Polar Actions on Hilbert Space

The Journal of Geometric Analysis Volume 5. Number l. 1995

Polar Actions on Hilbert Space

By Chuu-Lian Terng

ABSTRACT. An isometric H-action on a Riemannian manifold X is called polar if there exists a closed submanifold S of X that meets every H-orbit and always meets orbits orthogonally (S is called a section). Let G be a compact Lie group equipped with a biinvariant metric, H a closed subgroup of G x G, and let H act on G isometrically by (hi , h2) �9 x = hlxh2 I. Let P(G, H) denote the group of Hi-paths g : [0, 1] ~ G such that (g(0), g(1)) ~ H, and let P(G, H) act on the Hilbert space V = H~ 1], g) isometrically by g �9 u = gug -l - g,g-t . We prove that if the action of H on G is polar with a fiat section then the action of P (G, H) on V is polar. Principal orbits of polar actions on V are isoparametric submanifolds of V and are infinite-dimensional generalized real or complex flag manifolds. We also note that the adjoint actions of affine Kac-Moody groups and the isotropy action corresponding to an involution of an affine Kac-Moody group are special examples of P(G, H) actions for suitable choice of H and G.

Introduction

In finite dimensions, the theory of symmetric spaces, polar representations, and isoparametric submanifolds of Euclidean spaces are all closely related. For example, the following are known:

(1) The isotropy representation of symmetric space G / K at eK (which is called an s-

representation) is polar. Moreover, Dadok proved in [D] that up to orbital equivalence these are the only polar actions on Euclidean spaces.

(2) The orbits of s-representations are generalized full or partial flag manifolds.

(3) The principal orbits of a polar representation are isoparametric [Tell.

(4) Thorbergsson proved in [Th] that an irreducible, full, compact, isoparametric submanifold of codimension r > 3 in Euclidean space must be a principal orbit of some s-representation.

(5) There are many codimension-two isoparametric submanifolds [OT1,2, FKM] that are not orbits of any orthogonal representations. Nevertheless, these submanifolds share many of the

geometric and topological properties of flag manifolds.

It is therefore natural to ask whether there are analoguous results in infinite dimensions. A start in this direction is made in [Te2], where we proved an infinite-dimensional analogue of (3). In this paper,

we give a general construction for polar actions on Hiibert space, study the submanifold geometry of

the infinite-dimensional orbits of these actions, and discuss their relations to "infinite-dimenisonal

symmetric spaces."

Math Subject Classification 57S20, 22E65. Key Words and Phrases Hilbert space, isoparametric submanifolds, polar actions. Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-lnstitut ftir Mathematik in Bonn.

�9 1995 The Journal of Geometric Analysis ISSN 1050-6926

130 Chuu-Lian Terng

Let us first recall some definitions from transformation group theory [PT2]. A smooth isometric

action of a Hilbert Lie group G on a Hilbert manifold M iscalledproper if gn'xn ~ Y0 and Xn --+ XO

in M implies that {gn} has a convergent subsequence in G; it is called Fredholm if given any x E M

the orbit map G --~ M defined by g ~ g - x is a Fredholm map. Note that isotropy subgroups of proper actions are compact, and the orbits of Fredholm actions are of finite codimension. An

isometric, proper Fredholm action is called polar if there exists a closed submanifold S of M, which

meets all orbits and meets them orthogonally. Such an S is called a section for the action. We call

N ( S , G) = {g ~ G I g . S C S}, Z ( S , G) = {g ~ G l g . s = s Vs ~ S}

the normalizer and centralizer of S in G, respectively. The quotient group

w ( s , = N ( S ,

is a discrete group, called the generalized Weyl group associated to the polar action.

In [Te2], we obtained an infinite-dimensional analogue of (3) and also showed that the orbits of

polar actions on Hilbert spaces have many of the special properties of the finite-dimensional complex

or real flag manifolds. In particular, for a polar action of a Hilbert Lie group G on a Hilbert space V, we proved that the generalized Weyl group W of the action is an affine Weyl group, the focal points of a principal orbit are just the points on singular orbits, and the focal set meets the section in the union

of the reflection hyperplanes of W. The focal multiplicity of points on a reflection hyperplane is

invariant under W, so we can use them as markings for the Dynkin diagram of W, thereby associating to each polar action on a Hilbert space a marked affine Dynkin diagram. Moreover, we also proved

the following:

(i) If M is a principal orbit in V, then M is an isoparametric submanifold of V, i.e., the

normal bundle v ( M ) is globally fiat, and the shape operators A~x) and Av(y) are conjugate along any parallel normal field v.

(ii) The restriction of the distance function fa (x) = IIx - a II 2 to any G-orbit is a perfect Morse function, and the homology of an orbit can be computed directly from W and its multiplicities

in the same way as in [BS] for the finite-dimensional case.

There are two known families of polar actions on Hilbert space. The first family is given in [Te2]: the action of the Hilbert Lie group H 1 (S 1, G) of H 1 -loops of a compact connected, semisimple Lie

group G on the Hilbert space H ~ 1 , g) given by the gauge transformation

g �9 u = g u g -1 -- g , g - I

is polar. These orbits can be viewed as an infinite-dimenisonal analogue of complex full or partial

flag manifolds [PS]. A second family was constructed by Pinkall and Thorbergsson [PIT]: Let G / K be a symmetric space of compact type, and

= {g E HI( [0 , zr], G) I g(0), g(yr) E K}.

Polar Actions on Hilbert Space 131

Then the action of the Hilbert Lie group K on H~ Jr], ~) by gauge transformations is again polar. The orbits can be viewed as infinite-dimensional analogues of real flag manifolds (see [Te3]).

Note that the first example is related to the Adjoint action of G on G, and the second example

is related to the K • K-action on G. Since both the Adjoint action and the K • K-action on G are polar with fiat sections, this leads naturally to a general construction of polar actions on

Hilbert spaces. Namely, if H is a closed subgroup of G x G, and the action of H on G defined

by (hi, h2) . x = hlxh21 is polar with flat sections, then the action of the group P(G, H) of

HI-paths g in G with (g(0), g ( l ) ) E H on V = H~ 1], g) by gauge transformations

g �9 u = gug -t - g,g-a

is polar.

Next we discuss the relation between polar actions on Hilbert space and involutions of affine Kac-Moody groups. Let L(G) = HI(S 1, G) denote the Hilbert Lie group of Sobelov Hi-loops

in a compact, connected, simple Lie group G. Then the affine Kac-Moody group L ( G ) of type 1 is a 2-torus bundle 7r : L(G) ~ L(G) (see [PS]). There is a nondegenerate, Ad-invariant

bilinear form on the affine algebra L(g) [K,PS], and the Adjoint representation of L(G) on its

Lie algebra gives rise to the first example of polar action on Hilbert space above. We show that every involution p on G induces an involution p* on L(G) such that K is the fixed point set of

p* and the Adjoint representation of I ( on ~ gives rise to the second example of polar action on

Hilbert space above, where K is the fixed point set of p and ~ is the --1 eigenspace of dp* on L(g). Let cr be an outer automorphism of G of order k, and let L(G, or) denote the subgroup of

L (G) consisting of loops g that are Zk-equivariant, i.e., satisfy g (t + 2yr / k) = ty (g (t)). Then the subgroup L(G, ty) = rr-I (L(G, tr)) is the affine Kac-Moody group of type k. Moreover, we show

that the Adjoint actions of affine groups of type k on their Lie algebras give rise to the polar action

of P(G, H) on H~ 1], g), where H = {(x, ~r (x)) 1 x E G}. Following the terminology from the theory of finite-dimenisonal symmetric spaces, we call L ( G ) and L ( G , o') symmetric spaces

of type II, and L (G) / I ( a symmetric space of type I.

