Introduction Table of Contents DEFORMATIONS OF LIE SUBGROUPS AND THE VARIATION OF ISOTROPY SUBGROUPS

DEFORMATIONS OF LIE SUBGROUPS AND THE VARIATION OF ISOTROPY SUBGROUPS

BY

R . W . R I C H A R D S O N , J R .

University of Warwick Coventry, England and University of Washington, Seattle, Wash., U.S.A. (1)

Table of Contents

I n t r o d u c t i o n . . . 3 5 w 0. P r e l i m i n a r i e s . . . 3 7 C I t A ~ a I. DEFO~ATIONS OF LIE SUBGROUPS

w ]. A n a l y t i c f a m i l i e s of Lie s u b g r o u p s . . . 38 2.

3.

4. :Relation of T h e o r e m 3.1 t o r e s u l t s o n d e f o r m a t i o n s of s u b a l g e b r a s of Lie a l g e b r a s . 5. V a r i a t i o n s o n T h e o r e m 3.1 . . . 6. A l g e b r a i c f a m i l i e s of a l g e b r a i c s u b g r o u p s . . .

T h e n o r m a l d i s p l a c e m e n t of a n a n a l y t i c f a m i l y . . . 41 P r o o f of T h e o r e m 3.1 . . . 46 51 52 54 w 7. C o m p l e x - a n a l y t i c f a m i l i e s of s u b g r o u p s of c o m p l e x Lie g r o u p s . . . 58 C K A ~ T E ~ II. T H E V A I ~ I A T I O N O F I S O T I ~ O P Y S U B r

w 8. T h e a n a l y t i c f a m i l y G * . . . 6 0 w 9. A l g e b r a i c t r a n s f o r m a t i o n g r o u p s . . . 6 3 w 10. A n a l y t i c t r a n s f o r m a t i o n g r o u p s . . . 6 5 w 11. l ~ e l a t i o n b e t w e e n a n a l y t i c f a m i l i e s of s u b g r o u p s a n d f a m i l i e s of i s o t r o p y s u b g r o u p s . 66 w 12. E x a m p l e s . . . 68

Introduction

L e t t h e ( r e a l o r c o m p l e x ) L i e g r o u p G a c t a n a l y t i c a l l y o n t h e c o n n e c t e d ( r e a l o r c o m - p l e x ) a n a l y t i c m a n i f o l d M . I n t h i s p a p e r w e s h a l l s t u d y t h e b e h a v i o u r of t h e i s o t r o p y s u b g r o u p s Gx a s a f u n c t i o n of x E M . I f m = m i n x r d i m Gx, i t is t r i v i a l t o s h o w t h a t M 0 =

(1) P a r t i a l s u p p o r t received from N S F G r a n t GP-21504.

36 R . W . RICHARDSON, JR.

{x E M I dim Gx = m} is a dense open subset of M, the complement of an analytic subset of M. We shall restrict attention to the case in which M = M0, i.e. we assume t h a t all orbits of G on M have the same dimension. This case is well understood if G is a compact con- nected Lie group (even for the case of continuous G-actions). There exists a dense open subset U of M such t h a t G~ is conjugate to Gy for x, y E U. Moreover, M - U is of codimen- sion I> 2 in M and, if x E U and y E (M - U), t h e n G x is conjugate to a subgroup of G~. Thus, modulo conjugacy, all isotropy subgroups are the same on U and on ( M - U ) all t h a t happens is t h a t the isotropy subgroups pick up extra components. I n particular, the n u m b e r of components of the isotropy subgroup G x is an upper semi-continuous function of x E M .

The situation is quite different in the case of non-compact G. For example, there exists a n analytic action of G = S L 4 ( R ) on an analytic manifold M such t h a t all orbits of G on M have codimension 1 and such that, for x, y E M, G x is non-isomorphic to Gy unless x and y lie on the same orbit under G. Similar examples exist for semi-simple algebraic groups acting algebraically on quasi-affine algebraic varieties. To give another example, consider the irreducible representation of G = SLy(C) on (~4 and let U = {xEC4IGx is finite}; U is a non- e m p t y Zariski-open subset of C 4. There exists a Zariski closed subset A of U such t h a t G x = ( e ) for x E A and Gx is of order three if x E ( U - C ) . Thus, for the action of G on U, the n u m b e r of components of G~ is a lower semi-continuous function of x but is not upper semi- continuous. F o r all of these examples, see w 12.

I n order to s t u d y the behaviour of the isotropy subgroups, we shall s t u d y the more general problem of "deformations" of subgroups of a (real or complex) Lie group G. Roughly, an analytic family o / L i e subgroups o / G , parametrized b y an analytic manifold M, is an analytic submanifold H of G • M such t h a t the projection :7~M" II-->M is a submersion and, for every t E M , the fibre 7~M--l(t) is of the form Ht • {t), where Ht is a Lie subgroup of G.

Our basic result concerning such analytic families is the following:

THEOREM 3.1. L e t

H=(Ht)teM

be an analytic family o / L i e subgroups o/G, let toEM and let H =Hto. Let K be a Lie subgroup o / H such that the component group K / K ~ is ]initely generated and such that the Lie group cohomology space Hi(K, ~/~) vanishes. Then there exists an open neighborhood U o] t o in M and an analytic map ~: U ~ G with ~(to)=e such that H t ~ ~(t)K~(t) -1 /or every t E U.

Here $ (resp. 3) denotes the Lie algebra of G (resp. H) and the K-module structure of fl/~ is determined b y the adjoint representation of K on g.

Theorem 3.1 generalizes the result of A. Weil [24, p. 152] which states t h a t if F is a discrete, finitely generated subgroup of G such t h a t Hi(F, g ) = 0 , then F is "rigid". I t also

D E F O R M A T I O N S O F L I E S U B G R O U P S A N D T H E Y A R I A T I O N O F I S O T I ~ O P Y S U B G R O U P S ~

generalizes results of the author [17], [19] on deformations of subalgebras of Lie algebras to t h e case of Lie subgroups. Theorem 3.1 makes sense in the f r a m e w o r k of G~176 and we conjecture t h a t it holds there. However, our proof relies heavily on the assumption of analyticity. I n particular we require a theorem of M. Artin [1] on replacing formal power series solutions of analytic equations b y convergent power series solutions.

If, in particular, K = H is a reductive Lie group, t h e n Hi(K, 0/~)=0 and Theorem 3.1 applies.

L e t G act analytically on M such t h a t all orbits have the same dimension and let G = {(g, t)E G • M lg fi Gt}. I n general G = (Gt)tEM is not an analytic family of Lie subgroups of G. We say t h a t (g, t) fi G is a regular point of G if there exists a neighborhood U of t in M and an analytic m a p s: U-+ G with s(t) = g such t h a t s(u) E G= for every u e U. L e t O* be the set of regular points of G and let G* = {g 6 Gt[(g, t)eG*}. Then V*t is a Lie subgroup of G containing the identity component of Gt and G*=

((~)tEM

is an analytic family of Lie subgroups of G. Thus Theorem 3.1 can be applied to the analytic family G*. I f all orbits of G on M are locally closed, it follows from a result of J. Glimm [7] t h a t there exists a dense open subset U of M such t h a t Gt = G~ for every t E U.

We obtain our sharpest results in the case of algebraic transformation groups (over C).

I n this case, our m a i n result is

TH]~OREM 9.3.1. Let (G, X ) be an algebraic trans]ormation space with G an a]]ine alge- braic group. For each t E X , let U t denote the unipotent radical o / G t and let L t be a Levi sub- group o/Gt. Then there exists a ]inite [amily X 1 .... , X n o/non-singular Zariski-locaUy closed subsets o / X such that the ]ollowing conditions hold:

(a) X = Uj%Xr

(b) _For each j, X j is a Zariski open subset o] X - ( ~-1 (c) I f x, y E X j , then L~ and L~ are conjugate.

(d) _For each ], the/amily (Ut)t~x ~ is an algebraic/amily o/algebraic subgroups o/G.

Many of the results of this paper were announced in [20].

w 0. Preliminaries

0.1. As usual, N, Z, R and C denote respectively the set of natural numbers, the ring of integers, and the fields of real and complex numbers.

