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 the same

i

result since the representing matrix of is I aswehave noted ab ove In this basis X has the description

A

t

since X GLn E Thus the three conditions come down to A A showing that

A

GLn C Un E

In general a Cbilinear map h V V E on a free Emo dul V of rank n is called a Kahlerian structure

if it is a nondegenerate hermitean form on V

Prop osition Let V b e a free Emo dul of rank n and let h be a Kahlerian structure on V Then

there exists an Ebasis v v so that h v v

n

Pro of Since h and the p olarization formula we nd a vector v V so that hv v Now

hv v hv v so hv v CE Thus wecan rescale v V so that hv v Nowthe hermitean

complement v fw j hw v g is again a free Emo dul In fact it is at least a Cvector space of

dimension n and it is J invariant where J is the exceptional structure on V But J jv is again an

exceptional structure since J v jvj v JvandJ v jvj v Jvandthus the multiplicities of i

and i of J jv coincide Furthermore the restriction of h on v is again a Kahlerian structure whichisnow

left to the reader So v is a free Emo dul of rank n withKahlerian structure hjv and the induction

hyp othesis applies

Calibrated exceptional structures Let V be a complex symplectic vector space A

calibrated exceptional structure is given by an exceptional structure J so that J SpV ie Jv Jw

v w for all v w V The corresp onding euclidean structure is dened by g V V C

g v w v J w

and the corresp onding Kahlerian structure h V V E is given by

hv wg v wjv w

The triple V Jh is then a Kahlerian vector space In fact it is straightforward to see that g is nondegen

erate and moreover g is also J invariant ie g Jv Jwg v w for all v w V

Recall that in the real case a complex structure on a real symplectic vector space V is called

calibrated if the corresp onding bilinear form g is in addition p ositive denite However there isnt suchan

assumption in the complex case

A fundamental example Consider now a nitedimensional complex Lie algebra L with

Lie bracket The innitesimal adjoint action of L is given by the Lie homomorphism ad L glL

adab a b For a given a L the orbit La fadab j b Lg is in one to one corresp ondence

with the vector space V LC a where C aker ada is the centralizer of a In fact C a is nothing

a



else than the isotropy algebra of the Laction in the p oint a Consider next the dual vector space L of L





together with its innitesimal coadjoint action of L ie ad L glL



had abi h adabi h a bi



for L and a b L The coadjoint orbit through is then identied with the vector space LI

where

I fa L jh a bi b Lg

since I isobviously the isotropy algebra of L in Therefore the skewsymmetric form L L C

a bh a bi

induces a symplectic structure on the vector space LI It is called the KirillovKostant structure

Next assume that the Lie algebra L is semisimple ie the symmetric bilinear form B L L C

B a btr adaadb

is nondegenerate ie it denes a euclidean structure on L B is the Killing form on L Thus it denes an



isomorphism L L by

habi B a b

h is in addition equivariant with resp ect to the innitesimal adjoint action of L on L and the innitesimal whic



coadjoint action of L on L



adab ad a b

This follows from the relation

B a bcB a b c

for the Killing form B coming from the Jacobi identity for In particular the isotropy algebras corre

sp onds and wehave a canonical symplectic structure on V LC a induced from L L C

a a

b cB a b c

a

We are now going to dene an exceptional structure on V calibrated with resp ect to ifa is a semisimple

a a

element ie ada L L is semisimple Accordingly this will give V the structure of a Kahlerian complex

a

vector space in the sense of section To this end decomp ose L into the eigenspaces of the semisimple

endomorphism ada Thusifwe denote by L Eigada and observe that L C a weobtain

X

V C a L



where C denotes the nonzero eigenvalues of ada A fundamental observation is the relation

L L L

for C since ada is a derivation ie b c b cb c coming from the Jacobi identity

of This implies that L is orthogonal to L if since for a L b L we have that

ada adbL L and therefore B a b trada adb Now using that B is nondegenerate wend

that implies and moreover dim L dimL since B induces a nondegenerate pairing on

L L C Thus the nonzero eigenvalues come in pairs and cho osing a decomp osition of into a

p ositive and negative part



ie if and only gives now the decomp osition

V C aN N

where

X X

N L N L

 

