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Home , Adjoint representation

The Erwin Schrodinger International Boltzmanngasse

ESI Institute for Mathematical Physics A Wien Austria

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DV Alekseevsky

Vienna Preprint ESI January

Supp orted by Federal Ministry of Science and Research Austria

Available via httpwwwesiacat

FLAG MANIFOLDS

DVAlekseevsky

Contents

I Basic facts ab out Lie groups and Lie algebras

Lie groups and Lie algebras Functor Lie

Basic results ab out structure of Lie groups and Lie algebras

Structure of complex semisimple Lie algebras

Real forms of a complex semisimple

Parab olic subalgebras of a complex

I I Homogeneous manifolds

I I I Flag manifolds as homogeneous spaces of a complex semisimple

Flag manifolds of classical groups SL C SO C S p C

n n n

Flag manifolds of a semisimple Lie group G

The action of a maximal compact subgroup of G on a ag M GP

IV Flag manifolds as a quotient of a compact Lie group

T ro ots T chamb ers and T of a ag manifold

Realization of a ag manifold by an adjoint orbit

Closed invariant forms on a ag manifold

Decomp osition of the isotropy representation into irreducible submo dules

Invariant complex structures and Kahler metrics

Invariant KahlerEinstein metrics

Painted Dynkin graphs and classication of ag manifolds

Painted Dynkin graphs T chamb ers and Koszul numb ers

Examples

Typ eset by A ST X

M E

DVALEKSEEVSKY

I Basic facts ab out Lie groups and Lie algebras

Lie groups and Lie algebras Functor Lie

Recall that a Lie group is a smo oth manifold G together with structure of a group such that the

group op erations

G G G g g g g

i G G g g

are smo oth mappings

Here we will assume that either all ob jects are real and then G is a real Lie group or all ob jects

are complex and holomorphic then G is a complex Lie group

An example of Lie group is the general GLV AutV ie the group of all

n n

automorphisms of a nite dimensional V R or C If V is the arithmetic vector space

n n

k k R or C then the group GLk is denoted also by GL k and it is called the general matrix

n

group

Any closed subgroup G of GLV is a Lie group called a linear group

Example of linear groups are the following classical Lie groups

The group

SLV fA GLV det A g

of unimo dular transformations and the group

Aut V b fA GLV A b bA A bg

of automorphisms of a non degenerate bilinear form b If b is a skewsymmetric then the group

l n

AutV S p V S p C V C is called and it is connected For V k

l

n m the standard notation for symplectic group is S p k If b g is a symmetric form then the

m

group Aut V g O V is called the The connected comp onent of the unity is

g g

denoted by

SO V fA O V det A g

g g

n

or by SO C if V C

n

Lie algebra real or complex is a vector space over k R C with a bilinear op eration

g g X Y X Y g

which satises the Jacobi identity

X Y Z Y Z X Z X Y X Y Z g

Let G b e a Lie group Denote by L the left action of G on G that is the homomorphism

L G Di G g L L g g g g g G

g g

The space

L

G

X X g G X XG L g Lie G XG

g

of all L invariant vector elds on G is a subalgebra of the Lie algebra XG of vector elds with resp ect

G

to the Lie bracket

X Y X Y X Y Y X

Here a vector elds are considered as derivations of the algebra C G of smo oth functions on

G and o means the comp osition of derivations Lie algebra of a Lie group G is called the tangent Lie

algebra g of G

FLAG MANIFOLDS

Any L invariant vector eld X is dened by its value X in the p oint e G

G e

X X g R X g G

g e g e

where R g g g is the right multiplication In the case of a linear Lie group G X X g where

g g e

L

G

T G which dot stands for The map X X denes isomorphism XG

e e

L

G

with the T G If G GLV is a linear allows identify the tangent Lie algebra XG

e

Lie group then the Lie bracket on g T G is the

e

X Y X Y Y X X Y g l V End V

The map G g Lie G denes a functor from category of Lie groups to category of Lie algebras It

is called Lie functor and is very closed to b e an isomorphism of categories Indeed any Lie algebra g

is the tangent Lie algebra of some simply connected Lie group G dened up to an isomorphism Any

Lie group G with tangent Lie algebra g is isomorphic to the quotient G of G by a central discrete

subgroup of G All Lie groups with the same tangent Lie algebra are lo cally isomorphic ie they have

the same group op erations in some neighb orho o d of the unity Many prop erties of a Lie group G can b e

describ ed in terms of prop erties of the tangent Lie algebra g

For example Lie group G is solvable nilpotent commutative i the tangent Lie algebra g is solvable

nilpotent commutative Lie group G is simple ie has no non discrete normal subgroup semisimple ie

has no non discrete commutative normal subgroup reductive ie has a normal semisimple subgroup S

with commutative quotient GS i the tangent Lie algebra g is simple has no prop er ideal semisimple

has no prop er commutative ideal or reductive is a direct sum of commutative and semisimple Lie

algebras

There are corresp ondence b etween subalgebras resp ideals of a Lie algebra g and virtual

subgroups resp virtual normal subgroups of a Lie group G with Lie G g Subgroup H of a Lie group

G is called virtual if it is an immersed submanifold of G Such subgroup H has a structure of Lie group

but H is not necessary a closed subgroup of G even if it is simple

Exercise Prove that the classical Lie groups SLV SO V S p V has tangent Lie algebras

g

sl V fA g l V tr A g

so V fA sl V g Ax y g x Ay x y V g

g

sp V fA sl V Ax y x Ay x y V g

Prove that all of them are simple with the exception SO V where V C or V R and g has

g

signature or

Basic results ab out structure of Lie groups and Lie algebras

We collect here some useful general results ab out Lie groups and Lie algebras

LeviMaltsev theorem Any Lie algebra g can b e decomp osed into a sum

g s r s r

of a maximal semisimple subalgebra s and the radical r maximal solvable ideal

Any two maximal semisimple subalgebras are conjugated by an automorphism of g

Any connected Lie group G can b e decomp osed into a pro duct G S R of a maximal semisimple

subgroup S and the radical R ie the maximal solvable normal subgroup such that S R is a discrete

subgroup of G Any two maximal semisimple subgroups of G are conjugated by an automorphism of G

Any semisimple resp ectively reductive Lie algebra g is the direct sum of noncommutative

simple ideals

g g g

k

DVALEKSEEVSKY

resp ectively simple ideals and the

Any simply connected connected semisimple Lie group G is the direct pro duct of simple connected

normal subgroups

G G G

k

Any simply connected connected reductive Lie group G is the direct pro duct of simple Lie groups

p q

G i and the connected comp onent of the center connected Lie group Z G R T where

i

q

T S S

z

q

is the q torus

G Z G G G

k

Ado theorem Any Lie algebra g over R C admits exact linear representation that is a

homomorphism g g l V into general linear Lie algebra g l V End V with trivial

For any simply connected connected Lie group G there exist a discrete central subgroup such

that G admits an exact linear representation G GLV monomorphism into GLV

Let G b e a Lie group and denote by Ad the natural homomorphism of G into the group AutG

of automorphisms of G given by

Ad g g g g g g G

g

Since any automorphism of G preserves the unity e the group Ad AutG acts on the tangent space

