47.10 47. Manifolds for algebraic topology 521

47.9. Theorem. Let g be a Lie subalgebra of gl(∞). Then there is a smoothly

arcwise connected splitting regular Lie subgroup G of GL(∞) whose Lie algebra is

g. The exponential mapping of GL(∞) restricts to that of G, which is a local real

analytic di¬eomorphism near zero. The Campbell-Baker-Hausdor¬ formula gives a

real analytic mapping near 0 and has the usual properties, also on G.

Proof. Let gn := g©gl(n), a ¬nite dimensional Lie subalgebra of g. Then gn = g.

The convenient structure g = ’ n gn coincides with the structure inherited as a

lim

’

complemented subspace, since gl(∞) carries the ¬nest locally convex structure.

So for each n there is a connected Lie subgroup Gn ‚ GL(n) with Lie algebra gn .

Since gn ‚ gn+1 we have Gn ‚ Gn+1 , and we may consider G := n Gn ‚ GL(∞).

Each g ∈ G lies in some Gn and may be connected to Id via a smooth curve there,

which is also smooth curve in G, so G is smoothly arcwise connected.

All mappings exp |gn : gn ’ Gn are local real analytic di¬eomorphisms near 0, so

exp : g ’ G is also a local real analytic di¬eomorphism near zero onto an open

neighborhood of the identity in G. A similar argument applies to evol so that G is

regular. The rest is clear.

47.10. Examples. In the following we list some of the well known examples of

simple in¬nite dimensional Lie groups which ¬t into the picture treated in this

section. The reader can easily continue this list, especially by complex versions.

The Lie group SL(∞) is the inductive limit

SL(∞) = {A ∈ GL(∞) : det(A) = 1}

= lim SL(n) ‚ GL(∞),

’’

n’∞

the connected Lie subgroup with Lie algebra sl(∞) = {X ∈ gl(∞) : tr(X) = 0}.

The Lie group SO(∞, R) is the inductive limit

SO(∞) = {A ∈ GL(∞) : Ax, Ay = x, y for all x, y ∈ R(N) and det(A) = 1}

= lim SO(n) ‚ GL(∞),

’’

n’∞

the connected Lie subgroup of GL(∞) with the Lie algebra o(∞) = {X ∈ gl(∞) :

X t = ’X} of skew elements.

The Lie group O(∞) is the inductive limit

O(∞) = {A ∈ GL(∞) : Ax, Ay = x, y for all x, y ∈ R(N) }

= lim O(n) ‚ GL(∞).

’’

n’∞

It has two connected components, that of the identity is SO(∞).

47.10

522 Chapter X. Further Applications 48.1

The Lie group Sp(∞, R) is the inductive limit

Sp(∞, R) = {A ∈ GL(∞) : At JA = J}

= lim Sp(2n, R) ‚ GL(∞), where

’’

n’∞

01

«

¬ ’1 0 ·

01 · ∈ L(R(N) , R(N) ).

¬ ·

J =¬

¬

’1 0

·

..

.

It is the connected Lie subgroup of GL(∞) with the Lie algebra sp(∞, R) = {X ∈

gl(∞) : X t J + JX = 0} of symplectically skew elements.

47.11. Theorem. The following manifolds are real analytically di¬eomorphic to

the homogeneous spaces indicated:

L(Rk , R∞’k )

Idk

GL(k, ∞) ∼ GL(∞) ,

=

GL(∞ ’ k)

0

O(k, ∞) ∼ O(∞)/(Idk —O(∞ ’ k)),

=

G(k, ∞) ∼ O(∞)/(O(k) — O(∞ ’ k)).

=

The universal vector bundle (E(k, ∞), π, G(k, ∞), Rk ) is de¬ned as the associated

bundle

E(k, ∞) = O(k, ∞)[Rk ]

= {(Q, x) : x ∈ Q} ‚ G(k, ∞) — R(N) .

The tangent bundle of the Grassmannian is then given by

T G(k, ∞) = L(E(k, ∞), E(k, ∞)⊥ ).

Proof. This is a direct consequence of the chart construction of G(k, ∞).

48. Weak Symplectic Manifolds

48.1. Review. For a ¬nite dimensional symplectic manifold (M, σ) we have the

following exact sequence of Lie algebras, see also (45.7):

gradσ γ

0 ’ H 0 (M ) ’ C ∞ (M, R) ’ ’ X(M, σ) ’ H 1 (M ) ’ 0.

’’ ’

Here H — (M ) is the real De Rham cohomology of M , the space C ∞ (M, R) is

equipped with the Poisson bracket { , }, X(M, σ) consists of all vector ¬elds ξ

with Lξ σ = 0 (the locally Hamiltonian vector ¬elds), which is a Lie algebra for the

48.1

48.3 48. Weak symplectic manifolds 523

Lie bracket. Furthermore, gradσ f is the Hamiltonian vector ¬eld for f ∈ C ∞ (M, R)

given by i(gradσ f )σ = df and γ(ξ) = [iξ σ]. The spaces H 0 (M ) and H 1 (M ) are

equipped with the zero bracket.

Given a symplectic left action : G — M ’ M of a connected Lie group G on M ,

the ¬rst partial derivative of gives a mapping : g ’ X(M, σ) which sends each

element X of the Lie algebra g of G to the fundamental vector ¬eld. This is a Lie

algebra homomorphism.

wC w X(M, σ) w H (M )

gradσ

x

x

γ

i ∞

0 1

H (M ) (M, R)

χx

g

A linear lift χ : g ’ C ∞ (M, R) of with gradσ —¦χ = exists if and only if γ —¦ = 0

in H 1 (M ). This lift χ may be changed to a Lie algebra homomorphism if and only

if the 2-cocycle χ : g — g ’ H 0 (M ), given by (i —¦ χ)(X, Y ) = {χ(X), χ(Y )} ’

¯ ¯

χ([X, Y ]), vanishes in H 2 (g, H 0 (M )), for if χ = δ± then χ ’ i —¦ ± is a Lie algebra

¯

homomorphism.

If χ : g ’ C ∞ (M, R) is a Lie algebra homomorphism, we may associate the moment

mapping µ : M ’ g = L(g, R) to it, which is given by µ(x)(X) = χ(X)(x) for

x ∈ M and X ∈ g. It is G-equivariant for a suitably chosen (in general a¬ne)

action of G on g . See [Weinstein, 1977] or [Libermann, Marle, 1987] for all this.

48.2. We now want to carry over to in¬nite dimensional manifolds the procedure

of (48.1). First we need the appropriate notions in in¬nite dimensions. So let M

be a manifold, which in general is in¬nite dimensional.

A 2-form σ ∈ „¦2 (M ) is called a weak symplectic structure on M if it is closed

(dσ = 0) and if its associated vector bundle homomorphism σ ∨ : T M ’ T — M is

injective.

A 2-form σ ∈ „¦2 (M ) is called a strong symplectic structure on M if it is closed (dσ =

0) and if its associated vector bundle homomorphism σ ∨ : T M ’ T — M is invertible

with smooth inverse. In this case, the vector bundle T M has re¬‚exive ¬bers Tx M :

Let i : Tx M ’ (Tx M ) be the canonical mapping onto the bidual. Skew symmetry

of σ is equivalent to the fact that the transposed (σ ∨ )t = (σ ∨ )— —¦ i : Tx M ’ (Tx M )

satis¬es (σ ∨ )t = ’σ ∨ . Thus, i = ’((σ ∨ )’1 )— —¦ σ ∨ is an isomorphism.

48.3. Every cotangent bundle T — M , viewed as a manifold, carries a canonical

weak symplectic structure σM ∈ „¦2 (T — M ), which is de¬ned as follows (see (43.9)

for the ¬nite dimensional version). Let πM : T — M ’ M be the projection. Then

—

the Liouville form θM ∈ „¦1 (T — M ) is given by θM (X) = πT — M (X), T (πM )(X)—

for X ∈ T (T — M ), where denotes the duality pairing T — M —M T M ’ R.

,

Then the symplectic structure on T — M is given by σM = ’dθM , which of course

in a local chart looks like σE ((v, v ), (w, w )) = w , v E ’ v , w E . The associated

mapping σ ∨ : T(0,0) (E — E ) = E — E ’ E — E is given by (v, v ) ’ (’v , iE (v)),

48.3

524 Chapter X. Further Applications 48.6

where iE : E ’ E is the embedding into the bidual. So the canonical symplectic

structure on T — M is strong if and only if all model spaces of the manifold M are

re¬‚exive.

48.4. Let M be a weak symplectic manifold. The ¬rst thing to note is that the

hamiltonian mapping gradσ : C ∞ (M, R) ’ X(M, σ) does not make sense in general,

since σ ∨ : T M ’ T — M is not invertible. Namely, gradσ f = (σ ∨ )’1 —¦ df is de¬ned

only for those f ∈ C ∞ (M, R) with df (x) in the image of σ ∨ for all x ∈ M . A

similar di¬culty arises for the de¬nition of the Poisson bracket on C ∞ (M, R).