0,2. Throughout this paper, F will denote either R or C and, unless stated otherwise, all Lie groups and analytic manifolds will be t a k e n over F. All Lie groups and analytic manifolds are assumed p a r a c o m p a c t and Hausdorff. Our basic reference for Lie groups and analytic manifolds is [22] and we shall follow the terminology therein, except t h a t we

3 8 It. W . R I C H A I ~ D S O N , J R .

denote by

dfx:

Tx(M)-+ Tj(x)(A r) the differential (tangent linear mapping in the terminology of [22]) at x E M of an analytic m a p / : M - ~ N of analytic manifolds. I n particular, we shall consistently use the terminology and elementary results mentioned in [22] on immersions, submersions, subimmersions and fibre products.

0.3. The Lie algebra of a Lie group G, H, etc., will be denoted b y the corresponding lower ease German letter g, ~, etc. If V is a finite dimensional vector space over F, then GL(V) denotes the Lie group of automorphisms of V and End(V) denotes the F-algebra of endomorphisms of V. If G is a Lie group, then e denotes the identity element of G, G o the connected component of e in G, and expo: g-+ G and Ada: G-> GL(fl) denote respectively the exponential map of G and the adjoint representation of G. B y a linear representation of G, we shall always mean an

analytic

homomorphism G-+ GL(V), where V is

[inite-dimensional

over F. Lie subgroups are as defined in [9] (for the real ease; we take the analogous definition in the complex case); in particular, a Lie subgroup of G is not necessarily an analytic sub- manifold of G and is not required to have the topology induced as a subset of G. If the Lie group O acts as a transformation group on a topological space M and if t~M, then

Gt=

{gEGlg.t=t}

^{is the}

isotropy subgroup

of G at t and the Lie algebra gt of

Gt

^{is the}

isotropy subalgebra

of g at t.

A n o p e n i n F is a s e t of t h e f o r m . . . eF l Iz l < a for j = l . . . . , where a is a positive real number. If V is an r-dimensionaI vector space over F, an open box in V is the image in V of an open box in F ~ under a linear isomorphism q): F r-~ V.

Chapter I. Deformations of Lie subgroups w 1. Analytic families of Lie subgroups

Many of our definitions in this section are special cases of a more general situation considered by Douady and Lazard [6], that of an analytic family of Lie groups (or, in the terminology of D o u a d y - L a z a r d , Lie groups over M). I n contrast to [6], we require all mani- folds to be tIausdorff.

1.1.

De[inition.

Let G be a Lie group and 21/an analytic manifold. An

analytic [amily o/

Lie subgroups

of G, parametrized b y M, is an analytic manifold H satisfying the following conditions:

(a) H c G x M and the inclusion map i: H-+G • M is an immersion.

(b) If ~ru: H - ~ M denotes the composite map pru o i, where prM denotes the projec- tion G •

M-+M,

then ~u is surjective and is a submersion.

D E F O I : t M A T I O I q S O:F L I E S U B G R O U P S _AND T H E Y A R I A T I O ~ O F ISOTROX~Y S ~ J B G R O U ] P S 39 (c) F o r every t EM, the fibre z ~ ( t ) is of t h e f o r m H, • {t}, where Ht is a subgroup of G.

(d) L e t I t • MII denote t h e fibre p r o d u c t of I t with itself over M with respect to

=M a n d let #: t I • M t t - + I t a n d j: H - ~ H be defined b y #((x, t), (y, t)) = (xy, t) a n d j(x, t) = (x-% t). T h e n ,u a n d j are analytic maps.

1.2. Remarks. (a) W e do n o t require t h a t I t be an analytic submanifold of G x M. I n particular, t h e t o p o l o g y on t I is n o t necessarily t h e i n d u c e d t o p o l o g y as a subset of G • M.

(b) If H is an analytic submanifold of G • M, t h e n condition 1.1. (d) is a consequence of 1.1. (a)-(c).

(c) I t follows f r o m the definition t h a t each H t ( t E M ) is a Lie subgroup of G. Conse- q u e n t l y I I can be considered as a family of Lie subgroups of G, which depend analytically on the p a r a m e t e r t E M. We shall f r e q u e n t l y d e n o t e H b y (Ht)teM.

(d) I t follows f r o m 1.1. (b) t h a t t ~-~ dim H t is c o n s t a n t on each c o m p o n e n t of M. W e shall assume t h a t t ~-> d i m H~ is c o n s t a n t on all of M.

(e) For t E M, let c(t) be t h e n u m b e r of c o m p o n e n t s of Hr. T h e f u n c t i o n t ~->c(t) is n o t necessarily either u p p e r semi-continuous or lower semi-continuous. F o r examples, see w 12.

(f) A n u m b e r of non-triviM examples of analytic families of Lie subgroups are given in w 12.

1.3. Sections. L e t I t = (Ht)te M be an analytic f a m i l y of Lie subgroups of G a n d let U be an open subset of M. A n analytic section of H over U is an analytic m a p s: U->G such t h a t (i) s(t) EH t for e v e r y tE U a n d (ii) t h e m a p t~-> (s(t), t) is an analytic m a p of U into It.

I f I I is an analytic submanifold of G • t h e n condition (ii) follows f r o m condition (i), b u t this is n o t the case if I t is n o t an analytic submanifold. I t follows easily f r o m 1.1. (b) t h a t for every (x, t) EIt, there exists a n o p e n n e i g h b o r h o o d U of t in M a n d an anMytic section s: U-> G of H over U such t h a t s(t) = x. Moreover, if U is open in M, t h e n t h e c o n s t a n t m a p t ~-> e is an analytic section over U and, if sl, s 2 are analytic sections over U, t h e n t ~-+ sx(t ) s2(t) -1 is an analytic section over U.

1.4. The Lie algebra o] tI. F o r each t E M , let IJt denote the Lie algebra o f / / t a n d let

= {(X, t) E g • M I X E~t}. T h e n ~ is a sub-bundle of the trivial v e c t o r bundle g • M-+M;

is an analytic v e c t o r bundle of Lie algebras over M. W e shall call ~ t h e Lie algebra of the analytic f a m i l y tI. Define expri: ~ - * I t b y expH(X, t) = (expGX , t). I t is n o t difficult to show t h a t expH is an analytic map. I f toEM, the differential of e x p n at (0, to) is a linear isomorphism. I t follows f r o m the inverse function t h e o r e m t h a t expH defines an analytic manifold isomorphism of a n open n e i g h b o r h o o d of (0, to) in ~ onto an o p e n n e i g h b o r h o o d of (e, 0) in tI.

40 R . W . RICHARDSON, J R .

1.5.

Coordinates on the Grassmann mani/old.

L e t H = (Ht)teM be as above, let

toEM,

let ~ = ~t0 and let q = dim ~. I n order to m a k e computations involving the family ~ =

(~t)teM

of subalgebras of g, we shall need to introduce "coordinates" on the Grassmann manifold Fq(~) of q-dimensional vector subspaces of g. L e t W be a vector subspace of g such t h a t g is the vector space direct sum of ~ and W and let Fw be the open submanifold of l~q(g) con- sisting of all q-dimensional subspaces of g whose intersection with W is {0}. F o r each TEHomF(~, W), let

r 2 4 7

Then (I)(T)EFw and (I): Homy(~,

W ) ~ F w

is an isomorphism of analytic manifolds. I f F = (~, t h e n (D is an isomorphism of algebraic varieties. We denote b y O: Fw~Hom~(~), W) the inverse isomorphism. Classically, Fw is the big Schubert cell on Fq(~) and 0 defines Schubert coordinates on Fw.

F o r each t EM, the Lie algebra ~t m a y be considered as a point of Fq(g). I t follows easily from the definitions t h a t t ~ ~t is an analytic m a p of M into Fq(g). Choose an open neigh- borhood U of t o in M such t h a t ~tEFw for every tE U and let ~: U-+HomF(~, W) be defined b y ~(t) = O(~t). Then ~ is an analytic map.

1.6.

The pull-back o/an analytic/amily.

L e t

It

= ( H t ) ~ M be as above and l e t / : N - ~ M be an analytic m a p of analytic manifolds. We wish to define an analytic family/*(H) of Lie subgroups of G parametrized b y N, the pull-back of H b y / . We set

/*(H)

= {(x, s)eG x N]xEgns)}.

