Polar Actions on Hilbert Space

CHUU-LIAN TERNG *Department of Mathematics, Northeastern University

Boston, MA 02115, USA

§0. Introduction.

In finite dimensions, the theory of symmetric spaces, polar representations, andisoparametric submanifolds of Euclidean spaces are all closely related. For example, thefollowing 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 equiv-alence 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 ([Te1]).(4) Thorbergsson proved in [Th] that an irreducible, full, compact, isoparametric sub-

manifold of codimension r ≥ 3 in Euclidean space must be a principal orbit ofsome s-representation.

(5) There are many codimension two isoparametric submanifolds ([OT1,2], [FKM]) thatare not orbits of any orthogonal representations. Nevertheless these submanifoldsshare many of the geometric and topological properties of flag manifolds.

So it is natural to ask whether there are analoguous results in infinite dimensions. Astart in this direction is made in [Te2], where we proved an infinite dimensional analogueof (3). In this paper, we give a general construction for polar actions on Hilbert space,study the submanifold geometry of the infinite dimensional orbits of these actions, anddiscuss their relations to “infinite dimenisonal symmetric spaces”.

Let us first recall some definitions from transformation group theory ([PT2]). Asmooth isometric action of a Hilbert Lie group G on a Hilbert manifold M is called properif gn ·xn → y0 and xn → x0 in M implies that gn has a convergent subsequence in G;and it is called Fredholm if given any x ∈M the orbit map G→M defined by g → g ·xis a Fredholm map. Note that isotropy subgroups of proper actions are compact, andthe orbits of Fredholm actions are of finite codimension. An isometric, proper Fredholmaction is called polar if there exists a closed submanifold S of M , which meets all orbitsand meets them orthogonally. Such an S is called a section for the action. We call

N(S, G) = g ∈ G | g · S ⊂ S, Z(S, G) = g ∈ G | g · s = s ∀s ∈ Sthe normalizer and centralizer of S in G respectively. The quotient group

W (S, G) = N(S, G)/Z(S, G),

* Work supported partially by NSF Grant DMS 8903237 and by The Max-Planck-Institut fur Mathematik

in Bonn.

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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 ofthe finite dimensional complex or real flag manifolds. In particular, for a polar actionof a Hilbert Lie group G on a Hilbert space V , we proved that the generalized Weylgroup W of the action is an affine Weyl group, the focal points of a principal orbit arejust the points on singular orbits, and the focal set meets the section in the union of thereflection hyperplanes of W . The focal multiplicity of points on a reflection hyperplaneis 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 Dynkindiagram. Moreover, we also proved that:

(i) If M is a principal orbit in V then M is an isoparametric submanifold of V , i.e.,the normal bundle ν(M) is globally flat, and the shape operators Av(x) and Av(y)are conjugate along any parallel normal field v.

(ii) the restriction of the distance function fa(x) = ‖x−a ‖ 2 to any G-orbit is a perfectMorse function, and the homology of an orbit can be computed directly from Wand 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 isgiven in [Te2]: the action of the Hilbert Lie group H1(S1, G) of H1-loops of a compactconnected, semi-simple Lie group G on the Hilbert space H0(S1, g) given by the gaugetransformation

g · u = gug−1 − g′g−1

is polar. These orbits can be viewed as infinite dimenisonal analogue of complex fullor partial flag manifolds ([PS]). A second family was constructed by Pinkall and Thor-bergsson ([PiT]): Let G/K be a symmetric space of compact type, and

K = g ∈ H1([0, π], G) | g(0), g(π) ∈ K.

Then the action of the Hilbert Lie group K on H0([0, π], g) by gauge transformationsis again polar. The orbits can be viewed as infinite dimensional analogues of real flagmanifolds (see [Te3]).

Note that the first example is related to the Adjoint action of G on G, and thesecond example is related to the K × K-action on G. Since both the Adjoint actionand the K ×K-action on G are polar with flat sections, this leads naturally to a generalconstruction of polar actions on Hilbert spaces. Namely, if H is a closed subgroup ofG × G, and the action of H on G defined by (h1, h2) · x = h1xh−1

2 is polar with flatsections, then the action of the group P (G, H) of H1-paths g in G with (g(0), g(1)) ∈ Hon V = H0([0, 1], g) by gauge transformations

g · u = gug−1 − g′g−1

is polar.

Next we dicuss the relation between polar actions on Hilbert space and involutionsof affine Kac-Moody groups. Let L(G) = H1(S1, G) denote the Hilbert Lie group of

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Sobelov H1-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 π : L(G)→ L(G) (see [PS]). There is anon-degenerate, Ad-invariant bilinear form on the affine algebra L(g) ([K],[PS]), and theAdjoint representation of L(G) on its Lie algebra gives rise to the first example of polaraction on Hilbert space above. We show that every involutionρonG induces an involutionρ∗ on L(G) such that K is the fixed point set of ρ∗ and the Adjoint representation of Kon p gives rise to the second example of polar action on Hilbert space above, where K

is the fixed point set of ρ and p is the −1 eigenspace of dρ∗ on L(g). Let σ be an outerautomorphism of G of order k, and let L(G, σ) denote the subgroup of L(G) consistingof loops g that are Zk-equivariant, i.e., satisfy g(t+2π/k) = σ(g(t)). Then the subgroupL(G, σ) = π−1(L(G, σ)) is the affine Kac-Moody group of type k. Moreover, we showthat the Adjoint actions of affine groups of type k on their Lie algebras give rise to thepolar action of P (G, H) on H0([0, 1], g), where H = (x, σ(x)) |x ∈ G. Followingthe terminology from the theory of finite dimenisonal symmetric spaces, we call L(G)and L(G, σ) symmetric spaces of type II, and L(G)/K a symmetric space of type I.

The above discussion suggests that, as in finite dimensions, a close relationship doesexist between polar actions on Hilbert spaces and isotropy representations of symmetricspaces. 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 spaceH0([0, 1], g), and we study the relation between the H-action and the associated infinitedimensional action. In section 2, we give description of the principal curvatures of theprincipal 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 ofinfinite 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 spacesfrom polar actions on compact Lie groups with flat sections.

