Berezin symbols and spectral measures of representation operators
aa r X i v : . [ m a t h . SP ] S e p BEREZIN SYMBOLS AND SPECTRAL MEASURES OFREPRESENTATION OPERATORS
BENJAMIN CAHEN
Abstract.
Let G be a Lie group with Lie algebra g and let π be a unitary representationof G realized on a reproducing kernel Hilbert space. We use Berezin quantization inorder to study spectral measures associated with operators − idπ ( X ) for X ∈ g . As anapplication, we show how results about contractions of Lie group representations giverise to results on convergence of sequences of spectral measures. We give some examplesincluding contractions of SU (1 ,
1) and SU (2) to the Heisenberg group. Introduction
In the years 1970-1975, a general theory of quantization on homogeneous K¨ahler man-ifolds was developed by F. A. Berezin in [4], [5], [6]. In this theory, an important toolis the notion of covariant symbol of an operator acting on a reproducing kernel Hilbertspace of square-integrable holomorphic functions on a K¨ahler manifold [5]. In fact, thisnotion of covariant symbol has its own interest and, in particular, it appeared early thatBerezin symbols could be helpful to study spectral properties of operators on reproducingkernel Hilbert spaces, see [4].Let G be a Lie group and let π be a unitary representation of G on a reproducingkernel Hilbert space H consisting of functions on a homogeneous G -manifold M . Denoteby g the Lie algebra of G and by dπ the differential of π . In this paper, we are concernedwith self-adjoint operators of the form − idπ ( X ) where X ∈ g . When G is a simple Liegroup and X generates a non-compact one-parameter subgroup of G then C. C. Moorehas shown that the projection-valued measure E λ associated with − idπ ( X ) is absolutelycontinuous with respect to the Lebesgue measure and that the spectrum of − idπ ( X ) iseither R or half-line (0 , ∞ ) or ( −∞ , − idπ ( X ) is R unless G is the group of a bounded symmetric domain and,in that case, half-line occurs only for π in the holomorphic discrete series of G and forcertain X , see [28], [29].The aim of this paper is to show how Berezin symbols can be used to study the spectraldistribution d ( h E λ f, f i ) of − idπ ( X ) for some X ∈ g and f ∈ H . We give in particularsome integral formulas for spectral measures allowing explicit computations in some cases.This is illustrated by various examples including the non-degenerated unitary irreducible Mathematics Subject Classification.
Key words and phrases.
Berezin quantization; Berezin symbols; unitary representations; Lie grouprepresentations; reproducing kernel Hilbert space; coherent states; spectral measures; contractions ofrepresentations. representations of the Heisenberg group as well as the holomorphic discrete series repre-sentations of SU (1 , G is the contraction of a Lie group G , that is, G is the limit case of a sequence of Liegroups isomorph to G , then it often happens that the unitary irreducible representationsof G are also limits in some sense of sequences of unitary irreducible representations of G , see [21], [14]. Moreover, it appears that such contractions of Lie group representa-tions can be connected to convergence of Berezin (covariant) symbols of representationoperators, see for instance [9, 11, 12]. Thus by combining results on contractions of Liegroup representations with the above mentioned formulas for spectral measures of rep-resentation operators, we obtain here some results about convergence of these spectralmeasures. In particular we consider the contraction of the discrete series representationsof SU (1 ,
