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Journal of Lie Theory

Volume ?? (??) ??–??

c

?? Heldermann Verlag

Version of November 7, 2018

Moment map and Gelfand transform for the enveloping

algebra

Didier Arnal and Mohamed Selmi

Abstract. Describing the Gelfand construction for the analytic states on an

universal enveloping algebra, we characterize pure states and re-ﬁnd the main result

of a preceding work with L. Abdelmoula and J. Ludwig on the separation of unitary

irreducible representations of a connected Lie group by their generalized moment

sets.

Mathematics Subject Classiﬁcation 2010: 22D10, 22D20, 52A05.

Key Words and Phrases: Universal enveloping algebra; Analytic states; Moment

set of a representation.

1. Introduction

There is a direct relationship between representations of the C?-algebra of a locally

compact group Gand unitary representations of G. The Gelfand construction is

the main tool to describe the cyclic and irreducible representations of a C?-algebra

A, starting with a state or a pure state on A(see [D1] for instance). In this paper,

we consider a real connected and simply connected Lie group G, and the complex

enveloping associative algebra A(g) of its Lie algebra g. We then deﬁne the notion

of analytic state ϕon A(g), and perform the Gelfand construction, starting with ϕ.

This construction was described by K. H. Neeb in [Ne] for the large class of BCH-Lie

algebras. We present it here in the really simpler setting of ﬁnite dimensional Lie

algebras. The result is a unitary representations of Gitself. This representation is

irreducible if and only if ϕis a pure state.

Conversely, associated to any unitary representation πof G, there is a notion

of generalized moment map and set (see [W, BLS, ABLS, AALS]): each smooth

vector fin the space Hπof the representation πdeﬁnes naturally a state Φπ(f)

(see Section 2 for deﬁnition). It is proved in [AALS] that, if πis irreducible, the

convex hull Jπ,Cof this set of states characterizes the representation π.

In the present paper, we re-ﬁnd this result, by using the Gelfand construction

when fis an analytic vector, and proving that, if πis irreducible, the set of Φπ(f)

(fanalytic) is the subset of analytic pure states in Jπ,Cand the disjointness of the

sets {Φπ(f) : fanalytic}for two non-equivalent irreducible representations of G.

The starting point of this work is a personal remark of Jean Ludwig. We

thank him for his ideas and suggestions.

2Arnal and Selmi

2. The moment map

In this section, we recall deﬁnitions and main results for the moment map of

a unitary representation.

Let Gbe a connected and simply connected real Lie group, and πa unitary

representation of Gon an Hilbert space Hπ. Let us denote by gthe (real) Lie algebra

of Gand A(g) the complex universal enveloping algebra of g. The associative algebra

A(g) acts naturally on the space H∞

πof C∞vectors in Hπ, we denote by dπ the

corresponding representation.

Deﬁnition 2.1. Let fbe a non vanishing C∞vector in Hπ. The moment of π

in fis by deﬁnition the element Ψπ(f) of the real linear dual of A(g) deﬁned by:

Ψπ(f)(A) = Re1

ihdπ(A)f, f i

hf, f i.

The map Ψπ:H∞

π\ {0} → (A(g))?

Ris the moment map of π.

The range Iπof the map Ψπis the moment set of π, we denote by Jπthe

convex hull of Iπ.

Since each complex linear form ϕon A(g) can be written as ϕ(A) = ψ(iA) +

iψ(A), where ψis the real linear map ψ(A) = Im(ϕ(A)), we put also:

Deﬁnition 2.2. Let fbe a non vanishing C∞vector in Hπ. The complex

moment of πin fis by deﬁnition the element Φπ(f) of the linear dual (A(g))?

deﬁned by:

Φπ(f)(A) = hdπ(A)f, f i

hf, f i.

The map Φπ:H∞

π\ {0} → (A(g))?is the complex moment map of the

representation π.

The range Iπ,Cof the map Φπis the complex moment set of π, we denote by

Jπ,Cthe convex hull of Iπ,C.

These objects were studied in [BLS, ABLS, AALS, ABB]. The main result is:

Theorem 2.3. ([AALS]) Let πand ρbe two irreducible unitary representations

of G, then πand ρare equivalent if and only if Jπ=Jρ, if and only if Jπ,C=Jρ,C.

