Chapter XII Operation of a Lie Algebra Pair

Chapter XI1

Operation of a Lie Algebra Pair $1. Basic properties 12.1. Definition: Let ( E , F ) be a reductive Lie algebra pair (cf. sec. 10.1) and assume that ( E , i, 8, R, S,) is an operation of E. This operation restricts to an operation ( F , i, 8, R , 6,) of F (cf. Example 3, sec. 7.4). T h e corresponding invariant subalgebras of R will be denoted respectively by Re,=o and ReFX0. We shall say that ( E , F, i, 8, R, 8,) is an operation of the pair (E, F ) in the graded dzflerential algebra ( R , 8,) if the inclusion map Re,=, + ReF=,induces an isomorphism H(Re,-o)

--%H ( R O F - o ) *

Given such an operation, we adopt the following notation conventions,

to remain in force for the entire chapter:

(i) w R denotes the degree involution of R : wRz = ( - 1 ) p z , z E RP. (ii) T h e horizontal and invariant subalgebras for the underlying operations of E and F are denoted, respectively, by

(iii) The basic subalgebras for the operations are written

(iv) T h e obvious inclusions are denoted by

e E :B E + ReE=,,,

e F :B ,

+

R,,o

and

e: BE -+ BF.

and H(ReF=o) are identified (v) T h e cohomology algebras H(RoE=o) via the isomorphism induced by the inclusion map, and are denoted 498

1. Basic properties

499

simply by H(Reeo). I n particular, we have the commutative diagram

(vi) If the algebra H(R,=o) is connected, then the fibre projections associated with the operations of E and F are denoted respectively by eB:

-

H(ROxo)

(AE"),+o

and

-

&: H(ReSo)

(AF*),=,

(cf. sec. 7.10). (vii) T h e algebras APE, (AE*),=o, and H*(E) (respectively, APF, (AFY;)o,O,and H " ( F ) ) are identified via the isomorphisms X E and x i (respectively, xF and xf) of sec. 5.18 and sec. 5.19. (viii) The inclusion map of F into E is written j : F + E. It induces as described in sec. 10.1. homomorphisms j", j$=,,jv,and T h e basic subalgebra for the operation (F, iF, O F , AE", 6,) is (ix) denoted by (AEIF)(F,O,eF,O and its cohomology algebra is written H ( E / F ) . The inclusion map (AE*)iF=o,eF=o + AE" is denoted by k.

It is the purpose of this chapter to express H(BF) in terms of H ( B E ) and other invariants. 12.2. The associated semisimple operation. Consider an operation

(E, F, i, 8, R, 6,) of a reductive pair (E, F ) . T h e operation of E in (R, 6,) determines the associated semisimple operation ( E , i, 8, R, , 6,) as constructed in sec. 7.5. I n particular, 8 is a semisimple representation of E in R,. Since F is reductive in E, 8 restricts to a semisimple representation of F in R s , as follows from the definition of Rs and Proposition 111, sec.

4.7. Thus the inclusions

induce isomorphisms of cohomology (cf. Proposition I, sec. 7.3). Hence

500

XII. Operation of a Lie Algebra Pair

the same holds for the inclusion (Rs)e,-o -+ (Rs)e,-o, and so ( E , F, i, 6, Rs, 6,) is an operation of the pair (E, F). Finally consider the inclusion map Rs + R as a homomorphism of operations of F. Proposition I: Assume that the operation of F in Rs is regular. Then the inclusion induces an isomorphism

between the cohomology sequences of the two operations. Proof: In view of the corollary to Theorem 111, sec. 9.8, it is sufficient to show that the inclusion (Rs)e,-o -+ReF,, induces an isomorphism of cohomology. But this follows from the observation that ReE,, = (Rs)e,-o and the resulting commutative diagram

12.3. The structure operation. Let (E, F, i, 8, R , 6,) be an operation of a reductive pair (E, F). Recall from sec. 7.7 the definition of the structure operation

(E, iR@E

9

6R@E 9

8 AE",

associated with the operation of E in (R,6,).

dR@B)

so1

1. Basic properties

-

Proposition 11: The inclusion map

( ( R@ AE")o,=o

9

ROE)

( ( R0AE")o,=o

P

sRc3E)

induces an isomorphism of cohomology algebras; i.e., the structure operation is an operation of the pair ( E , F ) . Proof: Define an operator 6 in R @ AE" by setting

In sec. 7.7 we constructed an isomorphism of graded differential algebras @: ( R 0AE", 6)

( R @ AE",

BRBE)

satisfying @ 0 eROE(x)= ORBE(x)0 @,x E E (cf. Proposition V, sec. 7.7). Thus it is sufficient to show that the inclusion map induces an isomorphism

H ( ( R 0AE"),,,,

6) 5 H ( ( R 0AE")op=o,6).

which is induced by the obvious inclusions. In view of Proposition IV, sec. 7.6, the vertical arrows are isomorphisms. Our hypothesis on R implies that the upper horizontal arrow is an isomorphism. Hence so is the lower horizontal arrow. Q.E.D. 12.4. Fibre projection. Let ( E , F, i, 8, R, S,) be an operation of a reductive pair and assume that H(R,,,) is connected. In this section we shall define a homomorphism

to be called the fibre projection for the operation of the pair ( E , F ) .

502

XII. Operation of a Lie Algebra Pair

First, consider the inclusion map g:

Since

0(AE")iF=o,eF=o

[R 0(AE")iF=o]eF=o.

-+

SROE = 6~ 01

(cf. sec. 12.3 and sec. 7.7), and

60

+ +

WR

0SE

og = 0, it follows that

SROEog=go(81(@1+

WROBE).

Hence g induces a homomorphism

Lemma I: The homomorphism g # is an isomorphism. Proof: Filter the differential algebras by the ideals

Fp= and

C R0E-o O (AjE")iR=o,~F-o

j2P

pp = C [R O (A'E")i,-o]eF-o. j2P

Then g is filtration preserving, and so it induces homomorphisms gi:

(Ei, 4 ) -* (EL , 4

between the corresponding spectral sequences. Now consider the commutative diagram

H { [ RO (AE")i,=oIe,=o,

SR

O11 7 4

where A1 and 1, denote the obvious inclusions (cf. formula 1.8, sec. 1.7). Our hypothesis on R asserts that 1T is an isomorphism. Proposition IV,

1. Basic properties

503

sec. 7.6 (applied with A4 = (AE"),,=, and ,S = 0) shows that A$ is an isomorphism. Hence g, is an isomorphism and so by Theorem I, sec. 1.14, g# is an isomorphism.

Q.E.D.

Next, recall the structure homomorphism y R : R + R @ AE" defined in sec. 7.8. Since y R is a homomorphism of operations, it restricts to a homomorphism of graded differential algebras

For the sake of brevity, this map will be denoted simply by y R I p . On the other hand, since H(R0,,,) is connected, we have the canonical projection zR:

H(&=o)

0H ( E / F )+ H(E/F).

Definition: The jibre projection for the operation ( E , F, i, 8, R, SR) is the homomorphism

H(E/F)

P R : H(BF)

given by P R = XR

(g")-'

y$P*

12.5. Projectable operations. An operation ( E , F, i, 0, R,S R ) of a reductive pair ( E , F ) is called projectable if the underlying operation of E is projectable (cf. sec. 7.11). Let (E, F, i, 8, R, SR) be a projectable operation with projection q: R + Then the linear map

r.

q @ 1 : R @ (AE"),,,

+

(AE*)i,=o

induces a map

(cf. the relations above Proposition VII in sec. 7.11). Proposition 111: T h e fibre projection of a projectable operation is given by p R =

(q @ '!)e#,=O

Y&F*

504

XII. Operation of a Lie Algebra Pair

Proof: Consider the restriction qeF-o:ReF,, -+ T.Then ZdR

0L = (q 0L)&O

= q&=o

og#.

