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