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Differential Geometry, Eger (Hungary), 1989 (J. Szenthe, L. Tam´assy, eds.)

arXiv:math/9204222v1 [math.RT] 1 Apr 1992Differential Geometry, Eger (Hungary), 1989 (J. Szenthe, L. Tam´assy, eds. )Colloquia Mathematica Societatis J´anos Bolyai, 56.J´anos Bolyai Math.

Soc. and Elsevier, 1992, 477–489ALL UNITARY REPRESENTATIONSADMIT MOMENT MAPPINGSPeter W. Michor1.

Calculus of smooth mappings1.1. The traditional differential calculus works well for finite dimen-sional vector spaces and for Banach spaces.

For more general locallyconvex spaces a whole flock of different theories were developed, each ofthem rather complicated and none really convincing. The main difficultyis that the composition of linear mappings stops to be jointly continuousat the level of Banach spaces, for any compatible topology.

This wasthe original motivation for the development of a whole new field withingeneral topology, convergence spaces.Then in 1982, Alfred Fr¨olicher and Andreas Kriegl presented inde-pendently the solution to the question for the right differential calculusin infinite dimensions. They joined forces in the further development ofthe theory and the (up to now) final outcome is the book [F-K].In this section I will sketch the basic definitions and the most impor-tant results of the Fr¨olicher-Kriegl calculus.1.2.The c∞-topology.

Let E be a locally convex vector space. Acurve c : R →E is called smooth or C∞if all derivatives exist andare continuous - this is a concept without problems.

Let C∞(R, E) bethe space of smooth functions. It can be shown that C∞(R, E) doesnot depend on the locally convex topology of E, only on its associatedbornology (system of bounded sets).The final topologies with respect to the following sets of mappingsinto E coincide:(1) C∞(R, E).

(2) Lipschitz curves (so that { c(t)−c(s)t−s: t ̸= s} is bounded in E).Typeset by AMS-TEX1

2PETER W. MICHOR(3) {EB →E : B bounded absolutely convex in E}, where EB isthe linear span of B equipped with the Minkowski functionalpB(x) := inf{λ > 0 : x ∈λB}. (4) Mackey-convergent sequences xn →x (there exists a sequence0 < λn ր ∞with λn(xn −x) bounded).This topology is called the c∞-topology on E and we write c∞E for theresulting topological space.

In general (on the space D of test functionsfor example) it is finer than the given locally convex topology, it is nota vector space topology, since scalar multiplication is no longer jointlycontinuous. The finest among all locally convex topologies on E whichare coarser than c∞E is the bornologification of the given locally convextopology.

If E is a Fr´echet space, then c∞E = E.1.3.Convenient vector spaces. Let E be a locally convex vectorspace.

E is said to be a convenient vector space if one of the followingequivalent (completeness) conditions is satisfied:(1) Any Mackey-Cauchy-sequence (so that (xn −xm) is Mackey con-vergent to 0) converges. This is also called c∞-complete.

(2) If B is bounded closed absolutely convex, then EB is a Banachspace. (3) Any Lipschitz curve in E is locally Riemann integrable.

(4) For any c1 ∈C∞(R, E) there is c2 ∈C∞(R, E) with c′1 = c2(existence of antiderivative).1.4.Lemma. Let E be a locally convex space.Then the followingproperties are equivalent:(1) E is c∞-complete.

(2) If f : Rk →E is scalarwise Lipk, then f is Lipk, for k > 1. (3) If f : R →E is scalarwise C∞then f is differentiable at 0.

(4) If f : R →E is scalarwise C∞then f is C∞.Here a mapping f : Rk →E is called Lipk if all partial derivativesup to order k exist and are Lipschitz, locally on Rn. f scalarwise C∞means that λ ◦f is C∞for all continuous linear functionals on E.This lemma says that a convenient vector space one can recognizesmooth curves by investigating compositions with continuous linear func-tionals.1.5.