The proposition follows.

Q.E.D.

Example: Consider the operation (F,i ~ , AE*, d ~ ) where , (E,F ) is a reductive pair. Since 8B7,

(AE*)e-o EZ H((AE*)+o, )6,

EZ H*(E),

it follows that this is an operation of the pair ( E , F ) in (AE", dE). In this case BE = 'I and BF = (AE*)+O,eF=O. Moreover, AEX is connected and so the operation is projectable. Now we show that the fibre projection PMr:

H ( E / F )-+ H ( E / F )

is the identity map. In fact, according to Example 1, sec. 7.8, the structure homomorphism y for AE* satisfies y(@) - (1 0@) E

Hence,

(A+E*) @ AE',

@ E AE*.

(40L)(Y@) = @,

where q : AE* + r denotes the projection. Restricting this equation to (AE*)iF-O,BF-o yields

(48' ) e F - o

yAE*/F

= '*

Now pass to cohomology and apply Proposition 111, above. 12.6. Algebraic connections. Suppose ( E , F, i, 8, R, S,) is an operation of a reductive pair, such that the underlying operation of E admits an algebraic connection XE: E* -+ R'. Let X : F* + E* be an algebraic connection for the operation (F, iF, O F , AE*, 6,) (cf. sec. 10.5). Then the linear map Xp=

XEo

X :F"+R'

is an algebraic connection for the operation (F, i, 8, R, dR).

505

1. Basic properties

Definition: T h e triple (X, XE , X,) will be called an algebraic connection for the operation of the pair ( E , F ) .

I n view of Theorem I, sec. 8.4, the algebraic connections XF determine isomorphisms of graded algebras

f

=L

5 AE",

@ X, : (AE")+,o @ AF*

f E =L

@ ( X E ) , : Ri,,o @ AE"

f p =L

@ (XF), : Ri,,, @ AF"

X , X E , and

5R,

and

-

R.

Moreover, fE restricts to an isomorphism

and, clearly, the diagram

RiE..O @ (AE")iF=o@ AF" fE@t

I

=fE

-

RiE=o@ AE"

= N

fF

commutes. Thus an isomorphism

is given by f E , p

$of

N

RiF,, @ AF"

fE,p

-

I

f,

* R

: RiE=O@ (AE")iF=o@ AF" -3R 0 (1

Of) =fB7

0

(fE @ L).

Lemma 11: Let ( X , X,, X,) be an algebraic connection for the operation of ( E , F ) . Then the curvatures of X , X E , and X F are related by

xF(y*) =

zE(

Xy")

+(

X E ) ~ z.Y"), (

y"

F**

XII. Operation of a Lie Algebra Pair

506

If we identify RiEZo@ (AE*)+, 0AF" with R via the isomorphism formula in Lemma I1 reads

fE,F, the

ZF(Y") = z ~ ( X y *@ ) 1 01

+ 1 @ Zy" @ 1,

y*

E

F".

(12.1)

An operation ( E , F , i, 8, R, 6,) is called regular if the pair (E, F ) is reductive, H(Ro=o)is connected, and the operation admits an algebraic connection. Thus an operation of a reductive pair ( E , F ) is regular if and only if the underlying operation of E is regular (cf. sec. 8.21). Proposition IV: Let ( E , F, i, 8, R, 6,) be a projectable regular operation, with algebraic connection (X, Xg , X F ) . Assume @ E (AE")i,=o,e,=o and Q E BF are homogeneous elements of the same degree, such that

(1) and

6E@ = 0

and

6R(Q

+

(XE)A@)

(2) fg!~(Q) E RtE=O@ (AE")i,=o

= 0,

01.

Let E E H(B,) be the class represented by Q represents p R ( & ) .

+ (XE),,@.

Proof: It follows respectively from the definition of corollary to Proposition IV, sec. 8.10, that

fE,F

Then @ and the

YnmQ E (R+0(AE*)i,=o)o,=o

and

Y R / F ( ( 'E)A@)

Hence, if q : R

--+

-

0@

(R+0(AE*)iF=O)@F=O'

r is the projection

Now apply Proposition 111, sec. 12.5.

Q.E.D.

12.7. The cohomology diagram. Let ( E , F , i, 8, R, 6,) be a regular operation. Then the Weil homomorphisms

-

zg: (VE*)@=o H(B,), and

z:

(VP),,,

Z*: (VF")o=o -+ H ( E / F )

-

H(B,)

1. Basic properties

507

are defined. These, together with the homomorphisms introduced in sec. 12.1 and sec. 12.4 yield the diagram (VE")O=O

xH

H(BE)

It is called the cohomology diagram of the regular operation (E, F, i, 8 , R, b). The cohomology diagram combines e*

(1) the sequence H(BE) -+ H(B,) ..% H ( E / F ) , (2) the cohomology sequence for the operation of E in R, (3) the cohomology sequence for the operation of F in R, (4) the cohomology sequence for the pair ( E , F ) . In sec. 12.21 it will be shown that the cohomology diagram commutes. 12.8. Homomorphisms. Let ( E , F, i, 8,R, 6,) and ( E , F, i, 8, R, 6 ~ )

denote operations of the pair ( E , F ) . A homomorphism, v: R + R, of operations of E is automatically a homomorphism o f operations of F, and will be called a homomorphism of operations of the pair ( E , F ) . Such a homomorphism restricts to homomorphisms v B E : BE

+

8,

and

pBF

BB*+ B F

Moreover, if (E, F ) is reductive and H(R,=,) is connected, then the diagram

H(BF)

'"

H(&F)

commutes, as follows from the definitions.

508

XII. Operation of a Lie Algebra Pair

Finally, let ( X , XE , XF) be an algebraic connection for the first operation and set iE= pl 0 XB and = pl 0 X F . Then ( X , i E , i F ) is an algebraic connection for the second operation. In particular, a homomorphism between regular operations induces a homomorphism between the corresponding cohomology diagrams.

§2. The cohomology of BF 12.9. Introduction. In this article (E, F ) is a reductive Lie algebra pair, z: PE +. (VP),,, is a transgression (cf. sec. 6.13), and 'V

'G

0 =/,3=o

:P

E

* (VF*),9=0.

Then ((VF*),=,; a) is a PE-algebra, and the cohomology algebra H((VF*),=o @ A P E , --F) is isomorphic to H ( E / F ) (cf. Theorem 111, sec. 10.8). Let (E, F, i, 8, R, S,) be a regular operation with algebraic connection ( X , X E , X p ) (cf. sec. 12.6). Consider the map tR:P E + B E given by zR

=

(xE)v,O=O

t.

Then ( B E ,6,; zR) is a ( P E ,6)-algebra and the cohomology of the corresponding Koszul complex ( B E @ APE, VB) is isomorphic to H(Re=o) (cf. Theorem I, sec. 9.3). Now consider the tensor difference ( B E0(VF"),,,, BR; zR 0a), (cf. sec. 3.7). Its Koszul complex is given by

@ (VF*)i3=0 8

(BE

where

l7 = SR

with O,,(Z

1

@1

17),

+ K R - 17,

0Y / @ GoA - - - A QP) = (-1)s

and

0

9

c P

t,(@i)

i-0

ro(Z0p @ GoA = (-1)g

c P

i=O

* * *

A

- X @ p @00

A

A

. * -

@i

GP,

A

* * -

DP)

X @ a(@$)V p @ Q0A

z E B%,

A * * *

@i

Y E (VF*)e=.O,

* * .