The above discussion suggests that, as in finite dimensions, a close relationship does exist between polar actions on Hilbert spaces and isotropy representations of symmetric spaces. However, the precise nature of this relation is still not well understood.

The paper is organized as follows. In Section 1, we associate to each polar H-action with flat sections on a compact Lie group G, a polar action on the Hilbert space H ~ ([0, 1 ], g), and we study the relation between the H-action and the associated infinite-dimensional action. In Section 2, we give description of the principal curvatures of the principal orbits of the associated infinite-dimenisonal action in terms of the H-action. Finally, in Section 3, we discuss the relation between the isotropy

representations of infinite-dimensional symmetric spaces and polar actions on Hilbert spaces.

1. P(G, H)-actions

In this section, we give a general construction of polar actions on Hilbert spaces from polar

actions on compact Lie groups with flat sections.

132 Chuu-Lian Terng

We first establish notation. Let G be a connected, compact, semisimple Lie group, equipped with a biinvariant metric. Let

= Ht( [0 , 1], G), V = H~ 1], ~I)

denote the Hilbert Lie group of all Sobolev HI-paths from [0, 1] to G and the Hilbert space of H ~ maps from [0, 1 ] to g, respectively. (One should think of elements of V as representing connections (automatically fiat) for the product principal G-bundle over [0, 113 Let G act on V isometrically via gauge transformations:

g �9 u = g u g -J - g ,g-1.

It is obvious that given any u E V there exists g E G such that u = _ g , g - l . This proves that acts transitively on V. Using the same proof as in [Te2], we see that this action is proper and

Fredholm. So one way to produce interesting proper Fredholm, isometric actions on Hilbert space is to find closed, finite codimension subgroups of G, whose orbits in V give interesting geometric and topological manifolds. In the following, we will give a method of finding subgroups of G, whose action on V is polar.

Let

Let H be a closed, connected subgroup of G x G, acting on G by

(hi , h2) " g = h igh21 .

P ( G , H ) = {g E G : HI([O, II, G) I ( g ( 0 ) , g ( l ) ) E H } .

Note that the homomorphism

qJ " G ---~ G x G, g w-~ (g(0), g(1))

is a submersion and P ( G , H) = ~ - l ( H ) . It follows that P ( G , H ) is closed and has finite codimension in G, hence the action of P ( G , H ) is also proper and Fredholm.

Next we define two operators

E : H~ 1], 0) ~ HI([0, 1], G), qb : H~ 1], 9) --+ G

as follows: let E (u) (t) be the parallel translation in the trivial principal bundle I x G over I = [0, 1] defined by the connection u ( t ) d t , and qb(u) the holonomy, i.e., E ( u ) : [0, 1] --~ G is the unique solution of the initial value problem:

E - 1 E ' = u

E(O) = e,

and

qb(u) = E(u)(1) .

Polar Actions on Hilbert Space

P r o p o s i t i o n 1.1. With notation as above,

(i) E ( g . u) = g(O)E(u)g -~, qb(g. u) = (g(0) , g ( l ) ) - qb(u),

(ii) v E P (G , H) �9 u if andonlyi fd~(v) E H �9 Cb(u).

133

P roo f . It is easy to prove (i) by a direct computation. To see (ii), let v ~ P ( G , H ) �9 u. Then it follows from (i) that ~ ( v ) E H �9 qb(u). Conversely, suppose qb(v) = hlCb(u)h2 j for some (hi , h2) E H. Let g( t ) = E ( v ) - t ( t ) h t E ( u ) ( t ) . Then (g(0) , g(1)) = (ht, hz), and

E ( g . u) = g(O)E(u)g -l = h l E ( u ) E ( u ) - l h ~ l E (v ) = E(v ) ,

which implies that g �9 u = v. [ ]

We recall [PT1 ] that a section for a polar action is automatically totally geodesic. Moreover, it is well known that a compact, flat, totally geodesic submanifold of G containing e is a torus subgroup.

T h e o r e m 1.2. Let the notation be as above, and suppose the H-action on G is polar with flat sections. Let A be a torus section through e and let ct denote its Lie algebra. Then:

(1) the P (G , H)-action on V is polar, and the space fi = {fi I a E a} is a section, where f i : [ 0 , 1] ~ g denotes the constant map with value a;

(2) the generalized Weyl group Iv]/ = W(fi, P ( G , H) ) is an affine Weyl group;

(3) if d2 : P (G , H) --+ H is the homomorphism defined by qJ(g) = (g(0) , g(1)) , then

*(N(a, P ( G , H) ) = N ( A , H), qJ maps Z(fi, P ( G , H) ) isomorphically onto Z ( A , H), and

A = Ker(q~) 71N(fi, P(G , H ) ) = {g(t) = exp(,kt) I ~- E A(A)} ,

where A ( A ) is the unit lattice of A:

A ( A ) = {• E a I exp(~.) = e};

(4) qJ induces a surjective homomorphism from I~V to W = W ( A, H ), W is isomorphic to

I/VIA, and ~V is the semidirect product of W with A;

(5) qJ maps the isotropy subgroup P (G , H)a isomorphically to Hexp(a); in fact, its inverse

is the map ~o : Hexo(a) ~ P ( G , H)a defined by (hi, h2) w-~ g( t ) = e x p ( - a t ) h l exp(at);

(6) u E V is a singular point of the P (G, H)-action if and only if ~ (u) E G is a singular

point of the H-action.

134 Chuu-Lian Terng

Proo f . We prove each statement of the theorem below.

(1) To see that fi meets every P(G, H)-orbit , we let u �9 V. Since A meets every H-orbit,

there is a �9 a such that dO(u) �9 H . exp(a). But E ( h ) = exp(ta) , so dO(h) = exp(a) . By 1.1(ii),

we have h �9 P(G, H) �9 u. To prove fi is orthogonal to P(G, H) �9 h for all a �9 a, we note that

T ( P ( G , H ) . h ) ~ = { [ u , h ] - u ' l u �9 H l ( [ 0 , 1 ] , g), ( u ( 0 ) , u ( 1 ) ) �9

For b �9 a, we have [a, b] = 0 and

([u, h] - u ' , b) = ([u, al - u', b) dt

= (u, [a, bl) - (u, b) 'dt = 0 - (u(1) - u(O), b) = 0,

where the last equality follows from the fact that

T ( H . e)e = [x - y I (x, y) �9 I~} _L a.

(2) It was proved in [Te2] that the generalized Weyl group of a polar action on Hilbert space is

an affine Weyl group.