We first set up notations. Let G be a connected, compact, semi-simple Lie group,equipped with a bi-invariant metric. Let

G = H1([0, 1], G), V = H0([0, 1], g)

denote the Hilbert Lie group of all Sobolev H1-paths from [0, 1] to G and the Hilbertspace of H0-maps from [0, 1] to g respectively. (One should think of elements of Vas representing connections (automatically flat) for the product principal G-bundle over[0, 1]). Let G act on V isometrically via gauge transformations:

g · u = gug−1 − g′g−1.

It is obvious that given any u ∈ V there exists g ∈ G such that u = −g′g−1. This provesthat G acts transitively on V . Using the same proof as in [Te2], we see that this action

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

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

(h1, h2) · g = h1gh−12 .

LetP (G, H) = g ∈ G = H1([0, 1], G) | (g(0), g(1)) ∈ H.

Note that the homomorphism

Ψ : G→ G×G, g → (g(0), g(1))

is a submersion and P (G, H) = Ψ−1(H). It follows that P (G, H) is closed and hasfinite codimension in G, hence the action of P (G, H) is also proper and Fredholm.

Next we define two operators

E : H0([0, 1], g)→ H1([0, 1], G), Φ : H0([0, 1], g)→ G

as follows: let E(u)(t) be the parallel translation in the trivial principal bundle I × Gover I = [0, 1] defined by the connection u(t)dt, and Φ(u) the holonomy; i.e., E(u) :[0, 1]→ G is the unique solution of the initial value problem:

E−1E ′ = u

E(0) = e,

andΦ(u) = E(u)(1).

1.1 Proposition. With notations as above,(i) E(g · u) = g(0)E(u)g−1, Φ(g · u) = (g(0), g(1)) · Φ(u),

(ii) v ∈ P (G, H) · u if and only if Φ(v) ∈ H · Φ(u).

Proof. It is easy to prove (i) by a direct computation. To see (ii), let v ∈P (G, H) · u. Then it follows from (i) that Φ(v) ∈ H · Φ(u). Conversely, supposeΦ(v) = h1Φ(u)h−1

2 for some (h1, h2) ∈ H . Let g(t) = E(v)−1(t)h1E(u)(t). Then(g(0), g(1)) = (h1, h2), and

E(g · u) = g(0)E(u)g−1 = h1E(u)E(u)−1h−11 E(v) = E(v),

which implies that g · 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 con-taining e is a torus subgroup.

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1.2 Theorem. Let the notation be as above, and suppose the H-action on G is polarwith flat sections. Let A be a torus section through e and let a denote its Lie algebra.Then:(1) the P (G, H)-action on V is polar, and the space a = a | a ∈ a is a section,

where a : [0, 1]→ g denotes the constant map with value a,(2) the generalized Weyl group W = W ( a, P (G, H)) is an affine Weyl group,(3) if Ψ : P (G, H)→ H is the homomorphism defined by Ψ(g) = (g(0), g(1)), then

Ψ(N( a, P (G, H)) = N(A, H), Ψ maps Z( a, P (G, H)) isomorphically ontoZ(A, H), and

Λ = Ker(Ψ) ∩N( a, P (G, H)) = g(t) = exp(λt) |λ ∈ Λ(A),

where Λ(A) is the unit lattice of A:

Λ(A) = λ ∈ a | exp(λ) = e,

(4) Ψ induces a surjective homomorphism from W to W = W (A, H), W is isomorphicto W/Λ, and W is the semi-direct product of W with Λ,

(5) Ψ maps the isotropy subgroup P (G, H)a isomorphically to Hexp(a); in fact itsinverse is the map ϕ : Hexp(a) → P (G, H)a defined by (h1, h2) → g(t) =exp(−at)h1 exp(at),

(6) u ∈ V is a singular point of the P (G, H)-action if and only if Φ(u) ∈ G is asingular point of the H-action.

Proof. We prove each statement of the theorem seperately below.

(1) To see that a meets every P (G, H)-orbit, we let u ∈ V . Since A meetsevery H-orbit, there is a ∈ a such that Φ(u) ∈ H · exp(a). But E(a) = exp(ta), soΦ(a) = exp(a). By 1.1 (ii), we have a ∈ P (G, H) · u. To prove a is orthogonal toP (G, H) · a for all a ∈ a, we note that

T (P (G, H) · a)a = [u, a]− u′ |u ∈ H1([0, 1], g), (u(0), u(1)) ∈ h.

For b ∈ a, we have [a, b] = 0 and

〈[u, a]− u′, b〉 =∫ 1

0([u, a]− u′, b) dt

=∫ 1

0(u, [a, b])− (u, b)′dt = 0− (u(1)− u(0), b) = 0,

where the last equality follows from the fact that

T (H · e)e = x− y | (x, y) ∈ h ⊥ a.

(2) It was proved in [Te2] that the generalized Weyl group of a polar action onHilbert space is an affine Weyl group.

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(3) First we prove Ψ(N( a, P (G, H)) ⊂ N(A, H). Let g ∈ N( a, P (G, H)).Then for a ∈ a there exists b ∈ a such that g · a = b. Using 1.1 (i) and the fact thatE(a) = exp(ta), E(b) = exp(bt), we have exp(b) = g(0) exp(a)g(1)−1. This provesthat (g(0), g(1)) ∈ N(A, H).

Next we prove that N(A, H) ⊂ Ψ(N( a, P (G, H)). Let (h1, h2) ∈ N(A, H).Then h1Ah−1

2 = A. In particular, this implies that h1h−12 ∈ A, h1A = Ah2, h1Ah−1

1 =Ah2h

−11 = A, and h1ah

−12 = a. Now given a1 ∈ a, there exists b1 ∈ a such that

h1 exp(a1)h−12 = exp(b1). Set

g(t) = exp(−b1t)h1 exp(a1t).Then it is easily seen that (g(0), g(1)) = (h1, h2), and a direct computation shows that

g · a = b1 + exp(−b1t)h1(a− a1)h−11 exp(b1t).

Since h1ah−11 = a and a is abelian, g · a = b1 + h1(a− a1)h−1

1 ∈ a.To compute the intersection of Ker(Ψ) and N( a, P (G, H)), we assume that g ∈

N( a, P (G, H)) and g(0) = g(1) = e. Then for a ∈ a, there exists b ∈ a such thatg · a = b, so 1.1 (i) implies that

g(t) = E(b)−1g(0)E(a) = exp((a− b)t).But g(1) = exp(a − b) = e, i.e., (a − b) ∈ Λ. Conversely, let λ ∈ Λ(A), andg(t) = exp(λt). Then g ∈ P (G, H), g ∈ Ker(Ψ), and g · a is the constant map withconstant equal to a− λ. This proves that g ∈ Ker(Ψ) ∩N(a, P (G, H)).