G

g T G by linear transformations and this action preserves the Lie bracket on g

e

Ad X Ad Y Ad X Y g G X Y T M

g g g e

If G GLV is a linear Lie group

Ad X g X g

g

where denotes the matrix multiplication

We get a representation

Ad g Ad Aut g GLg

g

which is called the adjoint representation of a Lie group G

It induces the adjoint representation ad of the tangent Lie algebra g Lie G by derivations of g

ad g Derg X ad ad Y X Y X Y g

X X

Remark that the kernel of Ad coincides with Z G the center of G and ker ad Z g the center of g

Dene a symmetric bilinear form B on a Lie algebra g by

B X Y tr ad ad X Y g

X Y

It is called the Kil ling form The adjoint action of a Lie group G with Lie G g preserves B

B Ad X Ad Y B X Y g G X Y g

g g

In terms of the adjoint action of the Lie algebra this condition can b e written as

B ad X Y B X ad Y X Y Z g

Z Z

FLAG MANIFOLDS

This condition follows from previous relation if we remark that any vector Z g generate an parametric

d

subgroup g exp tZ of G with tangent vector Z at e g Z Then to get it is sucient to

t t

dt

t

dierentiate with g g with resp ect to t and put t

t

The following useful criterion of solvability and semisimplicity of a Lie algebra g is due to Cartan

A Lie algebra g is solvable if the commutant

g g span fX Y X Y gg

is in the kernel of B ie

B X Y Z X Y Z g

A Lie algebra g is semisimple if the B is non degenerate

Structure of complex semisimple Lie algebras

Cartan decomp osition of a semisimple Lie algebra

Let now g b e a semisimple Lie algebra Since Z g the adjoint representation ad X ad

X

ad Y X Y is exact and we can identify g with linear Lie algebra ad derg glg Remark

X g

that the connected group of automorphisms Ad G Aut g has g as Lie algebra It is called the adjoint

group

Dene a h of g as a maximal subalgebra such that the linear subalgebra ad

h

ad g is diagonalizable ie with resp ect to some of g all from ad are represented

h

by diagonal matrices

Cartan subalgebra h always exist and coincides with its normalizer

N h fx g x h hg

g

and any two Cartan subalgebras are conjugated by an automorphism of g

A linear form on Cartan subalgebra h is called a root if the corresp onding root space

g fx g ad x hx h hg

h

is not zero A non zero vector from g is called a root vector Denote by R the nite set of all ro ots

Then we have the following Cartan decomp osition of g into direct sum of subspaces

X

g h g

R

The main prop erties of such decomp osition are

g g g where g if R

B h g B g g if

R R

B j is non degenerate and dene isomorphism

h

B h h u B

Put

E R span R h R B E

R

Then h hR ihR and B j is p ositively dened

hR

DVALEKSEEVSKY

We will denote the corresp onding scalar pro duct on E and hR by

dim g

Cho ose a vector E g for all R such that

B E E

and put H E E Then

H E E

and g C H g g is a subalgebra isomorphic to sl C The isomorphism is given by

H E E

There is some freedom in choice of ro ot vector E Using this freedom one can cho ose E R such

that the following commutator relations hold

if R

E E

p E if R

where p is the maximal integer such that p R

Moreover there exist an algorithm for determination the sign in this formula Tits

As a corollary we have

Prop osition The system of ro ots R of a semi g determines g up to an isomorphism

Cartan decomp osition and ro ot system of classical complex Lie algebras

How we describ e the Cartan decomp osition of the classical Lie algebras

A sl C B so C C sp C D so C

l l l l l l l

which are tangent Lie algebras of the classical groups

SL C SO C S p C S O C

l l l l

l

A Denote by e i l the standard basis of the vector space V C and identify the

l i

Lie algebra g l V of endomorphisms with V V The Lie algebra A sl C consists of all

l l

traceless The subalgebra

l

X X

x e e x h h

i i i

i

i

of all diagonal with resp ect to the basis fe g elements of A is a Cartan subalgebra of A Set

i l l

E e e and denote by h the linear form on h dened by h x Then

ij i i i i

j

R f i j g

i j

FLAG MANIFOLDS

is the system of ro ots and E is the ro ot vector with ro ot

ij i j

The Cartan decomp osition of A is given by

l

X

A h C E

l ij

i j

B C D To describ e the Lie algebras B C D in a unied way we denote by b a non degenerate bilinear

l l l l l l

n

form in the space V C which is either symmetric b g or skewsymmetric b and by

autV b fA g l V b A b A g

the Lie algebra of endomorphisms which preserve b Then

l l l

B aut C g C aut C D aut C g

l l l

n n

We cho ose a basis e e e i l of C for n l and a basis e e i l of C

i i i i

for n l such that the form b is given by

l

X

e e B b g e e

l

i i

i

l

X

e e C b

l

i i

i

l

X

e e D b g

l

i i

i

where e is the dual basis of V and

i

x y x y y x x y x y y x

are wedge and symmetric pro duct

Such basis is called the standard basis of V We identify the dual space V with V by

means of the bilinear form b

V V x bx b x x V

Then the Lie algebras B D are identied with the space V of bivectors and C with the

l l l

space V of symmetric tensors For example a decomp osable bivector x y denes the

endomorphism

z x y z g y z x g x z y

which b elongs to autV g

The set of all diagonal with resp ect to the standard basis elements forms a Cartan subalgebra h

of autV b More precisely we have

l

X

x e e B h h

i i i l

i

l

X

x e e C h h

i i i l

i

l

X

x e e D h h

i i i l

i

DVALEKSEEVSKY

The corresp onding Cartan decomp osition is given by

l

X X X

C e e B h C e e C e e

i j l i j i j

ij

ij ij

l l l

X X X

C e e C e e C e e C h

i j i j i j l

ij ij ij

l

X X X

C e e D h C e e C e e

i j l i j i j

ij

ij ij

Denote by i l the standard basis of h dened by

i

h x

i i

Then the ro ots and corresp onding ro ot vectors are given in the following table

Table

g ro ots ro ots vectors

i j i j e e

i j i j

B i e e

l i i

i j i j e e

i j i j

i j i j e e

i j i j

i j e e

i j i j

C i j e e

l i j i j

e e

i i i

i j e e

i j i j

i j e e

i j i j

D i j e e

l i j i j

i j e e

i j i j

Ro ot systems

Let R E span R h b e the system of ro ots of a semisimple Lie algebra g with resp ect to a

R

Cartan subalgebra h Denote by G the dimensional subgroup g of the Aut g

asso ciated with the Lie subalgebra g C H g g R Studying the adjoint action of this

group on ro ot vectors one can establish the following fundamental prop erties of the set R

Z i R hj i

ii R S R R

where S h ji is the reection with resp ect to hyp erplane in E

iii R for R implies

Denition A nite set R of vectors in Euclidean space E is cal led an abstract reduced if

R generates E and satises iiii

From the denition one can derive the following two additional prop erties of ro ot system R

FLAG MANIFOLDS

Let R b e ro ots such that j j jj and Denote by the angle b etween

Then all p ossibilities for hj i h ji j j jj are describ ed in the following table