σ

De¬nition. For a weak symplectic manifold (M, σ) let Tx M denote the real linear

∨ —

σ

subspace Tx M = σx (Tx M ) ‚ Tx M = L(Tx M, R), and let us call it the smooth

cotangent space with respect to the symplectic structure σ of M at x in view of

the embedding of test functions into distributions. These vector spaces ¬t together

to form a subbundle of T — M which is isomorphic to the tangent bundle T M via

σ ∨ : T M ’ T σ M ⊆ T — M . It is in general not a splitting subbundle.

48.5. De¬nition. For a weak symplectic vector space (E, σ) let

Cσ (E, R) ‚ C ∞ (E, R)

∞

denote the linear subspace consisting of all smooth functions f : E ’ R such that

each iterated derivative dk f (x) ∈ Lk (E; R) has the property that

sym

dk f (x)( , y2 , . . . , yk ) ∈ E σ

is actually in the smooth dual E σ ‚ E for all x, y2 , . . . , yk ∈ E, and that the

mapping

k

E’E

(x, y2 , . . . , yk ) ’ (σ ∨ )’1 (df (x)( , y2 , . . . , yk ))

is smooth. By the symmetry of higher derivatives, this is then true for all entries

of dk f (x), for all x.

48.6. Lemma. For f ∈ C ∞ (E, R) the following assertions are equivalent:

(1) df : E ’ E factors to a smooth mapping E ’ E σ .

(2) f has a smooth σ-gradient gradσ f ∈ X(E) = C ∞ (E, E) which satis¬es

df (x)y = σ(gradσ f (x), y).

∞

(3) f ∈ Cσ (E, R).

Proof. Clearly, (3) ’ (2) ” (1). We have to show that (2) ’ (3).

Suppose that f : E ’ R is smooth and df (x)y = σ(gradσ f (x), y). Then

dk f (x)(y1 , . . . , yk ) = dk f (x)(y2 , . . . , yk , y1 )

= (dk’1 (df ))(x)(y2 , . . . , yk )(y1 )

= σ dk’1 (gradσ f )(x)(y2 , . . . , yk ), y1 .

48.6

48.8 48. Weak symplectic manifolds 525

48.7. For a weak symplectic manifold (M, σ) let

Cσ (M, R) ‚ C ∞ (M, R)

∞

denote the linear subspace consisting of all smooth functions f : M ’ R such

that the di¬erential df : M ’ T — M factors to a smooth mapping M ’ T σ M . In

view of lemma (48.6) these are exactly those smooth functions on M which admit

a smooth σ-gradient gradσ f ∈ X(M ). Also the condition (48.6.1) translates to a

∞

local di¬erential condition describing the functions in Cσ (M, R).

48.8. Theorem. The Hamiltonian mapping gradσ : Cσ (M, R) ’ X(M, σ), which

∞

is given by

gradσ f := (σ ∨ )’1 —¦ df

igradσ f σ = df or

is well de¬ned. Also the Poisson bracket

∞ ∞ ∞

{ } : Cσ (M, R) — Cσ (M, R) ’ Cσ (M, R),

,

{f, g} := igradσ f igradσ g σ = σ(gradσ g, gradσ f ) =

= dg(gradσ f ) = (gradσ f )(g)

∞

is well de¬ned and gives a Lie algebra structure to the space Cσ (M, R), which also

ful¬lls

{f, gh} = {f, g}h + g{f, h}.

We have the following long exact sequence of Lie algebras and Lie algebra homo-

morphisms:

gradσ γ

∞

0 ’ H 0 (M ) ’ Cσ (M, R) ’ ’ X(M, σ) ’ Hσ (M ) ’ 0,

’1

’’

where H 0 (M ) is the space of locally constant functions, and

{• ∈ C ∞ (M ← T σ M ) : d• = 0}

1

Hσ (M ) = ∞

{df : f ∈ Cσ (M, R)}

is the ¬rst symplectic cohomology space of (M, σ), a linear subspace of the De Rham

cohomology space H 1 (M ).

Proof. It is clear from lemma (48.6), that the Hamiltonian mapping gradσ is well

de¬ned and has values in X(M, σ), since by (33.18.6) we have

Lgradσ f σ = igradσ f dσ + digradσ f σ = ddf = 0.

By (33.18.7), the space X(M, σ) is a Lie subalgebra of X(M ). The Poisson bracket

is well de¬ned as a mapping { , } : Cσ (M, R) — Cσ (M, R) ’ C ∞ (M, R), and

∞ ∞

∞

it only remains to check that it has values in the subspace Cσ (M, R).

48.8

526 Chapter X. Further Applications 48.8

This is a local question, so we may assume that M is an open subset of a convenient

vector space equipped with a (non-constant) weak symplectic structure. So let f ,

g ∈ Cσ (M, R), then we have {f, g}(x) = dg(x)(gradσ f (x)), and we have

∞

)y)(x). gradσ f (x) + dg(x)(d(gradσ f )(x)y)

d({f, g})(x)y = d(dg(

= d(σ(gradσ g( ), y)(x). gradσ f (x) + σ gradσ g(x), d(gradσ f )(x)y

= σ d(gradσ g)(x)(gradσ f (x)) ’ d(gradσ f )(x)(gradσ g(x)), y ,

since gradσ f ∈ X(M, σ) and for any X ∈ X(M, σ) the condition LX σ = 0 im-

plies σ(dX(x)y1 , y2 ) = ’σ(y1 , dX(x)y2 ). So (48.6.2) is satis¬ed, and thus {f, g} ∈

∞

Cσ (M, R).

If X ∈ X(M, σ) then diX σ = LX σ = 0, so [iX σ] ∈ H 1 (M ) is well de¬ned, and by

iX σ = σ ∨ oX we even have γ(X) := [iX σ] ∈ Hs i(M ), so γ is well de¬ned.

1

Now we show that the sequence is exact. Obviously, it is exact at H 0 (M ) and at

Cσ (M, R), since the kernel of gradσ consists of the locally constant functions. If

∞

γ(X) = 0 then σ ∨ oX = iX σ = df for f ∈ Cσ (M, R), and clearly X = gradσ f .

∞

Now let us suppose that • ∈ C ∞ (M ← T σ M ) ‚ „¦1 (M ) with d• = 0. Then X :=

(σ ∨ )’1 —¦ • ∈ X(M ) is well de¬ned and LX σ = diX σ = d• = 0, so X ∈ X(M, σ)

and γ(X) = [•].

Moreover, Hσ (M ) is a linear subspace of H 1 (M ) since for • ∈ C ∞ (M ← T σ M ) ‚

1

„¦1 (M ) with • = df for f ∈ C ∞ (M, R) the vector ¬eld X := (σ ∨ )’1 —¦ • ∈ X(M ) is

well de¬ned, and since σ ∨ oX = • = df by (48.6.1) we have f ∈ Cσ (M, R) with

∞

X = gradσ f .

The mapping gradσ maps the Poisson bracket into the Lie bracket, since by (33.18)

we have

igradσ {f,g} σ = d{f, g} = dLgradσ f g = Lgradσ f dg =

= Lgradσ f igradσ g σ ’ igradσ g Lgradσ f σ

= [Lgradσ f , igradσ g ]σ = i[gradσ f,gradσ g] σ.

Let us now check the properties of the Poisson bracket. By de¬nition, it is skew

symmetric, and we have

{{f, g}, h} = Lgradσ {f,g} h = L[gradσ f,gradσ g] h = [Lgradσ f , Lgradσ g ]h =

= Lgradσ f Lgradσ g h ’ Lgradσ g Lgradσ f h = {f, {g, h}} ’ {g, {f, h}},

{f, gh} = Lgradσ f (gh) = (Lgradσ f g)h + gLgradσ f h =

= {f, g}h + g{f, h}.

Finally, it remains to show that all mappings in the sequence are Lie algebra homo-

morphisms, where we put the zero bracket on both cohomology spaces. For locally

constant functions we have {c1 , c2 } = Lgradσ c1 c2 = 0. We have already checked

that gradσ is a Lie algebra homomorphism. For X, Y ∈ X(M, σ)

i[X,Y ] σ = [LX , iY ]σ = LX iY σ + 0 = diX iY σ + iX LY σ = diX iY σ

is exact.

48.8

48.9 48. Weak symplectic manifolds 527

48.9. Symplectic cohomology. The reader might be curious whether there ex-

1

ists a symplectic cohomology in all degrees extending Hσ (M ) which appeared in

theorem (48.8). We should start by constructing a graded di¬erential subalgebra

of „¦(M ) leading to this cohomology. Let (M, σ) be a weak symplectic manifold.

The ¬rst space to be considered is C ∞ (Lk (T M, R)σ ), the space of smooth sec-

alt

tions of the vector bundle with ¬ber Lalt (Tx M, R)σx consisting of all bounded skew

k

σ

symmetric forms ω with ω( , X2 , . . . , Xk ) ∈ Tx M for all Xi ∈ Tx M . But these

spaces of sections are not stable under the exterior derivative d, one should con-

∞

sider Cσ -sections of vector bundles. For trivial bundles these could be de¬ned

∞

as those sections which lie in Cσ (M, R) after composition with a bounded linear

functional. However, this de¬nition is not invariant under arbitrary vector bundle

isomorphisms, one should require that the transition functions are also in some

∞ ∞

sense Cσ . So ¬nally M should have, in some sense, Cσ chart changings.