Now let g ∈ Z( a, P (G, H)). Since g · 0 = 0, g′g−1 = 0. So g(t) = h1 is aconstant map. But g · a = h1ah−1

1 = a. This implies that Ψ(g) = (h1, h1) ∈ Z(A, H),and Ψ maps Z( a, P (G, H)) injectively into Z(A, H). To prove surjectivity we let(h1, h2) ∈ Z(A, H). Then h1eh

−12 = e implies that h1 = h2. Set g(t) = h1. Since

h1ah−11 = a, g · a = a for all a ∈ a.(4) is a consequence of (3).(5) If g ∈ P (G, H)a, then from E(a)(t) = exp(ta) and 1.1 we have exp(at) =

g(0) exp(at)g−1. So exp(a) = g(0) exp(a)g(1)−1, i.e., Ψ(g) ∈ Hexp(a). Conversely,if h1 exp(a)h−1

2 = exp(a), then set g(t) = exp(−at)h1 exp(at), a direct computationas above implies that g · a = a.(6) follows from (5).

Let N be a submanifold of G, and Ω(G, e, N) the set of all H1-paths in G suchthat γ(0) = e and γ(1) ∈ N . It is known that (cf. [P1]) Ω(G, e, N) is a RiemannianHilbert manifold,

T (Ω(G, e, N))γ = uγ |u ∈ H1([0, 1], g),and the Riemannian inner product on T (Ω(G, e, N))γ is

(uγ, vγ) =∫ 1

0(u′(t), v′(t)) dt.

If N is an H-orbit in G, then P (G, H) acts on Ω(G, e, N) by g ∗ γ = g(0)γg−1, andthis action is isometric and transitive.

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1.3 Corollary. With the same notation and assumption as in 1.2, let N be the H-orbitthrough exp(a) in G, and

F : Ω(G, e, N)→ H0([0, 1], g), F (γ) = γ−1γ′.

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

1.4 Remark. Let M be a Riemannian manifold, G a compact Lie group acting on Misometrically, p ∈M a regular point, and N a G-orbit in M . Let

E : Ω(M, p, N)→ R, E(γ) =∫ 1

0‖ γ′(t) ‖ 2 dt

denote the energy functional. Bott and Samelson proved that (cf. [BS]) if the G-actionon M is variationally complete, then E is a perfect Morse function, and the homologyof Ω(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 = (G) = (g, g) | g ∈ G on G, i.e., theAdjoint action of G on G,

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

By results of Hermann ([He]) and Conlon ([C1]) there are two more families of variationalcomplete actions on G:(iii) the action of H = K1 × K2 on G, where both G/K1 and G/K2 are symmetric

spaces,(iv) the action of H = G(σ) = (g, σ(g)) | g ∈ G on G, where σ is an automorphism

of G; the action of G(σ) on G will be called the σ-action.

Conlon noted that the above four families of actions are polar with flat sections, and heproved that in general polar actions with flat sections are variationally complete (Conloncalled these sections K-transversal domains [C2]).

1.5 Examples. Applying 1.2 to the examples (i)–(iv) in 1.4 gives many polar actionson Hilbert spaces. In fact, the first and second families of examples described in theintroduction are the P (G, H)-action on V corresponding to examples (i) and (ii) in 1.4respectively. Note also that under the isometric embedding F in 1.3, the path spaceΩ(G, e, H · exp(a)) is embedded in the Hilbert space V = H0([0, 1], g) as a tautsubmanifold with constant principal curvatures, and the energy functional E correspondsto the square of the norm in the Hilbert space V , i.e., E(g) = ‖F (g) ‖ 2.

1.6 Theorem. (Conlon [C2]) Let M be a simply connected, complete Riemannian man-ifold, K a compact Lie group acting on M isometrically, and Ms the set of singularpoints. Suppose the K-action on M is polar with a flat section Σ, and the K-action onM is not transitive. Then

(i) Ms = ∅,(ii) Ms ∩ Σ is the union of finitely many totally geodesic hypersurfaces P1, . . . , Pr,

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(iii) the generalized Weyl group W (Σ, K) is a Coxeter group generated by the reflectionsof Σ in the Pi.

In the following, we will discuss further structures of the generalized Weyl groupfor 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 . LetW ⊂ Iso(Rk) be a Coxeter group, /i | i ∈ I the set of reflection hyperplanes of W ,and ui ∈ Rk the unit normal to /i. Then the rank of W is the dimension of the linearspan of ui | i ∈ I. In particular, the Dynkin diagram of a rank k, finite Coxeter grouphas k vertices, and that of a rank k infinite Coxeter group has k + 1 vertices. Recall alsothat the codimension of the principal orbits of an action is called the cohomogeneity. Fora polar action the cohomogeneity is the dimension of a section [PT1].

1.7 Theorem. Let the assumptions and notations be as in Theorem 1.2.(i) the rank of the affine Weyl group W = W ( a, P (G, H)) is equal to the cohomo-

geneity 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) · o is a principal orbit. Supposethat the rank of W is less than the dimension of a. Then there exists 0 = b ∈ a such thatthe line Rb does not meet any reflection hyperplane of W . So there is no focal point forM on the line Rb. This implies that the shape operator Ab = 0. But

TM0 = u′ |u ∈ H1([0, 1], g), (u(0), u(1)) ∈ h,

Ab(u′) = [u, b] = 0.

So [u, b] = 0 for all u ∈ TM0, which implies that [x, b] = 0 for all x ∈ g. Since g issemi-simple, b = 0, which is a contradiction. This proves (i).

(ii) follows directly from the proof of (i). Since the set of focal points of M is theset of singular points for the P (G, H)-action ([Te2]), if the action has no singular pointsthen M has no focal points. This implies that all shape operators Ab = 0 for b ∈ ν(M)0.Then the above computation implies that b = 0. Hence dim( a) = 0, i.e., the H-actionis transitive. This proves (ii).