1) and the contraction of the unitary irreducible representations of SU (2) tounitary irreducible representations of the Heisenberg group. Of course, we can hope forfurther results concerning unitary representations of other Lie groups.This paper is organized as follows. In Section 2 and Section 3, we recall some basicfacts on Berezin symbols of operators acting on reproducing kernel Hilbert spaces. Integralformulas for spectral measures of − idπ ( X ) are given in Section 4 and then illustrated inSection 5 by the case of the Heisenberg group. Finally, applications of contraction resultsto spectral measures are presented in Section 6 and Section 7.2. Generalities on Berezin quantization
In this section, we review some facts on Berezin quantization [5], [6]. We follow moreand less the presentation of [3] and [10].Let G be a Lie group and let M be a G -homogeneous space. Let µ be a G -invariantmeasure on M . Let K be a measurable function on M such that K ( x ) > µ be the measure on M defined by d ˜ µ ( x ) = K ( x ) − dµ ( x ). Let H be areproducing kernel Hilbert space of square integrable functions on M with respect to ˜ µ .This means that H is a Hilbert space with respect to the L -norm and, for each x ∈ M ,the evaluation map H ∋ f → f ( x ) is continuous. Then, for each x ∈ M , there exists aunique function e x ∈ H (called a coherent state ) such that f ( x ) = h f, e x i for each f ∈ H .The function k ( x, y ) := e x ( y ) = h e y , e x i is then called the reproducing kernel of H .Let α : G × M → C ∗ be a function such that α ( g g , x ) = α ( g , g · x ) α ( g , x )for each g , g ∈ G and x ∈ M . Then we can define an action π of G on the space of allfunctions on M , according to the formula( π ( g ) f )( x ) = α ( g − , x ) f ( g − · x ) . Assume that π ( g )( f ) ∈ H for each g ∈ G and f ∈ H . Then π induces a representation of G on H . Proposition 2.1. [3] , [10] . (1) If α and K are compatible in the sense that we have K ( g · x ) = | α ( g, x ) | − K ( x ) , g ∈ G, x ∈ M, EREZIN SYMBOLS... 3 then the representation π is unitary. (2) If the representation π is unitary then we have π ( g ) e x = α ( g, x ) e g · x , g ∈ G, x ∈ M, and k ( g · x, g · y ) = α ( g, x ) − α ( g, y ) − k ( x, y ) , g ∈ G, x, y ∈ M. (3) Moreover, in this case, µ can be normalized so that k ( x, x ) = K ( x ) ( x ∈ M ). In the rest of the section, we assume that the conditions introduced in the previousproposition are fulfilled.Now, let A be an operator on H . The Berezin (covariant) symbol of A is the function S ( A ) defined on M by(2.1) S ( A )( x ) = h A e x , e x ih e x , e x i and the double Berezin symbol of A is the function defined by(2.2) s ( A )( x, y ) = h A e y , e x ih e y , e x i for each x, y ∈ M such that h e x , e y i 6 = 0, see [5] for instance. We can easily recover A from s ( A ). Indeed, we have A f ( x ) = h A f , e x i = h f , A ∗ e x i = Z M f ( y ) A ∗ e x ( y ) K ( y ) − dµ ( y )= Z M f ( y ) h A ∗ e x , e y i K ( y ) − dµ ( y )= Z M f ( y ) s ( A )( x, y ) h e y , e x i K ( y ) − dµ ( y ) . Then we see that the map A → s ( A ) is injective and that the kernel of A is the function(2.3) k A ( x, y ) = h Ae y , e x i = s ( A )( x, y ) h e y , e x i . Moreover, we have the following result.
Proposition 2.2. [5] , [10] , [13](1) If the operator A on H has adjoint A ∗ , then we have S ( A ∗ ) = S ( A ) . (2) For A operator on H and g ∈ G , we have S ( π ( g ) − Aπ ( g ))( x ) = S ( A )( g · x ) , g ∈ G, x, y ∈ M. Hilbert spaces of holomorphic functions
Important examples of reproducing kernel Hilbert spaces are Hilbert spaces of holo-morphic functions, see [18]. Such spaces naturally appear in Harmonic Analysis as, forinstance, carrying spaces of the holomorphic discrete series representations of some semi-simple Lie groups, see [22] or, more generally, of holomorphic representations of quasi-Hermitian Lie groups, see [26].