We try now a simpler approach. Let

„¦k (M ) := M ) := {ω ∈ C ∞ (Lk (T M, R)σ ) : dω ∈ C ∞ (Lk+1 (T M, R)σ )}.

σ alt alt

Since d2 = 0 and the wedge product of σ-dual forms is again a σ-dual form, we get

a graded di¬erential subalgebra („¦σ (M ), d), whose cohomology will be denoted by

k

Hσ (M ). Note that we have

{ω ∈ „¦k (M ) : dω = 0} = {ω ∈ C ∞ (Lk (T M, R)σ ) : dω = 0},

σ alt

∞

„¦0 (M ) = Cσ (M, R),

σ

1

so that Hσ (M ) is the same space as in theorem (48.8).

Theorem. If (M, σ) is a smooth weakly symplectic manifold which admits smooth

∞

partitions of unity in Cσ (M, R), and which admits ˜Darboux charts™, then the sym-

plectic cohomology equals the De Rham cohomology: Hσ (M ) = H k (M ).

k

Proof. We use theorem (34.6) and its method of proof. We have to check that

the sheaf „¦σ satis¬es the lemma of Poincar´ and admits partitions of unity. The

e

second requirement is immediate from the assumption. For the lemma of Poincar´ e

let ω ∈ „¦k+1 (M ) with dω = 0, and let u : U ’ u(U ) ‚ E be a smooth chart of M

σ

with u(U ) a disked c∞ -open set in a convenient vector space E. We may push σ

to this subset of E and thus assume that U equals this subset. By the Lemma of

Poincar´ (33.20), we get ω = d• where

e

1

tk ω(tx)(x, v1 , . . . , vk )dt,

•(x)(v1 , . . . , vk ) =

0

which is in „¦k (M ) if σ is a constant weak symplectic form on u(U ). This is the

σ

case if we may choose a ˜Darboux chart™ on M .

48.9

528 Chapter X. Further Applications 49.2

49. Applications to Representations of Lie Groups

This section is based on [Michor, 1990], see also [Michor, 1992]

49.1. Representations. Let G be any ¬nite or in¬nite dimensional smooth real

Lie group, and let E be a convenient vector space. Recall that L(E, E), the space

of all bounded linear mappings, is a convenient vector space, whose bornology

is generated by the topology of pointwise convergence for any compatible locally

convex topology on E, see for example (5.18). We shall need an explicit topology

below in order to de¬ne representations, so we shall use on L(E, E) the topology of

pointwise convergence with respect to the bornological topology on E, that of bE.

Let us call this topology the strong operator topology on L(E, E), since this is the

usual name if E is a Banach space.

A representation of G in E is a mapping ρ from G into the space of all linear

mappings from E into E which satis¬es ρ(g.h) = ρ(g).ρ(h) for all g, h ∈ G and

ρ(e) = IdE , and which ful¬lls the following equivalent ˜continuity requirements™:

(1) ρ has values in L(E, E) and is continuous from the c∞ -topology on G into

the strong operator topology on L(E, E).

(2) The associated mapping ρ§ : G — bE ’ bE is separately continuous.

The equivalence of (1) and (2) is due to the fact that L(E, E) consists of all con-

tinuous linear mappings bE ’ bE.

Lemma. If G and bE are metrizable, and if ρ locally in G takes values in uniformly

continuous subsets of L(bE, bE), then the continuity requirements are equivalent to

(3) ρ§ : G — bE ’ bE is (jointly) continuous.

A unitary representation of a metrizable Lie group on a Hilbert space H satis¬es

the requirements of the lemma.

Proof. We only have to show that (1) implies (3). Since on uniformly continu-

ous subsets of L(bE, bE) the strong operator topology coincides with the compact

open topology, ρ is continuous G ’ L(bE, bE)co . By cartesian closedness of the

category of compactly generated topological spaces (see [Brown, 1964], [Steenrod,

1967], or [Engelking, 1989]), ρ§ is continuous from the Kelley-¬cation k(G — bE)

(compare (4.7)) of the topological product to bE. Since G — bE is metrizable it is

compactly generated, so ρ§ is continuous on the topological product, which inci-

dentally coincides with the manifold topology of the product manifold G — E, see

(27.3).

49.2. The Space of Smooth Vectors. Let ρ : G ’ L(E, E) be a representation.

A vector x ∈ E is called smooth if the mapping G ’ E given by g ’ ρ(g)x is

smooth. Let us denote by E∞ the linear subspace of all smooth vectors in E. Then

we have an injection j : E∞ ’ C ∞ (G, E), given by x ’ (g ’ ρ(g)x). We equip

C ∞ (G, E) with the structure of a convenient vector space as described in (27.17),

c—

i.e., the initial structure with respect to the cone C (M, E) ’ C ∞ (R, E) for all

∞

’

c ∈ C ∞ (R, G).

49.2

49.4 49. Applications to representations of Lie groups 529

49.3. Lemma.

(1) The image of the embedding j : E∞ ’ C ∞ (G, E) is the closed subspace

C ∞ (G, E)G = {f ∈ C ∞ (G, E) : f —¦ µg = ρ(g) —¦ f for all g ∈ G}

of all G-equivariant mappings. So with the induced structure E∞ becomes

a convenient vector space.

(2) The space of smooth vectors E∞ is an invariant linear subspace of E, and

we have j(ρ(g)x) = j(x) —¦ µg , or j —¦ ρ(g) = (µg )— —¦ j, where µg is the right

translation on G.

Proof. For x ∈ E∞ and g, h ∈ G we have j(x)µg (h) = j(x)(gh) = ρ(gh)x =

ρ(g)ρ(h)x = ρ(g)j(x)(h), so j(x) ∈ C ∞ (G, E)G . If conversely f ∈ C ∞ (G, E)G then

f (g) = ρ(g)f (e) = j(f (e))(g). Moreover, for x ∈ E∞ the mapping h ’ ρ(h)ρ(g)x =

ρ(hg)x is smooth, so ρ(g)x ∈ E∞ , and we have j(ρ(g)x)(h) = ρ(h)ρ(g)x = ρ(hg)x =

j(x)(hg) = j(x)(µg (h)).

49.4. Theorem. If the Lie group G is ¬nite dimensional and separable, and if

the bornologi¬cation bE of the representation space E is sequentially complete, the

space of smooth vectors E∞ is dense in bE.

Proof. Let x ∈ E, a continuous seminorm p on bE, and µ > 0 be given. Let

U = {g ∈ G : p(ρ(g)x ’ x) < µ}, an open neighborhood of the identity in G.

Let fU ∈ C ∞ (G, R) be a nonnegative smooth function with support in U with

f (g)dL g = 1, where dL denotes left the Haar measure on G. We consider the

GU

element G fU (g)ρ(g)xdL g ∈ bE. Note that this Riemann integral converges since

bE is sequentially complete. We have

fU (g)ρ(g)xdL g ’ x ¤ fU (g)p(ρ(g)x ’ x)dL g

p

G G

¤µ fU (g)dL g = µ.

G

fU (g)ρ(g)xdL g ∈ E∞ . We have

So it remains to show that G

j fU (g)ρ(g)xdL g (h) = ρ(h) fU (g)ρ(g)xdL g

G G

= fU (g)ρ(h)ρ(g)xdL g = fU (g)ρ(hg)xdL g

G G

fU (h’1 g)ρ(g)xdL g,

=

G

which is smooth as a function of h since we may view the last integral as having

values in the vector space C ∞ (G, bE) with a sequentially complete topology. The

integral converges there, since g ’ (h ’ fU (h’1 g)) is smooth, thus continuous

G ’ C ∞ (G, R), and we multiply it by the continuous mapping g ’ ρ(g)x, G ’ bE.

It is easy to check that multiplication is continuous C ∞ (G, R) — bE ’ C ∞ (G, bE)

for the topologies of compact convergence in all derivatives separately of composites

with smooth curves, which is again sequentially complete. One may also use the

compact C ∞ -topology.

49.4

530 Chapter X. Further Applications 49.6

49.5. Theorem. The mappings

ρ§ : G — E ∞ ’ E ∞ ,

ρ : G ’ L(E∞ , E∞ )

are smooth.

A proof analogous to that of (49.10) below would also work here.

Proof. We ¬rst show that ρ§ is smooth. By lemma (49.3), it su¬ces to show that

G—C ∞ (G, E)G ’ C ∞ (G, E)G ’ C ∞ (G, E)

(g, f ) ’ f —¦ µg

is smooth. This is the restriction of the mapping

G — C ∞ (G, E) ’ C ∞ (G, E)

(g, f ) ’ f —¦ µg ,

which by cartesian closedness (27.17) is smooth if and only if the canonically asso-

ciated mapping

G — C ∞ (G, E) — G ’ E

(g, f, h) ’ f (hg) = ev(f, µ(h, g))

is smooth. But this is holds by (3.13), extended to the manifold G. So ρ§ is smooth.

By cartesian closedness (27.17) again (ρ§ )∨ : G ’ C ∞ (E∞ , E∞ ) is smooth, and

takes values in the closed linear subspace L(E∞ , E∞ ). So ρ : G ’ L(E∞ , E∞ ) is

smooth, too.