1.8 Theorem. Let the assumptions and notations be as in Theorem 1.2. The slice repre-sentation of the H-action at exp(a) and the slice representation of the P (G, H)-actionat a are equivalent polar representations. In fact,(1) the slice representation of the H-action at exp(a) is given by

(h1, h2) · (b exp(a)) = (h1bh−11 ) exp(a),

(2) the normal plane ν(P (G, H) · a)a is

v(t) = exp(−at)b exp(at) | b exp(a) ∈ ν(H · exp(a))exp(a),

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and the slice representation at a is given by g · v = gvg−1,(3) f : ν(H · exp(a))exp(a) → ν(P (G, H) · a)a defined by

b exp(a) → v(t) = exp(−at)b exp(at)

is an equivariant isomorphism, i.e.,

ϕ((h1, h2)) · f(b exp(a)) = f((h1, h2) · b exp(a)),

where ϕ is as in 1.2 (5).

Proof. It is known ([PT1]) that the slice representations of a polar action arealso polar. The proof of (1)–(3) will be given seperately below.

(1) Given (h1, h2) ∈ Hexp(a), then we have h1 exp(a)h−12 = exp(a). So if

b exp(a) ∈ ν(H · exp(a))exp(a), then we have

(h1, h2) · b exp(a) = h1b exp(a)h−12 = h1bh

−11 exp(a).

(2) Let M = P (G, H) · a ⊂ V = H0([0, 1], g). By a direct computation, weobtain that

TMa = [u, a]− u′ |u ∈ H1([0, 1], g), (u(0), u(1)) ∈ h,

ν(M)a = v ∈ V | v′ = [v, a], (v(1), y) = (v(0), x) ∀(x, y) ∈ h.

It is easy to see that ifv′ = [v, a] then there exists b ∈ g such thatv(t) = exp(−at)b exp(at).In order for v ∈ ν(M)a, v must also satisfy the condition (v(0), x) = (v(1), y) for all(x, y) ∈ h, so

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

= (v(0), x) = (b, x) = (b exp(a), x exp(a)).

It follows that

(b exp(a), x exp(a)− exp(a)y) = 0, ∀(x, y) ∈ h.

ButT (H · exp(a))exp(a) = x exp(a)− exp(a)y | (x, y) ∈ h,

so (2) follows.

(3) is a consequence of (1) and (2).

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1.9 Theorem. Let the assumptions and notations be as in Theorem 1.2. Then,(1) there exists a ∈ a such that the affine Weyl group W = W ( a, P (G, H)) is the

semi-direct product of the finite Weyl group Wa and a lattice group Λ1, where Wa

is isotropy subgroup of W at a, and Λ1 is a lattice group containing the unit latticeΛ(A) and is invariant under Wa,

(2) the Weyl group of the slice representation of the H-action at α = exp(a) is theisotropy subgroup Wα of W = W (A, H), and the Weyl group of the slice repre-sentation of the P (G, H)-action at a is Wa,

(3) Wa is isomorphic to Wα,(4) the generalized Weyl group W (A, H) of the H-action on G is the semi-direct

product of Wα and the abelian group Λ1/Λ(A).

Proof. It follows from the standard theory of affine Weyl groups ([Bo]) thatthere exists a ∈ a such that

(i) W is the semi-direct product of the isotropy subgroup Wa and a lattice group Λ1,(ii) if W is of rank k, then Wa is a finite Weyl group of rank k,

(iii) a is a vertex of a Weyl chamber of W , and Wa is maximal among all isotropysubgroups of W

It is known that ([PT1]) the slice representation of a polar action is polar, and the Weylgroup of the slice representation at x is equal to the isotropy subgroup at x of thegeneralized Weyl group of the polar action. So Wα and Wa are the Weyl groups of theslice representations of the H-action and P (G, H)-action at α and a respectively. Itfollows from 1.2 (3)–(5) that Ψ(g) = (g(0), g(1)) induces an isomorphism from Wa toWα. 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 (possibly with bound-ary) of X is called a fundamental domain for the K-action if it meets each K-orbit inexactly one point. For example, a Weyl chamber in a is a fundamental domain of theaction of the affine Weyl group W on a.

1.10 Proposition. With the same notation and assumption as in 1.2. Let τ be a Weylchamber in a. Then(1) τ is a fundamental domain for the P (G, H)-action on V ,(2) τ and exp(τ) are isometric,(3) exp(τ) is a fundamental domain for the H-action on G.

Proof. The orbit space of a polar action is isometric to the orbit space of itssection under the action of the generalized Weyl group ([Te2]). Since τ is a fundamentaldomain for the W -action on a, (1) follows.

Let Φ be as in 1.1 and Ψ be as in 1.2. Then Φ(a) = exp(a). It follows form 1.1that Φ | τ is injective, and Φ(τ) = exp(τ) meets every H-orbit. This proves (3). SinceA is flat, exp : a→ A is a local isometry. So τ is isometric to exp(τ).

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1.11 Proposition. 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) If k = 1, then W (A, H) is Z2 or a dihedral group.(2) If k > 1, then W (A, H) is a rank k or k + 1 crystallographic group (i.e., a Weyl

group).

Proof. By 1.7 (i), the affine Weyl group W = W ( a, P (G, H)) has rank k, soW is generated by (k+1) reflections. By 1.2 (4), Ψ induces a homomorphism Ψ from Wonto W (A, H), so W (A, H) is a finite group generated by (k + 1) order two elements,which implies that it is a Coxeter group of rank at most k + 1. But 1.9 (1) and (2) implythat the rank of W (A, H) is at least k. So the rank of W (A, H) is either k or k +1. ButW is an affine Weyl group, so W satisfies the crystallographic condition if k > 1. Hencethe image Ψ(W ) = W (A, H) also satisfies the crystallographic condition if k > 1.

We end of this section, with two conjectures:1.12 Conjecture. If the H-action on G is of cohomogeneity 1, then the generalized Weylgroup 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.

1.13 Conjecture. 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)-action, i.e., there exists an isometry

from V to H0([0, 1], g) that maps L-orbits onto the P (G, H)-orbits.