Especially, if Gis solvable, for each regular, integral coadjoint orbit Ois

associated a family of irreducible unitary representations πO,χ (see [AK, B]). In

[ABB], the authors present a direct construction, allowing to re-ﬁnd Oand χfrom

IπO,χ .

Since g⊂A(g), we can restrict the real forms Ψπ(f) to g. Then [f]7→

Ψπ(f)|gis the moment map for the strongly Hamiltonian action of Gon the natural

symplectic manifold P(H∞) (see [W, ASZ]). The denomination ‘moment map’ for

the map Ψπis coming from this observation.

Arnal and Selmi 3

3. Gelfand construction for A(g)

Let (X1, . . . , Xd) be a basis of the Lie algebra g. Recall that each unitary represen-

tation πof Ghas a dense subspace Hω

πof analytic vectors (see [N]), that is vectors

ffor which there is r > 0, C > 0 such that for each n, each 1 ≤i1, . . . , in≤d,

kdπ(Xi1. . . Xin)fk ≤ rC nn! (1)

From now one, we restrict ourselves to the space Hω

π,1of analytic unit vectors

f:f∈ Hω,kfk= 1. We thus put

Φω

π= Φπ|Hω

π,1, Iω

π,C= Φω

π(Hω

π,1),and Jω

π,C= Conv(Iω

π,C).

Observe that if π0is a representation unitarily equivalent to π, there is a

unitary operator U:Hπ→ Hπ0such that π0(x) = U◦π(x)◦U−1, and for each

f∈ Hω

π,1,U f is in Hω

π0,1and

Φω

π0(Uf )(A) = hdπ0(A)U f, U fi=hU dπ(A)f, U fi= Φω

π(f)(A).

Therefore Φω

π0(Uf )=Φω

π(f) and Iω

π0,C=Iω

π,C.

Now A(g) is an involutive algebra, when we equip it with the involution (called

principal anti-isomorphism in [D2]):

(αY1. . . Yn)?= (−1)nαYn. . . Y1,

if Yi∈g,α∈C.

Therefore we shall now repeat the usual Gelfand construction for the involu-

tive algebra (A(g), ?) as for C?-algebras (see for instance [D1]).

Deﬁnition 3.1.

1. A linear form ϕ:A(g)→Cis positive if ϕ(A?A)≥0, for each A∈A(g) .

2. A state is a positive linear form ϕon A(g) , such that ϕ(1) = 1. We denote

by S(A(g)) the set of states of A(g) .

3. A state ϕis analytic if there is a basis (X1, . . . , Xd) of g, and positive constants

r,Csuch that, for any n, any 1 ≤i1, . . . , in≤d,

|ϕ(Xi1. . . Xin)| ≤ rCnn! (2)

Fix C, the set of states ϕfor which there is r=rϕ>0 such that the relation

(2) holds for any n,i1, . . . , inis denoted by Sω

C. The set of analytic states of

A(g) is denoted by Sω(A(g)): Sω(A(g)) = ∪C>0Sω

C. Clearly, the set Sω(A(g))

does not depend of the choice of the basis.

4. A state ϕis pure if the only state ψsuch that there exists a > 0 such that

aψ(A?A)≤ϕ(A?A) for each A∈A(g) is ϕitself. Denote by P(A(g)) (resp.

Pω(A(g))) the set of pure states (resp. of pure analytic states).

4Arnal and Selmi

Relation with the Neeb theory

Recall that in [Ne], K. H. Neeb studies systematically the integrability problem

for functional ϕon A(g), where gis a BCH-Lie algebra. To solve this problem, he

deﬁnes analytic functional as a linear form ϕsuch that Pn

ϕ(Xn)

n!converges for each

Xin a 0-neighborhood in g. If gis ﬁnite dimensional, a functional ϕis analytic if

and only if the relation (2) holds.

An important result of Neeb is the following: let ϕbe a functional on A(g),

pa sub-multiplicative semi-norm on g(p([X, Y ]) ≤p(X)p(Y)). Put:

kϕs

nkp= sup n|1

n!X

σ∈Sn

ϕ(Xσ(1) . . . Xσ(n))|:p(Xi)≤1 (1 ≤i≤n)o.

Now Theorem 6.10 in [Ne] says that if Gis a BCH-Lie group with Lie algebra g, if ϕ

is an analytic functional which is positive and such that there is a sub-multiplicative

semi-norm pon gfor which Pn

kϕs

nkptn

n!converges for some t > 0, then there is a

unique unitary representation (πϕ,Hϕ) of Gand a vector fin Hϕsuch that

ϕ(A) = hdπϕ(A)f, f i.