L e t ZN denote the restriction to ]*(H) of the projection G x N - ~ N . For each s e N, ~ ( s ) = Hs(~) x {s}. I n order to define the analytic manifold structure o n / * ( H ) we consider the analytic m a p s 1 a x / : G x N-> G x M and i: H-~ G x M. Since ~M: I I - ~ M is a submersion, it follows easily t h a t 1~ x ] and i are transversal at each point of the fibre product E = (G x N) x(a• Thus E is a closed analytic submanifold of (G x N) x H. One checks easily t h a t the projection (G x N) x I I - ~ G x N m a p s E bijectively o n t o / * ( l I ) . We define an analytic manifold structure on ]*(It) b y requiring t h a t this bijection be an isomorphism of analytic manifolds. I t is easy to check t h a t / * ( I I ) , with this analytic manifold structure, is an analytic family of Lie subgroups of G.

1.7.

Expansion o] an analytic /amily.

L e t

II

= ( g t ) t e M be an analytic family of Lie subgroups of G. I t is convenient for technical reasons to " e x p a n d " the family t t so as to obtain a new family I~I which contains all conjugates of all

H~(tEM).

We define t I = {(x, t, y) E G x M •

G Ix E yHt y-~}.

L e t ~M• a : ITI -~ M x G denote the restriction to ]I of the projection G x (M x

G)-+M x G.

For each (t,

y ) e M x G, gM• y) =(yHty -1)

x {(t, y)}.

I n order to define the analytic manifold structure on I t we proceed as in the preceding paragraph. We define ~: G x M x G-+ G x M b y a(x, t, y) =

(y-lxy, t).

One checks easily

D E F O R M A T I O N S O F L I E S U B G R O U P S A N D T H E V A R I A T I O N O F I S O T R O P Y S U B G R O U P S 41 t h a t a is a submersion. We now form the fibre product L of G • M • G and H over G • M, corresponding to or: G x M • G->G • M and i: l t - + G • M. Since ~ is a submersion, it fol- lows immediately t h a t ~ and i are transversal at each point of the fibre product L. Thus L is a closed analytic submanifold of (G • M • G) • It. The projection (G • M • G) • H - + G • M • G m a p s L bijectively onto ~I; we define an analytic manifold structure on ~I b y requiring t h a t this bijection be an isomorphism of analytic manifolds. One checks easily t h a t I~, with the above analytic manifold structure, is an analytic family of Lie subgroups of G. We write ~t=

(H(t,y))(t.~)GM•

where H(t.y)=yHty -1.

1.8. The analytic family H ~ L e t H ~ ((x, t)EHIxEH~

1.8.1. L:~M~A. H ~ is an open subset o / H .

Proof. L e t (x0, to) EH ~ We m a y write x o = (expaX1) ... (expaX~), where X 1 ... Xn ~1) = Ilt o. L e t W, U and ~o: U-* HomF(~, W) be as in 1.5. L e t ~ = { ( / , t) e 9 ]t e U} and define an analytic m a p ~: ~ - > I I b y

~(X, t) = ((expaX) (expv(X1 + ~ ( t ) . X1) ) ... (expa(X~ + ~(t)" X~)), t);

the analyticity of a follows from the analyticity of expR and the fact t h a t the group opera- tions on t t are analytic. We have :r to) = (x0, to) and ~(~u) c l t o ; moreover, it is easy to check t h a t the differential of ~ at (0, to) is a linear isomorphism. I t follows t h a t ~(~v) is a neighborhood of (x0, to) in H. This proves 1.8.1.

1.8.2. COROLLARY. H ~ is an analytic family o/Lie subgroups o I G.

1.8.3. Remarlr I n general, it is not true t h a t t l ~ is a connected component of It. See w 12.2.

w 2. The normal displacement of an analytic f a m i l y

Throughout this section H ⁼

(Ht)tEM

denotes an analytic family of Lie subgroups of t h e Lie group G. Since we are only interested in the local deformation theory, we assume throughout t h a t M is an open neighborhood of 0 in F r and we set H = H 0. L e t W be a vector subspace of ~ such t h a t ~ is the vector space direct sum of ~ and W. The normal displacement of the family H at 0 is an analytic m a p p i n g ~ of a neighborhood ~ of H • (0} in H • M into W. The normal displacement determines the family H locally. L e t K be a Lie subgroup of H. The main result of this section, Proposition 2.7.1, says t h a t if we take the Taylor series expansion of the restriction of ~ to ~ • (K • M), then the lowest order non-vanish- ing terms in this expansion can be interpreted as a one-cocycle of K with coefficients in an appropiate K-module.

42 n . "w. R I C H A R D S O N , J R .

2.1. Normal displacement/unctions.

2.1.1. Definition. A normal displacement/unction for t h e analytic family tt=(Ht)t~M at 0 is an analytic m a p y~ of an open n e i g h b o r h o o d ~ of H • {0} in H x M into W such t h a t t h e following conditions hold:

(a) ~f(x, 0) = 0 for e v e r y xEH;

(b) F o r every (x, t ) E ~ , ( e x p ~ ( x , t))xEHt a n d t h e m a p y: ~ - > H defined b y 7(x, t)=

((expa~f(x, t))x, t) is an analytic map;

(c) F o r e v e r y x E H, there exists an open n e i g h b o r h o o d Bz of x in H a n d an open neigh- b o r h o o d C~ of 0 in M such t h a t B~ • C~ c ~ / a n d such t h a t y defines an analytic manifold isomorphism of B~ • Cx onto an open n e i g h b o r h o o d of (x, 0) in It.

R o u g h l y speaking, a n o r m a l displacement function measures t h e n o r m a l variation o f t h e family I t a t 0. T h e existence of n o r m a l displacement functions is demo~lstrated in 2.3.

I t is n o t difficult to show t h a t t w o n o r m a l displacement functions agree in a n e i g h b o r h o o d of H • {0).

L e t ~v: ~ - ~ W be a n o r m a l displacement function for the family H at O, let N be an open n e i g h b o r h o o d of 0 in F a a n d let ]: N - ~ M be an analytic m a p such t h a t riO) = 0 . L e t

~/~ = ((x, t) E H • ~N] (x, ](t)) E ~/) a n d let ~v~: ~ - * W be defined b y ~ ( x , t) -~v(x,/(t)). T h e n it follows i m m e d i a t e l y f r o m t h e definitions t h a t y~ is a n o r m a l displacement f u n c t i o n for t h e pull-back analytic f a m i l y / * ( H ) =

(Hf(t))teN.

2.2. Local parametrization o / a / a m i l y o/submanifolds. L e t A, B a n d C be open boxes in (resp.) F ~, F q a n d P and let X be a closed (q § analytic submanifold of A • B • C. L e t ~c denote t h e restriction to X of t h e projection ,4 • B • C->C. Assume t h a t ar c is surjec~ive a n d is a submersion. I t follows t h a t each fibre 7rcl(z) (z E C) is a closed q- dimensional submanifold of A • B of the f o r m X~ • {z}, where X z is a closed q-dimensional analytic submanifold of A • J~. Thus X can be considered as a p a r a m e t r i z e d f a m i l y (X~)~c, of analytic submanifolds of A • B. W e assume f u r t h e r t h a t X 0 = {0} • J~.

2.2.1. L ~ A . There exist open boxes A l C A, BI ~ B and Cl C C and an analytic map Y: BI• C I ~ A 1 , with y(y, 0 ) = 0 /or yEB1, such that the map (y, z)~-~(y(y, z),y, z) is an a~mlytic mani/old isomorphism o/ B 1 • C I onto an open neighborhood of (0, O, O) in X .

Proo]. L e t arB• X ~ B x U denote t h e restriction to X of t h e projection A • B x C-~

B x C. Since ~c is a submersion a n d X 0 = {0} x B, t h e differential of :ZB • c at (0, 0, 0) is a linear isomorphism. T h e proof of 2.2.1. n o w follows easily f r o m t h e inverse function theorem.

DEFORI~IATIO:NS OF L I E SUBGROUI~S A N D T H E V A R I A T I O ] ~ OF I S O T R O F Y S U B G R O U F S 43 2.3. Existence o] normal displacement ]unctions. W e define an analytic m a p T: W x H x M - + G x M b y T(w, x, t)=((exp~w)x, t). If x e H , t h e n t h e differential of ~ at (0, x, 0) is a linear isomorphism. H e n c e T m a p s a n e i g h b o r h o o d of (0, x, 0 ) i s o m o r p h i c a l t y onto a n e i g h b o r h o o d of (x, 0) in G • M.