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

It is a consequence of section 1 that the submanifold geometry of the orbits of apolar H-action on G with flat sections determines the submanifold geometry of orbitsof the associated P (G, H)-action in the Hilbert space V = Ho([0, 1], g). For example,the curvature distributions, the curvature normals, and the marked Dynkin diagram ofM can be explicitly computed from the data of the H-action on G. These geometricdata of M have been given in [Te2] when the H-action is the Adjoint action on G, andin [PiT] when the H-action is the K ×K-action on G (here K is the fixed point set ofan involution on G). They are all obtained from well-known root space decompositonswith respect to the flat section. In the following, we will give the relation between thegeometry of M and the H-action.

To simplify the notation, we may assume that e ∈ G is a regular point for theH-action. So M = P (G, H) · 0 is a principal orbit, and is isoparametric in V ([Te2]).Note that ν(M)o = a, and given b ∈ a the vector field b(g · o) = gbg−1 is a parallelnormal field on M . Moreover, there exist a smooth subbundle E0 of TM , finite ranksmooth vector subbundles Ei | i ∈ I (the E ′is are called the curvature distributions),vi | i ∈ I ⊂ a, and parallel normal fields vi | i ∈ I (vi’s are called curvaturenormals) such that

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(1) TM = E0 ⊕ ΣEi | i ∈ I,(2) the shape operator Ab |Ei = (b, vi) idEi

for i ∈ I , and Ab |E0 = 0,(3) E0 is integrable, and leaves of E0 are affine subspaces of V ,(4) each Ei, i ∈ I , is integrable, and the leaf Si(x) of Ei through x ∈M is a standard

sphere, called the curvature sphere for Ei,(5) if /i = b ∈ a | (b, vi) = 1, then /i | i ∈ I is the set of reflection hyperplanes

of the affine Weyl group W for the P (G, H)-action, (this determines vi in terms ofW ),

(6) if mi denotes the rank of Ei, then the mi’s are invariant under W ,(7) the marked affine Dynkin diagram associated to M is the diagram for W with the

vertex corresponding to a simple root /i marked by mi = rank Ei.

Let τ be a Weyl chamber in a for the W -action on a. Then τ determines /i | i ∈I. By 1.10, τ is isometric to a fundamental domain of the H-action on G. So /i | i ∈I and vi | i ∈ I can be computed explicitly from the Euclidean geometry of thefundamental domain of H-action on G, which is a k-simplex (k = dim(a)). SinceEi(g · 0) = gEi(0)g−1, to find the relation between Ei and the H-action it suffices tofind the relation between Ei(0) and the H-action.

2.1 Proposition. With the same notation as above, choose ai ∈ a such that ai ∈ /i andai does not lie in /j for any j ∈ I , j = i. Let αi = exp(ai). Then(1) he ⊂ hαi

= (x, α−1i xαi) |x ∈ g ∩ h,

(2) for i ∈ I , the map (x, α−1i xαi) → v(t) = exp(−ait)[x, ai] exp(ait) gives a

well-defined isomorphism from hαi/he to Ei(0),

(3) [he, a] = 0,

(4) E0(0) = u ∈ H0([0, 1], z0) |∫ 1

0 u(t)dt ∈ T (He)e, where zo is the centralizerof a in g, i.e., z0 = x ∈ g | [x, a] = 0.Proof. Since e is assumed to be a regular point for the H-action, He fixes A

pointwise. So we obtain (3), and he ⊂ hα for any α ∈ A. Note that (h1, h2) ∈ Hα ifand only if h2 = α−1h1α. This proves (1).

It is known that (cf. [Te2]) P (G, H)ai· 0 is the curvature sphere Si(o) of Ei through

0, so Ei(0) = T (P (G, H)ai· 0)o. By (1) and 1.2 (5), we have

P (G, H)a = g(t) = exp(−at)h1 exp(at) | (h1, exp(−a)h1 exp(a)) ∈ H,

an (2) follows by a straightforward computation.To prove (4), we recall that

TMo = u′ |u ∈ H1([0, 1], g), (u(0), u(1)) ∈ h,

= v ∈ Ho([0, 1], g) |∫ 1

0v(t)dt ∈ T (He)e,

Ab(u′) = [u, b].

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So for v = u′ ∈ TMo, we have v ∈ E0 if and only if [u, b] = 0 for all b ∈ a. Thisimplies that u(t) ∈ z0, which proves (4).

In the following we give more detailed geometric description for the cohomogeneityone actions, and for the σ-actions.

2.2 Cohomogeneity one actions.

Suppose the H-action on G is of cohomogeneity 1, W (A, H) is the dihedral groupof order 2n, and A is a circle of length /. Let b ∈ a be a unit vector such that exp(/b) = e.We may assume that / = 1 and e is a singular point for the H-action. Then

βj = exp( jb/2n) | 0 ≤ j < 2n

is the set of singular points on A, exp(tb) | 0 ≤ t ≤ 1/2n is a fundamental domain forthe H-action, and the affine Weyl group W = W (a, P (G, H)) is the semi-direct productof Z2 with the lattice group Λ1 = jb/n | j ∈ Z. For 0 ≤ t ≤ 1/2n, let Mt denote theP (G, H)-orbit through the constant path tb. Let m1 and m2 denote the dimension ofHe/Hα and Hβ1/Hα respectively, where α = exp(tb) for some 0 < t < 1/2n. ThenMt | 0 < t < 1/2n is a family of isoparametric hypersurfaces of V , and the markedDynkin diagram associated to Mt is A1 marked with multiplicity (m1, m2), the non-zeroprincipal curvatures of Mt are λj = ((j/2n) − t)−1 | j ∈ Z, and the multiplicity ofλj is m1 if j is even, and is m2 if j is odd. Moreover,(1) M0 and M1/2n are smooth submanifolds of V with codimension 1+m1 and 1+m2

respectively, and M0∪M1/2n is the set of focal points of Mt in V for 0 < t < 1/2n,(2) given 0 < t < 1/2n, V can be written as the union of B0 ∪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 M1/2n in V ,

(iii) B0 ∩B1 = ∂B0 = ∂B1 = Mt,(iv) let bi = dim(Hi(Mt, Z2)), and p(x) =

∑bix

i the Poincare series of M , then

p(x) =(1 + xm1 )(1 + xm2 )

1− xm1+m2

=∞∑

k=0

xk(m1+m2)(1 + xm1 + xm2 + xm1+m2 ).