Table

j j j j jj

Let R The series of ro ots containing is dened as the set of all ro ots of the

form k k Z

series of ro ots containing has the form f k p k q g where p q and p q h ji

In particular if then R if R and R then

We asso ciated with a semisimple Lie algebra g a ro ot system R Two natural questions arrise

Are two semisimple Lie algebras with isomorphic ro ot system isomorphic

Is it true that any abstract ro ot system R is the ro ot system of some semisimple Lie group

The answer for b oth questions is p ositive

Theorem Let g resp g are semisimple Lie algebras and R resp R are root system of g resp g

with respect to a Cartan subalgebras h h Then any isomorphism of Euclidean vector spaces R

R which maps R onto R can be extended to an isomorphism g g of Lie algebras

Any abstract root system R is isomorphic to the root system of some semisimple Lie algebra

The semisimple Lie algebra g with given ro ot system R E may b e constructed as follows

C C

the dual vector space Let E b e the complexication of the vector space E and h E

Consider direct sum of vector spaces

X

g h C E

R

where E is a basis of dimensional vector space C E asso ciated with a ro ot R

g b ecome a semisimple Lie algebra with Cartan subalgebra h and ro ot system R if the Lie bracket

is dened by

h h

h E hE h h R

if R

E E

p E if R

where series of ro ots containing is given by f p q g An algorithm for determination

of the sign is given in Tits

System of simple ro ots

The classication of ro ot systems can b e reduced to classication of some sp ecial basises of the

Euclidean vector space E

DVALEKSEEVSKY

Denition Let R be a root system in Euclidean vector space E A set f g or roots is cal led

l

a basis of R or a system of simple roots if is a basis of E and any root R has integer coordinates

with respect to of the same sign

l

X

k k Z

i i i

i

where either k or k for i l If k resp k then the root is cal led positive

i i i i

resp negative

Hence a basis denes a decomp osition R R R of the ro ot system into disjoint sum of

p ositive ro ots R and negative ro ots R R

To construct a system of simple ro ots we dene the set E E n f g of regular

R

reg

elements in E as the set of vectors from E on which all ro ots take non zero values A connected

comp onent C of E is called a Weyl chamb er

reg

Any Weyl chamb er C is dened by inequalities

C f g

l

where are some ro ots These ro ots form a basis

i

f g

l

of R and any basis of R can b e obtained in such way The nite group

W S R generated by reections S the Weyl group of a ro ot system R acts simply

transitively on the set of Weyl chamb ers and hence basises of R A practical way for constructing a

basis of R is the following

Cho ose a regular element h E and dene the set of p ositive ro ots R as the set of ro ots which have

reg

p ositive value on h

R f R h g

A p ositive ro ot R is called simple if it is not a sum of two p ositive ro ots The set f g

l

of simple ro ots is a basis of R

Due to the formula

p q j

for series of ro ots f k p k q g a ro ot system R can b e reconstructed from simple ro ot system

inductively starting from simple ro ots At rst we determine all ro ots which are sum of two simple

ro ots then the ro ots which are sum of three simple ro ots etc

Similar to ro ot systems one can give an intrinsic characterization of a simple ro ot system as

follows

l

A basis f g of the Euclidean vector space E is called a simple ro ot system if

l

i j

a j

ij i j

j j

for any i j is a non p ositive integer

The matrix A ka k a j is called theCartan matrix of the simple ro ot system

ij ij i j

It determines up to an isometry and is characterized by the following prop erties

i a Z a a for i j

ij ii ij

FLAG MANIFOLDS

ii a a

ij j i

iii m a a for i j

ij ij j i

p

iv The matrix GA kg k g g m i j is p ositively dened

ij ii ij ij

The last prop erties follows from the fact that G is the Gram matrix of the basis f k kg

i i

The nice way for visualization of a simple ro ot system and its A was prop osed

by EB Dynkin He asso ciate with a graph which is called Dynkin graph by the following

rules

Any simple ro ot is represented by a vertex of

i

two verteces are jointed by m a a j j lines

i j ij ij j i i j j i

if j j j j and and hence a h j i a h j i we draw arrow which

i j i j ij i j j i j i

indicate the direction from long ro ot to short ro ot

i j

The determines simple ro ot system up to an isometry

Remark that if R E R E are ro ot systems in spaces E E resp ectively then R

R R is a ro ot system in the space E E E Such system R is called decomp osable

ant it corresp onds to a semisimple Lie algebra which is the direct sum of two semisimple ideals Ro ot

system is indecomp osable i the asso ciated Dynkin graph is connected It is equivalent to the condition

that the asso ciated Lie algebra is simple or in terms of Cartan matrix A that the Cartan matrix A is

indecomp osable

The last condition means that A can not b e transform into blo ck diagonal form by means of a

p ermutation of rows and the same p ermutation of columns

The classication of indecomp osable Cartan matrices leads to the following result

Classication theorem Besides the root systems A B C D of classical complex Lie algebras

l l l l

sl C so C sp C so there exist only in idencomposable root systems G F E E E

l l l l

The low index indicate the rank of the system ie the dimension l of the corresp onding Euclidean space

l

E

These ro ot systems and the corresp onding Lie algebra are called exceptional The exceptional

ro ot systems are describ ed in the following table taken from GOV

Table

DVALEKSEEVSKY

Typ e dim G R W

i

i i i

i j

E Aut R

i j k

i

i i i

i j

E Aut R

i j k l

i

i i i

i j

E Aut R W fg

i j k

i j i

F Aut R

G Aut R

i j i

The following notations are used in this table For F i is an orthonormal basis

i

of the dimensional Euclidean space E

For all other exceptional ro ot systems of rank l is the standard basis of the Euclidean

l

l

vector space R restricted to the hyp ersurface

l

X X

l

x E x

i i i

i

In particular

l

i j

i i i j

l l

For E is the vector with orthogonal to all vectors

i

We indicate the dimension of the corresp onding exceptional Lie algebra g system of simple ro ots

f g and the maximal ro ot and give a description of the Weyl group W

l

For completeness we give also similar description for classical ro ot system Here we indicate also

the ro ot vectors

Table

FLAG MANIFOLDS

Ro ot Ro ot

Typ e g dim g h W

vectors system R

l

X

x e e

e e

i i

i

i

j

A sl C l l S

l l i j i i i l

i j

X

x

i

e e

i j i j

i i i

e e

P

i j i j

l

l

i l

x e e B so C l l S Z

i i i l l l

i

e e

i j i j

l l

e e

i j i

e e

i j i j i i i

P

l

l

e e i l

x e e C sp C l l S Z

i j i j

i i i l l l

i

e e

l l i j i j

e e

i j i j i i i

P

l

l

e e i l

D so C l l S Z x e e

i j i j

l l l i i i

i

e e

l l l i j i j

Parab olic subalgebras of complex semisimple Lie algebra

Regular subalgebra

P

C E b e a Cartan decomp osition of a complex semisimple Lie algebra g Let g h

R

Denition

Subalgebra k of g is cal led to be regular if h k k

Subset Q R is cal led to be closed or a root subsystem if

Q Q R Q

ie Q R Q

Subset Q R is cal led to be symmetric resp asymetric i Q Q resp Q Q

s a s

Any Q R can b e decomp osed into a disjoint sum Q Q Q where Q Q Q resp

a s

Q Q n Q are symmetric and asymetric parts of Q

s a

Lemma If Q is closed then Q Q are closed

Prop osition A closed set Q R denes a regular subalgebra

X X

s

gQ h E E H Q i C E h C E

Q

Q Q

Its LeviMaltcev decomposition is

s a

gQ gQ gQ

DVALEKSEEVSKY

Conversely any regular Lie algebra k of g has the form

k h gQ

where h is a subalgebra of h and Q is a closed subset of R

Example A system of p ositive resp ectivelynegative ro ots R respR are closed and asymetric

