaction, acts on the K -theory K i ( F G (2 n )red ( F v × F v )) of the reduciblefunctional representation space F G (2 n )red ( F v × F v ) of the bilinear semigroup GL n ( F v × F v ) inone-to-one correspondence with the biaction of the cohomotopy bisemigroup Π k (BFGL( IR × IR ) + ) of the “plus” classifying bisemisheaf BFGL( IR × IR ) + . roof : This mixed higher bilinear (algebraic) KK -theory is directly related to the com-mutative diagram: K i − k ( F G (2 n )red ( F v ⊗ IR ) × ( F v ⊗ IR )) Π i [2 k ] (BFGL(( F v ⊗ IR ) × ( F v ⊗ IR ) + )) H i − k ( F G (2 n )red ( F v ⊗ IR ) × ( F v ⊗ IR )) mixed higher −− bilinear K -theoryinverse Chern mixedhigher restricted character inverse restricted higherΠ-homology-Π-cohomology referring to the preceding sections. The bilinear version of the total mixed higher algebraic KK -theory of Quillenadapted to the Langlands dynamical reducible global program of Langlands is: K ∗ ( F G (2 n )red ( IR × IR )) × K ∗ ( F G (2 n )red ( F v × F v )) = Π ∗ (BFGL( IR × IR ) + ) × Π ∗ (BFGL( F v × F v ) + ) . Proof : This total mixed higher algebraic KK -theory has to be related to the commuta-tive diagram: K ∗− ∗ ( F G (2 n )red ( F v ⊗ IR ) × ( F v ⊗ IR )) Π x [2 x ] (BFGL(( F v ⊗ IR ) × ( F v ⊗ IR )) + ) H ∗− ∗ ( F G (2 n )red ( F v ⊗ IR ) × ( F v ⊗ IR ))) total mixed higher −− bilinear KK -theoryinverse Chern totalmixed higher character inverse higherΠ-homology-Π-cohomology eferences [A-H] M. Atiyah, F. Hirzebruch,
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