Smooth mappings. Let E and F be locally convex vector spaces.A mapping f : E →F is called smooth or C∞, if f ◦c ∈C∞(R, F) for

ALL UNITARY REPRESENTATIONS ADMIT MOMENT MAPPINGS3all c ∈C∞(R, E); so f∗: C∞(R, E) →C∞(R, F) makes sense.LetC∞(E, F) denote the space of all smooth mapping from E to F.For E and F finite dimensional this gives the usual notion of smoothmappings: this has been first proved in [Bo]. Constant mappings aresmooth.

Multilinear mappings are smooth if and only if they are boun-ded. Therefore we denote by L(E, F) the space of all bounded linearmappings from E to F.1.6.

Structure on C∞(E, F). We equip the space C∞(R, E) with thebornologification of the topology of uniform convergence on compactsets, in all derivatives separately.

Then we equip the space C∞(E, F)with the bornologification of the initial topology with respect to all map-pings c∗: C∞(E, F) →C∞(R, F), c∗(f) := f ◦c, for all c ∈C∞(R, E).1.7. Lemma.

For locally convex spaces E and F we have:(1) If F is convenient, then also C∞(E, F) is convenient, for anyE. The space L(E, F) is a closed linear subspace of C∞(E, F),so it also convenient.

(2) If E is convenient, then a curve c : R →L(E, F) is smooth ifand only if t 7→c(t)(x) is a smooth curve in F for all x ∈E.1.8. Theorem.

The category of convenient vector spaces and smoothmappings is cartesian closed. So we have a natural bijectionC∞(E × F, G) ∼= C∞(E, C∞(F, G)),which is even a diffeomorphism.Of coarse this statement is also true for c∞-open subsets of convenientvector spaces.1.9.

Corollary. Let all spaces be convenient vector spaces.

Then thefollowing canonical mappings are smooth.ev : C∞(E, F) × E →F,ev(f, x) = f(x)ins : E →C∞(F, E × F),ins(x)(y) = (x, y)()∧: C∞(E, C∞(F, G)) →C∞(E × F, G)()∨: C∞(E × F, G) →C∞(E, C∞(F, G))comp : C∞(F, G) × C∞(E, F) →C∞(E, G)C∞(,) : C∞(F, F ′) × C∞(E′, E) →C∞(C∞(E, F), C∞(E′, F ′))(f, g) 7→(h 7→f ◦h ◦g)Y:YC∞(Ei, Fi) →C∞(YEi,YFi)

4PETER W. MICHOR1.10. Theorem.

Let E and F be convenient vector spaces. Then thedifferential operatord : C∞(E, F) →C∞(E, L(E, F)),df(x)v := limt→0f(x + tv) −f(x)t,exists and is linear and bounded (smooth).

Also the chain rule holds:d(f ◦g)(x)v = df(g(x))dg(x)v.1.11. Remarks.

Note that the conclusion of theorem 1.8 is the startingpoint of the classical calculus of variations, where a smooth curve in aspace of functions was assumed to be just a smooth function in onevariable more.If one wants theorem 1.8 to be true and assumes some other obviousproperties, then the calculus of smooth functions is already uniquelydetermined.There are, however, smooth mappings which are not continuous. Thisis unavoidable and not so horrible as it might appear at first sight.

Forexample the evaluation E × E′ →R is jointly continuous if and only ifE is normable, but it is always smooth. Clearly smooth mappings arecontinuous for the c∞-topology.For Fr´echet spaces smoothness in the sense described here coincideswith the notion C∞cof [Ke].

This is the differential calculus used by[Mic1], [Mil], and [P-S].A prevalent opinion in contemporary mathematics is, that for infinitedimensional calculus each serious application needs its own foundation.By a serious application one obviously means some application of a hardinverse function theorem. These theorems can be proved, if by assumingenough a priori estimates one creates enough Banach space situation forsome modified iteration procedure to converge.Many authors try tobuild their platonic idea of an a priori estimate into their differentialcalculus.