It is not diﬃcult to prove that if Gis a ﬁnite dimensional Lie group, the above

condition holds for any analytic state ϕin the sense of Deﬁnition 3.1, therefore the

next theorem is a direct consequence of Theorem 6.10 in [Ne]. However, to be

complete, we present here a direct proof of this theorem.

Theorem 3.2. If πis a unitary representation of Gand fa analytic unit vector

for π, then: Φω

π(f) : A7→ hdπ(A)f, f iis an analytic state for A(g).

Conversely, let ϕbe an analytic state on A(g), then there is a unitary repre-

sentation τof Gand an analytic vector f∈ Hω

τ,1such that:

ϕ= Φω

τ(f).

Proof. Let πbe a unitary representation of Gand f∈ Hω

π,1. By deﬁnition,

Φω

π(f) is a positive form: Φω

π(f)(A?A) = hdπ(A?A)f, f i=kdπ(A)fk2≥0, and a

state since Φω

π(f)(1) = hf, f i= 1. Now since fis analytic, Formula (1) says that

there is r > 0, C > 0 such that:

|Φω

π(f)(Xi1. . . Xin)|=|hdπ(Xi1. . . Xin)f, f i| ≤ kdπ(Xi1. . . Xin)fk ≤ rCnn!

Thus Φω

π(f)∈Sω(A(g)).

Conversely, if ϕ∈Sω(A(g)), put Annϕ={A∈A(g) : ϕ(A?A) = 0}. Since

ϕis positive, the relation b(A, B) = ϕ(B?A) deﬁnes a positive form b, linear in the

argument A, anti-linear in the argument B. By the Cauchy-Schwarz inequality:

|b(A, B)| ≤ (b(A, A)) 1

2(b(B, B)) 1

2.

Therefore, Annϕis the kernel of b: Annϕ={A∈A(g) : ϕ(B?A) = 0 (B∈A(g))},

it is a left ideal. Put V=A(g)/Annϕ, and let dτ be the natural A(g)-action on V:

dτ(A)[B]=[AB].

Arnal and Selmi 5

The form bdeﬁnes a scalar product ([A]|[B]) = b(A, B) = ϕ(B?A) on V. Com-

pleting V, we get an Hilbert space H=V. Let A=Xj1. . . Xja. For any n, any

1≤i1. . . in≤d, the square of the norm of the vector dτ (Xi1. . . Xin)[A] is:

k[Xi1. . . XinA]k2=|ϕ(A?Xin. . . Xi1Xi1. . . XinA)| ≤ rC2n+2a(2n+ 2a)!

Using the Stirling formula: n!∼√2πn n

en, if C0> C , we get:

C2n+2a(2n+ 2a)!

((C0)nn!)2∼1

√π

1

√n

C2a

e2a2n+ 2a

2n2n

(2n+ 2a)2aC

C02n

22n.

Since 2n+2a

2n2n→e2aand (2n+ 2a)2aC

C02n→0, there is rA>0 such that:

k[Xi1. . . XinA]k2≤rA(2C0)nn!2.(3)

This implies that the vectors in Vare analytic for the representation of g, thus

Theorem 1 in [FSSS] proves there is a unique unitary representation τof Gon H

such that V⊂ H∞, and dτ(X)[A] = [X A]: on V,dτ is the representation of A(g)

associated to τby diﬀerentiation.

Finally [1] is an analytic vector for τ(take A= 1 in the preceding computa-

tion), and by deﬁnition:

Φω

τ([1])(A) = hdτ(A)[1],[1]i= ([A]|[1]) = ϕ(A),

for each A∈A(g) , this ﬁnishes the proof.

The relation between pure states and irreducible representations of Gis:

Proposition 3.3.

1. Let πbe an irreducible unitary representation of G, and f∈ Hω

π,1an analytic

unit vector for π, then the state Φω

π(f)is pure.

2. Conversely, if ϕis a pure state, then the representation τdeﬁned in Theorem

3.2 is irreducible.

Proof.