49.6. Theorem. Let ρ : G ’ L(E, E) be a representation of a Lie group G. Then

the semidirect product

E∞ ρ G

from (38.9) is a Lie group and is regular if G is regular. Its evolution operator is

given by

1

ρ(Evolr (X)(s)’1 ).Y (s) ds, evolr (X)

evolr ∞ G (Y, X) ρ(evolr (X))

=

E G G G

0

for (Y, X) ∈ C ∞ (R, E∞ — g).

Proof. This follows directly from (38.9) and (38.10).

49.6

49.9 49. Applications to representations of Lie groups 531

49.7. The space of analytic vectors. Let G now be a real analytic ¬nite or

in¬nite dimensional Lie group, let again ρ : G ’ L(E, E) be a representation as in

in (49.1). A vector x ∈ E is called real analytic if the mapping G ’ E given by

g ’ ρ(g)x is real analytic.

Let Eω denote the vector space of all real analytic vectors in E. Then we have

a linear embedding j : Eω ’ C ω (G, E) into the space of real analytic mappings,

given by x ’ (g ’ ρ(g)x). We equip C ω (G, E) with the convenient vector space

structure described in (27.17).

49.8. Lemma.

(1) The image of the embedding j : Eω ’ C ω (G, E) is the space

C ω (G, E)G = {f ∈ C ω (G, E) : f —¦ µg = ρ(g) —¦ f for all g ∈ G}

of all G-equivariant mappings, and with the induced structure Eω becomes

a convenient vector space.

(2) The space of analytic vectors Eω is an invariant linear subspace of E, and

we have j(ρ(g)x) = j(x) —¦ µg , or j —¦ ρ(g) = (µg )— —¦ j, where µg is the right

translation on G.

Proof. This is a transcription of the proof of lemma (49.3), replacing smooth by

real analytic.

49.9. Theorem. If the Lie group G is ¬nite dimensional and separable and if

the bornologi¬cation bE of the representation space E is sequentially complete, the

space of real analytic vectors Eω is dense in bE.

See [Warner, 1972, 4.4.5.7].

Proof. Let x ∈ E, a continuous seminorm p on bE, and µ > 0 be given. Let

U = {g ∈ G : p(ρ(g)x ’ x) < µ}, an open neighborhood of the identity in G. Let

• ∈ C(G, R) be a continuous positive function such that G •(g)dL (g) = 2, where

dL denotes left Haar measure on G, and G\U •(g)p(ρ(g)x ’ x)dL (g) < µ.

Let f ∈ C ω (G, R) be a real analytic function with 1 •(g) < f (g) < •(g) for all

2

g ∈ G, which exists by [Grauert, 1958]. Then 1 < G f (g)dL (g) < 2, so if we replace

f by f /( G f (g)dL (g)) we get G f (g)dL (g) = 1 and G\U f (g)p(ρ(g)x’x)dL (g) < µ.

We consider the element G f (g)ρ(g)xdL g ∈ bE. This Riemann integral converges

since bE is sequentially complete. We have

f (g)ρ(g)xdL g ’ x ¤ f (g)p(ρ(g)x ’ x)dL g + f (g)p(ρ(g)x ’ x)dL g

p

G U G\U

¤µ f (g)dL g + µ < 2µ.

G

49.9

532 Chapter X. Further Applications 49.10

f (g)ρ(g)xdL g ∈ Eω . We have

So it remains to show that G

j f (g)ρ(g)xdL g (h) = ρ(h) f (g)ρ(g)xdL g

G G

= f (g)ρ(h)ρ(g)xdL g = f (g)ρ(hg)xdL g

G G

f (h’1 g)ρ(g)xdL g,

=

G

which is real analytic as a function of h, by the following argument: We have to

check that the composition with any continuous linear functional on E maps this to

a real analytic function on G, which is now a question of ¬nite dimensional analysis.

We could also apply here the method of proof used at the end of (49.4), but de-

scribing a sequentially complete compatible topology on C ω (G, bE) requires some

e¬ort.

49.10. Theorem. The mapping ρ§ : G — Eω ’ Eω is real analytic.

We could also use a method analogous to that of (49.5), but we rather give a variant.

Proof. By cartesian closedness of the calculus (11.18) and (27.17), it su¬ces to

show that the canonically associated mapping

ρ§∨ : G ’ C ω (Eω , Eω )

is real analytic. It takes values in the closed linear subspace L(Eω , Eω ) of all boun-

ded linear operators. So it su¬ces to check that the mapping ρ : G ’ L(Eω , Eω ) is

real analytic. Since Eω is a convenient space, by the real analytic uniform bound-

edness principle (11.12), it su¬ces to show that

ρ ev

’x

G ’ L(Eω , Eω ) ’ ’ Eω

’

is real analytic for each x ∈ Eω . Since the structure on Eω is induced by the

embedding into C ω (G, E), we have to check, that

ρ j

ev

G ’ L(Eω , Eω ) ’ ’ Eω ’ C ω (G, E),

’x

’ ’

g ’ ρ(g) ’ ρ(g)x ’ (h ’ ρ(h)ρ(g)x),

is real analytic for each x ∈ Eω . Again by cartesian closedness (11.18), it su¬ces

that the associated mapping

G—G’E

(g, h) ’ ρ(h)ρ(g)x = ρ(hg)x

is real analytic, and this is the case since x is a real analytic vector.

49.10

49.13 49. Applications to representations of Lie groups 533

49.11. The model for the moment mapping. Let now ρ : G ’ U (H) be a

unitary representation of a Lie group G on a Hilbert space H. We consider the space

of smooth vectors H∞ as a weak symplectic Fr´chet manifold, equipped with the

e

symplectic structure σ, the restriction of the imaginary part of the Hermitian inner

product , on H. See section (48) for the general notion of weak symplectic

manifolds. So σ ∈ „¦2 (H∞ ) is a closed 2-form which is non degenerate in the sense

that

σ ∨ : T H ∞ = H∞ — H ∞ ’ T — H∞ = H ∞ — H∞

is injective (but not surjective), where H∞ = L(H∞ , R) denotes the real topological

dual space. This is the meaning of ˜weak™ above.

√

49.12. Let x, y = Re x, y + ’1σ(x, y) be the decomposition of the Hermitian

√

inner product into real and imaginary parts. Then Re x, y = σ( ’1x, y), thus the

real linear subspaces σ ∨ (H∞ ) = σ(H∞ , ) and Re H∞ , of H∞ = L(H∞ , R)

coincide.

σ

Following (48.4), we let H∞ denote the real linear subspace

σ

H∞ = σ(H∞ , ) = Re H∞ ,

of H∞ = L(H∞ , R), the smooth dual of H∞ with respect to the weak symplectic

structure σ. We have two canonical isomorphisms H∞ ∼ H∞ induced by σ and

σ

=

Re , , respectively. Both induce the same structure of a convenient vector

σ

space on H∞ , which we ¬x from now on.

Following (48.7), we have the subspace Cσ (H∞ , R) ‚ C ∞ (H∞ , R) consisting of all

∞

smooth functions f : H∞ ’ R admitting smooth σ-gradients gradσ f , see (48.6).

Then by (48.8) the Poisson bracket

∞ ∞ ∞

{, } : Cσ (H∞ , R) — Cσ (H∞ , R) ’ Cσ (H∞ , R),

{f, g} := igradσ f igradσ g σ = σ(gradσ g, gradσ f ) =

= (gradσ f )(g) = dg(gradσ f )

∞

is well de¬ned and describes a Lie algebra structure on the space Cσ (H∞ , R).

There is the long exact sequence of Lie algebras and Lie algebra homomorphisms:

gradσ γ

∞

0

X(H∞ , σ) ’ H 1 (H∞ ) = 0.

0 ’ H (H∞ ) ’ Cσ (H∞ , R) ’ ’

’’ ’

49.13. We consider now like in (49.2) a unitary representation ρ : G ’ U (H). By

theorem (49.5), the associated mapping ρ§ : G — H∞ ’ H∞ is smooth, so we have

the in¬nitesimal mapping ρ : g ’ X(H∞ ), given by ρ (X)(x) = Te (ρ§ ( , x))X

for X ∈ g and x ∈ H∞ . Since ρ is a unitary representation, the mapping ρ has

values in the Lie subalgebra of all linear Hamiltonian vector ¬elds ξ ∈ X(H∞ ) which

respect the symplectic form σ, i.e. ξ : H∞ ’ H∞ is linear and Lξ σ = 0.

49.13

534 Chapter X. Further Applications 49.16

∞

49.14. Lemma. The mapping χ : g ’ Cσ (H∞ , R) which is given by χ(X)(x) =

1

2 σ(ρ (X)(x), x) for X ∈ g and x ∈ H∞ is a Lie algebra homomorphism, and we

have gradσ —¦χ = ρ .

For g ∈ G we have ρ(g)— χ(X) = χ(X)—¦ρ(g) = χ(Ad(g ’1 )X), so χ is G-equivariant.