I think that this makes the calculus inapplicable and hides theorigin of the a priori estimates. I believe, that the calculus itself shouldbe as easy to use as possible, and that all further assumptions (whichmost often come from ellipticity of some nonlinear partial differentialequation of geometric origin) should be treated separately, in a settingdepending on the specific problem.I am sure that in this sense the

ALL UNITARY REPRESENTATIONS ADMIT MOMENT MAPPINGS5Fr¨olicher-Kriegl calculus as presented here and its holomorphic and realanalytic offsprings in sections 2 and 3 below are universally usable formost applications.Let me point out as a final remark, that also the cartesian closedcalculus for holomorphic mappings along the same lines is available in [K-N], and recently the cartesian closed calculus for real analytic mappingwas developed in [K-M].2. The moment mapping for unitary representationsThe following is a review of the results obtained in [Mic2].

We includeonly one proof, the central application of the Fr¨olicher-Kriegl calculus.2.1. Let G be any (finite dimensional second countable) real Lie group,and let ρ : G →U(H) be a unitary representation on a Hilbert space H.Then the associated mapping ˆρ : G × H →H is in general not jointlycontinuous, it is only separately continuous, so that g 7→ρ(g)x, G →H,is continuous for any x ∈H.Definition.

A vector x ∈H is called smooth (or real analytic) if themapping g 7→ρ(g)x, G →H is smooth (or real analytic). Let us denoteby H∞the linear subspace of all smooth vectors in H. Then we have anembedding j : H∞→C∞(G, H), given by x 7→(g 7→ρ(g)x).

We equipC∞(G, H) with the compact C∞-topology (of uniform convergence oncompact subsets of G, in all derivatives separately). Then it is easily seen(and proved in [Wa, p 253]) that H∞is a closed linear subspace.

So withthe induced topology H∞becomes a Fr`echet space. Clearly H∞is alsoan invariant subspace, so we have a representation ρ : G →L(H∞, H∞).For more detailed information on H∞see [Wa, chapt.4.4.] or [Kn,chapt.

III.].2.2. Theorem.

The mapping ˆρ : G × H∞→H∞is smooth.Proof. By cartesian closedness of the Fr¨olicher-Kriegl calculus 1.8 it suf-fices to show that the canonically associated mappingˆρ∨: G →C∞(H∞, H∞)is smooth; but it takes values in the closed subspace L(H∞, H∞) of allbounded linear operators.

So by it suffices to show that the mappingρ : G →L(H∞, H∞) is smooth. But for that, since H∞is a Fr`echet

6PETER W. MICHORspace, thus convenient in the sense of Fr¨olicher-Kriegl, by 1.7(2) it sufficesto show thatGρ−→L(H∞, H∞)evx−−→H∞is smooth for each x ∈H∞. This requirement means that g 7→ρ(g)x,G →H∞, is smooth.

For this it suffices to show thatG →H∞j−→C∞(G, H),g 7→ρ(g)x 7→(h 7→ρ(h)(g)x),is smooth. But again by cartesian closedness it suffices to show that theassociated mappingG × G →H,(g, h) 7→ρ(h)(g)x = ρ(hg)x,is smooth.

And this is the case since x is a smooth vector.□2.3. we now consider H∞as a ”weak” symplectic Fr`echet manifold,equipped with the symplectic structure Ω, the restriction of the imag-inary part of the Hermitian inner product ⟨,⟩on H. Then Ω∈Ω2(H∞) is a closed 2-form which is non degenerate in the sense thatˇΩ: TH∞= H∞× H∞→T ∗H∞= H∞× H∞′is injective (but not surjective), where H∞′ = L(H∞, R) denotes thereal topological dual space.

This is the meaning of ”weak” above.2.4. Review.

For a finite dimensional symplectic manifold (M, Ω) wehave the following exact sequence of Lie algebras:0 →H0(M) →C∞(M)gradΩ−−−−→XΩ(M)γ−→H1(M) →0Here H∗(M) is the real De Rham cohomology of M, the space C∞(M) isequipped with the Poisson bracket {,}, XΩ(M) consists of all vectorfields ξ with LξΩ= 0 (the locally Hamiltonian vector fields), which is aLie algebra for the Lie bracket. gradΩf is the Hamiltonian vector fieldfor f ∈C∞(M) given by i(gradΩf)Ω= df, and γ(ξ) = [iξΩ].

