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On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical structures on a complex vector space of dimension n giving rise to isometry groups which are the Lie theoretic complexications of the classical groups While the pro cedure is quite clear for c c the orthogonal group On COn R and the symplectic group Spn CSpn R it turns out that the socalled exceptional numbers E cf Wildb erger are quite useful to describ e the complexications of GLn CGLn R and accordingly of Un C In fact one nds that c GLn C GLn C GLn C GLn E and c Un E Un C GLn C After clarifying this task of so to say elementary linear algebra over the exceptional numbers we come to the relative notions of these groups over a given complex manifold The corresp onding notions are called a euclidean symplectic or exceptional structure on a complex manifold of dimension n They intersect in the notion of a Kahler manifold The question of existence of such structures are briey discussed The main p ointhowever is the nal example Consider a semisimple complex Lie algebra L and let a L be a semisimple element Denote further by G IntLGLL the adjoint group of L It is shown that the adjoint orbit GaL carries the structure of a Kahler manifold Elementary theory of exceptional vector spaces Euclidean structures Consider a complex vector space V of dimension n A euclidean structure on V is a symmetric and nondegenerate bilinear form g V V C Prop osition If g is a euclidean structure on V there exists an orthonormal basis v v on V n ie g v v Pro of Since g we can cho ose a vector v V so that g v v By rescaling v observe that z z is surjectiveonCwemay assume that g v v The orthogonal complement W v fw V j g v wg is a euclidean subspace of V ie the restriction of g to W W remains nondegenerate In fact if w W and supp ose that W w then since v w one nds that V w implying w since g is nondegenerate on V Nowby induction on the dimension the result follows Remark Recall that the rescaling argument do es not apply in the real ie in the classical case In fact in this case a nondegenerate bilinear form has an additional invariant its index that is the dimension of a maximal subspace of V on which g is p ositive denite n Let V C and dene the standard euclidean structure by n X g z w z w n Then any euclidean vector space is equivalenttoC g In particular the complex orthogonal group t On CfA GLn C j A A g isup to isomorphythe isometry group of such a structure Exceptional structures Consider a complex vector space V of nite dimension and an endo morphism J V V satisfying J Since the minimal p olynomial of J divides X X iX i J is necessarily semisimple ie V splits into the ieigenspaces V EigJi EigJ i WecallJ an exceptional structure on V ifthemultiplicities of the i and ieigenspace of J are equal dim EigJi dim EigJ i Prop osition Let J be an exceptional structure on a nite dimensional complex vector space V w w of V such Then V is even dimensional say of dimension n and there exists a basis v v n n that the matrix of J with resp ect to this basis is n I n i n Pro of Let v v w w be a basis such that the corresp onding matrix of J is n n i n Then the basis v v w w v w satises the assertion Recall the basic construction of the exceptional numb ers see Wildb erger Here E C C together with its vectorspace structure ov er C and its multiplicative structure resulting from j where j E The null numb ers were dened as the set of noninvertible elements E ie ze with z C or ze with z C Here e ij e ij The Calgebra E is isomorphic to the direct pro duct of Calgebras of C with itself via C E e e since e e e e and e e asoneveries immediately Observe furthermore that multiplication by j E gives E its canonical exceptional structure since je ie je ie n Nowwe can identify our complex vector space of dimension nwiththe Emo dul E in the obvious way Set for z w C and v V z jw v zv wJ v n Then it follows immediately that V E Remark Recall that in the real case a complex structure on a real vector space is dened only byan endomorphism J V V with J satisfying no additional prop erties on its eigenvalue multiplicities c c c c c Denoting by V the complexication of V ie V V C and by J V V the complexication R c c of J ie its unique complex linear extension from V to V then the eigenvalue multiplicities of J are automatically equal In fact the complex conjugation conj z z on C gives rise to an Rlinear map c c c c bar conj V V which is an isomorphism Since bar EigJ i EigJ i the multiplicities of i and i coincide n n n Let us briey lo ok at the isometries of C concerning the standard exceptional structure J C C given by the matrix I n with resp ect to the canonical basis of C GLn EfA GLn C j AI IAg n Lo oking at A GLn EasanElinear endomorphism F of E and representing it with resp ect to the null i shows that the matrix of F has to commute with ie it has b o ckform basis e e ineach factor E i A with A A GLn C This shows that A GLn E GLn C GLn C Symplectic structures Consider a complex vector space V of nite dimension A symplectic structure on V is a nondegenerate and skewsymmetric bilinear form V V C Prop osition Let b e a symplectic structure on V Then V is necessarily even dimensionalsay v v w w and of dimension nand there exists a symplectic basis v w of V ie n v w Pro of Since there exist vectors v w V so that v w One can assume that v w Let H spanv w Now considering the symplectic complement H fx V j x v x w g we see that V H H is direct sum decomp osition of symplectic vector spaces In fact H is obviously a symplectic subspace ie jH H is nondegenerate implying H H and therefore V H H as vector spaces for dimension reasons and nally H is also symplectic Indeed if x H and H x then since H x we nd that V x implying x Nowby induction the prop osition follows n The standard symplectic structure on C is given by x y x y g x y g x y n where g is the standard euclidean structure on C The complex symmetric group is t Sp n CfA GLn C j A IA I g and the prop osition shows that the isometry group of a complex symplectic vector space of dimension n is isomorphic to Spn C The unitary group over the exceptional numb ers Nowwewant to compute the pairwise intersection of the complex groups On C GL n E and Spn C in GLn C To that end consider n E E the complex linear conjugation z jw z jw on E Observe the free Emo dul E Denote by n n thate e ande e The canonical hermitean form h i E E E is dened by n X h i Decomp osition in complex and ctitious part yields the equation h i g j n n where g and are the standard euclidean and symplectic structures on E C A hermitean form h n is by denition a sesquilinear form on V E ie a Cbilinear form satisfying hv whv w and hw v hv w for all E and v w V According to the standard Cbasis e e je je of n n n n C E the matrices of g J and are resp ectively On the other hand the matrices according to the null basis e e e e n n is i i i i n n Dene nowbyUn E the complexlinear transformations of C E resp ecting the standard hermitean n form on E ie t Un EfA GLn E j A A g In fact such a transformation is necessary Elinear ie the matrix A commutes with I Since t t t A On C wehave A A and since A Sp n C wend A IA I ie IA A I AI We havenow seen that Un ESp n C GLn E On C On the other hand the equations h i g j g J showthatSpn C GLn E On C GLn ESpn C On CUn E The three complex n n E geometries on C intersect in the exceptional unitary group U A B To identify Un E let us nally recall what it means that a matrix X in GLn C is C D t symplectic ie X IX X t t i A C C A t t ii B D D B t t iii A D C B n n Representing the linear transformation X E E with resp ect to the null basis yields
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