A

@i

@,

E

PE.

12.10. The main theorem. In this section we state the main theorem of this chapter. It contains Theorem I, sec. 9.3, and Theorem 111, sec. 10.8, as special cases. 509

XII. Operation of a Lie Algebra Pair

510

Theorem I: Let ( E , F, i, 0, R, 6,) be a regular operation, with algebraic connection ( X , X E , X,) and let t be a transgression in W(E),=o. Then there are homomorphisms of graded differential algebras y : (BE @ 6R:

and Q,:

((VJ#),=o

(VF9)8=0

(BE 0APE

-

8APE v ) 9

3

G)

63 APE, -%)

(BF

9

(Re,=o d ~ ) , 9

((AE8)iF=o,eF=o, d~),

with the following properties : (1) The induced homomorphisms y # , 8; , and p$ are isomorphisms of graded algebras. (2) The isomorphisms y # , t9;, p$, and the isomorphism t,:VPE

'c1

(VB")e=o

determine an isomorphism between the cohomology diagram

of the tensor difference, and the subdiagram

of the cohomology diagram of the operation.

2. The cohomology of BF

511

Remark: T h e rest of this article is devoted to the construction of 6, and y and the proof that 6; and y" are isomorphisms. T h e rest of (l), and (2), will be established in article 3. I n article 4, we shall derive corollaries and less immediate consequences of the theorem. In particular, the reader may skip directly to article 4,without going through the proof of Theorem I. T h e proof of the first part of the theorem is organized as follows. We construct a (noncommutative) diagram

BE

0(VF#),,o

@ APE

PP

BE @ APE

of graded differential algebras, which yields a commutative diagram of cohomology algebras. In particular, the lower half of the cohomology diagram is the commutative diagram in Corollary I , sec. 8.20. T h e difficulty is in the construction of 6 and 8;, this is done in sec. 12.13 and sec. 12.14. The homomorphism y is defined by y = aF 0 6, and the article concludes with the proof that y e is an isomorphism (sec. 12.16). 12.11. The algebra (( VF* 0R)oF=O, DF). We apply the results of secs. 8.17-8.20 to the operation of F in R. T h e antiderivation denoted there by D, will here be denoted by Dp:

DF = I, @ 6r( -

cP . s ( F ) 0

UP),

e

where fie, f, is a pair of dual bases for F" and F. According to sec. 8.17, ((VF" @ R)e,=o,D F ) is a graded differential algebra. Let c F : BF -+ ( V P @ R)eF=Obe the inclusion. I n sec. 8.19 we constructed a homomorphism (here denoted by aP) aF:

((vp"0R)BF=O D P ) 3

-

(BP

9

of graded differential algebras such that aF o cF = L. Moreover, by Theo-

XII. Operation of a Lie Algebra Pair

512

rem IV, sec. 8.17, and Proposition IX, sec. 8.19, a$ and isomorphisms. Finally, let

E$

are inverse

be the obvious inclusion and projection. Then Corollary I of sec. 8.20 yields the commutative diagram

(VJ'")e-o

rg

H((VF" @ R)e,=o)

G

H(Re,-o) (12.3)

12.12. The algebra VB" @ W(E). Consider the operation (E, i, O w , W(E),6,) (cf. Example 6 of sec. 7.4 and Chapter VI). Recall the de-

composition

-

dw = BE

+ 8, + h

and the projection nE:W ( E ) AE". Finally recall the canonical map (cf. sec. 6.7) @ E : (V+E")e=o ---* (A+E")e-o. Extend nE to a homomorphism nE:

VB" @ W ( E )-+ AE"

by setting nE(V+BW @ W ( E ) )= 0. Similarly define homomorphisms nj: VB" @ W ( E )

---f

and

W(E),

j = 1,2,

2. The cohomology of BF

513

T h e homomorphisms nE , ncl, and nz restrict to homomorphisms zE:

(VE" @ W(E)),=o+ (AE*),=,

and nj:(VE" @ W(E)),=,-+ W(E),=,,

j

=

1,2.

Now apply the results of secs. 8.17-8.20 to the operation (El i, O,,, W ( E ) ,dE). Consider the operator

in VE* @ W ( E )(P,e, a pair of dual bases for E" and E ) . According to sec. 8.17, Dw restricts to a differential operator in (VE" @ W(E)),=,. Moreover, combining Theorem IV, sec. 8.17, with Corollary I to Theorem V, sec. 8.20 yields the commutative diagram H((VE"

(Here E and

E

0w q ) o = o )

are the inclusion maps given by E ( Y )= 1 @ Y @ 1,

and

t(Y)= Y @ 1 @ 1

Recall from Theorem I, sec. 12.10 that

W E ) , = ,.

t

Y E (VE"),,,.)

denotes a transgression in

Lemma 111: There is a linear map, homogeneous of degree zero, s:

PE

+

(VE" @ W(E)),=,

with the following properties: For @ E P E ,

(1) Dw(s@)= 1 @ T@ @ 1 - t@ @ 1 @ 1. (2) dwnj(s@)= T@ 01, j = 1, 2. (3) n&@) = @.

XII. Operation of a Lie Algebra Pair

514

Proof: The diagram above shows that for @ E P E , 1 @ t @ @l t @ @ 1@

-

1 I m~D l v .

Hence there is a linear map, homogeneous of degree zero, s: P E

such that

(VE" @ w(E))o=o,

D,,(s@) = 1 @ t@ @ 1 - t@ @ 1 @ 1,

@E PE.

This establishes (1). A simple calculation shows that in (VE" @ W(E))o=o

n,Dlt-= Swn,

and

n,DIV= -&a2.

Substituting these relations in (1) we obtain (2). Finally, recall from Lemma V I I , sec. 6.13, that @ E 0 t = 1. In view of the definition of @ E it foIlows from (2) that (for @ E p E ) @

@Ex(@)

= n E ( n j ( s @ ) ) = nE($@),

j

=

1, 2.

Q.E.D.

12.13. The homomorphism 8. In this section we define a homomorphism of graded differential algebras

8:( B E0(VF"),,,

@ APE,

r)

+

((VF" @ R)oF=O,Dp)

First, consider the homomorphism

j" @ ( X E ) I ~ VE" : @ W ( E )+ VF" @ R, where ( X E ) I I . is the classifying homomorphism of the algebraic connection X E (cf. sec. 8.16). This homomorphism is F-linear (with respect to the obvious representations of F) and satisfies DF

(i'@ (XE)IIr) = (i'@ ( X E ) W )

DWa

Hence it restricts to a homomorphism

I : ((VE'

0W(E)),,=o, D,,.)

+

((VF"

0R)oF=O,DF)

of graded differential algebras. Lemma 111, (1) shows that

(DF o I ) ( s @ )= 1 @ tB@

-

00 @ 1,

@ E PE.

(12.4)

2. The cohornology of I?,

515

Now extend s to a homomorphism s,: APE

-+

(VEY @ W(E)),,=O,

and define 6 by S(Z 0Y @ @) = (Y @ 2) . I(s,,@),

xE

Y E (VF"),=o, @ E APE.

BE,

Then it follows at once from formula (12.4) that

60v

= D,

0

6,

and so 6 is a homomorphism of graded differential algebras. I n sec. 12.15 it will be shown that 6# is an isomorphism. 12.14. The homomorphism 4,. Let sl: PE + W(E),=obe the linear map given by s1 = 7c1 0 s. I n view of Lemma 111, sec. 12.12, we have drV(sl@) = T@ @ 1 and s ~ @ - 1 @ @ E ( V + P @ AEL),=o,

@ E PE.