(3) First we prove qJ(N(fi, P(G, H) ) C N ( A , H). Let g E N(a, P (G , H)) . Then for

a E athere exists b E a suchthat g-f i = /~ . Using 1.1(i) andthe fact that E ( h ) = exp( ta) , E(/~) =

exp(bt) , we have exp(b) = g(0) exp(a )g(1) - l . This proves that (g(0) , g (1) ) �9 N ( A , H) .

Next we prove that N ( A , H) C q,(N(a, P(G, H)) . Let (hi , h2) �9 N ( A , H) . Then hlAh21 = A. In particular, this implies that hlh21 E A, h lA = Ah2, h lAh~ 1 = Ahzh~ l = A, and hlah21 = a. Now given al �9 a, there exists bl �9 a such that h, exp(al)hf 1 = exp(bl). Set

g(t) = e x p ( - b l t ) h l exp(al t ) .

Then it is easily seen that (g(0) , g (1) ) = (hi , h2), and a direct computation shows that

g . h = bi + e x p ( - b l t ) h l ( a - at)h~ -1 exp(bl t ) .

Since hlah{ l = a and a is abelian, g �9 h = bl + hl(a -- ai)h~ 1 �9 ft.

To compute the intersection of Ker(qJ) and N(a, P(G, H)) , we assume that g E N(a, P(G, H ) ) and g(0) = g(1) = e. Then for a E a, there exists b E a such that g - h = b, so 1.1 (i) implies that

g(t) = E(b ) - l g(O)E(h) = exp((a - b)t) .

But g(1) = exp(a - b) = e, i.e., (a - b) E A. Conversely, let L E A ( A ) , and g(t ) = exp(Lt).


Then g E P(G, H) , g ~ Ker (~ ) , and g . f is the constant map with constant equal to a -- ,k. This proves that g E Ker(kO) fq N(fi, P(G, H)) .

Now let g ~ Z(~, P(G, H)) . Since g �9 0 = 0, g,g-E = 0. So g(t) = hi is a constant

map. But g �9 t~ = h l a h l I = a. This implies that qJ(g) = (hE, hi) E Z (A , H), and �9 maps

Z(fi, P(G, H)) injectively into Z(A , H). To prove surjectivity we let (hi , h2) E Z(A , H). Then hleh~ l = e implies that hE = h2. Set g(t) = hE. Since hlahT E = a, g �9 fi = ~ for all a c a.

(4) The statement is a consequence of (3).

(5) If g E P(G, H)a , then from E(h)( t ) = exp( ta) and 1.1 we have exp(at ) =

g(O)exp(at)g -I. So exp(a) = g(O)exp(a)g(l) -E, i.e., ~ ( g ) E Hexp(a). Conversely, if

h l exp(a)h2 -j = exp(a), then set g(t) = exp(--at)hE exp(at) , a direct computation as above

implies that g �9 fi = ft.

(6) The proof follows from (5). [ ]

Let N be a submanifold of G, and g2 (G, e, N ) the set of all H 1-paths in G such that y (0) = e

and F (1) E N. It is known (cf. [PI ]) that ~ (G, e, N ) is a Riemannian Hilbert manifold,

T(f2(G, e, N))• = {uy [ u ~ Hi( [0 , 11, g)},

and the Riemannian inner product on T(g2 (G, e, N ) ) y is

f0 1

(uF, vF) = (u'(t), v ' ( t ) )d t .

If N is an H-orbit in G, then P(G, H) acts on f l (G, e, N) by g * y = g(O)Fg -l, and this action

is isometric and transitive.

C o r o l l a r y 1.3. With the same notation and assumption as in 1.2, let N be the H-orbit through exp(a) in G, and

F : f2 (G, e, N ) --+ H~ 1], g), F ( y ) = F-1y,.

Then F is an isometric, equivariant embedding, and the image of F is equal to P (G, H ) �9 f .

R e m a r k 1.4. Let M be a Riemannian manifold, G a compact Lie group acting on M

isometrically, p E M a regular point, and N a G-orbit in M. Let

s : f2(M, p, N) -+ R, fo

E(y) = IIy'(t)ll2 dt

136 Chuu-Lian Terng

denote the energy functional. Bott and Samelson proved (cf. [BS]) that if the G-action on M is variationally complete, then g is a perfect Morse function, and the homology of ~2 (M, p, N) can be computed explicitly in terms of the singular data of the G-action. Moreover, they showed that the following H-actions on G are variationally complete:

(i) the action of the diagonal group H = ZX(G) = {(g, g) I g E G} on G, i.e., the Adjoint action of G on G;

(ii) the action of H = K x K on G, where G / K is a symmetric space.

By results of Hermann [He] and Conlon [C 1 ] there are two more families of variational complete actions on G:

(iii) the action of H = KI x K2 on G, where both G/K1 and G / K 2 are symmetric spaces,

(iv) the action of H = G(cr) = {(g, a ( g ) ) I g E G} on G, where a is an automorphism of G (the action of G(cr) on G will be called the cr-action).

Conlon noted that the above four families of actions are polar with flat sections, and he proved that

in general polar actions with fiat sections are variationally complete (Conlon called these sections K-transversal domains [C2]).

Example s 1.5. Applying Theorem 1.2 to the examples (i)-(iv) in Remark 1.4 gives many polar actions on Hilbert spaces. In fact, the first and second families of examples described in the introduction are the P(G, H)-action on V corresponding to examples (i) and (ii) in Remark 1.4 respectively. Note also that under the isometric embedding F in Corollary 1.3, the path space f2(G, e, H �9 exp(a)) is embedded in the Hilbert space V = H~ 1], g) as a taut submanifold with constant principal curvatures, and the energy functional s corresponds to the square of the norm in the Hilbert space V, i.e., s = IIF(g)ll z.

T h e o r e m 1.6. (Conlon [C2]). Let M be a simply connected, complete Riemannian mani-

fold, K a compact Lie group acting on M isometrically, and M s the set of singular points. Suppose the K-action on M is polar with a flat section ~, and the K-action on M is not transitive. Then

(i)

(ii)

(iii)

in the Pi.

M,r

M~ f3 N is the union of finitely many totally geodesic hypersurfaces { P~ . . . . . Pr },

the generalized Weyl group W (E , K ) is a finite group generated by the reflections of l~

In the following, we will discuss further structures of the generalized Weyl group for polar H-action on G with a flat section A (here G need not be simply connected). First we consider the rank of the affine Weyl group W of the P ( G , H)-action on V. Let W C Iso(R k) be a Coxeter group, {el I i E I} the set of reflection hyperplanes of W, and ui c=_ R k the unit normal to ei. Then the rank of W is the dimension of the linear span of {ui I i E I}. In particular, the Dynkin


diagram of a rank k, finite Coxeter group has k vertices, and that of a rank k infinite Coxeter group has k + 1 vertices. Recall also that the codimension of the principal orbits of an action is called the cohomogeneity. For a polar action the cohomogeneity is the dimension of a section [PT1].

Theorem 1.7. Let the assumptions and notation be as in Theorem 1.2.