2.3.1. LEMMA, Let x E H . There exists an open box A x in W, an open neighborhood B x o] x in H, an open box CxC M in F r and an analytic map ~fx: Bx • Cx~+ A x, with ~x(y, O) =0 /or every y E Bx, such that the ]ollowing conditions hold:

(a) T ~w~ps A~ • B~ • C~ isomorphically onto an open neighborhood o/(0, x, O) in G • M;

(b) Let Ex =-((w, y, t)CA x ^• Bx • C xlw =~x(Y, t)}. Then T maps the closed analytic sub- mani]old E~ isomorphically onto an open neighborhood L~ o/(x, 0) in It.

(c) The map t ~-~ (expa F~ (x, t) ) x o] Cx into G is an analytic section o / I t over Cx.

The proof of 2.3.1. follows readily from 2.2.1. The function F~ measures the n o r m a l displacement of the family I t in a neighborhood of (x, 0).

2,3.2. L]~MMA. Let x, y E H . Then %(b, t) =~,~(b, t) [or (b, t)E(B~ • C~,) N (B~ • C~).

Proo/. L e t z E (B z N Bv) a n d let V denote the connected c o m p o n e n t of z in B~ N Bv. We m a y choose open neighborhoods N(z) of z in V a n d N(0) of 0 in C~. N Cv such that, if (b, t) E N(z) • N(O), then v(~x(b, t), b, t) EL z ;1Ly a n d T(y3~(b, t), b, t) EL• N Ly. I t follows easily t h a t Fx(b, t) =~y(b, t) for (b, t) 6N(z) • ~V(0). Since V x (U~ N Cy) is connected a n d F~ a n d Fy are

analytic functions , Fx a n d F~ m u s t agree on V • (C~ N Cy). This proves 2.3.2.

L e t ~ / = [Jxe~(B~ • U~); ~/ is an open n e i g h b o r h o o d of H x (0} in H • M. We define an analytic function yJ: ~/-+ W as follows: L e t (b, t)E ~ . T h e n (b, t)E Bz x Cx for some x E H.

W e set y~(b, t)=F~(b, t). I t follows f r o m 2.3.2 t h a t F(b, t) is i n d e p e n d e n t of t h e choice of x E H . I t is an easy consequence of 2.3.1 t h a t F is a n o r m a l displacement function of t h e f a m i l y t l at O.

2.4. The in]initesimal displacement. N o w we w a n t to look at t h e power series expansion of the n o r m a l displacement function ~. First some notation. If ~ = ( ~ .... , ~r)E N r a n d t = ( t l , ..., t r ) E F r, we write t ~= t~ ~ ... tT, a n d l~] = ~ +... + ~r- F o r each x E H, the function t~-~f(x, t) can be e x p a n d e d in a power series F(x, t ) = ~ N ~ u ~ ( x ) t ~ (u~(x)EW) a b o u t 0, convergent in some open box a b o u t 0 in F r. Since y~(x, t) is analytic in b o t h x and t, it follows t h a t the functions x ~-~ u~ (x) are analytic functions defined on all of H with values in W.

F o r each hEN, we denote b y P~(F r, W) t h e v e c t o r space of homogeneous p o l y n o m i a l m a p s of degree n of F ~ into W. N o w let K be a Lie subgroup of H. F o r m E N a n d x E K we define Sm(x ) EPm(F r, W) to be the polynomial m a p t ~-> Xl~l=~u~(x) t~. T h e n S~ is an analytic m a p of K into P , , ( F ~, W). If S ~ 0 for all ?'<m, t h e n S~ is called t h e m - t h in/initesimal

44 R . W . R I C H A R D S O N ~ J R .

displacement

along K of the analytic family It. The i m p o r t a n t term, of course, is the first non-vanishing infinitesimal displacement along K.

I n the case K = H the first infinitesimal displacement coincides with the infnitesimal displacement defined b y K o d a i r a [14] for an analytic family of compact complex submani- folds of a complex manifold. To pass from K o d a i r a ' s definition to ours, one needs to note t h a t the normal bundle of H in G is trivial, so t h a t an analytic cross section of the normal bundle corresponds to an analytic m a p of H into W.

2.5.

On the cohomology o/Lie groups.

L e t L be a Lie group and let ~: L-~GL(V) be a linear representation of L. We define

ZI(L, V),

the space of one-cocyeles of L with coeffi- cients in the L-module V, to be the vector space of all analytic m a p s / : L-> V such t h a t

](xy) =/(x) + ~(x)./(y)

for all x, y EL. BI(L, V), the space of one-coboundaries, is defined to be the vector space of all analytic m a p s of L into V of the form

x~-->~(x).v-v

for some

vEV.

We have BI(L,

V)~ZI(L, V)

and the quotient vector space

Hi(L, V)=ZI(L, V)/

BI(L, V) is the first cohomology space of the Lie group L with coefficients in the L-module V.

I n this paper we shall be interested in the cohomology groups

Hi(K, W)

and H i ( K ,

P,n(F ~,

W)), where K and W are as above. First we need to define the K-module structure on W. L e t p:

W-->~/~

denote the restriction to W of the canonical projection g ~ g / ~ ; p is a vector space isomorphism. The restriction to K of the adjoint representation of G gives a representation of K on g, ~ is a K-submodule of 6 and thus we have a quotient K-module structure on g/~. We define the K-module structure of W b y transporting to W b y means of p the K-module structure of g/d, i.e. we require t h a t p:

W-->g/~

be a K-module isomor- phism. L e t ~: K - + G L ( W ) be the representation defining the K-module structure of W and let prw: g -~ W and pr~: g - ~ be the projections corresponding to the direct sum decomposi- tion f i = ~ + W. I f

x E K

and wE W, then

~(x).w=prw(Ada(x).w ).

We define a representation ~m of K on

Pm(F ~, W)

as follows: if

]EPm(F r, W)

and xE W, t h e n

Q~(x)./=Q(x)o/.

Equivalently, note t h a t

P~(F ~, W)

is canonically isomorhie to

P,~(F ~, F) | W,

where

P,~(F ~, F)

is the vector space of all homogeneous polynomial func- tions of degree m on F r. Then ~m is the tensor product of the trivial representation of K on

Pm(F ~, F)

and the representation ~ of K on W. I n particular, the K-module

P,n(F ~, W)

is the direct sum of K-modules isomorphic to W. Thus the vanishing of

Hi(K, W)

implies t h a t

Hi(K, P~(F r, W))

vanishes.

2.6.

Two technical lemmas.

2.6.1. L E ~ I A .

Let C~ M be an open box in F r and let/: C->g be an analytic map such

that ](t) e ~t /or every t E C. Expand / in a power series about O:/(t) = ~ c~t ~ (c~ e

~),

convergent

D E F O R M A T I O N S OF L I E S U B G R O U P S A N D T H E V A R I A T I O N O F I S O T R O P Y S U B G R O U P S 45 in some open box, and assume that the power series is oit order >1 m, i.e. c~ = 0 whenever I ~ [ < m.

T h e n c~E~ /or every ~ such that I~1 = m .

Prooit. Choose C small enough so t h a t

~eF~

for every tEC, let ~: C-+Homr(~, W) be as in 1.5 and assume t h a t the power series expansion of ~ converges in C. Since it(t)E~t for

teC,

we have it(t)=u(t)+q~(t).u(t), where u(t)=pr~/(t). T a k e power series expansions for u(t) and ~(t):

u ( t ) = ~ b~t ~ (b~E ~)

I~l~>m

~ ( t ) = ~ a~t ~ (a~E Homp(~, W)).

Thus it(t) = u(t) + ~(t). u(t) = ~. b~/~ § (higher order terms).

I~1 =rn

Consequently e~=b~E~, for ]~1 = m . This proves 2.6.1.

L e t D be an open box in g such t h a t expa defines an analytic manifold isomorphism of D onto an open neighborhood D 1 of e in G. Similarly, let E be an open neighborhood of (0, 0) in ~ such t h a t expti maps E isomorphically onto an open neighborhood E 1 of (e, 0)

i n H .

2.6.2. L E M M A. Let C c M be an open box and let ~: C ~ G be an analytic section o/ H over C such that ~(0)=e and ~ ( C ) c D 1. Let logo: D v + D denote the inverse o / e x p o and expand logo ~(t) in a power series about 0

logo~(t) = ~ a ~ t ~ (a~E ~),

convergent in some open box, and assume that this power series is o/ order >1 m. Then a~ E whenever I =m"

Proo/. We m a y assume C chosen small enough so t h a t (~(t), t ) E E 1 whenever t EC. I f t E C, it follows t h a t logo ~(t) E ~ . The conclusion now follows from 2.6.1.