1. Suppose a > 0 and ϕis a state such that aϕ(A?A)≤Φω

π(f). Since dπ(Xn)f∈

dπ(A(g))f, for any n, if gis a vector in (dπ(A(g))f)⊥, then g∈(π(G)f)⊥,

thus gis orthogonal to the closure Kof the vector space generated by π(G)f.

But Kis a closed vector subspace of Hπ, invariant under the action of π(G),

and containing f6= 0 , therefore K=Hπ,g= 0, and W=dπ(A(g))fis a

dense subspace in Hπ.

Now since kdπ(A)fk2= Φω

π(f)(A?A)≥aϕ(A?A),

Annf={A∈A(g) : dπ(A)f= 0}= AnnΦω

π(f)⊂Annϕ.

6Arnal and Selmi

Thus the surjective map A7→ dπ(A)finduces a bijective map [A]7→ dπ(A)f

between the spaces V=A(g)/Annfand W. Moreover the Hermitian form

b([A],[B]) = ϕ(B?A) is well deﬁned on Vand:

a2|b(A, B)|2≤a2b(A, A)b(B, B)≤ kdπ(A)fk2kdπ(B)fk2.

By passing to the completion of these spaces, there is a positive self-adjoint

operator Ton Hbounded by 1, such that, for any Aand Bin A(g) ,

ϕ(B?A) = hT dπ(A)f, dπ(B)fi.

But Tcommutes with dπ : indeed for each A,Band Cin A(g),

hT dπ(AB)f, dπ(C)fi=ϕ(C?AB) = ϕ((A?C)?B)

=hT dπ(B)f, dπ(A?C)fi

=hdπ(A)T dπ(B)f, dπ(C)fi.

thus by the density of W,T dπ(A) = dπ(A)Ton W, by T-continuity, for each

xin G,T π(x) = π(x)T, on W. This implies Tcommutes with π(x) on Hπ

for each x. By the Schur Lemma T=λId, but 1 = ϕ(1) = hT f, f i=λ,

ϕ= Φω

π(f), Φω

π(f) is a pure state.

2. Suppose now ϕis a pure analytic state, perform the Gelfand construction for

ϕ, as in Theorem 3.2. With the notation of the Theorem, the vector f= [1]

is analytic in Hτ, and ϕ= Φω

τ(f).

Let Kbe an invariant closed subspace of Hτ, let us decompose fon the direct

sum K ⊕ K⊥into f=f1+f2. If f1= 0, then Kbeing the closure of the

projection of dτ(A(g))fis vanishing. Suppose now f16= 0. Since Kand K⊥

are invariant, for any n, any 1 ≤i1, . . . , in≤d,

kdτ(Xi1. . . Xin)fk2=kdτ (Xi1. . . Xin)f1k2+kdτ (Xi1. . . Xin)f2k2≤rCnn!.

Thus f1is an analytic vector, and

Φω

τ(f1)(A?A) = hdτ(A?A)f1, f1i=kdτ (A)f1k2≤ kdτ (A)fk2=ϕ(A?A).

Since ϕis pure, this implies Φω

τ(f1) = ϕ, and choosing A= 1, f2= 0 . Thus

the only closed invariant subspaces of Hτare {0}and Hτ:τis irreducible.

Proposition 3.3 means that there is a canonical map F:Pω(A(g)) →b

G,

deﬁned by F(ϕ) = [τ] where τis deﬁned in Theorem 3.2. Since ϕ= Φτ([1]), this

map is onto. Now let [π] be in b

G, by deﬁnition, F−1([π]) = {Φπ(f) : f∈ Hω

π,1}:

F−1([π]) = Iω

π,C.

Corollary 3.4. Let πand ρtwo unitary irreducible representations of G. If π

and ρare equivalent, then:

Iω

π,C=Iω

ρ,Cand Iω

π=Iω

ρ.

If πand ρare not equivalent, then:

Iω

π,C∩Iω

ρ,C=∅and Iω

π∩Iω

ρ=∅.

Arnal and Selmi 7

4. Convex hulls

Let Xbe a convex subset in a real vector space. Recall that the extremal points in

Xare points vsuch that v=av1+ (1 −a)v2, with 0 ≤a≤1 and vi∈X,v16=v2,

implies a= 0 or a= 1. We denote by Ext(X) the set of extremal points in X.

Proposition 4.1. The subsets Sω

C,Sω(A(g)) (C > 0) of (A(g))?are convex,

and:

Ext(Sω(A(g))) = Pω(A(g)).