(i) Therankoftheaffine Weylgroup ~V = W(~, P ( G , H ) ) isequaltothecohomogeneity of the H-action on G, which is equal to the dimension of A.

(ii) If the H-action is not transitive, then the H-action on G always has singular points, or equivalently the P (G, H)-action has singular points.

Proof . We may assume that M = P ( G , H) �9 ~ is a principal orbit. Supposethat the rank of ~ / i s less than the dimension of a. Then there exists 0 # b E a such that the line Rb does not meet any reflection hyperplane of T~'. So there is no focal point for M on the line R/~. This implies that the shape operator A~; = 0. But

T M 6 = {u' l u ~ Hl([O, 1l, g), ( u ( 0 ) , u ( 1 ) ) E ~ } ,

A~,(u') = [u, b] =- O.

So [u, b] = 0 for all u E T M 6, which implies that Ix, b] = 0 for all x E 0. Since ~ is semisimple, b = 0, which is a contradiction. This proves (i).

The proof of (ii) follows directly from the proof of (i). Since the set of focal points of M is the set of singular points for the P (G, H)-action [Te2], if the action has no singular points then M has no focal points. This implies that all shape operators A~ = 0 for/9 c v ( M ) 6. Then the above computation implies that b = 0. Hence dim(fi) = 0, i.e., the H-action is transitive. This proves (ii).

[]

Theoreml.8. Let the assumptions and notation be as in Theorem l.2. The slice representation

of the H-action at exp(a) and the slice representation of the P ( G, H)-action at f are equivalent polar representations. In fact,

(1) the slice representation of the H-action at exp(a) is given by

(hi, h2)" (bexp(a)) = (h lbh~l )exp(a) ,

(2) the normalplane v ( P ( G , H ) �9 ~)~ is

{v(t) = e x p ( - a t ) b exp(at) I b exp(a) E v ( H . exp(a))exp(a) }

and the slice representation at Zt is given by g �9 v = gvg -l ,

13 8 Chuu-Lian Terng

(3) f : v ( n . exp(a))exp(a) ~ v ( P ( G , H) �9 fi)a definedby

b exp(a) w-~ v(t) = e x p ( - a t ) b exp(at)

is an equivariant isomorphism, i.e.,

go((hl, hz))- f ( b exp(a)) = f ( ( h l , h 2 ) - b exp(a)),

where q) is as in 1.2 (5).

Proof. It is known [PT1] that the slice representations of a polar action are also polar. The proof of (1)-(3) will be given separately below.

(1) Given (hi, h2) E Hexp(~), then we have hi exp(a)h21 = exp(a). So if bexp(a) E v ( H �9 exp(a))exp(~, then we have

( h i , hE) �9 b exp(a) = hlb exp(a)h~ -1 = hlbh-( 1 exp(a).

(2) Let M = P(G, H) �9 h C V = H~ 11, g). By a direct computation, we obtain that

TMa = {[u, f i ] - - u ' l u ~ HI(J0, 1],1~), (u(0) ,u(1)) E Ij},

v ( M ) a = { V E V l v ' = [ v , fi], ( v ( 1 ) , y ) = ( v ( O ) , x ) V(x ,y ) ~b}.

It is easy to see that if v' = [v, al then there exists b E g such that v(t) = e x p ( - a t ) b exp(at). In order for v ~ v(M)a, v must also satisfy the condition (v(O), x) = (v(1), y) for all (x, y) E I~, SO

(v(1), y) = (exp( -a )b exp(a), y) = (b exp(a), exp(a)y)

= (v(O), x) = (b, x) = (bexp(a), x exp(a)).

It follows that

But

(bexp(a), x e x p ( a ) - e x p ( a ) y ) = O , V(x, y) E b.

T ( H . exp(a))r ----- {x exp(a) - exp(a)y I (x, y) E l~},

so (2) follows.

(3) This statement is a consequence of (1) and (2). [ ]


T h e o r e m 1,9. Let the assumptions and notation be as in Theorem 1.2. Then:

(1) There exists a ~ n such that the affine Weyl group ~' ---- W(a, P ( G , H ) ) is the semidirect product of the finite Weyl group W~ and a lattice group A 1, where (Va is isotropy

subgroup of W at ~, and Ai is a lattice group containing the unit lattice A(A) and is invariant

under W~.

(2) The Weyl group of the slice representation of the H-action at a = exp(a) is the isotropy

subgroup W~ of W = W ( A, H ), and the Weyl group of the slice representation of the P ( G , H)-

action at ~ is WD.

(3) (Va is isomorphic to Wa.

(4) The generalized Weyl group W ( A , H ) of the H-action on G is the semidirect product

of W~ and the abelian group A i / A (A).

Proof . It follows from the standard theory of affine Weyl groups [Bo] that there exists t) E

such that

(i)

(ii)

(iii) ^

of W.

~ ' is the semidirect product of the isotropy subgroup Wa and a lattice group A l;

if W is of rank k, then l~r is a finite Weyl group of rank k;

is a vertex of a Weyl chamber of W, and l~'a is maximal among all isotropy subgroups

It is known [PT1 ] that the slice representation of a polar action is polar, and the Weyl group of the

slice representation at x isequal to the isotropy subgroup at x of the generalized Weyl group of the polar action. So W~ and W~ are the Weyl groups of the slice representations of the H-action and P ( G , H)-action at a and ~, respectively. It follows from 1.2 (3)-(5) that qJ(g) = (g(0), g(1)) induces an isomorphism from W~ to W~. This proves (1)-(3), and statement (4) is then a consequence of (1) and 1.2(4). [ ]

Let X be a Riemannian K-manifold. Then a submanifold a (possibly with boundary) of X is called a fundamental domain for the K-action if it meets each K-orbit in exactly one point. For example, a Weyl chamber in fi is a fundamental domain of the action of the affine Weyl group I~r

on G,

Propos i t ion 1.10. With the same notation and assumption as in 1.2, let r be a Weyl chamber

in a. Then

(1) r is a fundamental domain for the P ( G , H )-action on V,

(2) r and exp( r ) are isometric, and

(3) exp(r) is a fundamental domain for the H-action on G.

140 Chuu-Lian Terng

Proof. The orbit space of a polar action is isometric to the orbit space of its section under the ^

action of the generalized Weyl group [Te2]. Since r is a fundamental domain for the W-action on fi, (1) follows.

Let �9 be as in 1.1 and �9 as in 1.2. Then ~ ( f i ) = exp(a) . It follows from 1.1 that qb I r is

injective and alP(r) = exp ( r ) meets every H-orbit . This proves (3). Since A is flat, exp : a --+ A is a local isometry. So r is isometric to exp( r ) . [ ]

Proposition 1.11. Suppose the H-action on G is polar with a torus A as a section, and the action is not transitive. Let k denote the cohomogeneity of the H-action, i.e., k = dim(A).

(1) I fk = 1, then W(A , H) is Z2 oradihedralgroup.

(2) If k > 1, then the totally geodesic hyperplanes PI . . . . . Pr of Theorem 1.6 satisfy the crystallographic condition, i.e., the angle Oij between Pi and Pj is 7r / Pij with Pi~ 6 { 1, 2, 3, 4, 6}.