2.3 The σ-actions.

Let H = G(σ) = (g, σ(g)) | g ∈ G be as in 1.4 (iv), where σ is an outerautomorphism of order r on G (r = 2 or 3). Let K be the fixed point set of σ on g, andp = K⊥. Note that He is the diagonal group of K, and the slice representation at e is theAdjoint representation of K on K. Let a be a maximal abelian subalgebra of K. Then a

is a section of the P (G, H)-action.Since ad(a) : g → g | a ∈ a is a family of commuting skew-adjoint operators,

there exist ([H]) positive root systems 0 and 1 (subset of a∗), xα, yα ∈ K for α ∈

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0, rβ, sβ ∈ p for β ∈ 1, and a linear subspace p0 of p, which give the followingdecompositions of K and p:

K = a +∑α∈0

Rxα + Ryα, p = p0 +∑β∈1

Rrβ + Rsβ,

such that[a, xα] = −α(a)yα, [a, yα] = α(a)xα, ∀α ∈ 0,

[a, rβ ] = −β(a)sβ, [a, sα] = β(a)rβ, ∀β ∈ 1,

σ(uα) = uα, σ(pβ) = e2πi/rpβ,

(∗)

where uα = xα + iyα, pβ = rβ + isβ , and r = order (σ). Let M = P (G, H) · a be aprincipal orbit for some α ∈ a. Then the normal plane of M at a is a, and for b ∈ a,the shape operator of M in the direction of b is

Ab([u, a]− u′) = −[u, b].

Let a1, . . . , am be a basis for a. Then we have:

Case (1) order (σ) = 2. Let qj be a basis for p0. Then by (*) and a directcomputation, we see that the real and imaginary parts of uαe2nit, pβe(2n+1)it, aje

2nit, andqje

(2n+1)t form an eigenbasis for Ab with α(b)/(α(a) + 2n), β(b)/(β(a) + 2n + 1), 0and 0 as eigenvalues respectively. So the reflection hyperplanes of W ( a, P (G, H)) ina are the affine hyperplanes defined by α(t) = 2n, β(t) = 2n+1, for α ∈ 0, β ∈ 1and n ∈ Z.

Case (2) order(σ) = 3. By a direct computation we obtain that the real andimaginary parts of uαe3nit, pβe(3n+1)it, pβe(3n+2)it, aje

3nit, qje(3n+1)it, and qje

(3n+2)it

form an eigenbasis for Ab with α(b)/(α(a)+3n), β(b)/(β(a)+3n+1), β(b)/(β(a)+3n+2), 0, 0 and 0 as eigenvalues respectively, where qj, qj is a basis of the complexifyof (p0)C such that σ(qj) = e2πi/3qj (hence σ(qj) = e−2π/3qj). So the reflectionhyperplanes of W ( a, P (G, H)) in a are the affine hyperplanes defined by α(t) = 3n,β(t) = 3n + 1, β(t) = 3n + 2 for α ∈ 0, β ∈ 1 and n ∈ 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 givenautomorphisms 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 ( , ) the normalizedbi-invariant inner product on g, i.e., (hα, hα) = 2 for the shortest simple coroot ofg. Let L(g) = H0(S1, g), t the angle variable in S1, and let u′ denote du/dt. Let

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[u, v]o(t) = [u(t), v(t)]. Then L(g) is a Lie algebra. The affine Kac-Moody algebra oftype 1 is a two dimensional extension over L(g):

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

with the bracket operation defined by

[u, v] = [u, v]o + ω(u, v)c,

[d, u] = u′

[c, u] = [c, d] = 0,

where ω is the 2-cocyle defined on L(g) by

ω(u, v) =12π

∫ 2π

0(u(t), v′(t)) dt.

(Note that [ , ] is only defined on a dense subspace H1(S1, g) of L(g); see 3.4 for adiscussion of this point.) Let 〈 , 〉 denote the bi-linear form on L(g) defined as follows:

〈u, v〉 =∫ 2π

0(u(t), v(t)) dt,

〈c, d〉 = 1, 〈u, c〉 = 〈c, c〉 = 〈d, d〉 = 0.

Then 〈, 〉 is Ad-invariant, i.e.,

〈[ξ, η], ζ〉 = 〈ξ, [η, ζ]〉, ∀ξ, η, ζ ∈ L(g).

Next we construct two types of automorphisms on L(g) from automorphisms onG: Given σ : G → G, an automorphism of order k, we will use the symbol σ also todenote the induced automorphism on g. Let

σ : L(g)→ L(g)

be the linear map defined by

σ(u)(t) = σ(u(−2π/k + t)), σ(c) = c, σ(d) = d.

Let ρ be an involution on G, and

ρ∗ : L(g)→ L(g)

the linear map defined by

ρ∗(u)(t) = ρ(u(−t)), ρ∗(c) = −c, ρ∗(d) = −d.

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3.1 Proposition. Let σ : g→ g be an automorphism of order k on the simple Lie algebrag, and ( , ) the normalized Ad-invariant form on g. Then

(i) (σ(x), σ(y)) = (x, y),(ii) ω(σ(u), σ(v)) = ω(u, v),

(iii) ω(σ∗(u), σ∗(v)) = −ω(u, v), if σ is an involultion.

Proof. Since σ is an automorphism of g, 〈x, y〉 = (σ(x), σ(y)) is an Ad-invariant inner product on g. But g is simple implies that (σ(x), σ(x)) = c(x, x) forsome c > 0. Since σk = id, ck = 1. This implies c = 1, which proves (i). Using (i),the definition of the cocycle ω, and a direct computation, (ii) and (iii) follows.

As a consequence, we have

3.2 Corollary. If σ and ρ are automorphisms of g of order k and 2 respectively, then σand ρ∗ are automorphisms of L(g) of order k and 2 respectively.

The fixed point set L(g, σ) of σ is called the affine Kac-Moody algebra of type k ifσ is an order k outer automorphism of g. It is obvious that

L(g, σ) = L(g, σ) + Rc + Rd,

whereL(g, σ) = u ∈ L(g) |u(t) = σ(u(−2π/k + t)).

It can also be easily seen thatL(g) = K + p,

is the decomposition into the 1 and −1 eigenspaces of ρ∗, where

K = u ∈ L(g) |u(−t) = ρ(u(t)),p = u ∈ L(g) |u(−t) = −ρ(u(t))+ Rc + Rd.