Thespaces H0(M) and H1(M) are equipped with the zero bracket.Given a symplectic left action ℓ: G×M →M of a connected Lie groupG on M, the first partial derivative of ℓgives a mapping ℓ′ : g →XΩ(M)

ALL UNITARY REPRESENTATIONS ADMIT MOMENT MAPPINGS7which sends each element X of the Lie algebra g of G to the fundamentalvector field. This is a Lie algebra homomorphism.H0(M)i−−−−→C∞(M)gradΩ−−−−→XΩ(M)γ−−−−→H1(M)σxxℓ′ggA linear lift σ : g →C∞(M) of ℓ′ with gradΩ◦σ = ℓ′ exists if and onlyif γ ◦ℓ′ = 0 in H1(M).

This lift σ may be changed to a Lie algebrahomomorphism if and only if the 2-cocycle ¯σ : g × g →H0(M), givenby (i◦¯σ)(X, Y ) = {σ(X), σ(Y )} −σ([X, Y ]), vanishes in H2(g, H0(M)),for if ¯σ = δα then σ −i ◦α is a Lie algebra homomorphism.If σ : g →C∞(M) is a Lie algebra homomorphism, we may associatethe moment mapping µ : M →g′ = L(g, R) to it, which is given byµ(x)(X) = σ(X)(x) for x ∈M and X ∈g. It is G-equivariant for asuitably chosen (in general affine) action of G on g′.

See [We] or [L-M]for all this.2.5. We now want to carry over to the setting of 2.1 and 2.2 the pro-cedure of 2.4.

The first thing to note is that the hamiltonian mappinggradΩ: C∞(H∞) →XΩ(H∞) does not make sense in general, sinceˇΩ: H∞→H∞′ is not invertible: gradΩf = ˇΩ−1df is defined onlyfor those f ∈C∞(H∞) with df(x) in the image of ˇΩfor all x ∈H∞.A similar difficulty arises for the definition of the Poisson bracket onC∞(H∞).Let ⟨x, y⟩= Re⟨x, y⟩+ √−1Ω(x, y) be the decomposition of the her-mitian inner product into real and imaginary parts. Then Re⟨x, y⟩=Ω(√−1x, y), thus the real linear subspaces ˇΩ(H∞) = Ω(H∞,) andRe⟨H∞,⟩of H∞′ = L(H∞, R) coincide.2.6 Definition.

Let H∗∞denote the real linear subspaceH∗∞= Ω(H∞,) = Re⟨H∞,⟩of H∞′ = L(H∞, R), and let us call it the smooth dual of H∞in view ofthe embedding of test functions into distributions. We have two canoni-cal isomorphisms H∗∞∼= H∞induced by Ωand Re⟨,⟩, respectively.Both induce the same Fr´echet topology on H∗∞, which we fix from nowon.

8PETER W. MICHOR2.7 Definition. Let C∞∗(H∞, R) ⊂C∞(H∞, R) denote the linear sub-space consisting of all smooth functions f : H∞→R such that eachiterated derivative dkf(x) ∈Lksym(H∞, R) has the property thatdkf(x)(, y2, .

. .

, yk) ∈H∞∗is actually in the smooth dual H∗∞⊂H∞′ for all x, y2, . .

. , yk ∈H∞,and that the mappingkYH∞→H∞(x, y2, .

. .

, yk) 7→ˇΩ−1(df(x)(, y2, . .

. , yk))is smooth.

Note that we could also have used Re⟨,⟩instead of Ω.By the symmetry of higher derivatives this is then true for all entries ofdkf(x), for all x.2.8 Lemma. For f ∈C∞(H∞, R) the following assertions are equiva-lent:(1) df : H∞→H∞′ factors to a smooth mapping H∞→H∗∞.