Proof. By 1.7 (i), the affine Weyl group lviz = W(a , P(G, H)) has rank k, so Vr is generated by (k + 1) reflections. By 1.2 (4), qJ induces a homomorphism qJ from 1~r onto W(A, H), so W(A, H) is a finite group generated by (k + 1) order two elements. But IYr is an affine Weyl group,

so 14 z satisfies the crystallographic condition if k > 1. Hence the image ~ ( l ~ ' ) = W(A, H) also satisfies the crystallographic condition if k > 1. [ ]

We end of this section, with two conjectures:

Conjecture 1.12. If the H-act ion on G is of cohomogeneity 1, then the generalized Weyl group is crystallographic. More generally, if M is an isoparametric hypersurface of G, then the number g of distinct principal curvatures of M must be 1,2, 3, 4, or 6.

Conjecture 1.13. Given any polar action of a Hilbert Lie group L on a Hilbert space V, one can find some compact Lie group G and a closed subgroup H of G • G such that:

(1) the action of H on G is polar with flat sections;

(2) the L-action is orbital equivalent to the P(G, H)-act ion, i.e., there exists an isometry

from V to H~ 1], ~1) that maps L-orbits onto the P(G, H)-orbi ts .

2. The submanifold geometry of P(G, H)-orbits

It is a consequence of Section 1 that the submanifold geometry of the orbits of a polar H -

action on G with flat sections determines the submanifold geometry of orbits of the associated

P(G, H)-ac t ion in the Hilbert space V = H~ 1], ~). For example, the curvature distributions,

the curvature normals, and the marked Dynkin diagram of M can be explicitly computed from the


data of the H-act ion on G. These geometric data of M have been given in [Te2] when the H-ac t ion is the Adjoint action on G and in [PIT] when the H-act ion is the K x K-act ion on G (here K

is the fixed point set of an involution on G). They are all obtained from well-known root space

decompositons with respect to the fiat section. In the following, we will give the relation between

the geometry of M and the H-action.

To simplify the notation, we may assume that e E G is a regular point for the H-action. So M = P(G , H ) �9 () is a principal orbit, and is isoparametric in V [Te2]. Note that v(M)~ = ~, and given b E a the vector field b(g �9 ~) = g~g- l is a parallel normal field on M. Moreover, there

exist a smooth subbundle E0 of T M , finite rank smooth vector subbundles {El I i E I} (the E~s are called the curvature distributions), {vi I i E I} C a, and parallel normal fields {vi I i E I}

( ~ 's are called curvature normals) such that

(1) T M = Eo@ E{Ei l i E l};

(2) the shape operator At; I Ei = (b, v~) idE, for i E I , and Ab I E0 = 0;

(3) E0 is integrable, and leaves of E0 are affine subspaces of V;

(4) each El, i E I , is integrable, and the leaf Si(x) of Ei through x E M is a standard

sphere, called the curvature sphere for El;

(5) if/ i ---- a l (b, vi) = 1}, then {gi l i ~ I} is the set of reflection hyperplanes of the affine Weyl group I~ for the P ( G , H)-act ion, (this determines vi in terms of W);

(6) if mi denotes the rank of E~, then the m i ' s are invariant under lvis;

(7) the marked affine Dynkin diagram associated to M is the diagram for 1~ with the vertex

corresponding to a simple root ~, marked by mi = rank Ei.

Let r be a Weyl chamber in fi for the l~-action on ft. Then r determines {'~i I i E I}. By 1.10,

r is isometric to a fundamental domain of the H-act ion on G. So {gi I i E I} and {vi I i E I} can be computed explicitly from the Euclidean geometry of the fundamental domain of H-act ion on G, which is a k-simplex (k = dim(a)). Since Ei (g . ()) = gEi ( 6 ) g - l , to find the relation between Ei and the H-act ion it suffices to find the relation between Ei (l~) and the H-action.

P r o p o s i t i o n 2.1. With the same notation as above, choose ai C a such that cli E s and cti does not lie in ej for any j E I, j 7~ i. Let ot i = exp(ai) . Then

(1) 0e c Oct, = {(x, aT~xai) I x ~ ~} n O;

(2) for i ~ I, the map (x, otTl xoti) ~ v(t) = exp(--ait)[x, ai] exp(ait) gives a well- defined isomorphism from Octi/Oe to Ei (6);

(3) [0e, a] = 0;

(4) E0(0) = {u E H~ 1], 60) [ f l u ( t )d t E T(He)e} , where 3o is the centralizer of ain g, i.e., 30 = {x E 1~ ] [x, a] ---- 0}.

P r o o f . Since e is assumed to be a regular point for the H-act ion, He fixes A pointwise. So

we obtain (3), and 0e C Oct for any ot E A. Note that (hL, ha) E Hct if and only if h2 = ~ - l h l o t .

142 Chuu-Lian Terng

This proves (1).

It is known (cf. [Te2]) that P ( G , H)a i �9 () is the curvature sphere Si(b) of Ei through 0, so

Ei(()) = T ( P ( G , H)a~ �9 ())~. By (1) and 1.2 (5), we have

P ( G , H)a = {g(t) = exp(--at )hl exp(at) I (hi, e x p ( - a ) h l exp(a)) E H},

and (2) follows by straightforward computation.

To prove (4), we recall that

TM~ = { u ' l u c Hi([0 , 1],g), (u(0), u(1)) E I~},

{ f0 / = v E H~ l ] ,g ) ] v( t )d t E T ( H e ) e ,

Ag(u') = [u, b].

So for v = u' ~ TM~, we have v ~ E0 if and only if [u, b] = 0 for all b ~ a. This implies that

u( t ) E 30, which proves (4). [ ]

In the following we give more detailed geometric description for the cohomogeneity-one actions

and the tr-actions.

2.2. C o h o m o g e n e i t y - o n e act ions. Suppose the H-action on G is of cohomogeneity 1,

W ( A , H ) is the dihedral group of order 2n, and A is a circle of length s Let b ~ a be a unit vector such that exp(s = e. We may assume that s = 1 and e is a singular point for the H-action. Then

{~j = e x p ( j b / 2 n ) I 0 < j < 2n}

is the set of singular points on A, {exp(tb) ] 0 < t < 1/2n} is a fundamental domain for the H - action, and the affine Weyl group W = W(a, P ( G , H ) ) is the semidirect product of Z2 with the

lattice group A 1 = { j b / n I J E Z}. For 0 < t < 1/2n, let Mt denote the P (G, H)-orbi t through the constant path tb. Let m i and m2 denote the dimension of He / Ha and He, / Ha, respectively, where ot = exp(tb) for some 0 < t < 1/2n. Then {Mt I 0 < t < 1/2n} is a family of isoparametric

hypersurfaces of V, and the marked Dynkin diagram associated to Mt is .41 marked with multiplicity (ml, mz), the nonzero principal curvatures of Mt are {~.j = ( ( j / 2 n ) - t) -l I J ~ Z}, and the

multiplicity of ~.~ is ml if j is even, and is mz if j is odd. Moreover:

(1) M0 and M1/2n are smooth submanifolds of V with codimension 1 -4- ml and 1 -4- m2 respectively, and Mo U ml/zn is the set of focal points of Mt in V for 0 < t < l /2n .