Proof. Since Sω

C⊂Sω

C0if C≤C0, and Sω(A(g)) = ∪C>0Sω

C, the convexity of

Sω(A(g)) is a consequence of the convexity of each Sω

C. Let now 0 ≤a≤1 , ϕ,ψ

in Sω

C, then θ=aϕ + (1 −a)ψis clearly a state and

|θ(Xi1. . . Xin)| ≤ a|ϕ(Xi1. . . Xin)|+ (1 −a)|ψ(Xi1. . . Xin)| ≤ (arϕ+ (1 −a)rψ)Cnn!

Therefore Sω

Cis convex.

Let θbe an analytic state. Suppose ﬁrst θis not extremal: there is 0 <a<1

and ϕ6=ψin Sω(A(g)) such that θ=aϕ + (1 −a)ψ. Thus aϕ(A?A)≤θ(A?A), for

any A. This means that θis not a pure state.

Conversely, suppose that θis a non pure analytic state, more precisely,

suppose θ∈Sω

C. Let πbe the unitary representation of Gin Hπ=A(g)/Annθ

and f∈ Hω

πas in Theorem 3.2, such that θ= Φω

π(f). Since θis not pure, πis

not irreducible (Proposition 3.3): there is a non trivial invariant closed subspace K

in Hπ. Decompose fon Hπ=K ⊕ K⊥into f=f1+f2. Then fi6= 0. Indeed if

f2= 0, then dπ(A(g))f=dπ(A(g))f1⊂ K, taking the closure, we get: Hπ=K,

a contradiction. Let pbe the orthogonal projection onto K, since Kand K⊥are

invariant, pcommutes with π(x) for each x. Therefore p(Hω

π)⊂ Kω

π|K. Thus f1,f2

are analytic vectors in Hπ. Put ϕ= Φω

π(f1

kf1k), ψ= Φω

π(f2

kf2k), and a=kf1k2. Then:

θ= Φπ(kf1kf1

kf1k+kf2kf2

kf2k) = aϕ + (1 −a)ψ.

This ﬁnishes the proof of the proposition.

Let us now come back to the complex analytic moment set Iω

π,Cof a unitary

irreducible representation πand its convex hull, Jω

π,C. We have:

Proposition 4.2. The following holds:

Jω

π,C∩Pω(A(g)) = Jω

π,C∩Ext(Sω(A(g))) = Iω

π,C.

Proof. Let θbe in Jω

π,C∩Pω(A(g)), thus there are k,ϕi∈Iω

π,C,ai>0

(1 ≤i≤k), such that Piai= 1 and:

θ=X

i

aiϕi.

8Arnal and Selmi

Especially, ϕ1is a state such that a1ϕ1(A?A)≤θ(A?A). Since θis pure, this implies

ϕ1=θ,θ∈Iω

π,C.

Conversely, since πis irreducible, each state Φπ(f) in Iω

π,Cis pure and in

Jω

π,C. This proves the proposition.

Observe that Theorem 2.3 (see [AALS]) is now a direct consequence of Propo-

sition 4.2. Recall the notation Jπ= Conv(Iπ), Jπ,C= Conv(Iπ,C) .

Corollary 4.3. Let πand ρbe two unitary irreducible representations of a con-

nected Lie group G, then πand ρare equivalent if and only if Jπ=Jρ.

Proof. Let e

Gbe the universal covering group of Gand eπ,eρthe representations

of e

Gcorresponding to πand ρ. Then dπ =deπ,Iπ,C=Ieπ,C, and Jπ,C=Jeπ,C.

Let f∈ Hω

eπ=Hω

π, then

Φeπ(f)∈Jω

eπ,C∩Pω(A(g)) = Iω

eπ,C.

On the other hand, if Jπ=Jρ, then this analytic state is in Jeρ,C, thus we also have:

Φeπ(f)∈Jω

eρ,C∩Pω(A(g)) = Iω

eρ,C.

Therefore eπ∼eρ, and π∼ρ.

The converse being evident, this proves the corollary.

5. Complete convex sums

In [AALS], the complete moment set e

Jπ,Cof a unitary representation πis deﬁned

by:

e

Jπ,C=n∞

X

n=0

Φπ(fn) : fn∈ H∞

π,X

nkfnk2= 1,X

nkdπ(A)fnk2<∞for all A∈A(g)o.