Now define a homomorphism 6Rz

BE @ APE

+

Re,=,

by setting GR(Z

0@) = x

E BE,

' [(XE)IV(Si)~(@)],

@

Then GR is the Chevalley homomorphism associated with the algebraic connection XE and the linear map s1 (cf. sec. 9.3). InJarticular, 1 9 ~is a homomorphism of graded differential algebras. Moreover, by Theorem I, sec. 9.3, 6if is an isomorphism. 12.15. Proposition V:

8': H ( B E @ (VJ"),,o

The homomorphism

@ APE, V ) 4H ( ( V F " @ R)e,=o, D F )

induced by 6 is an isomorphism (cf. sec. 12.13). Proof: Filter the algebras BE @ (VF"),=o respectively, by the ideals

FP =

C B E @ (VFY)i=o@ APE

j2 P

and

0APE and ( V P @ R)o,=o, PP =

,x

3ZP

[(VF')' @ R]ep=O.

516

XII. Operation of a Lie Algebra Pair

Then 6 is filtration preserving and so we have an induced homomorphism of spectral sequences

6,:(Ei,di) -+ (Si, di),

-

i L 0.

In view of the comparison theorem (sec. 1.14) it is sufficient to show that

88: W E 0 , do)

f w o9

do)

is an isomorphism. First observe that

(Eo , do) = (BE 0( V F * ) e = o

O A P E , dB 01 06 + ER)

(cf. sec. 12.10), and that

commutes, where 1 is given by

and E ~ e2, are the obvious inclusions. Since 1 is an isomorphism, so is

According to sec. 12.14,

&?:

1.92 is an

isomorphism. By hypothesis

ff(K+=o)

-+

H(Re,-o)

is an isomorphism. Finally, Proposition IV, sec. 7.6, (applied with

2. The cohomology of B,

M

=

VF', 6,

(6

=

-

517

0, and the Lie algebra F ) shows that

@ E Z ) * : (VP"),=, 0H(R,,=o)

H((VF'

0R)B,=O, 1 08,)

is an isomorphism. I t follows that 68 (and hence 8#)is an isomorphism.

Q.E.D.

Lemma IV: The diagram (12.5) commutes. Proof: Let

be the homomorphism given by

Then diagram (12.5) commutes (obviously) if 8, is replaced by 8. Hence it is sufficient to show that 6o = ,8; equivalently, we must prove that

6 - 6 : B E @ (VF')i=o @ APE

+

C

j>P+l

( ( V P ) ' @ R)Bp=O.

Since 8 and 6 agree in BE @ (VF*)e=o@ 1, and since both are homomorphisms, it is sufficient to check that for @ E P E

( 8 - 6)(1 @ 1 @ @)

E

(V+F' @ R)B,=o.

(12.6)

But

(8 - @)(I @ 1 @ @)

= =

1 @ ( x E : ) W ( s l @ ) - (i"@ ( x E ) I Y ) ( s @ ' > 1 @ n l S @ - s@).

(Y @ ( x E ) , ) (

In view of the definition of 1 @ n1Q - 52 E V+E" @ W ( E ) ,

SZ E VE' @ W(E).

Hence, in particular,

( p@ ( x E ) , ) (

1@

- $0)

[p@ (xE)fi'](V+E' @ w(E))O~=O c

This establishes formula (12.6).

( V + P @ R)BF=O. Q.E.D.

XII. Operation of a Lie Algebra Pair

518

12.16. The homomorphism the homomorphisms CZF:(

9.

Recall from sec. 12.11 and sec. 12.13

V P @ R)e,=o -+ Bp

and @; BE

0(VF*),=o 0APE

+

(VF" @ R)e,=o.

Their composite will be denoted by y : (BE@ (VF")o=o @ APE

9

v)

+

( Bp 8,).

I t follows from sec. 12.11 and Proposition V, sec. 12.15, that y # is an isomorphism. Next, consider the diagram (12.2) of sec. 12.10. It is immediate from the definitions and formula (12.6), sec. 12.15, that the upper half commutes. Thus in view of the commutative diagram (12.3) of sec. 12.11 the diagram of cohomology algebras induced by (12.2) is commutative. In particular we obtain the commutative diagram

H(BE @ (VF")e=o @ APE)

2H(BE @ APB) sz 8%

.

(12.7)

$3. Isomorphism of the cohomology diagrams 12.17. T h e purpose of this article is to prove the rest of Theorem I, sec. 12.10. We carry over all the notation developed in article 2. Recall from sec. 12.14 and sec. 12.16 the isomorphisms

8:: H(BE @ APE) -5H(Re,o) and y # : H(BE @ (VF*),,o @ APE)

N_

H(B,).

T o finish the proof of Theorem I, we have to construct a homomorphism of graded differential algebras Q)F:

((VF*)e=o@ A P E , -K)

--t

((AE*)iF=o,eF-o, 6~1,

with the following properties: is an isomorphism. (2) The isomorphisms @, p f , y*, and t, define an isomorphism from the cohomology diagram of the tensor difference to the cohomology diagram of the operation (E, F,i, 8, R,6,). I n other words, the following diagrams commute: (1)

Q)$

vL?E

( 7 ~ ) :

7vlz

(VE*),=o

7v

I

G

__+

H(BE)

,I=

H(BE)

6

H(BE @ APE)

i

r 8;

eE'

4

1-

APE (12.8)

(AE*)e=o H(Re=o)7

=

519

9

520

XII. Operation of a Lie Algebra Pair

N

8;

(12.10)

.

(12.11)

First observe that, in view of the definition of t!lR(sec. 12.14), it follows from Theorem 11, sec. 9.7, that (12.8) commutes. That (12.10) commutes is established in sec. 12.16. Moreover, clearly y o mg = e, and so diagram (12.11) commutes. Finally, in sec. 12.18 we shall construct p F , prove that 93 is an isomorphism, and show that (12.9) commutes. I n sec. 12.19 we show that diagram (12.12) commutes. 12.18. The homomorphism cpF. Define s2: PE -+ W(E),=oby

(cf. sec. 12.12). Lemma I11 of sec. 12.12 implies that for @ E PE: 81ysz(@) = r@ 01

and

s,Q, -

1 0@ E ( V + P @ AE*),,,.

3. Isomorphism of the cohomology diagrams

521

The corresponding Chevalley homomorphism for the pair ( E , F ) (as defined in sec. 10.10) is the homomorphism

8,: (VF"),,, @ APE

+

(VF" @ AE")ep=o,

given by 8F(Y @ @) = (Y @ 1)

. ( j .@ L ) ( ( s ~ ) ~ @ ) , Y E (VF"),=o,

@ E APE.

Recall that (X,X E , X,) is a fixed algebraic connection for the operation of ( E , F ) in R. In sec. 10.9 we constructed a homomorphism a x :(VF"

@ AE*)ep,o

+

(AE*)iF=o,oF=o,

induced by the algebraic connection X. Now set V F = fiB7

0

ax : ((VF"),=o

@ A P E , -K)

-+

((AE")ip=o,oF=o,8 E ) .

It follows from Theorem 111, sec. 10.8 (as proved in sec. 10.11) that cpF is a homomorphism of graded differential algebras, and that q$ is an isomorphism making (12.9) commute. 12.19. Proposition

VI: The diagram (12.12) commutes; i.e., =V$OPif.

PROY*

Proof: Use the algebraic connection (X, X E , X,)

AE"

=

(AE"),,=, @ AF",

and BF

= (RiE=O

R

to write

= RiEZmo @ (AE")i,,,

@ (AE")iF

=O)Bp=O

@ AF"

8

as described in sec. 12.6. Define a homomorphism

$: B E @ (VF"),,o

@ APE

+

(Ri,=o @ (AE")i,=o)e,=o

by setting +(w @ Y @ @) = w @pp(Y @ @).