3.3 Construction of L(G).

We review briefly the construction of L(G) as given in Chapter 4 of [PS]. The firststep is to construct the central extension of the loop group L(G) = H1(S1, G) givenby the 2-cocycle ω. The left invariant 2-form on L(G) determined by the 2-cocycleω will also be denoted by ω. Since the inner product on g is normalized, the 2-form1

2πiω is an integral cohomology class of L(G), so there exists a principal S1-bundle,ϕ : P → L(G) with a connection 1-form β such that the curvature of β is ϕ∗ω, (i.e.,the Chern class of this principal bundle is [ω/2πi]). Then the group L(G) of bundleisomorphisms of P that preserve the connection β and cover a left translation /g for someg ∈ L(G) is the central extension by S1 given by the cocyle ω.

We fix a base point y0 ∈ ϕ−1(e). For a curve γ : [0, 1]→ L(G), let

Πγ : ϕ−1(γ(0))→ ϕ−1(γ(1))

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denote the parallel translation along γ given by the connection β. Suppose F ∈ L(G)and F covers the left translation /g on L(G). Since F ∗(β) = β,

F Πγ = Πgγ F.

So F is uniquely determined by g ∈ L(G) and F (y0), i.e., L(G) is the set of pairs (g, y),where g ∈ L(G) and y ∈ ϕ−1(g).

There is also an explicit construction of L(G) given in [PS] (p. 47), which we willuse to lift automorphisms σ and ρ∗ to L(G): Let (g, p, z) be a triple with g ∈ L(G), pa path in L(G) joining e to g and z ∈ S1. Then F can be constructed by using paralleltranslation. For y ∈ ϕ−1(h), we choose some curve γ in L(G) joining h to e, let

F (y) = Π−1gγ z(Πp(Πγ(y))).

This is well-defined and in L(G). For if γ1 and γ2 are curves joining x to e, then sinceL(G) is simply connected, γ−1

1 ∗ γ2 (the loop obtained by the inverse of γ1 followedby γ2) bounds a surface S in L(G). But S1 is abelian, so “curvature is infinitesimalholonomy”, i.e., we have C(γ−1

1 ∗ γ2)Πγ1 (y) = Πγ2 (y) for all y ∈ ϕ−1(x), where

C(γ−11 ∗ γ2) = exp (i

∫S

ω).

This also proves that (g1, p1, z1) and (g2, p2, z2) gives the same F if and only if

g1 = g2, z1 = C(p2 ∗ p−11 )z2;

and we call two such triples equivalent. Define multiplication of triples by

(g1, p1, z1) · (g2, p2, z2) = (g1g2, p1 ∗ (g1 · p2), z1z2).

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

It is easily seen that eis·(g, p, z) = (eis·g, eis·p, z) defines a continuous action of S1

on L(G), where (eis ·g)(t) = g(s+t) and eis ·p denotes the path (eis ·p)(r) = eis ·p(r).Then the semi-direct product L(G) = S1 |× L(G) given by this S1-action is a topologicalgroup, which is the group model for the affine algebra L(g).

3.4 Remark. Although L(G) is a Hilbert manifold, and a topological group, (i.e., thegroup multiplication is continuous), it is not a Hilbert Lie group, because the map

H1(S1, G)× S1 → H1(S1, G), (f, g) → f 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 adense subspace. It should also be noted that L(G) is a Hilbert manifold locally modeledon the product of the H1-space and R2, but the Ad-invariant metric is the product of theH0 metric on the H1-space and the Lorentz metric on R2, so L(G) is only a Lorentzmanifold in the weak sense. Our goal in reviewing this construction of affine groups isto suggest a good infinite dimensional analogue of s-representations and their relation topolar actions on Hilbert spaces. As we will now see, while the above problems make itdifficult to give a rigorous definition of infinite dimensional symmetric spaces, they donot interfere with the construction of the desired polar actions.

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3.5 The Adjoint action of L(G).

Letπ : L(G)→ L(G), (eis, (g, p, z)) → g

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

Ad(g)(u) = gug−1 + 〈gug−1, g′g−1〉 c,

Ad(g)(d) = −g′g−1 + d− 12〈g′g−1, g′g−1〉 c ,

if g = (1, (g, p, z)). Note that the intersection R∞ of the sphere of radius −1 with thehyperplane u + rc + sd ∈ L(g) | s = 1, i.e.,

R∞ = u + rc + d | u ∈ L(g), r = −(1 + 12〈u, u〉) ,

is a horosphere of the infinite dimensional hyperbolic space. Hence R∞ is invariant underthe Adjoint action of L(G) on L(g), and R∞ is isometric to the Hilbert space L(g) =H0(S1, g) via u + rc + d → u. Moreover, the action on R∞ factors through L(G), andthe corresponding action on L(g) is equivalent to the action of L(G) P (G,(G))on L(g) H0([0, 1], g) by gauge group transformations.

3.6 The automorphism σ on L(G).

Let σ be an automorphism of order k on G. Then

σ : L(G)→ L(G), σ(g)(t) = σ(g(−2π/k + t))

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

(eis, (g, p, z)) → (eis, (σ(g), σ(p), z))

is a well-defined automorphism on L(G), which will still be denoted by σ. Then thefixed point set L(G, σ) of σ is the affine group of type k, and L(g, σ) is its Lie algebra.Again

R∞σ = u + rc + d | u ∈ L(g, σ), r = −(1 + 12〈u, u〉)

is invariant under the Adjoint action of L(G, σ), and R∞σ is isometric to L(g, σ) viathe map u + rc + d → u. Moreover, the action on R∞σ factors through L(G, σ),and the corresponding action on L(g, σ) is equivalent to the action of L(G, σ) P (G, G(σ)) on L(g, σ) H0([0, 1], g) by gauge group transformations, where G(σ) =(x, σ(x)) |x ∈ G.

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3.7 Involution ρ∗ on L(G).

Let ρ be an involution on G, and K the fixed point set of ρ. Then

ρ∗ : L(G)→ L(G), ρ∗(g)(t) = ρ(g(−t))

is an involution on L(G). Note that ρ∗(eis · g) = e−is · ρ∗(g). Using this equality andthe construction of L(G) in 3.3, it follows that

(eis, (g, p, z)) → (e−is, (ρ∗(g), ρ∗(p), z−1))

is a well-defined involution on L(G), which will still be denoted by ρ∗. Then the fixedpoint set of ρ∗ on L(G) is

L(G)ρ = (1, (g, p, 1)) | ρ∗(g) = g, ρ∗(p(r)) = p(r) ,

which is isomorphic to

g ∈ L(G) | ρ(g(−t)) = g(t) P (G, K ×K).