The corresp onding Lie algebras

b h gR

are Borel subalgebras that is maximal solvable subalgebra of g Any two Borel subalgebras are conjugated

by an automorphism of g

Lemma A maximal asymetric closed subset Q of R is the set of al l positive roots with respect to some

Weyl chamber

Parab olic subalgebras

Parabolic subalgebra p is a subalgebra which contains a Borel subalgebra b

Construction of parab olic subalgebras Let is a basis of R R R R is the corresp onding

decomp osition of R and is a subset Dene

hi R hi R

a s

R n Hence and R R is a closed set Moreover R Then R R

0

0 0

a

z g g R h g R p

0 0

0

a

and semisimple part b h gR with the radical z gR is a parab olic subalgebra p

0

a

g

Here z is the borthogonal complement to the Cartan subalgebra h hH R i of g in

h We have

a a

g R z g z g g R

0 0

a

is a regular nilp otent subalgebra Moreover z g is a maximal reductive subalgebra of g and gR

0

of g

Prop ositionAny parabolic subalgebra is conjugated to a subalgebra of the form p

0

Proof The pro of is immediate Let k b e a parab olic subalgebra of g Since any two Borel subalgebras

are conjugated we may assume that k contains the Borel subalgebra b h gR Then k h gR

for some subset of where R R is a closed subset of R This implies that R R

0

Remark that any subset of dene a decomp osition of the Lie algebra g into sum of three

subalgebras

a a

k gR g gR

0 0

a

are nilp otent subalgebras where k h g z g is a reductive subalgebra and gR

0

This decomp osition is called generalized Gauss decomp osition

Remark also that the center z of k is always dierent from zero and k coins ides with the centralizer

of z

k Z z g

FLAG MANIFOLDS

I I Homogeneous manifolds

A homogeneous manifold is a manifold M together with a transitive action of a Lie group G on

M Transitivity means that for any x y M there exist g G such that g x y An other words G

has only one orbit on M

A homogeneous manifold M may b e identify with the coset space GK where K fg G g o og

is the stabilizer or stability subgroup of a p oint o M It is clear that K is closed but not necessary

connected subgroup of G Conversely any closed subgroup K of a Lie group G denes a homogeneous

manifold GK There exist unique smo oth structure on GK such that the natural action of G on GK

is smo oth

Assume for simplicity that the homogeneous space M GK is reductive that is Lie subalgebra

k Lie K of g Lie G admits Ad invariant complement m such that

K

g k m

is a Ad invariant decomp osition At is always the case if K is compact

K

Then we can identify m with the tangent space T M by

o

d

exp tX o m X

dt

t

where exp tX is the parametric subgroup of G generated by X

Under this identication the isotropy representation j of K is identied with the restriction of

the adjoint representation Ad j to m

K g

j K Ad K j

m

Many geometrical questions ab out homogeneous manifold M may b e reformulated in terms of the pair

G K of Lie group and then in terms of the corresp onding pair g Lie G k Lie K of Lie algebras

For example the classication of Ginvariant tensor elds of given typ e p q reduces to description of

p q tensors of the tangent space T M which are invariant under isotropy representation

o

j K GLT M

o

of the stabilizer Recall that j is dened by

d d

j k k k K

t t

dt dt

t t

where is a curve with

t

Hence the classication of Ginvariant tensor elds of typ e p q reduces to a description of the

p Ad K

space T m of Ad K invariant tensors of typ e p q on m

q

If the group K is connected this is the case of G is connected and M is simply connected then

ad p Ad p

k K

k Lie K m T m T

q q

where

p ad p

k

T m A T m ad A X k

X

q q

and problem reduces to a description of tensors on m invariant under the adjoint action of Lie algebra k

For example a homogeneous space M GK admits Ginvariant Riemannian metric g almost

complex structure J or almost symplectic structure i the representation Ad K jm is orthogonal ie

DVALEKSEEVSKY

preserves an Euclidean metric on m complex commute with a complex structure J or resp ectively

symplectic preserves some non degenerate skewsymmetric form

Remark that any linear representation of a compact Lie group K is orthogonal Hence any

homogeneous manifold M GK with compact subgroup K admits an invariant Riemannian metric

It is not dicult express the condition of integrability of an invariant almost complex structure

J or an invariant almost symplectic structure on a homogeneous manifold M GK in terms of Lie

algebras g k Recall that an almost complex structure J ie eld of endomorphisms J with J id is

integrable i the corresp onding complex eigendistribution

C

T M v T M TM C J v iv

of J with eigenvalue i is involutive ie the space of sections of T M is closed under Lie bracket

Let M GK b e a reductive homogeneous manifold with reductive decomp osition g k m

C C

Then we have natural identication T M m

Let J b e an invariant almost complex structure on M GK and J J j is the corresp onding

M M

complex structure on T M m

C

Denote by m and m the eigenspaces of J on m with eigenvalues i and i resp ectively They

are Ad K invariant and dene invariant eigendistributions T M and T M of J

M

One can easily check that almost complex structure J is integrable i

M

C

g k m

C

is a complex subalgebra of the complex Lie algebra g g C

Now we assume for simplicity that G is connected compact semisimple Lie group and describ e all

homogeneous spaces GK which admits an invariant symplectic structure ie non degenerate closed

form Let g k m b e a reductive decomp osition The value of at o eK is Ad invariant

o K

Ad

K

It may b e considered as form on g with Ker k nondegenerate form on m m

o

The closeness of means that is closed in the complex g of exterior forms on the Lie algebra g

o

d X Y Z cycl X Y Z X Y Z g

o

where cycl means the sum of cyclic p ermutations It is known that H g R for semisimple Lie

algebra g In other words any closed form is exact d where g is form and

o

d X Y X Y X Y g

This form is canonically dened Using nondegeneracy of Killing form B we can write X

as

X B h X X g

o

where h is some element of Lie algebra g Hence

o

X Y B h X Y B h X Y X Y g

and

k ker Z h fX g h X g

o g o o

This implies that K contains the centralizer

Z h fg G Ad h h g

G o g o o

FLAG MANIFOLDS

b ecause this centralizer is connected Since element h g is canonically dened by it is invariant

o o

under Ad Thus

K

K Z h

G o

Both inclusions imply equality K Z h We have

G o

Theorem KirillovKostantSouriau Let M GK be a homogeneous manifold of compact

semisimple Lie group G with invariant symplectic structure Then K Z h is the centralizer