Now let t E H ( B E @ (VF"),,, @ A P E ) . According to sec. 3.19, can be represented by a cocycle of the form

x = 1@ 0

+ 0,

C

XII. Operation of a Lie Algebra Pair

522

where

0 E (VP),,,

@ APE is a cocycle representing

0 E B i @ (VF"),,, Next, write y(z) = y(Q)

+ (y

-

pfit, and

@ APE.

00)+ 1 0V P Q .

+)(I

Then (by the definition of y )

Moreover, Lemma V, sec. 11.20, (below) shows that (Y'

-

$)(l @ 1'

LR$E=0

@ (AE")iF=O]oF=O'

Thus, Ijp the operation is projectable, Proposition IV, sec. 12.6 (applied with CD = v F ( 0 ) ) shows that plF(0) represents p R ( y # C ) .Hence

FP(P8C) = p R ( Y ' # C ) , and the proposition is proved in this case. Finally, suppose ( E , F , i, 8, R, 6,) is any regular operation. Let ( E , F , i, 8, R s , 6,) denote the associated semisimple operation (cf. sec. 12.2). Note that and

(Rs)O,=, = ROE=,

(Rs)jE=O,OE=O = BE.

Since XE(EQ)c R ; , X E may be regarded as an algebraic connection j E for the operation of E in Rs. Set

11,

=fE

0

x.

Next, observe that the inclusion map A: Rs

+R

is a homomorphism

of operations. Thus it restricts to a homomorphism

Since the operation of ( E , F ) in Rs is regular, the construction of articles 2 and 3 may be carried out with Rs replacing R and with (X, f E , replacing (X, X E , XF), but with the same linear map s: PE -+ (VE" @ W(E)),=o. This yields a homomorphism

xF)

@: BE @ (VP*)e=o

0APE

+

(Rs);,=o,e,=o.

3. Isomorphism of the cohomology diagrams

523

denote the fibre projection for the operation of (E, F ) in Rs. Then (cf. sec. 12.8) PR, = P R

~i#,=O,f3p=O.

Moreover, since the representation of E in R, is semisimple, the operation of ( E , F ) in R, is projectable. Thus, by the first part of the proof, P R s O P =

91$

O P B .

It follows that

This completes the proof of Proposition VI and hence the proof of Theorem I.

Q.E.D.

12.20. Lemma

V: Let I be the ideal in BF given by I

Then

=

C ( R t = o@ (A\Eb)ip,O)~F,O.

j 2 Z

Im(y

-

3) c I.

Proof: It is sufficient to establish the relations:

(12. 3)

(v - $)(z @ 1 @ 1) E I , (y' +)(1 @ Y 01) E I ,

zE

Y E (VP),,,.

(12. 4 )

(v-$)(1@1@@)EI,

@EPE.

(12. 5 )

-

BE.

It follows from the definitions that y and $ agree in BE @ 1 01, whence formula (12.13). Moreover if Y E(VF"),,,, then

(v-$)(1 @ y @ 1 ) = a F ( y @ l ) -

1 @)a,(p@1)

(12.16)

XII. Operation of a Lie Algebra Pair

524

(cf. sec. 12.11 and sec. 12.18). Further, a p and morphisms aF: VF" --+

and

Ri,,o @ (AE")i,=,

ax extend

a x :VF"

3

to the homo-

(AE")iF-o

given by and

ap(y") = TF(y+)

a x ( y * ) = T(y"),

y*

E

F".

Now formula (12.1), sec. 12.6, shows that

In view of this, relation (12.16) implies formula (12.14). T o prove (12.15) observe that s@

where

=

c !Pi i

xi!Pi@ Qi = s2@ and

1 @ @i

+

521,

SZ, E VE* @ V+BX@ AE". Since

( XE)W: V+B"

-+

C RiB-0,

j 22

it follows that 6(1 @ 1 0@) - XPYi @ 1 @ @i E VF" @ i

Next, write Qi = $i

4i E

+

&i,

(AE")iF=o@ 1

1 R!Ez,-,@ AE".

j 22

(12.17)

where and

&i

E

(AEX)iF=o @ AfF".

Then

Thus applying formula (12.17) we find

~ ( @1 1 @ @)

-

Ci [(2p),(j"!P{)]

*

(1 @ di) E I .

Now it follows from formula (12,1), sec. 12.6, that

(12.18)

3. Isomorphism of the cohomology diagrams

525

In view of formula (12.18), this implies that

y(lOIO@)-lOC(~vjY(Yi)).~i€z. i

But

$(I

01 0@) = 1 0[a,(? 0l)(h@)l = 1 @ax(CjYYi0Qi) a

=

and so (12.15) is established.

1

i

2gjYi)

*

di, Q.E.D.

$4. Applications of the fundamental theorem 12.21. Immediate consequences. Corollary I: The cohomology diagram of a regular operation of a reductive pair commutes. Proof: According to Proposition VI, sec. 3.20, the cohomology diagram of a tensor difference commutes. Thus, in view of Theorem I, (2) we have only to show that the diagram

H(RO=O) QR

-

H " ( E ) 7H " ( F )

commutes. R @ AF" be the structure homomorphism for the operaLet y g : R tion ( F , i, €',R, 8 R ) . Then, it follows from the definitions of y R (sec. 7.8) and /I (sec. 7.7) that the diagram

R

R @ AE"

3R @ AF"

commutes. This yields the commutative diagram

526

4. Applications of the fundamental theorem

527

Corollary 11: There is a spectral sequence converging to H(B,) whose &-term is given by

ESP#l IV = HP ( B E ) @ H 9 ( E / F ) . Proof:

Apply the results of sec. 3.9, recalling from sec. 10.8 that

H ( E / F )GZ H((VF*)O=o@ APE).

Q.E.D.

Corollary 111: There is a spectral sequence converging to H(B,) whose E,-term is given by

El'' Proof:

(VF*)S=o@ Hq(Re=o).

Apply the results of sec. 3.9, recalling from sec. 9.3 that

H(R8-0) EZ H ( B E @ APE). Corollary IV: Assume ( B E ,8,) alences

and

( H ( B E )O

Q.E.D.

is c-split. Then there are c-equiv-

, E:#,) 7(Re,=o , 8,)

(H(BE) @ (VF")O=O@ APE

9

v$

- 0,) 7 ( B F , 8 R ) *

They can be chosen so that the induced cohomology isomorphisms, together with the identity map of H(BE), define an isomorphism from the cohomology diagram of the operation to the diagram

Proof: This follows from the fundamental theorem, together with Proposition XI, sec. 3.29, applied to a c-splitting. (In applying Proposi-

528

XII. Operation of a Lie Algebra Pair

tion XI, copy the example of sec. 3.29, replacing BE @ APE by BE @ (VF*),,o @ APB, as in sec. 3.28.) Q.E.D. 12.22. N.c.z. operations of a pair. Let (E, F, i, 0, R, 8,) be a regular operation. Then (AE*)iF=o,eF=o is called noncohomologous to zero (n.c.z.) in BF if the fibre projection PR: H(BF)

H(E/F)

is surjective. Theorem 11: Let (E, F, i, 0, R, 6,) be a regular operation. Then (AE*)i,=o,e,=o is n.c.2. in BF if and only if there is a linear isomorphism of graded vector spaces

f:H(BE) @ H ( E / F )5 H(BF)

-

such that f(a @ /?) = e#(a) f(1 @ 8) and the diagram

commutes.