∞

Proof. First we have to check that χ(X) ∈ Cσ (H∞ , R). Since ρ (X) : H∞ ’ H∞

is smooth and linear, i.e. bounded linear, this follows from (48.6.2). Furthermore,

gradσ (χ(X))(x) = (σ ∨ )’1 (dχ(X)(x)) =

= 1 (σ ∨ )’1 (σ(ρ (X)( ), x) + σ(ρ (X)(x), )) =

2

= (σ ∨ )’1 (σ(ρ (X)(x), )) = ρ (X)(x),

since σ(ρ (X)(x), y) = σ(ρ (X)(y), x).

Clearly, χ([X, Y ])’{χ(X), χ(Y )} is a constant function by the long exact sequence.

∞

Since it also vanishes at 0 ∈ H∞ , the mapping χ : g ’ Cσ (H∞ ) is a Lie algebra

homomorphism.

For the last assertion we have

χ(X)(ρ(g)x) = 1 σ(ρ (X)(ρ(g)x), ρ(g)x)

2

= 1 (ρ(g)— σ)(ρ(g ’1 )ρ (X)(ρ(g)x), x)

2

= 1 σ(ρ (Ad(g ’1 )X)x, x) = χ(Ad(g ’1 )X)(x).

2

49.15. The moment mapping. For a unitary representation ρ : G ’ U (H) we

can now de¬ne the moment mapping

µ : H∞ ’ g = L(g, R),

µ(x)(X) := χ(X)(x) = 1 σ(ρ (X)x, x),

2

for x ∈ H∞ and X ∈ g.

49.16. Theorem. The moment mapping µ : H∞ ’ g has the following proper-

ties:

(1) We have (dµ(x)y)(X) = σ(ρ (X)x, y) for x, y ∈ H∞ and X ∈ g. Conse-

∞

quently, we have evX —¦µ ∈ Cσ (H∞ , R) for all X ∈ g.

(2) If G is a ¬nite dimensional Lie group, for x ∈ H∞ the image of dµ(x) :

H∞ ’ g is the annihilator g—¦ of the Lie algebra gx = {X ∈ g : ρ (X)(x) =

x

0} of the isotropy group Gx = {g ∈ G : ρ(g)x = x} in g . If G is in¬nite

dimensional we can only assert that dµ(x)(H∞ ) ⊆ g—¦ .

x

(3) For x ∈ H∞ the kernel of the di¬erential dµ(x) is (Tx (ρ(G)x))σ = {y ∈

H∞ : σ(y, Tx (ρ(G)x)) = 0}, the σ-annihilator of the ˜tangent space™ at x of

the G-orbit through x.

(4) The moment mapping is equivariant: Ad— (g) —¦ µ = µ —¦ ρ(g) for all g ∈ G,

where Ad— (g) = Ad(g ’1 )— : g ’ g is the coadjoint action.

49.16

49.16 49. Applications to representations of Lie groups 535

(5) If G is ¬nite dimensional the pullback operator

µ— : C ∞ (g , R) ’ C ∞ (H∞ , R)

∞

actually has values in the subspace Cσ (H∞ , R). It is also a Lie algebra

homomorphism for the Poisson brackets involved.

Proof. (1) Di¬erentiating the de¬ning equation, we get

(dµ(x)y)(X) = 1 σ(ρ (X)y, x) + 1 σ(ρ (X)x, y) = σ(ρ (X)x, y).

(a) 2 2

∞

From lemma (48.6) we see that evX —¦µ ∈ Cσ (H∞ , R) for all X ∈ g.

(2) and (3) are immediate consequences of this formula.

(4) We have

µ(ρ(g)x)(X) = χ(X)(ρ(g)x) = χ(Ad(g ’1 )X)(x) by lemma (49.14)

= µ(x)(Ad(g ’1 )X) = (Ad(g ’1 ) µ(x))(X).

(5) Take f ∈ C ∞ (g , R), then we have

d(µ— f )(x)y = d(f —¦ µ)(x)y = df (µ(x))dµ(x)y

(b)

= (dµ(x)y)(df (µ(x))) = σ(ρ (df (µ(x)))x, y)

by (a), which is smooth in x as a mapping into H∞ ∼ H∞ ‚ H∞ since g is ¬nite

=σ

∞

dimensional. From lemma (48.6) we have that f —¦ µ ∈ Cσ (H∞ , R).

σ(gradσ (µ— f )(x), y) = d(µ— f )(x)y = σ(ρ (df (µ(x)))x, y)

by (b), so gradσ (µ— f )(x) = ρ (df (µ(x)))x. The Poisson structure on g is given as

follows: We view the Lie bracket on g as a linear mapping Λ2 g ’ g. Its adjoint

P : g ’ Λ2 g is then a section of the bundle Λ2 T g ’ g , which is called the

Poisson structure on g . If for ± ∈ g we view df (±) ∈ L(g , R) as an element in g,

the Poisson bracket for fi ∈ C ∞ (g , R) is given by {f1 , f2 }g (±) = (df1 §df2 )(P )|± =

±([df1 (±), df2 (±)]). Then we may compute as follows.

(µ— {f1 , f2 }g )(x) = {f1 , f2 }g (µ(x))

= µ(x)([df1 (µ(x)), df2 (µ(x))])

= χ([df1 (µ(x)), df2 (µ(x))])(x)

= {χ(df1 (µ(x))), χ(df2 (µ(x)))}(x) by lemma (49.14)

= σ(gradσ χ(df2 (µ(x)))(x), gradσ χ(df1 (µ(x)))(x))

= σ(ρ (df2 (µ(x)))x, ρ (df1 (µ(x)))x)

= σ(gradσ (µ— f2 )(x), gradσ (µ— f1 )(x)) by (b)

= {µ— f1 , µ— f2 }H∞ (x).

49.16

536 Chapter X. Further Applications 50

Remark. Assertion (5) of the last theorem also remains true for in¬nite dimen-

sional Lie groups G, in the following sense:

We de¬ne Cσ (g , R) as the space of all f ∈ C ∞ (g , R) such that the following

∞

condition is satis¬ed (compare with lemma (48.6)):

ι

df : g ’ g factors to a smooth mapping g ’ g ’ g , where ι : g ’ g is

’

the canonical injection into the bidual.

∞

Then the Poisson bracket on Cσ (g , R) is de¬ned by {f, g}(±) = ±([df (±), dg(±)]),

and the pullback µ— : C ∞ (g , R) ’ C ∞ (H∞ , R) induces a Lie algebra homomor-

phism µ— : Cσ (g , R) ’ Cσ (H∞ , R) for the Poisson brackets involved. The proof

∞ ∞

is as above, with obvious changes.

49.17. Let now G be a real analytic Lie group, and let ρ : G ’ U (H) be a

unitary representation on a Hilbert space H. Again we consider Hω as a weak

symplectic real analytic manifold, equipped with the symplectic structure σ, the

restriction of the imaginary part of the Hermitian inner product , on H.

Then again σ ∈ „¦2 (Hω ) is a closed 2-form which is non degenerate in the sense

that σ ∨ : Hω ’ Hω = L(Hω , R) is injective. Let

Hω := σ ∨ (Hω ) = σ(Hω ,

—

‚ Hω = L(Hω , R)

) = Re Hω ,

denote the analytic dual of Hω , equipped with the topology induced by the isomor-

phism with Hω .

49.18. Remark. All the results leading to the smooth moment mapping can now

be carried over to the real analytic setting with no changes in the proofs. So all

statements from (49.12) to (49.16) are valid in the real analytic situation. We

summarize this in one more result:

49.19. Theorem. Consider the injective linear continuous G-equivariant mapping

i : Hω ’ H∞ . Then for the smooth moment mapping µ : H∞ ’ g from (49.16)

the composition µ —¦ i : Hω ’ H∞ ’ g is real analytic. It is called the real analytic

moment mapping.

Proof. It is immediately clear from (49.10) and the formula (49.15) for the smooth

moment mapping, that µ —¦ i is real analytic.

50. Applications to Perturbation Theory of Operators

The material of this section is mostly due to [Alekseevsky, Kriegl, Losik, Michor,

1997]. We want to show that relatively simple applications of the calculus developed

in the ¬rst part of this book can reproduce results which partly are even stronger

than the best results from [Kato, 1976]. We start with choosing roots of smoothly

parameterized polynomials in a smooth way. For more information on this see the

reference above. Let

P (t) = xn ’ a1 (t)xn’1 + · · · + (’1)n an (t)

50

50.1 50. Applications to perturbation theory of operators 537

be a polynomial with all roots real, smoothly parameterized by t near 0 in R. Can

we ¬nd n smooth functions x1 (t), . . . , xn (t) of the parameter t de¬ned near 0, which

are roots of P (t) for each t? We can reduce the problem to a1 = 0, replacing the

variable x by the variable y = x ’ a1 (t)/n. We will say that the curve (1) is

smoothly solvable near t = 0 if such smooth roots xi (t) exist.

50.1. Preliminaries. We recall some known facts on polynomials with real coef-

¬cients. Let

P (x) = xn ’ a1 xn’1 + · · · + (’1)n an

be a polynomial with real coe¬cients a1 , . . . , an and roots x1 , . . . , xn ∈ C. It

is known that ai = σi (x1 , . . . , xn ), where σi (i = 1, . . . , n) are the elementary

symmetric functions in n variables:

σi (x1 , . . . , xn ) = xj1 . . . xji .