2.7. T h e / i r s t non-vanishing in]initesimal displacement is a one-cocycle. L e t y~ and S~(] > 0) be as in 2.4. and let m be the smallest positive integer such t h a t S,~40; the analytic function S,n: K - + P m ( F r, W) is the first non-vanishing infinitesimal displacement of H along K.

2.7.1. PROPOSITION. S m is a one-cocycle.

Proo/. L e t x, y E K. ~Ve m u s t show t h a t

Sm(xy) = S,~(x) + ~(x) Sin(y).

For z E K and t E F r, let sm(z , t) =Sin(z) (t). I t suffices to show t h a t Sm(Xy, t) = SIn(X, t) + ~(X) Sin(y, t)

46 R . ~v. I ~ I C I t A R D S O I % J R .

for all t in some n e i g h b o r h o o d of 0 in F r. Choose an open box C in F ~ such t h a t C ~ C x N C~ N C~, where C~, C, a n d Cxy are as in Proposition 2.3.1. Define analytic m a p s ~ , % a n d ~ v of C into G b y : ~ ( t ) = (exp~ y~(x, t)) x; %(t) = (exp~ ~(y, t)) y; a n d ~y(t) = (exp~ ~(xy, t))xy. I t is a consequence of 2.3.1. (c) t h a t ~ , ~ a n d ~ are analytic sections of I t over C. Define an analytic section ~: C ~ G b y

~(t) =~(t)~y(t)~y(t) -1 = e x p ~ ~,(x, t) (x(exp~ ~(y, t) )x -~) exp~ -y~(xy, t)

= exp~ y~(x, t) expv ( A d v x . F(Y, t)) e x p ~ - F(xy, t).

W e h a v e ~(0) = e. F r o m the first order terms of t h e Campbell-Hausdorff formula, we see t h a t t h e power series expansion a b o u t 0 of log~ ~(t) has t h e f o r m

log~ ~(t) = s~(x, t)+ Ad~ x . s ~ (y, t ) - s m (xy, t)+ {terms of order > m in t).

This series converges in some open box C~ ~ C. I t follows f r o m L e m m a 2.6.2. t h a t 0 =prw(s~(x, t)+Ad~ x . s~(y, t)-Sm(Xy , t))=s~(x, t ) + @(x)s~(y, t)-sm(xy, t), for t~C~.

This proves 2.7.1.

w 3. Proof of Theorem 3.1

This section will be d e v o t e d to t h e proof of the following theorem.

3.1. T ~ E o l~Z~. Let H = ( H t ) t e M be an analytic/amily o/Lie subgroups o/G, let t o 6 M and let H =Hto. Let K be a Lie subgroup o/ H such that the component group K / K ~ is finitely generated and such that Hi(K, ~/~)=0. Then there exists an open neighborhood U o/ t o in M and an analytic map 9: U-+G such that ~(t)K~(t)-l~ Ht /or every tE U.

The proof of T h e o r e m 3.I falls into two parts. B y using Proposition 2.7.1 a n d t h e vanishing of Hi(K, ~/~), we show t h a t there exists a formal power series solution of the problem. T h e n we use a recent t h e o r e m of )5. A r t i n [1] to show t h a t t h e formal power series solution can be replaced b y a convergent power series solution.

W e assmne t h r o u g h o u t w 3 t h a t M is an open n e i g h b o r h o o d of O in F r a n d t h a t t o = 0.

3.2. Enlarging the /amily H. L e t I t = (H(t,~))(t.~)~M• be t h e " e x p a n d e d " f a m i l y de- fined in 1.7. Define an analytic m a p (~: M • W-+M • G b y a(t, w) = (t, expa w) a n d let 74 denote t h e pull-back family a*(~I). T h u s we m a y write 74=(H(~.w))(~.w)~M• where H(t,~) = (expG w)Ht(expG- w). W e let ~0 be a n e i g h b o r h o o d of H • {(0, 0)} in H • M • W a n d ~ : 720-+ W be a n o r m a l displacement function for t h e analytic family 74.

I n order to be able to use s t a n d a r d n o t a t i o n for the power series expansion of ~F we assume t h a t we are given a basis (X 1 ... X~) of W a n d we identify F p with W b y the linear isomorphism

I ) E F O B M A T n I O N S O F :LIE S U B G R O U P S A N D T t I E Y A I ~ I A T I O N O F I S O T R O P Y S U B G R O U P S 4 7

W = ( W l , . . . , W~) F W - W l X 1 -~ ... - } - w p X p .

W i t h this identification, we have the power series expansion

~ ( x , t, w) = ~ a(~,~)(x) t~w ~.

(~, fl)

H e r e t h e s u m is t a k e n over all (~, fl) EN r • N ~ and, for each (c~,/~), x~>a(~, fl) (x) is an ana- lytic m a p f r o m H into W. F o r each x EH, the corresponding power series in (t, w) is conver- gent in some open b o x in F" • F p.

3.2.1. L]~IMA. ~lZl=la(0 ' ~) (x) w ~ = w -- O(x). w.

Pro@ Define T: W ~ M • W b y T(w)=(O,w). T h e n y*(~)=(Hw)wEw, where H w = (expe w ) H ( e x p a - w ). L e t ~ [ = { ( y , w ) E H x W[(y,O, w)E~/0} a n d define ~: ~ - ~ W b y 9(y, w) = ~ ' ( y , 0, w); t h e n c f is a n o r m a l displacement function for the f a m i l y 7"(~/). L e t C be a (sufficiently small) open b o x in W, let x E H a n d define an analytic section ~: C-~G of

~,*(~/) over C b y

~(w) = (expe w ) x ( e x p a - w)x -1 expa ~(x, w).

\~'e h a v e ~(0)=e. T h e power series expansion of log G ~(w) a b o u t 0 has t h e f o r m log~ ~(w) = w - A d ~ ( x ) . w - ~ a(o.p)(x)ure + (E),

I~l =1

where (E) consists of terms of degree higher t h a n one in w. I t follows f r o m 2.6.2 t h a t 0 = prw (w - A d e (x). w - ~ a(0. ~)(x) w ~) = w - ~(x). w - F a(o.~) (x) w r

l~l • l ]flI~l

This proves 3.2.1.

3.2.2. Remarlc. E q u i v a l e n t l y , 3.2.1 says t h a t D3u~ (0, 0, 0), t h e partial derivative of lIZ with respect t o w at (0, 0, 0) is 1 w - ~ ( x ). (1W denotes the i d e n t i t y m a p of W.)

3.3. The [ormal power series solution.

3.3.1. PBOPOSITIOSr. There exists a sequence ( g j ) ~eN o/homogeneous polynomial maps gj EPj(F ~, W) such that, i / m EN and/m =go + "'" +gm, then the power series expansion about 0 o/the analytic/unction t ~+ ~F(x, t,/re(t)) is o/order >~ m + 1/or every x E K.

Pro@ B y i n d u c t i o n on m. I f we define go =/o = 0 , t h e n it is clear t h a t t-->~g2"(x, t,/o(t)) is of order > 1 for e v e r y x E K. Assume t h a t we h a v e h a v e defined gs EPJ( F~, W) for ] = 1 ... m satisfying t h e conditions of the proposition. L e t / m = g o + . . . + g m a n d let ~,~: M - , M x W denote t h e analytic m a p t~-~(t, /re(t)). L e t ~ m = { ( x , t ) ~ H • M l(x , t,/m(t)E~0} a n d let

~0m: Urn-+ W be defined b y ~om(x, t)=~F(x, t, fro(t)); Fm is is a n o r m a l displacemen~ function

48

R. W. RICHARDSON~ JR.

for t h e p u l l - b a c k f a m i l y ~0*(~H). T h u s we h a v e a p o w e r series e x p a n s i o n a b o u t 0 (in t)

~m(X, t)=~N,C~(X)t~;

for each :r ca is a n analytic m a p of H into W and, for each

xEH,

t h e series converges for t in some open b o x in F r. I t follows f r o m t h e induction h y p o t h e s i s t h a t , for each ~ such t h a t [ ~[ < m + 1, t h e restriction of % to K vanishes. We define q~+l:

H • F ' - + W b y

qm+~(x,

t) = ~l~l=~+l

c~(x)t:

a n d we let Qm+~:

K~P,~+I( F~, W)

be defined b y

Qm+~ (x)(t)=q,,+~(x, t).

B y Proposition 2.7.1, applied to the analytic f a m i l y ~*(~H), we see t h a t

Qm+x ~Z~(K, Pz+I(F ~, W)).