WBF)

Proof: Apply Theorem VIII, sec. 3.21.

Q.E.D.

Theorem 111: Let (E, F, i, 0, R, 6,) be a regular operation. Then:

(1) If H ( B E )has finite type, then so does W(BF)and the corresponding PoincarC series satisfy

I~H(BJH(EIF).

~ H ( B F )

Equality holds if and only if (AE*)i,-o,e,=o is n.c.2. in B,. (2) If H ( B E )has finite dimension then so does H(B,) and dim H(BF) 5 dim H ( B E )dim H ( E / F ) . Equality holds if and only if (AE*)iF-O,,F--O is n.c.2. in B p .

4. Applications of the fundamental theorem

529

(3) If H(B,) has finite dimension, then

XH(B,)

=

XH(B~)XH(B/F).

I n particular, the Euler-PoincarC characteristic of H ( B F )is zero unless (E, F ) is an equal rank pair. Proof: Apply Corollaries IV, V, and VI to Theorem VIII, sec. 3.21, together with Theorem XI, sec. 10.22. Q.E.D.

12.23. Algebra isomorphisms. Proposition VII: Let ( E , F, i, 8 , R, S R ) be a regular operation, and assume that (@)+ = 0. Then there are c-equivalences

( B E @ (AE*)ip=O,Bp=O, and

8R

@4

( B E @ APE? S R @ L ,

+

WR

@

7(BP?

7 ( R f 3 ~ = 0 ,S R ) *

Moreover, these can be chosen so the induced isomorphisms of cohomology make the diagrams

(12.19)

and

530

XII. Operation of a Lie Algebra Pair

Proof Since (Zg)+ = 0, the ( P E , 8)-algebras ( B E , 8,; 0) and ( B E ,8,; (ZE)"0 T ) are equivalent (cf. sec. 3.27). Hence (cf. the corollary to Proposition XI, sec. 3.29) they are c-equivalent. Now it follows from the remarks at the end of sec. 3.28 that there are

c-equivalences

such that the induced isomorphisms of cohomology define an isomorphism between the cohomology diagrams. The proposition follows now from Theorem I, sec. 12.10. Q.E.D. Theorem N : Let ( E , F, i, 8, R, 6,) be a regular operation. Then the following conditions are equivalent :

(1) There is a homomorphism A: (VF*),,, graded algebras, which makes the diagram

-+

H ( B E )0I m Z# of

commute. (2) There is a c-equivalence

such that the induced isomorphism of cohomology makes the diagram (12.19) commute.

4. Applications of the fundamental theorem

531

(3) There is an isomorphism of graded algebras H(BE)

8H ( E I F )

H(BF),

which makes the diagram (12.19) commute. Proof Apply Theorem IX, sec. 3.23, together with Theorem I, (Z), sec. 12.10. Q.E.D.

If ( X $ ) +

Corollary I:

= 0,

then the conditions of the theorem hold.

Corollary 11: If (BE, 6,) and ((AE*)lF=o,oF=o, 6,) are c-split, and if the conditions of the theorem hold, then ( B F ,6,) is c-split. Theorem V: Let (E, F , i, 8, R, 6,) be a regular operation. Assume F is abelian and E is semisimple. Then the following conditions are

equivalent :

( X $ ) + = 0. (2) There is an isomorphism H ( B E ) @ H ( E / F )-% H(B,) of graded algebras such that the diagram (12.19) commutes.

(1)

Proof Since F is abelian, (VF*),=, = VF" and so (VF*),=, is generated by elements of degree 2. Since E is semisimple, Pk = 0 for k < 3. Hence (VE")$=, = 0 for k < 4, and so

jL,(VE"),=, c (V+F'),,,

. (V+F"),=,.

This shows that ((VF*),=,; (T) is an essential symmetric PE-algebra. Now apply the corollary to Proposition VIII, sec. 3.25. Q.E.D. Theorem VI: Let ( E , F , i, 0, R, S,) be a regular operation which satisfies the conditions of Theorem IV. Let z be a transgression in W(E),=,and define a subspace P , of PE by

P,

=

{@ E Px

I je"=,(Z@)

c (V+F*),=,

*

(V+F"),=,}.

Then P , is contained in the Samelson subspace for the operation ( E , i, 8, R,

w.

XII. Operation of a Lie Algebra Pair

532

Proof: Apply Theorem X, sec. 3.26.

Q.E.D.

12.24. Special Cartan pairs. Let (E, F)be a Cartan pair with Samelson subspace P c PE. Let t be a transgression in W(E),,o and consider the linear map (T

*V

=je=o

0

t

-

: PE

(VF"),=o,

(cf. sec. 6.13 and sec. 10.8). The pair (E, F) will be called a special Cartan pair if u ( P ) c (Imji,o)+

Observe that if and so

t'

- (Imji=o)+.

is a second transgression, then

(t -

(12.21)

eE

0

(t- t') = 0

r')(PE) c (V+E")e=o * (V+B")e=o

(cf. Lemma VII, sec. 6.13, and Theorem 11, sec. 6.14). This shows that the condition (12.21) is independent of the choice of t. Clearly, an equal rank pair is a special Cartan pair. Moreover, Theorem X, (4) and ( 5 ) , sec. 10.19, show that if F is n.c.2. in E, then (E, F) is a special Cartan pair. Finally, Proposition VII, sec. 10.26, implies that a symmetric pair is a special Cartan pair. Lemma VI: Let (E, F) be a special Cartan pair. Then there is a transgression t in W(E),,, such that (jL0

0

.)(P)

= 0.

Proof: Let tl be any transgression in W(E),j=o. Then it follows from relation (12.21) that there is a linear map a: P

-+

(V+E*),,o

- (V+P),,

,

homogeneous of degree 1, such that

(]Looa)(@) Write PE

=

P @P

= ( j L o0 tl)(@),

and define

@ E P.

t:PE + (VE")e-o

by

4. Applications of the fundamental theorem

Then Theorem 11, sec. 6.14, yields ee 0 sec. 6.13, z is a transgression. Clearly,

(J-LO

0

t = 1;

533

thus, by Lemma VII,

z ) ( P ) = 0.

Q.E.D.

A transgression satisfying the condition of Lemma VI will be called adapted to the special Cartan pair ( E , F ) . Next, consider the Koszul complex, ((VF"),=, @ APE, -Vo), where a =jV,=, 0 z and z is adapted. Let P be a Samelson complement and let 5 denote the restriction of a to P. Then, since a(P) = 0,

((VF*),=o@ APE, -Vm) = ((VF'),,,

@ AP,

--E) @ (AP, 0).

Moreover, because (El F ) is a Cartan pair, we have

(VF*)o=O= VPF

and

dim P

=

dim P F .