Similarly, let us deﬁne now the complete states ψ:

Deﬁnition 5.1. An analytic state ψis the sum of a convex series of states if there

exists a sequence (ϕn) of states and a sequence (an) of positive real numbers such

that:

ψ=

∞

X

n=0

anϕn.

Lemma 5.2. If ψ=Pnanϕnis the sum of a convex series of states, then for

each n,an6= 0 implies that ϕnis an analytic state.

Proof. Fix a basis {X1, . . . , Xd}of g, since ψis analytic, there are r > 0 and

Arnal and Selmi 9

C > 0 such that, for any k, and any 1 ≤i1, . . . , ik≤d,

|ψ(Xik. . . Xi1Xi1. . . Xik)|= (−1)kψ(Xik. . . Xi1Xi1. . . Xik)

=X

n

an(−1)kϕn(Xik. . . Xi1Xi1. . . Xik)

≤rC2k(2k)!.

Thus if an>0 ,

|ϕn(Xik. . . Xi1Xi1. . . Xik)| ≤ r

an

C2k(2k)!.

Now since the form (A, B )7→ ϕn(B?A) on A(g) is positive, the Cauchy-Schwarz

inequality |ϕn(B?A)|2≤ϕn(B?B)ϕn(A?A) holds and:

|ϕn(Xi1. . . Xik)| ≤ rr

an

Ckp(2k)!.

Now, the Stirling formula implies:

(2k)! ∼√4πk 2k

e2k

∼2√πk 22kk!

√2πk 2

=22k

√πk (k!)2,

or p(2k)! ∼(πk)−1/42kk!, this proves that ϕnis an analytic state.

From now on, we suppose that if an= 0 , then ϕn(Xi1. . . Xik) = 0 for any

k > 0 (ϕnis the map called augmentation in [D2]).

Observe that, if τnis the unitary representation such that ϕn= Φω

τn(fn)

deﬁned in Theorem 3.2, then putting f0

n=√anfn, and denoting ⊕τnthe direct

sum of the τn, we get:

k⊕nf0

nk2=X

nkf0

nk2=X

n

an=ψ(1) = 1.

Moreover, for any Ain A(g),

k⊕ndτn(A)f0

nk2=X

nkdτn(A)f0

nk2=X

n

ankdτn(A)fnk2

=X

n

anΦω

τn(fn)(A?A) = ψ(A?A)<∞

Therefore ψ= Φω

⊕τn(⊕f0

n) belongs to Iω

⊕τn,C.

Proposition 5.3. Let ψ=Pnanϕnbe the sum of a convex series of analytic

states and π(resp. τn) the representation associated to ψ(resp. to ϕn) by Theorem

3.2. Then πis unitarily equivalent to a subrepresentation of ⊕nτn.

Proof. Recall that Hπis the completion of the pre-Hilbert space V=A(g)/Annψ,

equipped with the bilinear form:

([A]|[B]) = ψ(B?A).

10 Arnal and Selmi

As above, denote ϕn= Φτn(fn), and f0

n=√anfn. Deﬁne thus the map U:V→

⊕nHτnby

U([A]) = X

n

dτn(A)f0

n.

Observe that Uis well deﬁned because, if Abelongs to Annψ,

0 = ψ(A?A) = X

n

anhdτn(A?A)fn, fni=X

n

ankdτn(A)fnk2.

Thus U([A]) = 0. Moreover Uis linear and an isometry, since

kU([A])k2=X

n

ankdτn(A)fnk2=ψ(A?A) = k[A]k2.

We can thus extend Uto the space Hπ.

We saw that if Cis such that ψ∈Sω

C, then each ϕnis in Sω

2C. Fix the basis

{X1, . . . , Xd}of gand deﬁne the norm on gby kPixiXik= sup |xi|. Let Xbe in

gsuch that kXk<1

4Cd , then we claim that the following holds:

U(π(exp X)[A]) = X

k

1

k!U(dπ(Xk)[A]) = X

kX

n

1

k!dτn(XkA)f0

n

=X

nX

k

1

k!dτn(XkA)f0

n

=X

n

τn(exp X)dτn(A)f0

n= (⊕nτn)(exp X)U([A]).

Indeed, thanks to the inequality (3) for any C0such that 2C > C0> C , we saw

there is rA>0 such that:

kdτn(XkA)f0

nk ≤ X

i1,...ik|xi1. . . xik|kdτn(Xi1. . . XikA)f0

nk

≤rA(2C0)kk!dk

(4Cd)k≤rAk!C0

2Ck

.