Since

H~(K, Pm+I(F r, W))

= 0 , t h e r e exists

gz+~

ePm+l( F~, W) such t h a t

Qm+~(x)

=~(x)Ogm+~--gm+l for e v e r y

xeK.

Set ~z+~(x, t) =~F(x,

t, Ira(t)

+9m+l(t)), for (x, t) in a sufficiently small n e i g h b o r h o o d of H x {0} in H x M . L e t

yeK.

W e w a n t to show t h a t t h e p o w e r series expansion a b o u t 0 of

t ~'m+l (Y, t)

is of order > m + 1.

I f f l e N ~, we define

fl!=l-i~/=~fl.,!.

expansion:

3.3.2.

W e n e e d t h e m u l t i v a r i a b l e f o r m of t h e b i n o m i a l

(~) =(~!)(5!)-1((~--5)!) -1,

a n d 5~<fl

means fl-SEN ~.

H e r e

I f we use 3.2.2 a n d e x p a n d t-~Vm+l(y, t) in a p o w e r series a b o u t 0, we o b t a i n

~p,~+l (y,t) = ~ a(~.~)(y) (~) t~/m(t)~-~ gm+l (t)~ ; (~,~.~)

here t h e s u m is t a k e n over all triples (r j~, 5) E3N r x ~v x N p such t h a t 5 4;6. I f we s u m over all t e r m s with 5 = 0 , we o b t a i n t h e power series e x p a n s i o n of ~m(Y, t). T h u s we h a v e

~Pm+l(Y,t)=~m(Y,t)+ : a(~.~)(Y)(fl~)t~fm(t)~-~gm+1(t)~.

(r

Letcf(a.fl.6)(t)=(~)t~/m(t)~-~grn+l(t)~ ; (p(~,fl..)is

a p o l y n o m i a l in t. T h e order of ~(~.~.,)(as a formal power series in t) is >1 I~1 + I ~ - 51+ (m + 1)151. Thus, if ~ . 0 , ~,=. p.~) is of order

> ~ + 1 unless I ~1 = I ~ - 51 = 0 and 151 = 1. The homogeneous component of degree ~ + 1 in t h e power series e x p a n s i o n of ~ ( y , t) is

qz+l(Y, t) = Q(y).

gm+l (t) -g,~+~(t).

T h u s t h e homogeneous c o m p o n e n t of degree m + 1 in t h e p o w e r series expansion of ~2m+l(Y, t) is

If a, b E F p

a n d fl E N ~ t h e n

DWFO~ATIO~S O F L I E SU~Gl~OV~S A~D T~E V~IATIO~ OF ISOTROP~r SUBGROUPS 49 e(Y) "gin+l(t) -gin+l(t) + ~ a(o.~)(Y)gm+l(t)~;

l~[=l

it follows from L e m m a 3.2.1 t h a t this homogeneous component is 0. Thus the power series expansion of t->F,,+l(y , t) about 0 is of order > m § This completes the induction and concludes the proof of 3.3.1.

3.4. M . A r t i n ' s theorem. If the formal power series/(t) = ~ 0 gj (t) converges (where the gj(t) are chosen as in Proposition 3.3.1), then setting ~ ( t ) = e x p a - / ( t ) , the conclusion of Theorem 3.1 would be satisfied. We shall use a recent theorem of M. Artin [1] to show t h a t the gj(t) can be chosen so t h a t ~ l g j ( t ) converges in a neighborhood of 0. I n order to apply this theorem directly, however, we need to reformulate things slightly, so there is still more work to be done.

3.4.1. P ~ o P o s l T I O ~ (Artin). Let fj ( ] = 1 . . . m) be analytic functions defined in a neighborhood of (0, O) in F ~ x F p such t h a t / j ( 0 , 0 ) = 0 / o r ?'=1 .... , m. A s s u m e there exist formal power series y~ (i = 1, ..., p) in r variables with 0 constant term such that the formal power series fj (x, y(x)) (~ = 1 ... m) vanish identically (here x - (x 1 ... x~) and y(x) = (yl(x), .... y~ (x)). Then there exist analytic functions ~j (j ~ 1 ... p), defined in a neighborhood A of 0 in F ~, such that ]i(x, T(x)) = 0 for x E A and i = l .... , m. Moreover, one m a y require that the power series expansion of y j - z~ (] = 1 ... p) about 0 is of order > n, where n m a y be chosen arbitrarily.

3.5. A technical lemma. Asumme t h a t M m F ~ is chosen small enough t h a t ~)tEF W for every t e M , let 0: F w ~ H o m / ~ , W) be defined as in 1.5 and define the analytic m a p ~:

M - ~ H o m / ~ , W) b y ~(t) --| E x p a n d ~ in a power series about 0.

~ ( t ) = ~Ny~t ~ ( ~ e I-IomF (~, W)).

L e t r: HomF(I), W ) - - ~ H o m / L W) denote the restriction map. Then the power series expan- sion of r o ~ about 0 is given b y roq~(t) = ~ ( r o ~ ) t ~. If the terms of order less t h a n m in the power series expansion of r o ~ vanish, then the terms of order m in this expansion give the ruth order infinitesimal displacement along ~ of the analytic family ~ = ( ~ ) ~ of subs1- gebras of ~. I t is intuitively clear t h a t if the ruth order infinitesimal displacement along K of the analytic family H ⁼ ( H t ) t e M vanishes, then so does the ruth order infinitesimal dis- placement along ~ of the analytic family @ = (~t)teM. This is proved in the l e m m a below.

3.5.1. Lw~MA. Let F be a normal displacement function/or the a n a l y t i c / a m i l y H = ( H t ) t e M

and, /or ] ~ N , let S~: K--->P~(F ~, W) be defined as in 2.4. A s s u m e that Sr vanishes/or ] <~m.

Then the power series expansion o/roof about 0 is of order > m.

4 - - 722901 Acts mathematlca 129. I m p r i m 6 le 1 5 u i n 1972.

5 0 R . 'V~ ~. I%IOI-IA~DSO~, J R .

Proo].

For every X e ~ , let v(X)EEndr(~) be defined b y the convergent power series (ad X) j.

j=o (i+ l)!'

symbolically, one can write

r(X) = (e ~ z -

])/ad X. F o r x E G, let R(x): G-~ G denote right translation b y x,

y~->yx.

I t is well known t h a t

(3.5.2) d(R(exps - X) o exps)x = v(X)

(see, e.g. [9, p. 95]).

L e t s be the order of the power series expansion of r o ~ about 0 and assume t h a t s ~<m.

We shall show t h a t this leads to a contradiction. We m a y choose X C ~ such t h a t the fol- lowing conditions hold: (i) (X) is invertible; ( i i ) r ( X ) ( W ) N ~={0}; and (iii) there exists

~EN ~ such t h a t [c~I = s ~nd ~ ( X ) 4 0 . Let

QEP~(F ~, W)

be d d i n e d b y

Q(t)

= p ~ ( ~ ( x ) ( ~

~(x)t~).

lairs

I t follows from (i), (ii), (iii) above t h a t Q =~0.

L e t

C c M

be a (sufficiently small) open box in F ~ and define w: C->g b y

~(t) = logs (exps (X +~(t) X) e x p s - X ) .

Then v(0) = 0 and it is a consequence of (3.5.2) t h a t the power series expansion of T about 0 takes the form

(3.5.3)

T(t) =v(X)( ~ q~(X)t~)+(terms

of order :> s).

l~I=s

Define an analytic section ~:

C-+G

b y

~(t) = (exps

(X+cf(t) X)

(exps - X ) (exps -yJ (exps X, t)).

We have ~(0) = e. Since the power series expansion of t ~ ( e x p ~

X, t)

is of order > s, we see from (3.5.3) t h a t the power series expansion of logso ~ takes the form

log~ ~(t)= v(X) ( ~

cf=(X)t ~)

+ ( t e r m s of order > s).

lal ~s

I t follows from L e m m a 2.6.2 t h a t prw(v(X)~=(X))=0 for every c~ such t h a t I gl = s . This implies t h a t

Q(t)

= pr~(~(x) ( ~

~(x)t~))

I~l-s

vanishes identically in t, which gives a contradiction.

3.6. The

convergent power series solution.

L e t tF be as in 3.1, let

gjEPj(F ~, W)

(]E/~ ~) be as in Proposition 3.3.1 and let

](t)

denote the formal power series ~J~0

g~(t).