Thus Theorem VII, sec. 2.17 yields

H((Vp*),,o @ A P ) == Ho((VF")~=o @ AP) =

(VF*)o,o/(VF"),,,

o PE g

I m I*,

where

I*: (VF*),=o + H((VF*),=, @ APE) is the .homomorphism induced by the inclusion map. I t follows that

H((VF*),,, @ APE)= Im I# @ AP. Finally, consider the isomorphism

(cf. sec. 12.18). It determines (via the equation above) the isomorphism

yF: AP @ Im I # -ZH ( E / F ) given by

534

XII. Operation of a Lie Algebra Pair

Evidently, the diagram

AP @ I m 1*

commutes (cf. diagram (12.9) in sec. 12.17). 12.25. Operation of a special Cartan pair. Let ( E , F, i, 8, R, 6,) be a regular operation of a special Cartan pair. Assume that the (PE, 6)algebra (BE, 6,; t,) and the P,-algebra ((VF*),=,,; 0) are defined via an adapted transgression t (cf. sec. 12.9). Let p be the Samelson subspace for ( E , F) and let P be a Samelson complement. Observe that VB restricts to a differential operator V’ in BE 0AP. Moreover, the inclusion

i: (BE 0AP, V B ) + (BE 0APE, V’) is a homomorphism of graded differential algebras. On the other hand, since a(P) = 0, the inclusion

iF: (BE @ Ap, VB) + (BE@ (VF*),=,, @ APE’,V )

-

is also a homomorphism of graded differential algebras. Hence so is i F : (BE

0@, V B )

(BF

9

where y is the homomorphism in Theorem I, sec. 12.10. Now let a : Im 1# -+ (VF*),=, be a linear map, homogeneous of degree zero, such that 1# o a = 1. Thenp, 0 1$ 0 a = 1 as follows from the cohomology diagram in sec. 12.7. Theorem VII: Let ( E , F, i, 8, R, 6,) be a regular operation of a special Cartan pair. Then a linear isomorphism of graded spaces

f:H(BE0AP) 0I m 2# -3H ( B F )

4. Applications of the fundamental theorem

535

Moreover, this isomorphism makes the diagrams

H(BF) and

€€(BE 0AP)

,Ig

Im I#

PR

-

H(BE @ AP)

i*

=

W E )

I

€€(BE @ APE) eH

(12.23)

' H(Ro=o)

e;

commute (cf. sec. 12.24 for y F and sec. 12.14 for 82). Proof: T h e commutativity of the diagrams (12.22) and (12.23) is an immediate consequence of the definition off and Theorem I, sec. 12.10. It remains to be shown that f is a linear isomorphism. Identify I m X # with Im Z* via y$ (cf. sec. 12.24 and sec. 12.17). Then a becomes a linear map a : Im I*

--+

(VF*),,,

satisfying I # o w = L. Now define a linear map

g: H(BE @ A P ) 0I m I*

-+

H(BE @ (V8'*),=0

@ APE)

by

g( C

0q ) = i;( C ) . (mf

0

a)(q),

C E H(BE 0AP), q E 1x11 I # ,

(cf. sec. 12.10 for mg). I n view of Theorem I, sec. 12.10, we have

f = y # o g and so it is sufficient to show that g is a linear isomorphism.

XII. Operation of a Lie Algebra Pair

536

Consider the (P, d)-algebra (BE@ AP, t$;TR) and the P-algebra ((VP*),=o; 5 ) given by fR(@) = z R ( @ @ )

1

and

5 ( @ )= o(@),

@E

P.

The Koszul complex of their tensor difference

((BE @ A P ) @ (VF*)e,o

0AP, p)

coincides with the Koszul complex ( B E @ (VP),c,=o @ AP @ AP, V ) under the obvious identification (since a(P) = 0). With this identification, iFbecomes the base inclusion for the tensor difference. Since (VF*),zj,o

=

and

VPF

dim PF = dim P

it follows (as in sec. 12.24) that

H((VP),cj=o@ A P ) = Im T'"

=

I m I#.

Now Corollary I1 to Theorem VIII, sec. 3.21, shows that g is an isomorphism. Q.E.D. Again let (E, F, i, 8, R, 6,) be a regular operation of a special Cartan pair. Let I be the ideal in H ( B E @ AP) @ (VF")e-o generated by the elements of the form f$@@1-1@5@,

@€P,

and let z:

H ( B , @ AP) @ (VF")+o

+

(H(BE @ A P ) @ (VP)+o)/I

be the corresponding projection. Consider the homomorphism y : H(B.q

@ A P ) 0(VF*),,o

+

H(BF)

given by

r(C

y ) = (W

0

iF)*(C)

*

%(yU),

C E H(BE

AP),

E

(VF*)e=o.

Proposition VIII: With the hypotheses above, y factors over z to yield an isomorphism of graded algebras (H(BE

8AP) 0 (VF*)e-o)/I 2H(BF).

4. Applications of the fundamental theorem

537

Proof: This follows from Corollary I11 to Theorem VIII, sec. 3.21, in exactly the same way as Theorem VII followed from Corollary 11. Q.E.D. 12.26. Examples. 1. Equal rank pairs: Let ( E , F, i, 8, R, 6,) be a regular operation of an equal rank pair. Then according to Theorem XI sec. 10.22, X # is surjective. But

p,, X j = P. 0

Thus p R is surjective, and so (AE*)iF=O,BF=O is n.c.2. in B F . Moreover P = 0, and Theorem VII, sec. 12.25, provides a linear isomorphism

H(BE) @ H ( E / F )5 H(B,) (cf. also Theorem 11, sec. 12.22). Note that diagram (12.22) reduces to the diagram of Theorem I1 in this case. O n the other hand, diagram (12.23) yields the commutative diagram

This shows that Imeg

=

Imei.

(12.24)

Finally, Proposition VIII, sec. 12.25, yields an algebra isomorphism (H(BE)

@ (vF*)fJ=O)/I-% H(BP'),

where I denotes the ideal which is generated by the elements of the form t @ D @ l - I @ O @ , GEPE. 2. N.c.z. pairs: Let ( E , F , i , 8 , I?, 6,) be a regular operation of a reductive pair such that F is n.c.2. in E. Then, by Theorem IX, sec. 10.18, (%*)+= 0. Thus the isomorphism f of Theorem VII is given by

f = (W 0 i~)*: H(BE @ A P ) 5 H(B,),

538

XII. Operation of a Lie Algebra Pair

and so in this case f is an isomorphism of graded algebras induced from an isomorphism of graded differential algebras. I n particular,

(BE @ Ap,

PB)

7(BF,

dR)*

Moreover, diagrams (12.22) and (12.23) reduce to the commutative diagrams

H(BE @ AP) L A?'

and

'I-

5%

I

8;

12.27. Lie algebra triples. A reductive Lie algebra triple (L, E, F ) is a sequence of Lie algebras L 2 E 3 F such that

(1) L is reductive. (2) E is reductive in L. (3) F is reductive in E. Note that then F is reductive in L (cf. Proposition 111, sec. 4.7). Moreover, the commutative diagram

4. Applications of the fundamental theorem

539

implies that the vertical arrow is an isomorphism, and so (E, F, iE, AL", 6,) is a regular operation of the pair ( E , F ) . With the terminology of sec. 12.1 we have

BE = (AL*)ig=o,eg=o

and

BF

=

OE,

(ALx)~F=o,~F=o.

Thus, Theorem I, sec. 12.10, expresses H(L/F) in terms of H(L/E) and other invariants. Now assume that (E, F ) is a special Cartan pair, and let z be an adapted transgression in W(E)B=o.Then Theorem VII, sec. 12.25, yields a linear isomorphism

f:H((AL")i,=o,eE=o@ A P ) @ I m 1*"-

H(L/F).

Theorem VIII: Let (L, E, F ) be a reductive triple and assume that ( E , F ) is an equal rank pair. Then

def(L, E ) = def(L, F). In particular, (L, E ) is a Cartan pair if and only if (L, F ) is. Proof:

satisfy

In view of formula (12.24) in sec. 12.26, the inclusions

I m kg

=

Im k $ .

It follows that the Samelson subspaces PE and Pp for the pairs (L, E ) and (L, F ) coincide. Since by hypothesis dim PE = dim P F , it follows that def(L, E ) = dim PL - dim PE - dim PE = def(L, F).

Q.E.D. Corollary: Let H be a Cartan subalgebra of E. Then (L, E ) is a Cartan pair if and only if (L, H ) is. Proof: Apply Theorem XII, sec. 10.23.