G o

of some element h g Lie G and hence M can be identied with the adjoint orbit M Ad Gh of

o o

this element Moreover the value of the symplectic form in a point h M Ad Gh is given by

h o

X Y B h X Y X Y T M g

h h

Corollary There exist correspondence between invariant symplectic structures on a homogeneous

manifold M GK of a compact semisimple Lie group G and elements h g with the stabilizer K in

G

Such elements have the orbit isomorphic to GK

Homogeneous manifolds GK of compact semisimple Lie group G which are Gdieomorphic to

adjoint orbits of G are cal led ag manifolds

I I I Flag manifolds as homogeneous spaces of a complex semisimple Lie group

Flag manifolds of classical groups SL C SO C S p C

n n n

Now we describe the manifolds of complex ags and complex isotropic ags and identify them with

coset spaces of classical Lie groups modulo parabolic subgroups

n

Let V C be the complex vector space and p p p is an r tuple of natural numbers such

r

that p p n

r

Denition A ag of type p or pag is a system f V V V of subspaces of V such that

r

V V and dim V p

i i i i

The unimodular group

SL C fA GLC det A g

n

n

acts transitively on the set F V of al l pags Let e e e be the standard basis of V C We

p i n

wil l denote by e e the complex subspace of V spanned by e e The pag

k k

e e e e f e e

p p p

r 2 1

is cal led the standard pag or pag associated with the standard basis Its stabilizer P in SL C

p n

consists of al l upper blocktriangular matrices with the blocks of size p p p p p

Hence we may identify F V with the coset space SL C P

p n p

A ag of type n is cal led a ful l ag Remark that the stabilizer P of the standard

n

ful l ag f associated with the standard basis is the subgroup B of al l upper triangular matrices from

SL C that is a of SL C The stabilizer P of the standard ag of any type p contains

n n p

B

DVALEKSEEVSKY

Now we describes the manifolds of isotropic ags which are homogeneous spaces of other classical

Lie groups SO C S p C

n n

n

Assume that a non degenerate bilinear form b in V C is given symmetric b g or skew

symmetric b Last case is possible only if n l The classical groups SO C S p C are

n

n

dened as the connected component AutV b of the group AutV b of automorphisms of V which

preserves b As in ch section we identify the Lie algebras so V and sp V with the space V

g

of bivectors and with the space V of symmetric tensors

We wil l x a basis e e i l of V for n l and e e e i l for n l such

i i i i

that be e be e and al l other products are zero In such standard basis the form b is

i j ij

given by

X X

n l b g e e b e e

i i i i

X

n l b g e e e e

i i

Denitions A ag f V V of type p p p in the space V b is cal led isotropic if

r r

bj

V

r

n

In this case r l The group AutV b acts transitively on the manifold I F V of isotropic

p

pags p p p Moreover its connected component Aut V b acts on I F V transitively if

r p

b or b g and n l or b g n l and p l In the case b g n l p l the

r r

group Aut V g SO C D has two open orbits SO C f SO C f where f f are ags

l l l l

associated with the standard basis e e e e and the basis e e e e e e on the

l l l l

manifold I F V of ags of type p p p l see GOV

p r

Denote by

e e e e f e e

p p p

r 2 1

the standard isotropic pag in V b associated with the standard basis e e e

i i

of the point f Exercise Describe the stability subalgebra p

f

0

For example for D case b g n l it is given by

l

p he e i j e e st if p j p then i p i

p i j i j s s sj

Flag manifolds of a complex semisimple Lie group

Now we dene ag manifolds associated with any connected complex semisimple Lie group G

Recal l that a subgroup P of G is cal led parabolic if it contains a Borel subgroup ie a maximal solvable

subgroup B of G

Denition A ag manifold of a complex semisimple Lie group G is the quotient M GP of G by a

parabolic subgroup P

It is known that a parabolic subgroup P of G is always connected and coincides with its normalizer

N P P This implies that a ag manifold M GP is simply connected complex manifold

G

Moreover M GP can be realized as a closed hence compact orbit of G in the projective space PV

associated with some linear representation G GLV of G more precisely the orbit of the highest

weight vector

Flag manifolds of the classical Lie groups are exactly the manifolds of pags or isotropic pags

described above

FLAG MANIFOLDS

Lie algebra p of a parabolic subgroup P is a parabolic subalgebra Hence we can write

a

k g R p p

0

0

k z g

a

R n R

0

The generalized Gauss decomposition

a a

g n k n g R k g R

0 0

induces a decomposition of some open subset G of G into product of the corresponding subgroups

reg

G N K N K N N N feg

reg

a

Moreover P K N This decomposition shows that the where Lie K k Lie N g R

0

nilpotent subgroup N acts simply transitively on the open dense subset M G P of the ag

reg reg

manifold M GP Hence any complex coordinates on N for example independent matrix elements

of a matrix representation of N dene local complex coordinates on M In the case of Grassmanian

such coordinates are standard local complex coordinates of the Grassmanian

Remark that any chain of subsets

k

of a system of simple roots of the Lie algebra g Lie G denes a tower of holomorphic Gequivariant

berings

GP fp ointg GP GB GP

2 1

is the parabolic subgroup with tangent Lie algebra where P

i

R R n R h g R p

i i i

i i i

In particular if

f g f g

i i i i

then we have the tower

GB GP GP GP GG fp ointg

f g f g f g

1 1 2 1 i

Al l bers

P P

f g f g

1 i 1 i1

can be identied with a primitive ag manifold ie the quotient of a G modulo a

i

maximal parabolic subgroup P which corresponds to the Dynkin graph of G with one deleted root

i i i

The action of a maximal compact subgroup of G on a ag manifold M GP

Recal l that a ag manifold M GP of a connected complex semisimple Lie group G is simply

connected and compact Then by Mostov theorem a maximal compact subgroup G of G acts on M

transitively From the exact sequence of homotopy groups of the bration G M G G one

x

of a point x M is a connected subgroup of G Hence it is can derived easily that the stabilizer G

x

determined by the Lie algebra

g g p p fx p X X g x

DVALEKSEEVSKY

where g Lie G is the Lie algebra of the maximal compact subgroup G of G that is the xed point

set of the standard antiholomorphic involution We can write

a a

p Lie P z g g R k g R

0 0

a

is invariant under the antiholomorphic involution and Since the subalgebra k z g R

0

a a

g R g R

0 0

we get

a

g p k z g R

x

0

Hence the stability subalgebra g is a compact form of the subalgebra k Z z In particular g

g

x x

z is the centralizer of a commutative subalgebra z of g This subalgebra z generates a torus T in Z

g

a compact connected semisimple Lie group G At is known that the centralizer of a torus in a connected

compact semisimple Lie group is connected This implies

z Z T Z t G Z

G G G

x

where t z is an element generating a dense parametric subgroup of the torus T We get

Prop osition Any ag manifold M GP can be identied with the quotient G K of a semisimple

t of an element t g Lie G and compact Lie group G G modulo the centralizer K Z

G

t hence with an adjoint orbit Ad

G

IV Flag manifolds as a quotient of a compact Lie group

T ro ots T chamb ers and T Weyl group of a ag manifold

Let M GP be a ag manifold of a complex semisimple Lie group G We know that the Lie

algebra p of P has a semidirect decomposition

k n z g g R n p p

0

into a sum of a reductive subalgebra k and the maximal nilpotent ideal n It denes a decomposition