1¤j1 <···<ji ¤n

n i

Denote by si the Newton polynomials j=1 xj , which are related to the elementary

symmetric function by

sk ’ sk’1 σ1 + sk’2 σ2 + · · · + (’1)k’1 s1 σk’1 + (’1)k kσk = 0 (k ¤ n).

(1)

The corresponding mappings are related by a polynomial di¬eomorphism ψ n , given

by (1):

σ n := (σ1 , . . . , σn ) : Rn ’ Rn

sn := (s1 , . . . , sn ) : Rn ’ Rn

sn := ψ n —¦ σ n .

Note that the Jacobian (the determinant of the derivative) of sn is n! times the

Vandermond determinant: det(dsn (x)) = n! i>j (xi ’ xj ) =: n! Van(x), and even

the derivative itself d(sn )(x) equals the Vandermond matrix up to factors i in the

i-th row. We also have det(d(ψ n )(x)) = (’1)n(n+3)/2 n! = (’1)n(n’1)/2 n!, and

consequently det(dσ n (x)) = i>j (xj ’ xi ). We consider the so-called Bezoutiant

s0 s1 . . . sn’1

«

¬ s1 s2 . . . sn ·

B := ¬ . . ·.

.

. . .

. . .

sn’1 sn . . . s2n’2

Let Bk be the minor formed by the ¬rst k rows and columns of B. From

k’1

«1 x ... x1

1 1 ... 1

«

1

xk’1 ·

¬ x1 x2 . . . xn · ¬ 1 x2 . . . 2

. ··¬.

Bk (x) = .

¬. .

¬ ·

. .·

. . . . .

. . . . . .

k’1 k’1 k’1

xk’1

x1 x2 . . . xn 1 xn . . . n

it follows that

(xi1 ’ xi2 )2 . . . (xi1 ’ xin )2 . . . (xik’1 ’ xik )2 ,

(2) ∆k (x) := det(Bk (x)) =

i1 <i2 <···<ik

since for n — k-matrices A one has det(AA ) = i1 <···<ik det(Ai1 ,...,ik )2 , where

Ai1 ,...,ik is the minor of A with the indicated rows. Since the ∆k are symmetric we

˜ ˜ ˜

have ∆k = ∆k —¦ σ n for unique polynomials ∆k , and similarly we shall use B.

50.1

538 Chapter X. Further Applications 50.4

50.2. Result. [Sylvester, 1853, pp.511], [Procesi, 1978] The roots of P are all real

˜ ˜ ˜

if and only if the matrix B(P ) ≥ 0. Then we have ∆k (P ) := ∆k (a1 , . . . , an ) ≥ 0

˜

for 1 ¤ k ¤ n. The rank of B(P ) equals the number of distinct roots of P , and its

signature equals the number of distinct real roots.

50.3. Proposition. Let now P be a smooth curve of polynomials

P (t)(x) = xn ’ a1 (t)xn’1 + · · · + (’1)n an (t)

with all roots real and distinct for t = 0. Then P is smoothly solvable near 0.

This is also true in the real analytic case and for higher dimensional parameters,

and in the holomorphic case for complex roots.

d

Proof. The derivative dx P (0)(x) does not vanish at any root, since they are dis-

tinct. Thus, by the implicit function theorem we have local smooth solutions x(t)

of P (t, x) = P (t)(x) = 0.

50.4. Splitting Lemma. Let P0 be a polynomial

P0 (x) = xn ’ a1 xn’1 + · · · + (’1)n an .

If P0 = P1 · P2 , where P1 and P2 are polynomials with no common root. Then for P

near P0 we have P = P1 (P )·P2 (P ) for real analytic mappings of monic polynomials

P ’ P1 (P ) and P ’ P2 (P ), de¬ned for P near P0 , with the given initial values.

Proof. Let the polynomial P0 be represented as the product

P0 = P1 .P2 = (xp ’ b1 xp’1 + · · · + (’1)p bp )(xq ’ c1 xq’1 + · · · + (’1)q cq ).

Let xi for i = 1, . . . , n be the roots of P0 , ordered in such a way that for i = 1, . . . , p

we get the roots of P1 , and for i = p + 1, . . . , p + q = n we get those of P2 . Then

(a1 , . . . , an ) = φp,q (b1 , . . . , bp , c1 , . . . , cq ) for a polynomial mapping φp,q , and we get

σ n = φp,q —¦ (σ p — σ q ),

det(dσ n ) = det(dφp,q (b, c)) det(dσ p ) det(dσ q ).

From (50.1) we conclude

(xi ’ xj ) = det(dφp,q (b, c)) (xi ’ xj ) (xi ’ xj )

1¤i<j¤n 1¤i<j¤p p+1¤i<j¤n

which in turn implies

det(dφp,q (b, c)) = (xi ’ xj ) = 0,

1¤i¤p<j¤n

so that φp,q is a real analytic di¬eomorphism near (b, c).

50.4

50.7 50. Applications to perturbation theory of operators 539

50.5. For a continuous function f de¬ned near 0 in R let the multiplicity or order

of ¬‚atness m(f ) at 0 be the supremum of all integers p such that f (t) = tp g(t)

near 0 for a continuous function g. If f is C n and m(f ) < n then f (t) = tm(f ) g(t)

where now g is C n’m(f ) and g(0) = 0. If f is a continuous function on the space

of polynomials, then for a ¬xed continuous curve P of polynomials we will denote

by m(f ) the multiplicity at 0 of t ’ f (P (t)).

The splitting lemma (50.4) shows that for the problem of smooth solvability it is

enough to assume that all roots of P (0) are equal.

Proposition. Suppose that the smooth curve of polynomials

P (t)(x) = xn + a2 (t)xn’2 ’ · · · + (’1)n an (t)

is smoothly solvable with smooth roots t ’ xi (t), and that all roots of P (0) are

equal. Then for (k = 2, . . . , n)

˜

m(∆k ) ≥ k(k ’ 1) min m(xi ).

1¤i¤n

m(ak ) ≥ k min m(xi ).

1¤i¤n

This result also holds in the real analytic case and in the smooth case.

Proof. This follows by (50.1.2) for ∆k and by ak (t) = σk (x1 (t), . . . , xn (t)).

50.6. Lemma. Let P be a polynomial of degree n with all roots real. If a1 = a2 = 0

then all roots of P are equal to zero.

x2 = s2 (x) = σ1 (x) ’ 2σ2 (x) = a2 ’ 2a2 = 0.

2

Proof. From (50.1.1) we have 1

i

50.7. Multiplicity lemma. For an integer r ≥ 1 consider a C nr curve of poly-

nomials

P (t)(x) = xn + a2 (t)xn’2 ’ · · · + (’1)n an (t)

with all roots real. Then the following conditions are equivalent:

(1) m(ak ) ≥ kr for all 2 ¤ k ¤ n.

˜

(2) m(∆k ) ≥ k(k ’ 1)r for all 2 ¤ k ¤ n.

(3) m(a2 ) ≥ 2r.

Proof. We only have to treat r > 0.

(1) implies (2): From (50.1.1) we have m(˜k ) ≥ rk, and from the de¬nition of

s

˜ ˜

∆k = det(Bk ) we get (2).

˜

(2) implies (3) since ∆2 = ’2na2 .

(3) implies (1): From a2 (0) = 0 and lemma (50.6) it follows that all roots of the

polynomial P (0) are equal to zero and, then, a3 (0) = · · · = an (0) = 0. There-

fore, m(a3 ), . . . , m(an ) ≥ 1. Under these conditions, we have a2 (t) = t2r a2,2r (t)

and ak (t) = tmk ak,mk (t) for k = 3, . . . , n, where the mk are positive integers and

50.7

540 Chapter X. Further Applications 50.8

a2,2r , a3,m3 , . . . , an,mn are continuous functions, and where we may assume that

either mk = m(ak ) < ∞ or mk ≥ kr.

Suppose now indirectly that for some k > 2 we have mk = m(ak ) < kr. Then we

put

m := min r, m3 , . . . , mn < r.

3 n

We consider the following continuous curve of polynomials for t ≥ 0:

¯

Pm (t)(x) := xn + a2,2r (t)t2r’2m xn’2

’ a3,m3 (t)tm3 ’3m xn’3 + · · · + (’1)n an,mn (t)tmn ’nm .

¯

If x1 , . . . , xn are the real roots of P (t) then t’m x1 , . . . , t’m xn are the roots of Pm (t),

¯

for t > 0. So for t > 0, Pm (t) is a family of polynomials with all roots real. Since

¯

by theorem (50.2) the set of polynomials with all roots real is closed, Pm (0) is also

a polynomial with all roots real.

¯

By lemma (50.6), all roots of the polynomial Pm (0) are equal to zero, and for those

k with mk = km we have nr ’ mk ≥ kr ’ mk ≥ 1, thus ak,mk is C nr’mk ⊆ C 1 and

ak,mk (0) = 0, therefore m(ak ) > mk , a contradiction.