(2) Given 0 < t < 1/2n, V can be written as the union of B0 U B1 such that

(i) B0 is the normal disk bundle of radius t of M0 in V,

(ii) B1 is the normal disk bundle of radius 1/2n -- t of Mr/2n in V,

(iii) B0 f3 BI = OBo = OBl = Mt, and

(iv) then


letting bi = dim(Hi(Mr, Z2)), and p(x ) = ~_~ bix i the Poincar6 series of M,

p ( x ) = (1 "q- x m ' ) ( l "Jl- X m2 )

| - - x m l + m l

: ~ xk(m'+m2)(1 "-I- X ml + X m2 -~- x m l + m 2 ) .

k=O

2.3. The a-actions. Let H = G(o-) = {(g, o ( g ) ) I g �9 G} be as in 1.4(iv), where o is an outer automorphism of order r on G (r = 2 or 3). Let J~ be the fixed point set of o on [t, and p = J~• Note that He is the diagonal group of K , and the slice representation at e is the Adjoint

representation of K on .~. Let a be a maximal abelian subalgebra of ~. Then ~ is a section of the P ( G, H)-act ion.

Since {ad(a) " g + 0 I a E a} is a family of commuting skew-adjoint operators, there exist

[HI positive root systems A0 and A 1 (subset of a*), x~, y~ �9 Y for c~ �9 A0, r , , s , �9 p for ~ �9 A1, and a linear subspace P0 of p, which give the following decompositions of J~ and p:

J~=a+ ~ Rx~+Ry~, ctEA o

P = P 0 + ~ R r , + R s / 3 ,

such that

[a, x~] = --ot(a)y~, [a, y~] = et(a)x~,

[a, r,] = --~(a)s~, [a, s~] = f l (a)r , ,

o(u~) = u~, o ( p , ) = e2"i/r p/3,

Vot E Ao,

Vfl �9 A~,

(*)

where u~ = x~ + iy~, P/3 = r/~ + is , , and r = order (o ) . Let M = P(G, H) �9 fi be a principal

orbit for some ot E a. Then the normal plane of M at & is fi, and for/9 C a, the shape operator of M in the direction of/9 is

At;([u, f] - u') = - [ u , [~].

Let al . . . . . am be a basis for a. Then we have the following:

Case ( 1 ): order ( o ) = 2. Let {qj } be a basis for P0. Then by (*) and a direct computation, we see that the real and imaginary parts of uae 2nit, p~e (2n+l)it, aje 2nit, and qje (2n+l)t form an eigenbasis

for At; with ot(b)/(et(a) + 2n), ~(b) / ( f l (a) + 2n + 1), 0 and 0 as eigenvalues, respectively. So the

reflection hyperplanes of W(fi, P (G, H) ) in a are the affine hyperplanes defined by or(t) = 2n,

f l ( t ) = 2 n + 1, forot E Ao, fl E Al a n d n E Z.

Case (2): o rder (a ) = 3. By direct computation we obtain that the real and imaginary parts of ucte 3nit, p#e (3n+l)it, p~e (3n+2)it, a je 3nit, (]j e (3n+l)it, and q.e form an eigenbasis for At; with

ot(b) / (~(a) + 3n), ~ ( b ) / ( ~ ( a ) + 3n q- 1), ~ ( b ) / ( ~ ( a ) + 3n q- 2), 0, 0 and 0 as eigenvalues,

144 Chuu-Lian Terng

respectively, where {q j, ~j } is a basis of the complexify of (P0)c such that a (qj) = e2~ri/3qj (hence

a (61j) = e-2Jr/3qj) �9 So the reflection hyperplanes of W(fi, P(G, H)) in a are the affine hyperplanes

definedby a ( t ) = 3n, fl(t) = 3n + 1, f i( t) = 3n + 2 forot E A0, fl C /XL andn E Z.

3. Involutions of Kac-Moody algebras and the P (G, H)-actions

In this section, we review the construction of affine Kac-Moody algebras [K] and affine groups

[PS]. We also construct automorphisms on affine groups from given automorphisms on compact Lie groups and discuss their relations with the P (G, H)-actions of Section 2.

Let G be a compact, simply connected, simple Lie group, and let ( , ) be the normalized

biinvariant inner product on g, i.e., (ha, ha) = 2 for the shortest simple coroot of 9. Let L(9 ) =

H~ SL, 9), t be the angle variable in S 1, and u' denote du /d t . Let [u, rio(t) =- [u(t) , v(t)]. Then

L (1~) is a Lie algebra. The affine Kac-Moody algebra of type 1 is a two-dimensional extension over

L(ft):

L(g) = L(g) + Rc + Rd

with the bracket operation defined by

[u, v] = [u, V]o + oJ(u, v)c,

[d, u] = u',

[c, u] = [c ,d ] = O,

where o~ is the 2-cocyle defined on L(~) by

1f02 o9(u, v) = ~ (u(t) , v ' ( t ) )d t .

(Note that [, ] is only defined on a dense subspace H l (S l , f~) of L(~t); see 3.4 for a discussion of this point.) Let ( , ) denote the bilinear form on L(O) defined as follows:

(U, V)

(c, d)

f0 2~ = (u(t), v(t)) dt,

= 1, ( u , c ) = ( c , c ) = ( d , d ) = O .

Then ( , ) is Ad-invariant, i.e.,

([~, ,7], ~) = (~, [o, ~]), v~, ,1, r ~ L(g).

Next we construct two types of automorphisms on L(g) from automorphisms on G: Given

a : G --+ G, an automorphism of order k, we will use the symbol a also to denote the induced


automorphism on g. Let

c? -L(9) L(9)

be the linear map defined by

~(u ) ( t ) = a ( u ( - 2 y r / k + t)),

Let p be an involution on G, and

the linear map defined by

p ,

p*(u)( t ) = p ( u ( - t ) ) ,

( c ) = c ,

L(9)

(d) = d.

p*(c) = - c , p*(d) = - d .

P r o p o s i t i o n 3.1. Let cr : 1~ --+ 1~ be an automorphism of order k on the simple Lie algebra

9, and ( , ) the normalized Ad-invariant form on 9. Then

(i) (o'(x), a ( y ) ) = (x, y),

(ii) o ) (6 (u ) , ~ ' (v)) = co(u, v),

(iii) co(a*(u), a * ( v ) ) = --o)(u, v), if a is an involultion.

Proof i Since a is an automorphism of 9, (x, y) = (or(x), a ( y ) ) is an Ad-invariant inner

product on g. But g is simple implies that (a (x ) , a ( x ) ) = c(x , x) for some c > 0. Since a k = id,

c k = 1. This implies c = 1, which proves (i). Using (i), the definition of the cocycle co, and direct

computation, (ii) and (iii) follows. [ ]

As a consequence, we have

C o r o l l a r y 3.2. If a and p are automorphisms of 9 of order k and 2, respectively, then ~ and p* are automorphisms of L(g ) of order k and 2, respectively.