P K N of the Lie group P

Denote by the standard antiholomorphic involution of G which determines a compact form G of

G G is the xed point set of Then P K is a compact form of Lie group K and we can identify

M GP with G K We x a Cartan subalgebra h of k Lie K It is also a Cartan subalgebra of

g LieG We wil l denote by R resp R the root system of h k resp h g and set R R n R

We can write

X

k h C E z gR

R

0

where z Z k is the center of k

X

g k m m C E

0

R

C

M of the tangent space T M of M G K at the point We wil l identify m with the complexication T

eK Then m fx M x xg is identied with T M and the isotropy representation of K on

C

jg T M is identied with the restriction to m of the adjoint representation Ad

K

FLAG MANIFOLDS

As before we denote by hR the of the Cartan subalgebra h spanned by roots

hR B R

and we put

t z hR

Then z it

Remark that roots from R are real linear forms on hR

Denition The restriction a j of a root R R n R to t is cal led a T root

t

We wil l denote the set of al l T roots by R It is a nite set of nonzero vectors in the Euclidean

T

vector space t In general R is not a root system

T

Denition

An element t t is cal led to be regular if any T root has nonzero value on t

fa g of regular elements is cal led a A connected component C of the set t t n

reg R

T

T chamber

The subgroup

R

0

W fw W w R R g

of the Weyl group W of R which preserves R is cal led T Weyl group

R

0

acts on the set of T chambers but the action is not transitive in general T Weyl group W

Realization of a ag manifold by an adjoint orbit

We say that an element t g denes a realization of the ag manifold G K by an adjoint

Then we have canonical identication of the ag manifold G K orbit if it has stabilizer K in Ad

G

t such that the point o eK is identied with t with the orbit Ad

G

Prop osition

An element t g denes a realization of the ag manifold M G K i it t

reg

R

0

orbit Two elements t t with stabilizer K belong to one G orbit if they belong to one W

R

0

t w t w W

CorollaryThe set of adjoint orbits of G isomorphic to M G K is parameterized by elements of

R

0

t W

reg

Proof of Proposition

t Hence t b elongs to the center Let t g has stabilizer K Then k Z

o g

z Z k h h

of k and

R f R t g R t

DVALEKSEEVSKY

This shows that it t

reg

Let t t b e elements with the common stabilizer K which b elong to the same orbit Then

K The element Ad preserves the center z of k and transforms t Ad t for some g N

g g G

a Cartan subalgebra h z h of k into a Cartan subalgebra h z h where h h are

Cartan subalgebras of the semisimple part k k of k Using the fact that any two Cartan

0

subalgebras of k are conjugated we can change g to g g a for some a K such that Ad

g

R

0

0

j h w W preserves the Cartan subalgebra h and its decomp osition h z h Then Ad

g

On the other hand

0

t w t t Ad t Ad

g g

a

b ecause t t z and Ad j z id

a



Closed invariant forms on a ag manifold

Prop osition Let M G K be a ag manifold and k it gR the decomposition of the Lie

algebra of K into sum of the center it and the semisimple ideal gR

There exist natural onetoone correspondence between element from t and closed invariant forms

on M

G

t t M

t

where is G invariant form on M whose value at o eK is given by

t

i

j dB t m T M

t

Here m is the Ad K invariant complement to k in g identied with the tangent space T M B is the

Kil ling form of h and d g h is the exterior dierential

X Y X Y g g d X Y

Moreover symplectic ie non degenerated forms corresponds to regular elements t t

t reg

The pro of is straightforward Remark that a regular element t t denes an identication

reg

it M G K Ad

G

it is exactly the KirillovKostantSouriau form on M Ad The corresp onding symplectic form

G t

0

t of the orbit Ad

G

X Y B t X Y X Y T M g

t t

0

Decomp osition of the isotropy representation into irreducible submo dules

Now we x a system of simple ro ots of the ro ot system R and denote by R R the

corresp onding systems of p ositive and negative ro ots

DenitionA root R R n R is cal led K simple if R for any R

FLAG MANIFOLDS

Prop osition There exist natural onetoone correspondence between T roots irreducible Ad

K

P

C

C E m of the ag manifold G K submodules of the complexied tangent space m

0

R

and the set of K simple roots from R R n R given by

X

R m C E

T

0

R j

t

module m where is the lowest weight of irreducible Ad

K

For pro of see Sieb AlPer GOV

Invariant complex structures and Kahler metrics

A regular element t t denes a partial order in t We say that a T ro ot is positive resp

negative if t resp t

A p ositive T ro ot is called to b e simple if it is not a sum of two p ositive T ro ots

The set

f g

T m

of all simple T ro ots is called a T basis It is a basis of t such that any T ro ot can b e written as linear

combination

X

k

i i

with integer co ecients k of the same sign all k or all k The T chamb er C t which

i i i

contains t is dened by inequalities

C t f g

m

Denote by

f g

m m

the set of the corresp onding K simple ro ots Then is a system of simple ro ots of R It

contain and dep ends only on T chamb er C t

Prop osition AlPer There exist natural onetoone correspondence between

T basises f g

T m

T chambers C f g

m

systems f g of simple roots of R which contain xed system of simple

m

roots of R

decomposition R R R of R R n R into disjoint union of asymetric closed subsets R

and R R

invariant complex structures on ag manifold G K dened up to a sign

Proof Let f g b e a T basis Then C f g is the asso ciated T chamb er

T m m

The pro of is the same as for ordinary Weyl chamb er Now we put

m

DVALEKSEEVSKY

It follows from prop erties of weights of an irreducible representation of semisimple Lie algebra that

contains a basis of t Using the formula for series of ro ots one can prove that scalar

pro duct b etween dierent elements from is non p ositive This implies that is a basis of R

Denote by R R the corresp onding systems of p ositive and resp ectively negative

ro ots and put R R R R n Then it is clear that R R R are closed sets of ro ots

and R R R is the disjoint sum Now we dene a decomp osition of the complexied tangent space

C

T M m m m where

m gR m gR

invariant they can b e extended to two complex invariant Since the subspaces m m are Ad

K

distributions T M and T M We dene an invariant almost complex structure J on M G K

such that T M and T M are eigendistributions of J with eigenvalues i and i resp ectively Since

k m is a subalgebra of g J is integrable see section

Conversely any invariant complex structure J on M G K denes a decomp osition

m m m m m

invariant subspaces m m such that k m is a subalgebra Denote by of m into sum of two Ad

K

Q resp Q the set of ro ots asso ciated with ro ot vectors from m resp m Then Q Q and