(2) f has a smooth Ω-gradient gradΩf ∈X(H∞) = C∞(H∞, H∞)such that df(x)y = Ω(gradΩf(x), y). (3) f ∈C∞∗(H∞, R).2.9.

Theorem. The mappinggradΩ: C∞∗(H∞, R) →XΩ(H∞),gradΩf := ˇΩ−1 ◦df,is well defined; also the Poisson bracket{,} : C∞∗(H∞, R) × C∞∗(H∞, R) →C∞∗(H∞, R),{f, g} := i(gradΩf)i(gradΩg)Ω= Ω(gradΩg, gradΩf) == (gradΩf)(g) = dg(gradΩf)is well defined and gives a Lie algebra structure to the space C∞∗(H∞, R).We also have the following long exact sequence of Lie algebras andLie algebra homomorphisms:0 →H0(H∞) →C∞∗(H∞, R)gradΩ−−−−→XΩ(H∞)γ−→H1(H∞) = 0

ALL UNITARY REPRESENTATIONS ADMIT MOMENT MAPPINGS92.10. We consider now again as in 2.1 a unitary representation ρ : G →U(H).

By theorem 2.2 the associated mapping ˆρ : G × H∞→H∞issmooth, so we have the infinitesimal mapping ρ′ : g →X(H∞), given byρ′(X)(x) = Te(ˆρ(, x)) for X ∈g and x ∈H∞. Since ρ is a unitaryrepresentation, the mapping ρ′ has values in the Lie subalgebra of alllinear hamiltonian vector fields ξ ∈X(H∞) which respect the symplecticform Ω, i.e.

ξ : H∞→H∞is linear and LξΩ= 0.Now let us consider the mapping ˇΩ◦ρ′(X) : H∞→T(H∞) →T ∗(H∞). We have d( ˇΩ◦ρ′(X)) = d(iρ′(X)Ω) = Lρ′(X)Ω= 0, so thelinear 1-form ˇΩ◦ρ′(X) is closed, and since H1(H∞) = 0, it is exact.So there is a function σ(X) ∈C∞(H∞, R) with dσ(X) = ˇΩ◦ρ′(X),and σ(X) is uniquely determined up to addition of a constant.

If we re-quire σ(X)(0) = 0, then σ(X) is uniquely determined and is a quadraticfunction. In fact we have σ(X)(x) =Rcx ˇΩ◦ρ′(X), where cx(t) = tx.Thusσ(X)(x) =R 10 Ω(ρ′(X)(tx), ddttx)dt == Ω(ρ′(X)(x), x)R 10 dt= 12Ω(ρ′(X)(x), x).2.11.

Lemma. The mappingσ : g →C∞∗(H∞, R),σ(X)(x) = 12Ω(ρ′(X)(x), x)for X ∈g and x ∈H∞, is a Lie algebra homomorphism and gradΩ◦σ =ρ′.For g ∈G we have ρ(g)∗σ(X) = σ(X) ◦ρ(g) = σ(Ad(g−1)X), so σ isG-equivariant.2.12.

The moment mapping. For a unitary representation ρ : G →U(H) we can now define the moment mappingµ : H∞→g′ = L(g, R),µ(x)(X) := σ(X)(x) = 12Ω(ρ′(X)x, x),for x ∈H∞and X ∈g.

10PETER W. MICHOR2.13 Theorem. The moment mapping µ : H∞→g′ has the followingproperties:(1) (dµ(x)y)(X) = Ω(ρ′(X)x, y) for x, y ∈H∞and X ∈g, so µ ∈C∞∗(H∞, g′).

(2) For x ∈H∞the image of dµ(x) : H∞→g′ is the annihilator gΩxof the Lie algebra gx = {X ∈g : ρ′(X)(x) = 0} of the isotropygroup Gx = {g ∈G : ρ(g)x = x} in g′. (3) For x ∈H∞the kernel of dµ(x) is(Tx(ρ(G)x))Ω= {y ∈H∞: Ω(y, Tx(ρ(G)x)) = 0},the Ω-annihilator of the tangent space at x of the G-orbit throughx.