Let L(g) = K + p denote the 1,−1 eigendecomposition of ρ∗ on L(g) as before. ThenK is the Lie algebra of L(G)ρ, and the Adjoint action of L(G)ρ leaves p invariant, andinduces on p the isotropy representation of the “symmetric space” L(G)/L(G)ρ at theidentity coset. This isotropy action also leaves R∞ρ = R∞ ∩ p invariant. Moreover, R∞ρis isometric to u ∈ L(g) | ρ(u(−t)) = −u(t) via the map u + rc + d → u, and thethe action of L(G)ρ on R∞ρ is equivalent to the P (G, K ×K)-action on H0([0, 1], g).

3.8 Natural embedding of L(G)/L(G)ρ into L(G).

Let ρ be an involution on G, and K the fixed point set of ρ. Then the finitedimensional symmetric space G/K can be naturally embedded in G as the orbit Nthrough e of the action of G(ρ) = (x, ρ(x)) |x ∈ G on G, i.e., N = xρ(x)−1 |x ∈G. Similarly in infinite dimension, the involution ρ∗ induces an action of L(G) onL(G) via h ∗ y = hyρ∗(h)−1, and the orbit M of this action through the identity gives anatural embedding of L(G)/L(G)ρ into L(G). We also note that e ∈ M , the restrictionτ of the isometry x → x−1 on L(G) to M induces an isometry of M , τ(e) = e, and dτe

is−id on TMe. So if p = g ∗ e ∈ M , then τp = gτg−1 has the property that τp(p) = p

and d(τp)p is −id on TMp. This is the characteristic property of a globally symmetricspace. To describe M explicitly, we compute directly and see that

M = (eis, (g, p, z)) | g(t) = h(t)ρ(h(−s− t))−1 for some h ∈ L(G).

Note that for g(t) = h(t)ρ(h(−s−t))−1 we have g = e−is/2 ·((eis/2 ·h)ρ∗(eis/2 ·h)−1).So the loop part π(M) = S1 ·M , where

M =hρ∗(h)−1 | h ∈ L(G)

⊂ L(G).

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Next we claim that

M = g ∈ L(G) | g(t)−1 = ρ(g(−t)), g(0), g(π) ∈ N Ω(G, N, N),

where Ω(G, N, N) denote the Hilbert manifold of H1-paths in G with end points in N .It is easy to see that M is contained in the right hand side. Now suppose given g ∈ L(G)such that g(t)−1 = ρ(g(−t)) and g(0), g(π) ∈ N . Then there exist x, y ∈ G such thatxρ(x)−1 = g(0) and yρ(y)−1 = g(π). Let r : [0, π] → G be any H1-map such thatr(0) = x, r(π) = y, and define

h(t) =

r(t), if t ∈ [0, π];g(t)ρ(r(2π − t)), if t ∈ [π, 2π].

Then h ∈ L(G) and hρ∗(h)−1 = g, i.e., g ∈M .

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References

[BS] Bott, R and Samelson, H., Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80(1958), 964–1029.

[Bo] Bourbaki, N., Groupes et Algebres de Lie. Hermann, Paris, 1968.[C1] Conlon L., The topology of certain spaces of paths on a ocmpact symmetric space, Trans. Amer. Math.

Soc. 112 (1964), 228–248.[C2] Conlon L., Variational completeness and K-transversal domains, J. Differential Geom. 5 (1971), 135–147.[C3] Conlon L., A class of variationally complete representations, J. Differential Geom. 7 (1972), 149–160.[D] Dadok J., Polar coordinates induced by actions of compact Lie groups, Trans. Amer. Math. Soc. 288

(1985), 125–137.[FKM] Ferus, D., Karcher, H. and Munzner, H. F., Cliffordalgebren und neue isoparametrische hyperflachen,

Math. Z. 177 (1981), 479–502.[H] Helgason, S., Differential Geometry and Symmetric Spaces. Academic Press, 1978.[He] Hermann, R., Variational completeness for compact symmetric spaces, Proc. Amer. Math. Soc. 11 (1960),

554–546.[K] Kac, V.G., Infinite Dimensional Lie Algebras. Cambridge University Press, 1985.[OT1] Ozeki, H. andTakeuchi,M., On some types of isoparametric hypersurfaces in spheres, I, Tohoku Math. J.

127 (1975), 515–559.[OT2] Ozeki, H. andTakeuchi,M., On some types of isoparametric hypersurfaces in spheres, II, Tohoku Math. J.

28 (1976), 7–55.[P1] Palais, R.S., Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340.[P2] Palais, R.S., Foundations of Global Non-linear Analysis. Benjamin Co. NY, 1968.[PT1] Palais, R.S. and Terng, C.L., A general theory of canonical forms, Trans. Amer. Math. Soc. 300 (1987),

771–789.[PT2] Palais R.S., and Terng C.L., Critical Point Theory and Submanifold Geometry. Lecture Notes in Math.,

vol. 1353, Springer-Verlag, Berlin and New York, 1988.[PiT] Pinkall, U., and Thorbergsson, G., Examples of infinite dimensional isoparametric submanifolds, Math.

Z. 205 (1990), 279–286.[PS] Pressley, A. and Segal, G. B., Loop Groups. Oxford Science Publ., Clarendon Press, Oxford, 1986.[Te1] Terng, C.L., Isoparametric submanifolds and their Coxeter groups, J. Differential Geom. 21 (1985), 79–

107.[Te2] Terng, C.L., Proper Fredholm submanifolds of Hilbert spaces, J. Differential Geom. 29 (1989), 9–47.[Te3] Terng, C.L., Variational completeness and infinite dimensional geometry, Proceedings of Leeds Confer-

ence, edited by L. Verstraelen and A. West, Geometry and Topology of Submanifolds, III, World Scientific,Singapore (1991), 279–293.

[Th] Thorbergsson, G., Isoparametric foliations and their buildings, Annals of Math. 133 (1991), 429–446.

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