BENJAMIN CAHEN
In this section, we assume that, with the notation of Section 2, M is a domain of C n , G consists of holomorphic automorphisms of M and H consists of holomorphic functionson M .Let µ L be the Lebesgue measure on M ⊂ C n . Write µ = δ.µ L for the G -invariantmeasure on M where δ > M . In order to avoid technicalities,we also assume here that K is continuous,.Then H consists of all holomorphic functions f on M such that k f k H := Z M | f ( z ) | K ( z ) − δ ( z ) dµ L ( z ) < ∞ . We can easily see that the evaluation map f → f ( z ) is continuous. Indeed, given z ∈ M we can fix r > D r ( z ) := { w ∈ M : | w k − z k | < r, k =1 , , . . . , n } is contained in M . By the mean value property, we have f ( z ) = ( πr ) − n Z D r ( z ) f ( w ) dµ L ( w ) . Then, by the Cauchy-Schwarz equality, we get | f ( z ) | ≤ ( πr ) − n k f k H Z D r ( z ) K ( w ) δ ( w ) − dµ L ( w ) , hence the continuity of the map f → f ( z ).Note that by the same way we can show that the space O ( M ) of all holomorphicfunctions on M being endowed with uniform convergence on compact subsets, the naturalinjection H → O ( M ) is injective.In this context, the reproducing kernel ( z, w ) → k ( z, w ) = h e w , e z i H of H is holomor-phic in the variable z and anti-holomorphic in the variable w . More generally, let A bean operator on H . Then the function s ( A )( z, w ) is holomorphic in the variable z andanti-holomorphic in the variable w . Consequently, s ( A )-hence A -is determinated by itsrestriction to the diagonal of M × M , that is, by S ( A ).Note that, in many cases of interest, the polynomials are element of H , see [10], [12],[26]. 4. Berezin symbols and spectral measures
Here we retain the notation of Section 2 and we assume that we are in the settingof Section 3. Let us introduce some additional notation. Let S ( R ) be the space of allSchwartz functions on R and S ′ ( R ) be the space of all tempered distributions on R . Thenormalization of the Fourier transform F : S ( R ) → S ( R ) is taken here as follows. For φ ∈ S ( R ), we define ( F φ )( x ) = Z R e − itx φ ( t ) dt. The inverse Fourier transform is then( F − φ )( t ) = 12 π Z R e itx φ ( x ) dx. Recall that F can be extended to S ′ ( R ) via hF ( ν ) , φ i = h ν, F φ i EREZIN SYMBOLS... 5 for each ν ∈ S ′ ( R ) and each φ ∈ S ( R ). Similarly, one has hF − ( ν ) , φ i = h ν, F − φ i . Now, let A be a self-adjoint operator on H and let A = R R λ dE λ be the spectraldecomposition of A . Then we also have, for each t ∈ R ,exp( − itA ) = Z R e − itλ dE λ . In particular, if we take f ∈ H such that k f k H = 1 and we denote µ f := d ( h E λ f, f i H )then we have(4.1) h exp( − itA ) f, f i H = Z R e − itλ dµ f ( λ ) . Since k f k H = 1, µ f is a probability measure on R hence a tempered distribution on R .Then we have the following proposition. Proposition 4.1.
Let f ∈ H such that k f k H = 1 . (1) µ f is the inverse Fourier transform of the function F f : t → h exp( − itA ) f, f i H considered as a tempered distribution on R . (2) Suppose that F f is integrable on R . Then µ f is absolutely continuous with respectto the Lebesgue measure on R and its density ϕ f is given by ϕ f ( λ ) = 12 π Z R e itλ F f ( t ) dt. Proof. (1) For each φ ∈ S ( R ), we have hF µ f , φ i = h µ f , F φ i = Z R ( F φ )( λ ) dµ f ( λ )= Z R Z R e − itλ φ ( t ) dµ f ( λ ) dt = Z R F f ( t ) φ ( t ) dt by Eq. 4.1. Then we get F µ f = F f .(2) For each φ ∈ S ( R ), we have h µ f , φ i = hF − F f , φ i = h F f , F − φ i = Z R F f ( t )( F − φ )( t ) dt = 12 π Z R F f ( t ) (cid:18)Z R e itλ φ ( λ ) dλ (cid:19) dt = 12 π Z R (cid:18)Z R e itλ F f ( t ) dt (cid:19) φ ( λ ) dλ. Indeed, we can apply Fubini’s Theorem, since ( t, λ ) → F f ( t ) φ ( λ ) is integrable on R . Theresult follows. (cid:3) BENJAMIN CAHEN