The fixed point set L(9 , a ) of 6 is called the affine Kac-Moody algebra of type k if cr is an

order k outer automorphism of ft. It is obvious that

where

L(9, a ) = L(9, a ) + Rc + Rd ,

L ( g , a ) = {u c L(~) l u( t ) = a ( u ( - 2 z r / k + t))}.

146

It can also be easily seen that

Chuu-Lian Terng

= +

is the decomposition into the 1 and -- 1 eigenspaces of p*, where

= {u �9 L(~) ] u ( - t ) = p(u( t ) )} ,

= {u �9 L(g) I u ( - t ) = - p ( u ( t ) ) } + Rc + Rd.

3.3. C o n s t r u c t i o n o f L ( G ) . We review briefly the construction of L ( G ) as given in Chapter 4 of [PSI. The first step is to construct the central extension of the loop group L(G) = H l (S 1 , G)

given by the 2-cocycle 09. The left-invariant 2-form on L(G) determined by the 2-cocycle o9 will

also be denoted by o9. Since the inner produc(on 0 is normalized, the 2-form __109 is an integral 2zri

cohomology class of L ( G ) , so there exists a principal Sl-bundle, ~o : P --~ L (G) with a connection 1 -form/5 such that the curvature of fl is ~'09, (i.e., the Chern class of this principal bundle is [09/2rr i ]). Then the group L (G) of bundle isomorphisms of P that preserve the connection fl and cover a left

translation s for some g �9 L (G) is the central extension by S l given by the cocyle o9.

We fix a base point Y0 �9 ~ 0-1 (e). For a curve F : [0, 1] --+ L ( G ) , let

I-I v : ~ - l ( y ( O ) ) -+ f - l ( F ( 1 ) )

denote the parallel translation along F given by the connection/5. Suppose F �9 L (G) and F covers

the left translation s on L(G) . Since F*(f l ) = r ,

F o Hv = Hgv o F.

So F is uniquely determined by g �9 L (G) and F(yo), i.e., L(G) is the set of pairs (g, y), where g �9 L ( G ) and y �9 ~o-l(g).

There is also an explicit construction o f / , ( G ) given in [PS, p. 47], which we will use to lift

automorphisms ~ and p* to L ( G ) : Let (g, p , z) be a triple with g �9 L(G) , p a path in L(G) joining e to g and z �9 S 1 �9 Then F can be constructed by using parallel translation. For y �9 ~o - l (h),

we choose some curve F in L(G) joining h to e, let

F ( y ) = n ] r

This is well defined and in L ( G ) . For if F1 and F2 are curves joining x to e, then since L(G) is simply connected, Fi -1 �9 F2 (the loop obtained by the inverse of Fl followed by F2) bounds a surface S in L(G) . But S 1 is abelian, so "curvature is infinitesimal holonomy," i.e., we have

C(y1-1 �9 F2)I-I• (y) = H• ) for all y E qg-l(x), where

Polar Ac t ions on Hi lber t Space

This also proves that (gl, Pl, Zl) and (g2, P2, z2) gives the same F if and only if

gl : g2, zl = C(p2 * pll)Z2 �9

147

We call two such triples equivalent. Define multiplication of triples by

(gl, Pl, zl) �9 (g2, P2, z2) = (gig2, Pl * (gl �9 P2), z lz2) .

Then the group of equivalence classes of these triples is the Hilbert Lie group L (G ) , whose Lie algebra is the central extension/,(g) = L(g) + R c of L(g) by w.

It is easily seen that e is �9 (g, p, z) = ( e is �9 g, e is �9 p, z) defines a continuous action of S 1 on L(G) , where (e is �9 g ) ( t ) = g ( s W t) and e is �9 p denotes the path (e is �9 p ) ( r ) = e c~ . p ( r ) . Then

the semidirect product L ( G ) = S t ~< L ( G ) given by this Sl-action is a topological group, which

is the group model for the affine algebra L(g).

R e m a r k 3.4. Although L ( G ) is a Hilbert manifold, and a topological group (i.e., the group

multiplication is continuous), it is not a Hilbert Lie group, because the map

H l ( S I , G ) x S l - - - > H l ( S I , G ) , ( f , g )~ - -~ f o g

while smooth in the second variable, is only continuous in the first variable [P2,PS]. As a result, the Lie bracket of the corresponding "Lie algebra" L (g) is only defined on a dense subspace. It should also be noted that L ( G ) is a Hilbert manifold locally modeled on the product of the Hi-space and R z, but the Ad-invariant metric is the product of the H ~ metric on the HI-space and the Lorentz metric on R 2, so L ( G ) is only a Lorentz manifold in the weak sense. Our goal in reviewing this

construction of affine groups is to suggest a good infinite-dimensional analogue of s-representations and their relation to polar actions on Hilbert spaces. As we will now see, while the above problems make it difficult to give a rigorous definition of infinite-dimensional symmetric spaces, they do not interfere with the construction of the desired polar actions.

3.5. The Adjoint action of [,(G). Let

Jr : L ( G ) ---> L ( G ) , (e i'~, (g, p , z ) ) ~-> g

denote the natural projection. Then the Adjoint action of L ( G ) on its Lie algebra L(t~) [K,PS] is

given by

Ad(~)(u) : g u g - l + (gug -1, g , g - l ) c,

1 Ad(~)(d) : _ g , g - I q_ d - ~ ( g , g - 1 , g , g - l ) c ,

148 Chuu-Lian Terng

if ~ : (1, (g, p, Z)). Note that the intersection R ~ of the sphere of radius -- 1 with the hyperplane {u + rc + sd E L(0) I s -- 1}, i.e.,

R ~ = {u + r c + d I u 6 L(0), r = - (1 + �89 u))},

is a horosphere of the infinite-dimensional hyperbolic space. Hence R ~ is invariant under the Adjoint action of L(G) on L(9), and R ~ is isometric to the Hilbert space L(0) = H~ l, 9) via u q- rc -t- d ~ u. Moreover, the action on R ~176 factors through L(G) , and the corresponding action on L(0 ) is equivalent to the action of L(G) ~ P(G, A(G)) on L(0) --~ H~ 1], 0) by gauge group transformations.

3.6. The automorphism 6 on L(G) . Let ~r be an automorphism of order k on G. Then

: L (G) --+ L(G) , 6(g ) ( t ) : ~r (g( -2:r /k + t))

is an automorphism on L(G) . It follows from the construction of L(G) in 3.3 that

(e is, (g, p, z)) w-> (e is, (6(g) , ~ ( p ) , z))

is a well-defined automorphism on L (G), which will still be denoted by 6. Then the fixed point set L(G, or) ofo" is the affine group of type k, and L(0, or) is its Lie algebra. Again

R ~ : { u + r c + d [ u ~ L ( g , o - ) , r = - ( l + � 8 9

is invariant under the Adjoint action of L(G, or), and R~ is isometric to L(0, or) via the map u + r c + d w-> u. Moreover, the action on R~ factors through L (G, tr), and the corresponding action on L( 0, or) is equivalent to the action of L(G, tr) ~_ P(G, G(~r)) on L(0, or) _~ H~ 1], 0) by gauge group transformations, where G(o') : {(x, cr (x)) I x E G}.