Dening now J jN i id and J jN i id induces an exceptional structure J J V V

a a a

N N

since dim N dim N Weidentify V with N N Finally this exceptional structure is calibrated

a

with resp ect to the KirillovKostant structure as is seen by the following argumen t If b c N then

a

b c C a by relation so b c B a b c ib ic JbJc The same argument

a a a

applies for the case b c N If b N and c N then JbJc ib ic b c Thus we

a a a

conclude that

JbJc b c

a a

for all b c N N showing that J is a calibrated exceptional structure on V The asso ciated euclidean

a

structure g V V C is therefore dened by

a a a

g b c b J c

a a

and the asso ciated Kahlerian structure h V V E by h g j as usual

a a a a a a

Since B L L C is a nondegenerate pairing we can nd a basis E E of L r

r

L and E E of L so that B E E for r and Let dim

r

H E E C a and dene the complex numbers

B a H

Denoting by diag we see that the matrix describing the KirillovKostant structure on V with

a a

resp ect to E E is

Of course the matrix describing the exceptional structure J corresp onding to this basis is

a

i

i

implying that the matrix of g is

a

i

i

Remark Note that the centralizer C aL is itself a euclidean subspace with resp ect to the Killing

structure on L In fact since C a L the Killing form B induces a nondegenerate pairing of C a

with itself In particular the quotient V LC a inherits a euclidean structure B using the natural

a a

identication of V with C a Now C a N N L However bythechoice of the basis E E

a

the matrix of the euclidean structure B is given by

a

This shows that the Killing form is in general not the complex part of the Kahlerian structure on V describ ed

a

ab ove In fact this is the case if and only if i ie i for all and r

Remark Observe that the Kahlerian structure on V describ ed ab ove is not canonical in the sense that

a

wehavechosen a decomp osition of the nonzero ro ots of a into p ositive and negative ones However in

the real case when L is a real semisimple Lie algebra it is not dicult to see that for a semisimple a L

the ro ots of ada are all on the real or imaginary axes of C This gives then a canonical choice of p ositive

and negativeby declaring a ro ot to b e p ositive if R or iR On the other hand the induced

euclidean structure g on V is only p ositive denite ie the complex structure J is calibrated with resp ect

a a a

to if and only if all ro ots are on the imaginary axes ie the Lie algebra L is compact Thus in the

a

case of a compact semisimple Lie algebra the adjoint orbit V carries a canonical Kahler structure for any

a

a L since ev ery element is semisimple

Gstructures on complex manifolds

Denition of Gstructures Consider a complex manifold X Wedenoteby FX X the

bundle of linear frames on X Thus a p oint p in the bre xover x X is a basis of the tangentspace

n

TX of X in x It may b e seen as the linear isomorphism from the standard complex vector space C to

x

n

TX carrying the canonical basis e e ofC into the basis given by p

x n

n

p C TX

x

The Lie group GLn C acts naturally on FX by

gpz pg z

n

for p FX g GLn C and z C In fact FX is a principle GLn Cbundle ie there exists a

covering U ofX and equivariant holomorphic dieomorphisms U U GLn C where

GLn C acts only on the right factor

gpg p

Therefore for x U U the matrix p p GLn C do es not dep end on p x dening

the transition functions U U GLn C x p p for some arbitrary p x

Of course the f g characterize the principle bundle and in the case at hand they are nothing else than

the transition functions of the tangent bundle itself FX is the socalled asso ciated principle bundle for the

vector bundle TX see eg Steenro d

Now for any Gprinciple bundle P X G acomplexLie group one denes a reduction of P to a

closed subgroup H G to b e a homomorphism Q P of an H principle bundle Q to P ie

P Q

and hastobe H equivariant A Gprinciple bundle is thus reducible to the closed subgroup H if and only

if there exists a bundle covering U ofX so that the corresp onding transition functions take their

values not only in G but in H

Denition Let G be a closed subgroup of GLn C and X a complex manifold of dimension n A

Gstructure on X is a reduction of the bundle of linear frames to G

e call sometimes the submanifold B QFX If Q FX is the reduction homomorphism w

G

the Gstructure In particular for G On C B is called an almosteuclidean structure or Riemannian

G

structure for G Spn CGLn Canalmostsymplectic structure and for G GLn EGLn C

an almostexceptional structure on X

In general if P X is a Gprinciple bundle and Y is a complex Gmanifold meaning that G acts

holomorphically by biholomorphisms one can build the asso ciated bre bundle

P Y

G

which is the quotientofP Y by the diagonal action of G If in particular P is the principle GLn Cbundle

of linear frames over X and GGLn C is a closed subgroup then the coset space Y GLn CG is in

a natural way a GLn Cspace and thus we can build the asso ciated bre bundle

F FX GLn CG

GLnC

Now a reduction of P to G is the same as a holomorphic section of X in this bre bundle If GLn CG

is contractible then there exists always a continuous section of F This is the reason in this language

why on a real manifold there always exists a Riemannian structure since On RGLn R is maximal

compact ie GLn ROn Ris dieomorphic to a cell In a sense the obstruction for the existence of

a Gstructure is only in the top ology of GLn RG in the real case Similarly any symplectic manifold

carries an almostcomplex structure since Un CSpn R is again a maximal compact subgroup

Now in the complex analytic case wehave in addition to the top ological obstruction of the homogeneous

space GLn CG an analytical obstruction In fact the existence of a continuous section do es not at all

imply the existence of a holomorphic section in contrast to the dierentiable case However if X is a