(4) The moment mapping is equivariant: Ad′(g) ◦µ = µ ◦ρ(g) forall g ∈G, where Ad′(g) = Ad(g−1)′ : g′ →g′ is the coadjointaction. (5) The pullback operator µ∗: C∞(g, R) →C∞(H∞, R) actuallyhas values in the subspace C∞∗(H∞, R).

It also is a Lie algebrahomomorphism for the Poisson brackets involved.2.14. Let again ρ : G →U(H) be a unitary representation of a Liegroup G on a Hilbert space H.Definition.

A vector x ∈H is called it real analytic if the mappingg 7→ρ(g)x, G →H is a real analytic mapping, in the real analyticstructure of the Lie group G.We will use from now on the theory of real analytic mappings ininfinite dimensions as developed in [K-M]. So the following conditionson x ∈H are equivalent:(1) x is a real analytic vector.

(2) g ∋X 7→ρ(exp X)x is locally near 0 given by a converging powerseries. (3) For each y ∈H the mapping g ∋X 7→⟨ρ(exp X)x, y⟩∈C issmooth and real analytic along affine lines in g, locally near 0.The only nontrivial part is (3) ⇒(1), and this follows from [K-M, 1.6and 2.7] and the fact, that ρ is a representation.Let Hω denote the vector space of all real analytic vectors in H. Thenwe have a linear embedding j : Hω →Cω(G, H) into the space of realanalytic mappings, given by x 7→(g 7→ρ(g)x).

We equip Cω(G, H) with

ALL UNITARY REPRESENTATIONS ADMIT MOMENT MAPPINGS11the convenient vector space structure described in [K-M, 5.4, see also3.13]. Then Hω consists of all equivariant functions in Cω(G, H) and istherefore a closed subspace.

So it is a convenient vector space with theinduced structure.The space Hω is dense in the Hilbert space H by [Wa, 4.4.5.7] andan invariant subspace, so we have a representation ρ : G →L(Hω, Hω).2.15. Theorem.

The mapping ˆρ : G × Hω →Hω is real analytic inthe sense of [K-M].Proof. Similar to the proof of theorem 2.2.□2.16.

Again we consider now Hω as a ”weak” symplectic real analyticFr´echet manifold, equipped with the symplectic structure Ω, the restric-tion of the imaginary part of the hermitian inner product ⟨,⟩on H.Then again Ω∈Ω2(Hω) is a closed 2-form which is non degenerate inthe sense that ˇΩ: Hω →H′ω = L(Hω, R) is injective. LetH∗ω := ˇΩ(Hω) = Ω(Hω,) = Re⟨Hω,⟩⊂H′ω = L(Hω, R)again denote the analytic dual of Hω, equipped with the topology in-duced by the isomorphism with Hω.2.17 Remark.

All the results leading to the smooth moment mappingcan now be carried over to the real analytic setting with no changesin the proofs. So all statements from 2.9 to 2.13 are valid in the realanalytic situation.

We summarize this in one more result:2.18 Theorem. Consider the injective linear continuous G-equivariantmapping i : Hω →H∞.Then for the smooth moment mapping µ :H∞→g′ from 2.13 the composition µ ◦i : Hω →H∞→g′ is realanalytic.

It is called the real analytic moment mapping.2.19. Remarks.

It is my conjecture that for an irreducible represen-tation which is constructed by geometric quantization of an coadjointorbit (the Kirillov method), the restriction of the moment mapping tothe intersection of the unit sphere with the space of smooth vectors takesvalues exactly in the orbit one started with, if the construction is suitablynormalized.I have been unable to prove this conjecture in general, but HerbertWiklicky [Wi] has checked that this is true for the Heisenberg group.He also checked that this moment mapping produces the expectationvalue for the (angular) momentum in physically relevant situations and

12PETER W. MICHORhe claims that this moment mapping describes a sort of classical limit forthe quantum theory described by the unitary representation in question.Let me add some thoughts on the rˆole of the moment mapping inthe study of unitary representations. I think that its restriction to theintersection of the unit sphere with the space of smooth vectors mapsto one coadjoint orbit, if the representation is irreducible (I was unableto prove this).