50.8. Algorithm. Consider a smooth curve of polynomials

P (t)(x) = xn ’ a1 (t)xn’1 + a2 (t)xn’2 ’ · · · + (’1)n an (t)

with all roots real. The algorithm has the following steps:

(1) If all roots of P (0) are pairwise di¬erent, P is smoothly solvable for t near

0 by (50.3).

(2) If there are distinct roots at t = 0 we put them into two subsets which splits

P (t) = P1 (t).P2 (t) by the splitting lemma (50.4). We then feed Pi (t) (which

have lower degree) into the algorithm.

(3) All roots of P (0) are equal. We ¬rst reduce P (t) to the case a1 (t) = 0 by

replacing the variable x by y = x ’ a1 (t)/n. Then all roots are equal to 0,

so m(a2 ) > 0.

˜

(3a) If m(a2 ) is ¬nite then it is even since ∆2 = ’2na2 ≥ 0, m(a2 ) = 2r, and by

the multiplicity lemma (50.7) ai (t) = ai,ir (t)tir (i = 2, . . . , n) for smooth

ai,ir . Consider the following smooth curve of polynomials

Pr (t)(x) = xn + a2,2r (t)xn’2 ’ a3,3r (t)xn’3 + · · · + (’1)n an,nr (t).

If Pr (t) is smoothly solvable and xk (t) are its smooth roots, then xk (t)tr

are the roots of P (t), and the original curve P is smoothly solvable, too.

Since a2,2m (0) = 0, not all roots of Pr (0) are equal, and we may feed Pr

into step 2 of the algorithm.

(3b) If m(a2 ) is in¬nite and a2 = 0, then all roots are 0 by (50.6), and thus the

polynomial is solvable.

(3c) But if m(a2 ) is in¬nite and a2 = 0, then by the multiplicity lemma (50.7)

all m(ai ) for 2 ¤ i ¤ n are in¬nite. In this case we keep P (t) as factor of

50.8

50.9 50. Applications to perturbation theory of operators 541

the original curve of polynomials with all coe¬cients in¬nitely ¬‚at at t = 0

after forcing a1 = 0. This means that all roots of P (t) meet of in¬nite order

of ¬‚atness (see (50.5)) at t = 0 for any choice of the roots. This can be seen

as follows: If x(t) is any root of P (t) then y(t) := x(t)/tr is a root of Pr (t),

hence by (50.9) bounded, so x(t) = tr’1 .ty(t) and t ’ ty(t) is continuous

at t = 0.

This algorithm produces a splitting of the original polynomial

P (t) = P (∞) (t)P (s) (t),

where P (∞) has the property that each root meets another one of in¬nite order at

t = 0, and where P (s) (t) is smoothly solvable, and no two roots meet of in¬nite

order at t = 0, if they are not equal. Any two choices of smooth roots of P (s) di¬er

by a permutation.

The factor P (∞) may or may not be smoothly solvable. For a ¬‚at function f ≥ 0

consider:

x4 ’ (f (t) + t2 )x2 + t2 f (t) = (x2 ’ f (t)).(x ’ t)(x + t).

Here the algorithm produces this factorization. For f (t) = g(t)2 the polynomial

is smoothly solvable. There exist smooth functions f (see (25.3) or [Alekseevsky,

Kriegl, Losik, Michor, 1997, 2.4]) such that x2 = f (t) is not smoothly solvable, in

fact not C 2 -solvable. Moreover, in loc. cit. one ¬nds a polynomial x2 +a2 (t)x’a3 (t)

with smooth coe¬cient functions a2 and a3 which is not C 1 -solvable.

50.9. Lemma. For a polynomial

P (x) = xn ’ a1 (P )xn’1 + · · · + (’1)n an (P )

˜ ˜

with all roots real, i.e. ∆k (P ) = ∆k (a1 , . . . , an ) ≥ 0 for 1 ¤ k ¤ n, let

x1 (P ) ¤ x2 (P ) ¤ · · · ¤ xn (P )

be the roots, increasingly ordered.

Then all xi : σ n (Rn ) ’ R are continuous.

Proof. We show ¬rst that x1 is continuous. Let P0 ∈ σ n (Rn ) be arbitrary. We have

to show that for every µ > 0 there exists some δ > 0 such that for all |P ’ P0 | < δ

there is a root x(P ) of P with x(P ) < x1 (P0 )+µ and for all roots x(P ) of P we have

x(P ) > x1 (P0 ) ’ µ. Without loss of generality we may assume that x1 (P0 ) = 0.

We use induction on the degree n of P . By the splitting lemma (50.4) for the

C 0 -case, we may factorize P as P1 (P ) · P2 (P ), where P1 (P0 ) has all roots equal

to x1 = 0 and P2 (P0 ) has all roots greater than 0, and both polynomials have

coe¬cients which depend real analytically on P . The degree of P2 (P ) is now

smaller than n, so by induction the roots of P2 (P ) are continuous and thus larger

than x1 (P0 ) ’ µ for P near P0 .

50.9

542 Chapter X. Further Applications 50.10

Since 0 was the smallest root of P0 we have to show that for all µ > 0 there exists

a δ > 0 such that for |P ’ P0 | < δ any root x of P1 (P ) satis¬es |x| < µ. Suppose

there is a root x with |x| ≥ µ. Then we get a contradiction as follows, where n1 is

the degree of P1 . From

n1

(’1)k ak (P1 )xn1 ’k

’xn1 =

k=1

we have

n1 n1 n1

µk 1’k

k 1’k 1’k

µ ¤ |x| = ¤ |ak (P1 )| |x|

(’1) ak (P1 )x < µ = µ,

n1

k=1 k=1 k=1

provided that n1 |ak (P1 )| < µk , which is true for P1 near P0 , since ak (P0 ) = 0.

Thus, x1 is continuous.

Now we factorize P = (x ’ x1 (P )) · P2 (P ), where P2 (P ) has roots x2 (P ) ¤ · · · ¤

xn (P ). By Horner™s algorithm (an = bn’1 x1 , an’1 = bn’1 + bn’2 x1 , . . . , a2 =

b2 + b1 x1 , a1 = b1 + x1 ), the coe¬cients bk of P2 (P ) are continuous, and so we may

proceed by induction on the degree of P . Thus, the claim is proved.

50.10. Theorem. Consider a smooth curve of polynomials

P (t)(x) = xn + a2 (t)xn’2 ’ · · · + (’1)n an (t)

with all roots real, for t ∈ R. Let one of the two following equivalent conditions be

satis¬ed:

(1) If two of the increasingly ordered continuous roots meet of in¬nite order

somewhere then they are equal everywhere.

˜

(2) Let k be maximal with the property that ∆k (P ) does not vanish identically

˜

for all t. Then ∆k (P ) vanishes nowhere of in¬nite order.

Then the roots of P can be chosen smoothly, and any two choices di¬er by a per-

mutation of the roots.

Proof. The local situation. We claim that for any t0 , without loss of generality

t0 = 0, the following conditions are equivalent:

(1) If two of the increasingly ordered continuous roots meet of in¬nite order at

t = 0 then their germs at t = 0 are equal.

˜

(2) Let k be maximal with the property that the germ at t = 0 of ∆k (P ) is not

˜

0. Then ∆k (P ) is not in¬nitely ¬‚at at t = 0.

(3) The algorithm (50.8) never leads to step (3c).

(3) ’ (1) Suppose indirectly that two of the increasingly ordered continuous non-

equal roots meet of in¬nite order at t = 0. Then in each application of step (2) these

two roots stay with the same factor. After any application of step (3a) these two

roots lead to nonequal roots of the modi¬ed polynomial which still meet of in¬nite

50.10

50.11 50. Applications to perturbation theory of operators 543

order at t = 0. They never end up in a factor leading to step (3b) or step (1). So

they end up in a factor leading to step (3c).

(1) ’ (2) Let x1 (t) ¤ · · · ¤ xn (t) be the continuous roots of P (t). From (50.1.2)

we have

˜ (xi1 ’ xi2 )2 . . . (xi1 ’ xin )2 . . . (xik’1 ’ xik )2 .

(4) ∆k (P (t)) =

i1 <i2 <···<ik

˜ ˜

The germ of ∆k (P ) is not 0, so the germ of one summand is not 0. If ∆k (P ) were

in¬nitely ¬‚at at t = 0, then each summand was in¬nitely ¬‚at, there were two roots

among the xi which met of in¬nite order, thus by assumption their germs were

equal, so this summand vanished.

˜

(2) ’ (3) Since the leading ∆k (P ) vanishes only of ¬nite order at zero, P has

exactly k di¬erent roots o¬ 0. Suppose indirectly that the algorithm (50.8) leads to

step (3c). Then P = P (∞) P (s) for a nontrivial polynomial P (∞) . Let x1 (t) ¤ · · · ¤

xp (t) be the roots of P (∞) (t) and xp+1 (t) ¤ · · · ¤ xn (t) those of P (s) . We know

that each xi meets some xj of in¬nite order and does not meet any xl of in¬nite

order, for i, j ¤ p < l. Let k (∞) > 2 and k (s) be the number of generically di¬erent

roots of P (∞) and P (s) , respectively. Then k = k (∞) + k (s) , and an inspection of

˜

the formula for ∆k (P ) above leads to the fact that it must vanish of in¬nite order

at 0, since the only non-vanishing summands involve exactly k (∞) many generically

di¬erent roots from P (∞) .