Then it follows

D E F O R M A T I O N S O F L I E S U B G R O U P S A N D T t I E V A R I A T I O N O F I S O T R O F Y S U B G R O U P S 5 1

from Proposition 3.3.1 that, for every x 6 K, the formal power series t ~--->viZ(x, t,/(t)) vanishes identically. Choose elements x I ... x~ 6 K such t h a t the cosets x l K ~ ..., x n K ~ generate the component group K / K ~ L e t C c M and A be (sufficiently small) open boxes in (resp.) F ~ and W and define analytic function ~: C • A-~ W (j ~ 1 .... , n) b y ~(t, w) = ~F(xj, t, w).

Define an analytic m a p ~0: C • A - ~ H o m ~ (f, W) b y ~o(t, w) = ro| (exp~ w) (~t)); note t h a t ~ Ad a (exp~ w) (~t) if and only if ~t0(t , w) = 0 . I t follows immediately t h a t the formal power series )~j(t, /(t)) (j = 1 ... n) vanish identically. L e t m 6N, let /re(t)~go(t)+... +gin(t), let ~%: M ~ d k / • W be as in the proof of Proposition 3.3.1 and let ~/m(X, t)=IF(x, t,/re(t)) (~0m is defined in a neighborhood of H • {0} in H • M). I t follows from Proposition 3.3.1 and L e m m a 3.5.1, applied to the analytic family ~*(~/) with normal displacement function V,~, t h a t the power series expansion of t~20(t,/,~(t)) about 0 is of order ~> m + l . Thus the formal power series 20(t ,/(t)) vanishes identically. Hence we can a p p l y A r t i n ' s theorem to the family of analytic functions '~0, ..., ~ - Thus there exist an open box U c C irt F" and an analytic m a p fi: U , W such t h a t f i ( U ) c A and such t h a t 2j(t, f i ( t ) ) - 0 for t 6 U and j = 0 ... n. Define ~]: U ~ G b y ~ ( t ) = e x p a - f l ( t ) . Since 20(t, fl(t))=0, it follows t h a t c Ada (~(t) -~) (f)t), hence t h a t K ~ r](t)-~Ht~](t) or, equivalently, t h a t ~(t) K~ -~ c Hr.

Similarly, for ] > 0 , ,~(t, fl(t))~O implies t h a t x~6rt(t)-~Htr](t), or t h a t ~q(t)x~rl(t)-~6Ht.

Since K ~ 0 {x~ .... , xn} generates K, it follows t h a t ~)(t) K ~ (t) -~ ~ H t for t 6 U. This completes the proof of Theorem 3.1.

w 4. Relation of Theorem 3.1 to results on deformations on subalgebras of Lie algebras

L e t G be a Lie group with Lie algebra g, let q < d i m 9 and let Aq be the Zariski closed subset of Pq(.q) consisting of all q-dimensional subalgebras of g. The following result was proved in [19, w 9]:

4.L L e t '~ be a q-dimensional subalgebra of g and let ~ be a subalgebra of ~ such t h a t the Lie algebra cohomology space HI( 3, g/~) vanishes. Then there exists an open neigh- borhood V of ~ in Aq and an analytic map/~: V ~ G such t h a t Ade (/~(a)) (3) c a for every a 6 V.

Now let H : ( H t ) t e M be an analytic family of Lie subgroups of G, let t o s and let H =Hto. L e t K be a connected Lie subgroup of H such t h a t H 1 ({, ~/'0)=0. Then it follows from 4.1 t h a t there exists an open neighborhood U of t o in M and an analytic m a p ~: U-+G such t h a t Ad G (rt(t)) (3)c %t for every t 6 U. This implies t h a t ~/(t)K~l(t)-lc Ht for t 6 U. Thus, if K is u connected Lie subgroup and the Lie algrebra cohomology space H 1 ({, g/%) vanishes, the conclusion of Theorem 3.1 follows from 4.1.

I f K is connected and ~: K ~ G L ( V ) is a representation, then the Lie group cohomology space H 1 (K, V) can be canonically identified with a subspace of H 1 (f, V). Thus, H 1 (~, V) = 0

52 R. W . R I C H A I ~ D S O ~ J R .

implies t h a t H i ( K , V ) = 0 . The converse statement does not hold, e.g. for K a torus and the trivial representation. There are m a n y examples of analytic families H = (Ht)te M and connected K ~ H t , as above for which H i ( K , ~/~)=0, but HI(~, g/0)~=0. Consequently, even if K in Theorem 3.1 is assumed connected, the conclusion of Theorem 3.1 is stronger t h a n a n y corollaries one can derive from 4.1.

w 5. Variations on Theorem 3.1

5.1. A n extension o/Theorem 3.1. Let the Lie group G be a closed norm~l Lie subgroup of the Lie group L. I f x E L , then Intzx: L - * L denotes the inner antomorphism y~-~xyx-L I f x EG, then IntLx maps G into G and induces the identity isomorpism of L/G. Since AdLx is the differential of IntLx, it follows that AdLx induces the identity map of I/~; equivalent!y, if X E i, then (Adz x. X - X) E ~.

5.1. THEOREM. Let G be a closed normaILie subgroup o/ the Lie group L, let H ~ ( H t ) t e M be an analytic/amily o/ Lie subgroups o/ G, let t o E M and let H = Ht.. Let K be a Lie subgroup o] H such that the component group K / K ~ is ]initely generated. Assume that there exists a vector subspace V o / I such that the/oUowing condition holds: For every ~E ZI(K, ~/~), there exists vE V such that ~ ( x ) = 7 ~ r /or every x E K (here zg/~: g ~ ~/~ denotes the canonical projection). Then there exists an open neighborhood U o / t o in M and an analytic map ~ : U ~ V such that (exp~ ~](t) K(expL -- ~(t)) c H~ /or every t E U.

We shall only sketch the proof of Theorem 5.1, which is essentially the same as t h a t of Theorem 3.1. Let M be an open neighborhood of 0 in F ~ and let t o = 0. First define an expanded analytic family I ~ = (H(t.~))(t,~)e~• L as in 1.7 (with G replaced be L; I t is an ana- lytic family of Lie subgroups of G since G is normal in L). Let ~: M • V - ~ M • JL be the map (t, v)-+ (t, expL V) and let tI~ be a normal displacement function for the pull-back an- alytic family ~* (fI). As in w 3, one defines a formal power series/(t) = ~j=0gj( ~ t) (gj EP~ ( F ~, V)) such that, for every x EK, the formM power series t->tF(x, t,/(t)) vanishes identically.

Using Artin's theorem, one can replace /(t) by a convergent power series -~](t), and ~](t) satisfies the conclusion of Theorem 5.1.

5.2. Remark. Let the Lie group A act analytically on G by automorphisms and let L = G • A be the corresponding semi-direct product. The Lie algebra 1 can be identified with the vector space direct sum ~ C)a. Let H, H and K be as in Theorem 5.I and assume t h a t a satisfies the hypothesis on V in the statement of Theorem 5.1. I n this case, Theorem 5.1 can be regarded as the extension of Theorem 3.1 from the case of the group of inner automorphisms to tile case of an arbitrary Lie group of automorphisms of G.

D ] ~ F O R M A T I O N S O F L I E S U B G R O U P S A~*D T H E V A R I A T I O N O F ISOTRO~PY S U B G R O U P S 5 3

5.3. More on the cohomolgy o 1Lie groups. L e t H be a Lie group, let L be a closed normal subgroup of H a n d let V be un H-module. Set V L = {v E V [ g v = v for every g EL]. L e t Z~(H, L, V) -- {q~ EZ~(H, V) I~(L) == {0}} and let H~(H, L, V) denote the image of ZI(H, L, V) in Hi(H, V).

5.3.1. L E M ~ A . Let the notation be as above. Then H~(H, L, V) is canonically isomorphic to Hi(H/L, VL).

The proof of 5.3.1 is elementary and will be omitted.

5.4. Analytic/amilies containing a / i x e d normal subgroup.

5.4.1. PROPOSITION. Let H=(Ht)te ~ be an analytic /amily o / L i e subgroups o / G , let toEM and set H=Hto. Assume that the component group H / H ~ is linitely generated and that there exists a normal Lie subgroup L o / G such that L is a closed Lie subgroup o I H t/or every tE M and such that HI(H/L~ (~/~)L)~0. Then there exists an open neighborhood U o] t o in M and an analytic map ~] : U ~ G with ~](0) = e such that ~(t) H~](t)-l~ Ht lor every t E U.