R R Q Q The closed set R Q is parab olic in the sense of Bourbaki and hence it contains

some system of p ositive ro ots R such that R R Then it is clear that Q R Denote by the

system of simple ro ots from R Then and the restriction of ro ots from n to t denes

a T basis



CorollaryThere exist natural onetoone correspondence between regular points t t and invariant

associated with point t is given by on ag manifold M G K The metric g Kahler metrics g

t t

0 0

J g

t t

0 0

is the symplectic form associated with to and J is the invariant complex structure associated where

t

0

with T chamber C t

Remark If we change J to complex structure J asso ciated with other T chamb er then we get invariant

g

pseudoKahler metric g ie a pseudoRiemannian metric with LeviCivita connection r such that

g

r J

Invariant KahlerEinstein metrics

Let f g b e xed basis of the ro ot system R and f g is

k m

a basis of the ro ot system R which contains We will denote by the fundamental weights

m

asso ciated with simple ro ots ie

m

i j

j j

ij i j j i j

j i

The fundamental weights form a basis of the space t dual to t and isomorphic to t via Killing form

i

G

The natural isomorphism b etween t and the space M of invariant forms on M is given by

X X

i i

h ji d t k

i i

0

R +

FLAG MANIFOLDS

invariant form where are forms on m dual to E E One can check that is Ad

K

on m T M Hence it denes an invariant form on M G K

Twoform is integer ie denes an integer cohomological class i the co ordinates k are

i

integers

The form

X

0

R

+

is called Koszul form It is a linear combination of weights with integer p ositive co ecients n

i i

m

X

n

i i

i

The numb ers n are called Koszul numb ers

i

Prop osition AlPer The form associated with Koszul form is the invariant nondegenerate

closed form which represent rst Chern class of the invariant complex structure J associated with

Moreover the corresponding metric g J is an invariant KahlerEinstein metric

Painted Dynkin graphs and classication of ag manifolds

Denition

Let g be a complex semisimple Lie algebra and k a complex which is isomorphic

to centralizer of some element t g The Dynkin graph of the root system R of a g with some

vertices painted in black is cal led a painted Dynkin graph of type g k if after deleting black

vertices we get the Dynkin graph of the root system R of k

A bijection V between roots of some basis of R and vertices V of such that

becomes the Dynkin graph of is cal led an equipment of the painted Dynkin graph

We will denote by and the subset of white and resp ectively black ro ots from ie the

W B

ro ots asso ciated with white and resp ectively black vertices of

A painted Dynkin graph of typ e g k with an equipment denes a ag manifold M G K

where G is the compact form of the Lie group G with Lie algebra g and K is the subgroup of G

asso ciated with the compact form k of the Lie algebra

k h g

W W

Remark that the Weyl group W of R acts on the set of equipments of a painted Dynkin graph

on the right and equipments w w W dene Gisomorphic ag manifolds

Assume that there exist equipments of

V V

which dene non isomorphic ag manifolds that is the subalgebras k and k generated by

W

W

white ro ots are not conjugated Changing to w for appropriate w W we may assume that

Then where is an automorphism of the underline Dynkin graph of which is

not an automorphism of the painted graph that is This is p ossible only when a some

W W

connected comp onent of has typ e A D or E We have

l l

DVALEKSEEVSKY

Prop osition Let be a painted Dynkin graph and underlined Dynkin graph

If has no nontrivial automorphism or any automorphism of is an automorphism of ie it

preserves the set of white vertices then al l equipments of dene isomorphic ag manifolds

Let V be an equipment of which denes a ag manifold M and is an automorphism

of which is not an automorphism of Then the equipment

V

of white roots associated with and dene a ag manifold non isomorphic to M if the sets

W

W

are not conjugated by an element of the Weyl group Moreover any ag manifold associated with an

equipment of can be obtained by this way

Remark It is not dicult to give the necessary and sucient condition that the set of white

W

W

ro ots asso ciated with equipments are not conjugated For example for A this condition

l

is that not all white vertices of are isolated

Examples

The equipments

23 l1l 12 l1l 12 23

ij i j

dene non isomorphic ag manifolds pro jective space of lines and the dual pro jective space of hyp er

planes

The equipments

12 23 23 l1l 12 l1l

l

dene isomorphic ag manifolds F C

l

Denition Two painted Dynkin graphs of the same type g k are cal led to be equivalent if there

exist equipments V V which dene isomorphic ag manifolds

Lemma Two painted Dynkin graphs are equivalent i there exist equipments V

V with the same set of white roots

W

Proof It is clear that equipments with the same set of white ro ots dene the same stability

W

subalgebra

k k h g

W W

and hence the same ag manifold

Conversely if equipments dene conjugated subalgebras kP i k then there exists an

W

W

element n from the normalizer of the joint Cartan subalgebra h of k and k which transforms

W

W

onto Denote by w the corresp onding element of the Weyl group W N hT where T the

W G

W

of G with asso ciated with Lie algebra h Then the equipments w has the same set

of white ro ots



To state the main result which give a classication of ag manifolds of a given compact semisimple

Lie group G we need the following denition

FLAG MANIFOLDS

Denition Let G be a painted Dynkin graph of type D k for some k

l

We say that has class B resp WWB or WWW if at least one of the right end vertices

is black resp are white but is black or respectively a are white

l l l l l l l l

Theorem Two connected painted Dynkin graphs are equivalent i they have the same type g k

and in the case g D the same class

l

Proof Due to Lemma the pro of reduces to construction for any two connected painted Dynkin graphs

of the same typ e and class equipments

V V

of white ro ots At was done by M Nishiyama Nish with the same set

W

W

Examples The painted graph

corresp onds to two non isomorphic ag manifolds of the form SU T SU SU

They are dened by the following equipments of

56 23 34 45 56 45 34 23 12 12

Painted Dynkin graphs equivalent to are the following

Painted Dynkin graphs T chamb ers and Koszul numb ers

Now we x a ag manifold M G K and a Cartan decomp osition

X X

g h C E C E k h

R R

0

of the corresp onding complex Lie algebras g k We x also a basis of the ro ot system R Let b e

a painted Dynkin graph of typ e g k We say that an equipment is admissible if the white

ro ots of form the set Let

n f g

B n

Denote by the corresp onding fundamental weights

m

i j

h j i h j i

ij i i j

j j

Prop osition There exist natural onetoone correspondence between T chambers of a ag manifold

M G K and admissible equipments The T chamber associated with an equipment

is given by

m

X

x B x g C ft

i i i

i

DVALEKSEEVSKY

where B is the Kil ling form and are fundamental weights associated to the black roots n

i i B

The corresponding invariant complex structure J is dened by the decomposition of the complexied

tangent space

X

C

C E T M m

0

R RnR

0

given by

m m m m gR m gR

where

R R R R

The Chern form of J is given by

X

i

h ji

0

R

+

where are forms on m dual to E E and

m

X X

n

i i

0

i

R

+

is the Koszul form and n are some natural numbers Koszul numbers

i

If g is one of the classical Lie algebras A B C D the Koszul numb ers can b e calculated as

l l l L

follows AlPer

n b where b is the numb er of white ro ots which are connected with the black ro ot in

i i i i

the painted Dynkin graph by a chain of white ro ots

In the case g B the white long ro ots of the last right white chain are counted with multiplicity

l

two and last short ro ot is counted with multiplicity one

In the case g C white ro ots of the last right white chain are counted with multiplicity two

l

In the case g D last white chain which dene the ro ot system D is considered as chain of

l k

lenght k If one of the right end ro ots is black and the other is white then the co ecient b near

i

the black ro ot is equal k where k is the numb er of white ro ots connected with the black ro ot