The global situation. From the ¬rst part of the proof we see that the algorithm

(50.8) allows to choose the roots smoothly in a neighborhood of each point t ∈ R,

and that any two choices di¬er by a (constant) permutation of the roots. Thus, we

may glue the local solutions to a global solution.

50.11. Theorem. Consider a curve of polynomials

P (t)(x) = xn ’ a1 (t)xn’1 + · · · + (’1)n an (t), t ∈ R,

with all roots real, where all ai are of class C n . Then there is a di¬erentiable curve

x = (x1 , . . . , xn ) : R ’ Rn whose coe¬cients parameterize the roots.

That this result cannot be improved to C 1 -roots is shown in [Alekseevsky, Kriegl,

Losik, Michor, 1996, 2.4].

Proof. First we note that the multiplicity lemma (50.7) remains true in the C n -

case for n > 2 and r = 1 in the following sense, with the same proof:

If a1 = 0 then the following two conditions are equivalent

(1) ak (t) = tk ak,k (t) for a continuous function ak,k , for 2 ¤ k ¤ n.

(2) a2 (t) = t2 a2,2 (t) for a continuous function a2,2 .

In order to prove the theorem itself, we follow one step of the algorithm. First we

1

replace x by x + n a1 (t), or assume without loss of generality that a1 = 0. Then we

choose a ¬xed t, say t = 0.

50.11

544 Chapter X. Further Applications 50.12

If a2 (0) = 0 then it vanishes of second order at 0, for if it vanishes only of ¬rst order

˜

then ∆2 (P (t)) = ’2na2 (t) would change sign at t = 0, contrary to the assumption

that all roots of P (t) are real, by (50.2). Thus, a2 (t) = t2 a2,2 (t), so by the variant

of the multiplicity lemma (50.7) described above we have ak (t) = tk ak,k (t) for

continuous functions ak,k , for 2 ¤ k ¤ n. We consider the following continuous

curve of polynomials

P1 (t)(x) = xn + a2,2 (t)xn’2 ’ a3,3 (t)xn’3 · · · + (’1)n an,n (t).

with continuous roots z1 (t) ¤ · · · ¤ zn (t), by (50.9). Then xk (t) = zk (t)t are

di¬erentiable at 0 and are all roots of P , but note that xk (t) = yk (t) for t ≥ 0, but

xk (t) = yn’k (t) for t ¤ 0, where y1 (t) ¤ · · · ¤ yn (t) are the ordered roots of P (t).

This gives us one choice of di¬erentiable roots near t = 0. Any choice is then given

by this choice and applying afterwards any permutation of the set {1, . . . , n} which

keeps the function k ’ zk (0) invariant .

If a2 (0) = 0 then by the splitting lemma (50.4) for the C n -case we may factor

P (t) = P1 (t) . . . Pk (t), where the Pi (t) have again C n -coe¬cients, and where each

Pi (0) has all roots equal to ci , and where the ci are distinct. By the arguments

above, the roots of each Pi can be arranged di¬erentiably. Thus P has di¬erentiable

roots yk (t).

But note that we have to apply a permutation on one side of 0 to the original roots,

in the following case: Two roots xk and xl meet at zero with xk (t) ’ xl (t) = tckl (t)

with ckl (0) = 0 (we say that they meet slowly). We may apply to this choice an

arbitrary permutation of any two roots which meet with ckl (0) = 0 (i.e. at least of

second order), and we get thus every di¬erentiable choice near t = 0.

Now we show that we can choose the roots di¬erentiable on the whole domain R.

We start with the ordered continuous roots y1 (t) ¤ · · · ¤ yn (t). Then we put

xk (t) = yσ(t)(k) (t),

where the permutation σ(t) is given by

σ(t) = (1, 2)µ1,2 (t) . . . (1, n)µ1,n (t) (2, 3)µ2,3 (t) . . . (n ’ 1, n)µn’1,n (t) ,

and where µi,j (t) ∈ {0, 1} will be speci¬ed as follows: On the closed set Si,j of all t

where yi (t) and yj (t) meet of order at least 2 any choice is good. The complement

of Si,j is an at most countable union of open intervals, and in each interval we

choose a point, where we put µi,j = 0. Going right (and left) from this point we

change µi,j in each point where yi and yj meet slowly. These points accumulate

only in Si,j .

50.12. Theorem. The real analytic case. Let P be a real analytic curve of

polynomials

P (t)(x) = xn ’ a1 (t)xn’1 + · · · + (’1)n an (t), t ∈ R,

50.12

50.13 50. Applications to perturbation theory of operators 545

with all roots real.

Then P is real analytically solvable, globally on R. All solutions di¬er by permuta-

tions.

By a real analytic curve of polynomials we mean that all ai (t) are real analytic in t,

and real analytically solvable means that we may ¬nd xi (t) for i = 1, . . . , n which

are real analytic in t and are roots of P (t) for all t. The local existence part of this

theorem is due to [Rellich, 1937, Hilfssatz 2], his proof uses Puiseux-expansions.

Our proof is di¬erent and more elementary.

Proof. We ¬rst show that P is locally real analytically solvable near each point

t0 ∈ R. It su¬ces to consider t0 = 0. Using the transformation in the introduction

we ¬rst assume that a1 (t) = 0 for all t. We use induction on the degree n. If n = 1

the theorem holds. For n > 1 we consider several cases:

The case a2 (0) = 0. Here not all roots of P (0) are equal and zero, so by the

splitting lemma (50.4) we may factor P (t) = P1 (t).P2 (t) for real analytic curves of

polynomials of positive degree, which have both all roots real, and we have reduced

the problem to lower degree.

The case a2 (0) = 0. If a2 (t) = 0 for all t, then by (50.6) all roots of P (t) are 0,

and we are done. Otherwise 1 ¤ m(a2 ) < ∞ for the multiplicity of a2 at 0, and

˜

by (50.6) all roots of P (0) are 0. If m(a2 ) > 0 is odd, then ∆2 (P )(t) = ’2na2 (t)

changes sign at t = 0, so by (50.2) not all roots of P (t) are real for t on one side of 0.

This contradicts the assumption, so m(a2 ) = 2r is even. Then by the multiplicity

lemma (50.7) we have ai (t) = ai,ir (t)tir (i = 2, . . . , n) for real analytic ai,ir , and we

may consider the following real analytic curve of polynomials

Pr (t)(x) = xn + a2,2r (t)xn’2 ’ a3,3r (t)xn’3 · · · + (’1)n an,nr (t)

with all roots real. If Pr (t) is real analytically solvable and xk (t) are its real analytic

roots then xk (t)tr are the roots of P (t), and the original curve P is real analytically

solvable too. Now a2,2r (0) = 0 and we are done by the case above.

Claim. Let x = (x1 , . . . , xn ) : I ’ Rn be a real analytic curve of roots of P on an

open interval I ‚ R. Then any real analytic curve of roots of P on I is of the form

± —¦ x for some permutation ±.

Let y : I ’ Rn be another real analytic curve of roots of P . Let tk ’ t0 be a con-

vergent sequence of distinct points in I. Then y(tk ) = ±k (x(tk )) = (x±k 1 , . . . , x±k n )

for permutations ±k . By choosing a subsequence, we may assume that all ±k are

the same permutation ±. But then the real analytic curves y and ± —¦ x coincide on

a converging sequence, so they coincide on I and the claim follows.

Now from the local smooth solvability above and the uniqueness of smooth solutions

up to permutations we can glue a global smooth solution on the whole of R.

50.13. Now we consider the following situation: Let A(t) = (Aij (t)) be a smooth

(real analytic, holomorphic) curve of real (complex) (n — n)-matrices or operators,

50.13

546 Chapter X. Further Applications 50.14

depending on a real (complex) parameter t near 0. What can we say about the

eigenvalues and eigenfunctions of A(t)?

In the following theorem (50.14) the condition that A(t) is Hermitian cannot be

omitted. Consider the following example of real semisimple (not normal) matrices

2t + t3 t

A(t) := ,

’t 0

t2 t2

t

±t 1+

1+

t2

»± (t) = t + ± t2 1+ 4, x± (t) = ,

2 4

2 ’1

where at t = 0 we do not get a base of eigenvectors.

50.14. Theorem. Let A(t) = (Aij (t)) be a smooth curve of complex Hermitian

(n — n)-matrices, depending on a real parameter t ∈ R, acting on a Hermitian space

V = Cn , such that no two of the continuous eigenvalues meet of in¬nite order at

any t ∈ R if they are not equal for all t.

Then the eigenvalues and the eigenvectors can be chosen smoothly in t, on the whole

parameter domain R.

Let A(t) = (Aij (t)) be a real analytic curve of complex Hermitian (n — n)-matrices,

depending on a real parameter t ∈ R, acting on a Hermitian space V = Cn . Then

the eigenvalues and the eigenvectors can be chosen real analytically in t on the whole

parameter domain R.

The condition on meeting of eigenvalues permits that some eigenvalues agree for

all t ” we speak of higher ˜generic multiplicity™ in this situation.

The real analytic version of this theorem is due to [Rellich, 1940]. Our proof is

di¬erent.