Q.E.D.

85. Bundles with fibre a homogeneous space 12.28. The cohomology diagram. Let 9= (P, n,B, G) be a principal bundle with compact connected structure group G and assume that B is connected. Let K be a closed connected subgroup of G. Restricting the principal action of G on P to K yields an action of K. Let nl:P --t P / K be the canonical projection. Then P1 = ( P I ,nl, P / K , K ) is again a principal bundle (cf. sec. 5.7, volume 11). Moreover, n factors over nl to yield a smooth map n6:P / K B, and E = ( P / K , n6,B, G / K ) is a fibre bundle. Finally, we have the principal bundle 9K = (G, nK 9 G / K ,K ) . These bundles are combined in the commutative diagram --f

(12.25)

where, for z E P,

6,

+

PIK

n€

B,

(1) A, is the inclusion map a H z . a, a E G. (2) A, is the restriction of A , to K. (3) A, is the induced map. Note that A , , a,,and A, are the fibre inclusions for the bundles 9, pl, and 5'. Observe also that A, is a homomorphism of K-principal bundles. The homomorphism Af: H ( P / K )-+H ( G / K ) is independent of the choice of z in P. In fact, if zo E P and z1E P, choose a smooth path zt connecting zo and zl.Then the maps A,t define a homotopy from Azoto AL1,and thus A: = A$. We denote this common homomorphism by ee: H ( P / K ) H(GIK) +

and call it the jibre projection for the bundle 5. 540

5. Bundles with fibre a homogeneous space

T h e diagram (VE”),=,

541

‘sja

H(B)

(12.26)

is called the cohomology diagram corresponding to the diagram (12.25). Here el. = A,# and ,dP = a,”are the fibre projections for the bundles .P and T h e cohomology diagram commutes. In fact, it was shown in sec. 6.27, volume 11, that the upper square commutes. Since A, is a homomorphism of principal bundles, and since the Weil homomorphism is natural it follows that ec 0 hpl = h p K . Finally, the commutativity of the rest of the diagram follows directly from diagram (12.25).

e.

12.29. Induced operation of a pair. Let E and F denote the Lie algebras of G and K. Then we have the associated operations ( E , i, 8, A ( P ) ,6 ) and ( F , i, 8, A ( P ) ,6 ) (cf. sec. 8.22). Since the principal action of K on P is the restriction to K of the principal action of G, it follows that the second operation is the restriction of the first operation. I n view of Theorem I, sec. 4.3, volume 11, the inclusion maps

-

A(P)@,=, A ( P )

and

A(P),,=,

-+

A(P)

induce isomorphisms of cohomology Thus the inclusion map

induces an isomorphism of cohomology, and so (E, F, i, 8, A ( P ) , 6) is an operation of the pair (E, F ) . Now since G is compact, the pair (E, F) is reductive. Thus the “algebraic” fibre projection

PA,,): H ( ’ ( P ) ~ F = O , O F = O )

+

H(’/F)

512

XII. Operation of a Lie Algebra Pair

is defined (cf. sec. 12.4). On the other hand, isomorphism z:

can be regarded as an

7cf

A ( P / K )5 A(P)ip=O,eF=O ,

while in sec. 11.1 we defined an isomorphism

H ( E / F )-% H(G/K).

E & ~ :

Proposition IX: With the hypotheses and notation above, the diagram

aH ( E / F )

H(A(P)iF-o,@F-o) ((7I:)-l)*

I

=

H(PIK)

2

1

e&K

' H(G/K)

et

commutes. Proof: Consider the operation ( E , F, i, 8, A(G), &), where i(h) = i ( X h )and 8(h) = 8(Xh) ( X , is the left invariant vector field generated by h ) (cf. sec. 7.21). Since the map A,: G -+ P is G-equivariant,

A,*:A(G) t A ( P ) is a homomorphism of operations. On the other hand, in sec. 7.21 we defined a homomorphism of operations E ~ ( :E , F , i, 8 , AE*, S,) -+ ( E , F, i, 8, A(G), 6,) inducing an isomorphism in cohomology . Recall from the example of sec. 12.5 that the fibre projection PhE*

-

:H(E/F)

H(E/F)

is just the identity map. Now the naturality of the algebraic fibre projection gives the commutative diagram

connecting the various fibre projections.

5. Bundles with fibre

a homogeneous space

543

It follows that

12.30. The cohomology of P/K. Theorem IX: Let 9= ( P ,n,B, G ) be a principal bundle with compact connected structure group G and connected base. Let K be a closed connected subgroup of G. Then there are c-equivalences

( A ( P / K ) ,4 7 (W) 0(VF*),=llO APE, and

(A(P), 6, 7 ( A ( B )@ ( A ( G / W ,a) 7 ((Vp')e=o

v,

VB),

0APE -K). 9

The induced isomorphism of cohomology algebras determines an isomorphism from the cohomology diagram of sec. 12.10 to the cohomology diagram (12.26) (with BE replaced by A ( B ) and H(B,) replaced by H ( B ) ) . Proof: In view of the canonical isomorphisms

A ( P / K )-5~ ( ~ ) i p = o , f J p = o and

A ( B ) 2A(P)iE-O,eE-O,

the theorem follows from Theorem I, sec. 12.10, together with diagram 8.22, sec. 8.27, diagram 11.1, sec. 11.4, and Proposition IX, sec. 12.29.

Q.E.D.

E, G / K , is called noncohomologous is surjective. I n this case, Theorem IX,

12.31. N.c.z. fibres. The fibre of

to zero in P / K if the map .oc together with Theorem VIII, sec. 3.21, yield the commutative diagram

(12.27)

XII. Operation of a Lie Algebra Pair

544

where f is an isomorphism of graded vector spaces satisfying f(a

0B ) = (.Pa)

*

f(1 08))

a E H(B),

B E H(G/K).

In particular, if G and K have the same rank, then the map

h p K :(VF")e,o

-+

H(G/K)

is surjective (cf. sec. 11.7). The cohomology diagram (sec. 12.28) shows that in this case GIK is n.c.2. in PIK. On the other hand, we can apply Theorem IX, sec. 3.23, to obtain Theorem X: Let 9= (P, n,B , G), K , be as in Theorem IX. Then the following conditions are equivalent :

( 1 ) There is a homomorphism of graded algebras W: (VFY)e=o

-+

H ( B ) @ I m hpK

which makes the diagram

commute. (2) There is a c-equivalence

(A(B x GIK), 6) 7 A ( P / K ) such that the induced isomorphism of cohomology makes the diagram (12.27) commute. (3) There is an isomorphism of graded algebras

f:H ( B ) 0H ( G / K )-5H ( P / K ) which makes the diagram (12.27) commute.

5 . Bundles with fibre a homogeneous space

545

Finally, as in Theorem V, sec. 12.23, we have Theorem XI: Let 9=(P, n,B, G), K be as in Theorem IX. Assume the Lie algebra of G is semisimple and that K is a torus in G. Then the following conditions are equivalent:

(1) h& = 0. (2) The conditions of Theorem X hold. Example: Let 9= (P, n,B, G) be a principal bundle as in Theorem XI, and assume h& # 0. Let T be a maximal torus in G. Then rank T = rank G. Hence, as we have just seen, there is a linear isomorphism

f:H ( B ) 0H ( G / T )5 H ( P / T ) such that f(a @ B ) = (@a) .f ( l @ B), a c H ( B ) , @ E H ( G / T ) ,and the diagram (12.27) commutes. On the other hand, since h$ # 0, Theorem XI, together with Theorem X, (3), shows that f cannot be an algebra isomorphism.