It is known that not all irreducible representations comefrom line bundles over coadjoint orbits (alias geometric quantization),but there might be a higher dimensional vector bundle over this coadjointorbit, whose space of sections contains the space of smooth vectors assubspace of sections which are covariantly constant along some complexpolarization.References[A-K]Auslander, Louis; Kostant, Bertram, Polarization and unitary representationsof solvable Lie groups, Inventiones Math. 14 (1971), 255–354.

[Bo]Boman, Jan, Differentiability of a function and of its compositions with func-tions of one variable, Math. Scand.

20 (1967), 249–268. [F-K]Fr¨olicher, Alfred; Kriegl, Andreas, Linear spaces and differentiation theory,Pure and Applied Mathematics, J. Wiley, Chichester, 1988.

[L-M] Libermann, Paulette; Marle, C. M., Symplectic geometry and analytical me-chanics, Mathematics and its applications, D. Reidel, Dordrecht, 1987. [Ke]Keller, Hans H., Differential calculus in locally convex spaces, Springer Lec-ture Notes 417, 1974.

[Ki1]Kirillov, A. A., Elements of the theory of representations, Springer-Verlag,Berlin, 1976.

[[Ki2] Kirillov, A. A., Unitary representations of nilpotent Lie groups, Russian Math.Surveys 17 (1962), 53–104.

[Kn]Knapp, Anthony W., Representation theory of semisimple Lie groups, Prince-ton University Press, Princeton, 1986. [Ko]Kostant, Bertram, Quantization and unitary representations, Lecture Notesin Mathematics, Vol.

170,, Springer-Verlag, 1970, pp. 87–208.

[Kr1]Kriegl, Andreas, Die richtigen R¨aume f¨ur Analysis im Unendlich - Dimen-sionalen, Monatshefte Math. 94 (1982), 109–124.

[Kr2]Kriegl, Andreas, Eine kartesisch abgeschlossene Kategorie glatter Abbildun-gen zwischen beliebigen lokalkonvexen Vektorr¨aumen, Monatshefte f¨ur Math.95 (1983), 287–309. [K-M] Kriegl, Andreas; Michor, Peter W., The convenient setting for real analyticmappings, Acta Mathematica (1990).

[K-N]Kriegl, Andreas; Nel, Louis D., A convenient setting for holomorphy, CahiersTop. G´eo.

Diff. 26 (1985), 273–309.

[Mic1] Michor, Peter W., Manifolds of differentiable mappings, Shiva MathematicsSeries 3, Orpington, 1980.

ALL UNITARY REPRESENTATIONS ADMIT MOMENT MAPPINGS13[Mic2] Michor, Peter W., The moment mapping for unitary representations, J. GlobalAnal. Geo (1990).

[Mil]Milnor, John, Remarks on infinite dimensional Lie groups, Relativity, Groups,and Topology II, Les Houches, 1983, B.S. DeWitt, R. Stora, Eds., Elsevier,Amsterdam, 1984.

[P-S]Pressley, Andrew; Segal, Graeme, Loop groups, Oxford Mathematical Mono-graphs, Oxford University Press, 1986. [Wa]Warner, Garth, Harmonic analysis on semisimple Lie groups, Volume I,Springer-Verlag, New York, 1972.

[We]Weinstein, Alan, Lectures on symplectic manifolds, Regional conference seriesin mathematics 29 (1977), Amer. Math.

Soc..[Wi]Wiklicky, Herbert, Physical interpretations of the moment mapping for uni-tary representations, Diplomarbeit, Universit¨at Wien, 1989.Institut f¨ur Mathematik, Universit¨at Wien, Strudlhofgasse 4, A-1090Wien, Austria.

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