We shall only sketch the proof, which is similar to the proof of Theorem 3.1. L e t IUI = (H(t,x))(:,z)eM• G be the expanded analytic family, defined as in 1.7, let ~ = ~ | W as in w and let o~: M • WL-~M • G be defined b y c~(t, w) =(t, expzw). L e t ~F be a normal displace- m e n t function for the pull-back family ~*(tI). I t follows from the normality of L and the definition of I t t h a t L ~ H(t.u) for every (t, w) E M • W L. Therefore ~F(x, t, w) = 0 for every (x, t, w)E L • M • W L. Consequently, if }I' is any analytic family obtained b y pull-back from the family ~* (tI), it follows t h a t the first non-vanishing infinitesimal displacement along H of the family t I ' is an element of Z I ( H , L, ~/~). Si~ce H i ( H , L, ~/~)=

H i ( H / L , (~/~)L)=O one can, exactly as in 3.6, find a formal power s e r i e s / ( t ) = ~ o g j ( t ) (gjEPj(F ~, W r~) such that, for every x E H , the formal power series t-+~F(x, t,/(t)) vanishes identically. The rest of the proof is as in w 3.

5.4.2. COROLLARY. Let H, G and H be as in 5.4.1. Assume that M is connected, that H~ ~ = H ~ every t GM and that Ht/H ~ is o/(/inite) order m / o r every t EM. Then H t is conju- gate to H / o r every t E M.

Proo/. L e t iV be the normalizer of H ~ in G. Then _N is a closed Lie subgroup of G and H t c N for every t E M; thus fI is an analytic family of Lie subgroups of N. Therefore we m a y assume t h a t G = N , i.e. t h a t H ~ is normal in G. L e t s E M . Since H~/H ~ is finite, the cohomology group HI(Hs/H ~ (9/~)Ho) vanishes, and we m a y apply Proposition 5.4.1. This, plus the fact t h a t the order of Ht/H~ is independent of t E M, implies t h a t H t is conjugate to Hs 0 for every t in a neighborhood of s. Since M is connected, the conclusion ot 5.4.2 now follows.

5 4 : 1%. W . R I C H A R D S O l f f ~ J R .

w 6. Algebraic families of algebraic subgroups

I n this section we shall deal with (complex) algebraic families of algebraic subgroups of a (complex) algebraic group. Our methods involve a somewhat unsatisfactory mixture of transcendental and algebraic techniques.

6.1. Conventions. Our basic reference for algebraic g e o m e t r y and algebraic groups is [2] and we shall follow the notation and terminology therein. All algebraic varieties and algebraic groups are t a k e n over G. If X is an algebraic variety, then X has an underlying structure of (reduced) complex space, which we denote b y the same letter XI Thus there are two topologies given on X, the Zariski topology and the usual topology of X as a complex space; we shall refer to the latter topology as the Hausdorff topology of X. I f f: X-+ Y is a morphism of algebraic varieties, then ] is also a holomorphic m a p of the under- lying complex spaces. An algebraic subgroup of an algebraic group G is a subgroup of G which is a Zariski closed subset of G.

6.2. Algebraic families of algebraic subgroups.

6.2.1. Definition. L e t G be an algebraic group and M a non-singular algebraic variety.

An algebraic family of algebraic subgroups of G, parametrized b y M, is a non-singular Zariski locally closed subvariety t t of G x M satisfying the following conditions:

(a) L e t ~ : H - + M denote the restriction of the projection prM: G x M - + M. Then ~zr is surjective and is a submersion (i.e., the differential of ~M is surjective at each point of H).

(b) For every t CM, the fibre ~Ml(t) is of the form H t x {t}, where H t is a subgroup of G.

I t follows easily from the definitions t h a t an algebraic family of algebraic subgroups of G has an underlying structure of a complex analytic family of Lie subgroups of the complex Lie group G.

L e t H = (Ht)te M be an algebraic family of algebraic subgroups of the algebraic group G.

I t follows from the definition t h a t dim H t is constant on each component of M.

6.2.2. L~MMA. Let n ~ ( H t ) t e M be as above, assume that M is connected and let q be the common dimension o/ the H t ( t E M ). Then the map t-+~t o/ M into F~(~q) is a morphism of algebraic varieties.

The proof is elementary and will be omitted.

6.3. Some lemmas on algebraic families. L e t H = (Ht)te M be an algebraic family of alge- braic subgroups of the algebraic group G and let K be an algebraic subgroup of G.

6.3.1. L ]~ M MA. Let A = {t E M [ H t ~ K}. Then A is a Zarislci locally closed subset of M.

D / ] F O R M A T I O n S OF L I E S U B G R O ~ P S A N D T H E Y A R I A T I O ) 7 OF I S O T t r 1 6 5 SI~BGI~OU]?S 5 5

Proof. Since H , ~ K ~ if a n d

only if ~D

~, it follows i m m e d i a t e l y f r o m L e m m a 6.2.2 t h a t A 0 = {t E M IH t ~ K ~ is a Zariski closed subset of M. L e t Xl, . . . , X n be a complete set of representatives for t h e eosets of K modulo K 0 a n d let A j ~ {t E M ] xj E Ht} (j = i ... n). T h e n H N ({xj} x M ) = { x j } • A t is Zariski locally closed and hence A t is a Zariski locally closed subset of M. Thus A = I"1 ~=0 At is Zariski locally closed.

We recall t h a t a subset C of an algebraic v a r i e t y X is eonstructible if C is t h e u n i o n of a finite n u m b e r of Zariski locally closed subsets of X. A constructible subset of X contains a Zariski open, Zariski dense subset of its Zariski closure. If C is a constructible subset of X a n d cf: X - ~ Y is a m o r p h i s m of algebraic varieties, t h e n ~(C) is a constructible subset of Y.

6.3.2. LEMMA. Let B = { t E M [ H t contains a con/agate o / K } . Then B is a construetible subset o / M .

Proo/. Define ~: G • M x G -~ G • M b y o~(x, t, y) ~ (y-lxy, t), let I i = ~-I(H) a n d let

~ i • G : f t - > M x G denote the restriction to l i of the projection pr~• ~: G x (M • G)-> M • G.

T h e n t h e fibre 7~• t) has the f o r m (yHty -1) • {(t, y)}. L e t H(t.~)=yHty -l. One checks easily t h a t t I = (H(t. ~)) (t. ~) ~ ~• ~ is a n algebraic family of subgroups of G. B y L e m m a 6.3.1, A = {(t, y) E M • G I yHty-~ ~ K } is a Zariski locally closed subset of M x G. L e t prM: G X M - ~ M denote t h e projection. T h e n B = p r M ( A ) is a constructible subset of M.

6.3.3. LEMMA. Let X be an algebraic variety, let U be a subset of X which is open with respect to the Hausdor// topology o / X and let B be a construetible subset of X which contains U.

Then there exists a Zariski open subset V o / X such that U ~ V c B.

Proof. B y i n d u c t i o n on dim X (dimension as an algebraic variety). T h e l e m m a is trivial for dim X = 0 . L e t n = dim X a n d assume t h a t the result holds Ior varieties of dimension less t h a n n. L e t X1, ..., Xr be t h e irreducible c o m p o n e n t s of X which do n o t m e e t U a n d let X ' = X - ( J ~ I X j . Set B ' = X ' N B. T h e n X ' is Zariski o p e n i n X a n d B ' is a construetible subset of X ' . Replacing X a n d B b y (resp.) X ' a n d B', we m a y assume t h a t X m e e t s every irreducible c o m p o n e n t of X. I t is clear t h a t U, a n d hence B, is Zariski dense in X. T h u s B contains a Zariski open, Zariski dense subset V 0 of X. L e t C = X - V0. T h e n U N C ~ B N C a n d dim C < n . Thus, b y t h e inductive hypothesis, there exists a Zariski open subset V1 of C such t h a t U N C c V1~ B N C. L e t V = V0 0 V1. T h e n V is Zariski open in X a n d U ~ V ~ B.

6.4. A~plication of Theorem 3.1 to algebraic/amilies o/algebraic subgroups.

6.4.1. THEOREM. Let H = ( H t ) t ~ M be an algebraic family o/algebraicsubgroups o/the algebraic group G, let toEM and let H = H w Let K be an algebraic subgroup o / H such that H i ( K , ~/~)=0. Then there exists a Zariski open subset V o/ M containing t o such that Ht contains a conjugate o / K ]or every t E V.