Stein manifold a fundamen tal example of Grauert Grauerts Oka principle says that F has a continuous

section if and only if F has a holomorphic section Thus over a Stein manifold we conclude again that the

obstruction for the existence of a Gstructure is in a sense only in the top ology of GLn CG meaning the

existence of a section of the top ological bre bundle F X forgetting ab out the complex analytic structure

of X

Finally observe that all the homogeneous spaces GLn COn C GLn CGLn E GLn C

n

Spn C and Spn CUn E parametrizing the euclidean structures on C the exceptional structures

n n n

the symplectic structures on C and the calibrated exceptional structures on C with resp ect to on C

the standard symplectic structure are not contractible This shows that in general these structures do not

is Stein exist on a complex manifold even in the case when X

A fundamental question in the theory of Gstructures is When is a given Gstructure lo cally at

n n

That means that it is lo cally equivalent to the standard Gstructure on C or R in the real case If

G On C we call a lo cally at Gstructure a euclidean structure if G GLn EGLn C an

exceptional structure and if G Spn CGLn Cwe call a lo cally at structure a symplectic structure

on X A necessary condition in the case G On C is that the sectional curvature tensor R dened as

in the real casevanishes In the case G GLn E a necessary condition is that the torsion tensor of J

ie N J TX TX TX

N J J J J J J

vanishes Finally in the case G Spn C it is necessary that the almostsymplectic form is closed In

the real case these necessary conditions are also known to b e sucient Thus we dene an almosteuclidean

structure g on a complex manifold to b e euclidean if the asso ciated Riemannn curvature tensor vanishes

we dene an almostexceptional structure J on a ndimensional complex manifold to be exceptional if

the asso ciated torsion tensor vanishes and we dene an almostsymplectic structure on a ndimensional

complex manifold to b e symplectic if it is closed

If X is a symplectic manifold an exceptional structure J on X is called calibrated if it is p ointwise

calibrated with resp ect to Then the asso ciated Kahlerian structure is said to dene a Kahler structure

on X Nowwe think that the given denitions of euclidean exceptional and symplectic structure agree ie

we hop e that the answer of the following question is in the armative

and d equivalent to a lo cally at structure also Question Are the conditions R N J

in the complex case

A fundamental example Consider now a complex Lie group G and let g b e its Lie algebra



Then G op erates on g via its adjoint action and on g via its coadjoint action The tangent space of the

 

coadjoint orbit G g for some g is canonically identied with g g whereg fa g jh a bi

for all b g g It carries therefore the KirillovKostant structure discussed earlier Now it is easy to see

that the corresp onding form on X GisGinvariant and moreover closed In fact this follows from

Cartans formula for d ie

d X Y Z XY Z YX Z ZY Z

X Y Z X Z Y Y Z X

Thus for any complex Lie group the coadjoint orbit carries a natural structure of a symplectic complex

manifold

Consider now a semisimple complex Lie group G and let a g be a semisimple element Then

the ro ots of ada are the same as the ro ots of ad Ad g a for any g G In fact if one identies G via

Ad G Aut g GLg with AdGandg with adg glg then Ad is the conjugation action on g Thus

cho osing once a decomp osition of the nonzero ro ots of a into negative and p ositive ones



gives an almostexceptional structure J on the adjoint orbit Gag whichisclearlyGinvariantbycon

struction Tocheck that the torsion tensor vanishes ie

J X Y JXJY JXY X J Y

wemay assume that X adabandY ad acforb c N N and since the innitesimal action of g

on the vector elds of Ga is up to a minus sign a Lie homomorphism one has to check the ab ove relation

simply for X b and Y c in N N Now if b N and c N then the equation is obviously

fullled since J jN i id and J jN i id However to satisfy the relation also in the cases b c N

and b c N we need that the decomp osition has the prop erty

In this case the relation is true also in these cases An equivalentformulation is of course that N and N

are Lie subalgebras of g which are necessarily nilp otent then Wehaveproved now

Theorem Let Ga b e a semisimple adjoint orbit of a semisimple complex Lie group G Let C be

the nonzero eigenvalues of ada Then if and only if and anychoice of a decomp osition of

into negative and p ositive eigenvalues meaning that if and only if satisfying

C n induces a Kahler structure on X

n

As a particular case consider G SLn C and a sln C Then the adjoint orbit

n

of G is naturally identied with SLn CGLn C In view of the ab ove discussion wemay identify

n

this as SUn EUn E and call it the exceptional pro jective space P E The exceptional structure

comes from the natural decomp osition of sln C according to a

sln Cgln CN N

where

trb

gln C j b gln C

b

and

v

n
n

v C N j N j v C

v

n n

Since SUn Eacts transitively on fv E jhv vi g we conclude that we mayidentify P E

n  n 

with the quotientofE fv E jhv vi g by the natural diagonal action of E v v ie

n n  

P EE E

Adress Frank Lo ose Mathematisches Institut D Dusseldorf email lo osecsuniduesseldorfde