3.7. Involut ion p* on L(G) . Let p be an involution on G, and K the fixed point set of p. Then

p* : L (G) --~ L(G) , p*(g)(t) = p ( g ( - t ) )

is an involution on L(G) . Note that p*(e is �9 g) = e -is �9 p*(g). Using this equality and the construction of L (G) in 3.3, it follows that

(e is, (g, p, z)) ~ (e -is, (p*(g), p *(p), z - l ) )

is a well-defined involution on L(G) , which will still be denoted by p*. Then the fixed point set of p* on L(G) is

L(G)p = {(1, (g, p, 1)) I P*(g) = g, p*(p(r)) = p( r )} ,

Polar Ac t ions on Hi lber t Space | 49

which is isomorphic to

{g ~ L (G) I p ( g ( - t ) ) = g(t)} ~ P(G, K • K) .

Let L (9) -~ J~ + ~ denote the 1, - 1 eigendecomposition of p* on L (g) as before. Then J~ is the Lie

algebra of L(G)o, and the Adjoint action of L(G)p leaves ~ invariant, and induces on ~ the isotropy

representation of the "symmetric space" L ( G ) / L (G)R atthe identity coset. This isotropy action also

leaves R p = R ~ (q ~ invariant. Moreover, R p is isometric to {u E L(g) I p ( u ( - t ) ) = - u ( t ) }

via the map u + r c + d w-~ u, and the the action of L (G)p on R p is equivalent to the P (G, K • K) -

action on H~ 1], g).

3.8. Natural embedding of L ( G ) / L ( G ) p in to L ( G ) . Let p be an involution on G and K the fixed point set of p. Then the finite-dimensional symmetric space G / K can be naturally

embedded in G as the orbit N through e of the action of G(p) : {(x, p(x ) ) I x E G} on G, i.e., N = {xp(x) -l I x c G}. Similarly in infinite dimension, the involution p* induces an action of

(G) on L (G) via h * y = hyp* (h)- l , and the orbit ,(4 of this action through the identity gives a

natural embedding of L ( G ) / L ( G ) p into L(G) . We also note that e c M, the restriction r of the

isometry x w-~ x - l on L ( G ) to h~/induces an isometry of/~/, r ( e ) = e, and dze is - i d on T~le. So if p = g * e E M, then rp = grg -1 has the property that rp(p) = p and d(rp) e is --id on

T~/lp. This is the characteristic property of a globally symmetric space. To describe 374 explicitly,

we compute directly and see that

= {(e is, (g, p , z)) I g(t) = h ( t ) p ( h ( - s - t)) -l for some h c L(G)}.

Note that for g(t) = h ( t ) p ( h ( - s - t)) -l we have g = e -is/2 �9 ( ( e is~2 �9 h)p*(e is~2 �9 h) - l ) . So

the loop part 7r(M) = S 1 �9 M, where

M = {hp*(h)- ' I h E L(G)} C L(G) .

Next we claim that

M = {g ~ L (G) I g( t) -~ = p ( g ( - t ) ) , g(O), g(rr) ~ N} "~ ~ ( G , N, N) ,

where ~ ( G , N, N) denote the Hilbert manifold of HI-paths in G with endpoints in N. It is

easy to see that M is contained in the right-hand side. Now suppose given g E L (G) such that g(t) -l = p (g( - - t ) ) and g(0), g(zr) ~ N. Then there exist x, y E G such thatxp(x) -1 = g(O) and yp(y)-J = g(zr). Let r : [0, zr] ~ G be any HI -map such that r(0) = x, r(Jr) = y, and

define

r(t), ift ~ [0, Jr], h(t) = g(t)p(r(27r - t)) i f t E [Jr, 27r].

Then h E L(G) and hp*(h) -1 = g, i.e., g E M.

150 Chuu-Lian Terng

[BS]

[Bo] [Cl1

[C2l [C3] [D]

[FKM]

[H] [He] [K] [OTI]

[OT2]

[P1] [P2] [PT1] [PT2]

[PIT]

[PS] [Tel] [Te2] [Te3]

[Th]

R e f e r e n c e s

Bott, R., and Samelson, H., Applications of the theory of Morse to symmetric spaces. Amer. J. Math. 80, 964-1029 (1958). Bourbaki, N., Groupes et Algebres de Lie. Hermann, Paris, 1968. Conlon, L., The topology of certain spaces of paths on a ocmpact symmetric space. Trans. Amer. Math. Soc. 112, 228-248 (1964). Conlon, L., Variational completeness and K-transversal domains. J. Differential Geom. 5, 135-147 (1971). Conlon, L., A class of variationally complete representations. J. Differential Geom. 7, 149-160 (1972). Dadok, J., Polar coordinates induced by actions of compact Lie groups. Trans. Amer. Math. Soc. 288, 125-137 (1985). Ferus, D., Karcher, H., and MiJnzner, H. E, Cliffordalgebren and neue isoparametrische hyperfl~ichen. Math. Z. 177, 479-502 (1981). Helgason, S., Differential Geometry and Symmetric Spaces. Academic Press, 1978. Hermann, R., Variational completeness for compact symmetric spaces. Proc. Amer. Math. Soc. 11, 554-546 (1960). Kac, V. G., Infinite Dimensional Lie Algebras. Cambridge University Press, 1985. Ozeki, H., and Takeuchi, M., On some types of isoparametric hypersurfaces in spheres, I. Tohoku Math. J. 127, 515-559 (1975). Ozeki, H., and Takeuchi, M., On some types of isoparametric hypersurfaces in spheres, II. Tohoku Math. J. 28, 7-55 (1976). Palais, R. S., Morse theory on Hilbert manifolds. Topology 2, 299-340 (1963). Palais, R. S., Foundations of Global Non-linear Analysis. Benjamin Co., New York, 1968. Palais, R. S., and Terng, C. L., A general theory of canonical forms. Trans. Amer. Math. Soc. 3tl0, 771-789 (1987). Palais, R. S., and Terng, C. L., Critical Point Theory and Submanifold Geometry, Lecture Notes in Math., vol. 1353. Springer-Vedag, Berlin and New York, 1988. Pinkall, U., and Thorbergsson, G., Examples of infinite dimensional isoparametric submanifolds. Math. Z. 205, 279-286 (1990). Pressley, A., and Segal, G. B., Loop Groups. Oxford Science Publ., Clarendon Press, Oxford, 1986. Terng, C. L., Isoparametric submanifolds and their Coxeter groups. J. Differential Geom. 21, 79-107 (1985). Terng, C. L., Proper Fredholm submanifolds of Hilbert spaces. J. Differential Geom. 29, 9 4 7 (1989). Terng, C. L., Variational completeness and infinite dimensional geometry. Proceedings of Leeds Conference, edited by L. Verstraelen and A. West. Geometry and Topology ofSubmanifolds, III, pp. 279-293. World Scientific, Singapore, 1991. Thorbergsson, G., Isoparametric foliations and their buildings. Annals of Math. 133, 429-446 (1991).

Received August 6, 1992

Department of Mathematics, Northeastern University, Boston, MA 02115

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Polar Actions on Hilbert Space