Therefore by Lebesgue’s theorem of dominated convergence, we can exchange the

sums over kand nin the above computation, this proves our claim.

Since Gis connected and simply connected, this implies that for each xin

G, each f∈ Hπ,

U(π(x)f)=(⊕nτn)(x)(Uf ).

This proves the proposition.

As in [AALS], if πis a unitary representation of G, we denote by eπthe

representation ℵ0πsum of a countably many representations unitarily equivalent to

π. Moreover the proof of Proposition 3.2 in [AALS] says that e

Jπ,C=Ieπ,C. Therefore

Theorem 1.2 of [AALS] is a direct corollary of Proposition 5.3. Recall that two

unitary representations πand ρare called quasi equivalent if the representations eπ

and eρare unitarily equivalent.

Arnal and Selmi 11

Corollary 5.4. Two unitary representations πand ρof a connected Lie group G

are quasi equivalent if and only if e

Jπ,C=e

Jρ,C, if and only if their complete moment

sets Ieπand Ieρcoincide.

Proof. By considering the extensions of the representations πand ρto the

universal covering group of G, we can suppose that Gis simply connected.

If πand ρare quasi-equivalent, then eπand eρare equivalent and Ieπ=Ieρ.

Conversely, if Ieπ=Ieρ, then Iω

eπ,C=Iω

eρ,C. Pick f∈ Hω

π,1, then Φω

π(f) is in

Iω

π,C⊂Iω

eπ,C=Iω

eρ,C. This means there is a sequence (fn) of vectors in Hω

ρ,1, a sequence

(an) of positive numbers, such that:

ψ= Φω

π(f) = X

n

anΦω

ρ(fn) = Xanϕn.

For each ndeﬁne Hτnas the closure of the subspace dρ(A(g))fnof Hρ, and τn

as the restriction of ρto Hτn. Thus τnis the representation coming from ϕnby

Theorem 3.2. Now Proposition 5.3 implies that πis a subrepresentation of ⊕nτn,

thus a subrepresentation of eρ.

Therefore eπis a subrepresentation of e

eρ=eρ. Interchanging the roles of πand

ρ, we obtain that eρis a subrepresentation of eπ, an usual argument proves that eπ

and eρare equivalent (see [D1] or [AALS]).

References

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representations of connected Lie groups by their moment sets, J. Funct. Anal.,

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separability teorem for type 1 solvable Lie groups, Adv. Pure Appl. Math. 9

(2018), 247-277.

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representations of exponential Lie groups, J. Lie Theory, 10 (2000), 399–410.

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[D1] J. Dixmier, Les C?-Alg`ebres et leurs Repr´esentations, Cahiers scientiﬁques,

Gauthier-Villars, Paris (1957).

[D2] J. Dixmier, Alg`ebres Enveloppantes, Cahiers scientiﬁques, Gauthier-Villars,

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[FSSS] M. Flato, J. Simon, H. Snellman and D. Sternheimer, Simple facts about

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Ecole Norm. Sup. 4(1972), 423–

434.

[G] R. Goodman, Analytic and entire vectors for representations of Lie groups,

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[Ne] K. H. Neeb, On analytic vectors for unitary representations of inﬁnite dimen-

sional Lie groups, Ann. Inst. Fourier (Grenoble), 61 (2011), 18391874.

[N] E. Nelson, Analytic vectors, Ann. of Math. II, 70 (1959), 572–615.

[W] N. J. Wildberger, Convexity and unitary representations of nilpotent Lie groups,

Invent. Math., 98 (1989), 281–292.

Didier Arnal

Institut de Math´ematiques de Bour-

gogne UMR CNRS 5584

Universit´e de Bourgogne Franche

Comt´e

B.P. 47870

F-21078 Dijon Cedex

France

Didier.Arnal@u-bourgogne.fr

Mohamed Selmi

Laborattoire Physique Math´ematique,

Fonctions sp´eciales et Applications

LR 11 ES 35

Universit´e de Sousse

Ecole Sup´erieure des Sciences et de

Technologie de Hammam Sousse

Rue Lamine Abassi 4011 H.Sousse

Tunisie

Mohamed.Selmi@fss.rnu.tn