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