The calculation of Koszul numb ers for ag manifolds of exceptional Lie groups is an op en problem We

see that the Koszul numb ers of an invariant complex structure do not dep end on equipment of a painted

Dynkin graph but only on

The following prop osition give an explanation of this phenomenon

Prop osition Let be admissible equipments of painted Dynkin graphs

which dene a ag manifold M G K Denote by C C resp J J the associated T chambers

resp invariant complex structures on M

Then the fol lowing condition are equivalent

i the painted graphs are isomorphic

R

0

ii T chambers C C are equivalent with respect to T Weyl group W

R

0

C w C for some w W

iii the complex structures J J are Gdieomorphic ie there exist a dieomorphism f of M which

commutes with G such that f J J

FLAG MANIFOLDS

Proof Let b e a painted Dynkin graph and V V two admissible equipments

of which dene a ag manifold M G K Then using case by case study of the ro ot systems R

R

0

such that w for all simple Lie algebras g one can check that there exists an element w W

R

0

equivalent T chamb ers C C Hence admissible equipments of denes W

C w C

R R

0 0

equivalent T chamb ers Denote by b e two W Conversely let C C w C w W

B

B

the corresp onding systems of simple ro ots Then

w w w

B

B

R

0

This shows that the corresp onding painted Dynkin graphs are isomorphic It remains to show that W

equivalent T chamb ers C C w C corresp ond to Gdieomorphic invariant complex structures J J on

M G K

It is known that a Gdieomorphism of M which commutes with G is given by

n g K g K n g nK g K G K

k for n N

G

We may assume that Ad preserves the Cartan subalgebra h of k This means that the group

n

k K of Gdieomorphisms of M G K is identied with a subgroup of the T Weyl group N

G

R

0

h T AR R g fA W N W

G

Now the equivalency of ii and iii follows from the fact that T Weyl chamb er C and invariant complex

structure J determine each other

If is the system of simple ro ots asso ciated with C then m g is the ieigenspace

B B

for the corresp onding complex structure J Conversely if m is ieigenspace for an invariant complex

structure J then is the set of all simple ro ots from asymmetric closed system

B

R R E m

and

C ft t t g

B



Remark At is not dicult to prove that two invariant complex structures J J on a ag manifold

M G K are dieomorphic i the corresp onding T chamb er C C are Aut R R equivalent where

AutR R is the group of all automorphisms of R which preserves R

Some examples

Now we illustrate the general theory by some examples For simplicity we will consider only ag

manifolds of the group of typ e A

l

Any painted Dynkin graph of typ e A is equivalent to a painted graph of the following typ e l

DVALEKSEEVSKY

Besides the left chain of black vertices of length m which may b e absent all other black

vertices are isolated The lengths of the chains of white vertices are

n n n k

k

P

and l n m

i

Such graph will b e called standard It dene two non isomorphic ag manifolds of the typ e

m

M SU U S U n U n

l k

if has no non trivial automorphisms or not all white ro ots are isolated ie n In other cases

dene unique ag manifold M

To describ e all equipments of the describ ed standard graph and the graph equivalent to its it

l

is useful to change the notation for the vectors of the standard basis of the space R as follows We

denote by

k k

m

n n

k 1

l

the vectors of the standard orthonormal basis of the space R restricted to the hyp erplane

n o

X

l

hR x R x

i

Then the ro ot system R of A is given by

l

a a a ab a b

R

ij i j i i

j ij i j ij i j

To simplify the notation we also set

b ab b

n

a

We dene the standard equipment of the standard painted Dynkin diagram as follows

1 12 2 23 3 k1k k 1 2 k

mm

12 23

12 12 12 12 n 1n n 1n n 1n

1 1 2 2

k k

Now we describ e admissible equipments of all painted Dynkin graphs representing the ag mani

folds of the typ es

M SU T S U U and M SU T S U U

We consider only one of two such manifolds for each of the groups SU and SU The equipments

asso ciated with other manifold of such typ e can b e obtained by of Dynkin graph

Case of the manifold

M SU U S U U g A k C A A

1 12 2 2 1

1

12 12 13 1

2 2 21 1 1

1

12 23 12 2

FLAG MANIFOLDS

1 1 2 2 2

1 1

12 2 1 12 23

2 2 1 1 2

1 1

23 3 1 12 12

12 2 2 2 1

1

12 23 3 12

2 2 21 1 2

1

12 23 12 1

Hence on the ag manifold M SU U S U U there exist distinct invariant complex

structures They are mutually non Gdieomorphic but only three of them are mutually not dieomor

phic

Case of the manifold M SU U S U U g A k C A A

1 1 1 12 2

2 12

1 12 23 12

21 1 1 1 2

12 1

12 23 3 12

2 2 21 1 1

2 12

1 12 12 23

1 12 2 2 1

1 12

23 12 2 12

2 2 1 1 1 2

2 2 1

12 2 1 12 23 1

1 1 2 2 1

1 2 12 2

23 3 1 1 12

1 1 1 1 2 2

1 2 2

1 12 13 3 1 12

2 1 1 1 1 2

1 1 2

2 1 12 12 3 12

2 1 1 1 2

1 12 2

2 1 12 23 12

1 1 2 2 1

1 12 2

23 3 1 12 12

Other admissible equipments are obtained from describ ed ones by transp osition which

b elongs to the T Weyl group

Hence we have dierent painted Dynkin graphs which corresp onds to mutually non dieo

morphic invariant complex structures

DVALEKSEEVSKY

Each of these graphs has admissible equipments These four equipments determine four dierent

but dieomorphic invariant complex structures which is divided into two pairs of Gdieomorphic

complex structures The Gdieomorphic complex structures is asso ciated with two equipment of the

graph which are related by the transp osition

References

Al Alekseevsky DV Homogeneous Einstein metrics Di Geom and its Applic Pro ceed of the

Conf Brno

AlPer Alekseevsky DV Perelomov AM Invariant KahlerEinstein metrics on compact homogeneous

spaces Funct Anal and Applic

Bes Besse A Eistein manifolds SpringerVerlag

BFR Bordemann MForger M Romer H Paving the way towards supersymmetric sigmamodels Com

mun Math Phys v

Bor Borel A Kahlerian coset spaces of semisimple Lie group Pro c Nat Acad USA D N

BorHir Borel A Hirzebruch F Characteristic classes and homogeneous spaces Amer J Math

N

Bourb Bourbaki N Groupes et algebres de Lie ch Hermann Paris

GOV Gorbatzevich VV Onishchik AL Vinb erg EB Structure of Lie groups and Lie algebras Encycl

of MathSci v Lie groups and Lie algebras Springer Verlag

Kos Koszul JL Sur la forme hermitienne canonique des espace homogenes complexes Canad J

Math N

Nis Nishiyama M Classication of invariant complex structures on irreducible compact simply con

nected coset spaces Osaka JMath

Sieb Sieb enthal J Sur certains modules dans une algebre de Lie semisimple Comment Math Helv

N

Tits Tits I Sur les constantes de structure et le theoreme dexistence des algebres de Lie semisimples

Publ Math IHES v