Berezin symbols naturally appear when we consider the particular case where f = k e z k − H e z for z ∈ M . Indeed, in this case we have F f ( t ) = k e z k − H h exp( − itA ) e z , e z i H = S (exp( − itA ))( z )and µ f is then given by h µ f , φ i = h F f , F − φ i = Z R S (exp( − itA ))( z )( F − φ )( t ) dt for each φ ∈ S ( R ). In particular, if the function t → S (exp( − itA ))( z ) is integrable on R for each z ∈ M then µ f has density(4.2) ϕ f ( λ ) = 12 π Z R S (exp( − itA ))( z ) e itλ dt and we have the following formula for the Berezin symbol of E λ (which determines E λ ,see Section 3). Since S ( E λ )( z ) = h E λ f, f i H = µ f (] − ∞ , λ ]) = Z λ −∞ ϕ f ( x ) dx, we get S ( E λ )( z ) = 12 π Z R Z ] −∞ ,λ ] S (exp( − itA ))( z ) e ixt dt dx. In this paper, we focus on the case where A = − idπ ( X ) for X ∈ g . We have then S (exp( − itA ))( z ) = S ( π (exp( − tX )))( z ) . Thus we see that in this case µ f is closely connected to the function g → S ( π ( g )) which iscalled the star-exponential (since it is a convergent version of the formal star-exponentialwhich appears in Deformation Quantization, see [20], [1]) and played a central role in theconstruction of the generalized Fourier transform, [2], [31].5. Example: the Heisenberg group
Let G be the Heisenberg group and g be the Lie algebra of G . Let v , v , v be abasis of g in which the only non trivial brackets are [ v , v ] = v .For ( a , a , a ) ∈ R , we denote by [ a , a , a ] the element exp G ( a v + a v + a v ) of G . The multiplication of G is then given by[ a , a , a ] · [ b , b , b ] = [ a + b , a + b , a + b + ( a b − a b )]for ( a , a , a ) ∈ R and ( b , b , b ) ∈ R .We fix a real number γ > γ < G whose restriction to the center of G is the character [0 , , a ] → e iγa ,see [19]. We describe now the Bargmann-Fock realization π γ of this representation, seefor instance [30], [9].Let H γ be the Hilbert space of all holomorphic functions f on C such that k F k γ := Z C | f ( z ) | e −| z | / γ dµ γ ( z ) < ∞ where dµ γ ( z ) := (2 πλ ) − dx dy . Here z = x + iy with x, y ∈ R . EREZIN SYMBOLS... 7
Let us consider the action of of G on C defined by g · z := z + λ ( a − ia ) for g =[ a , a , a ] ∈ G and z ∈ C . Then π γ is the representation of G on H γ given by( π γ ( g ) f )( z ) = α ( g − , z ) f ( g − · z )where α is defined by α ( g, z ) = exp ( − ia γ + (1 / a + a i )( − z + γ ( − a + a i )))for g = [ a , a , a ] ∈ G and z ∈ C .The differential of π γ is given by ( dπ γ ( v ) f )( z ) = 12 izf ( z ) + γif ′ ( z )( dπ γ ( v ) f )( z ) = 12 zf ( z ) − γf ′ ( z )( dπ γ ( v ) f )( z ) = iγf ( z ) . The coherent states are given by e γz ( w ) = exp ¯ zw/ γ . Then we have the reproducingproperty f ( z ) = h f, e z i γ for each f ∈ H γ where h· , ·i γ denotes the scalar product on H γ .Note also that an orthonormal basis of H γ is the family f γp ( z ) = (2 p γ p p !) − / z p for p ∈ N .Then we can easily verify that for each g = [ a , a , a ] ∈ G , the Berezin symbol of π γ ( g ) is S γ ( π γ ( g ))( z ) = exp (cid:0) iγa − γ ( a + a ) + ( a + ia ) z + ( − a + ia )¯ z (cid:1) . Now, let X = a v + a v + a v ∈ g with ( a , a ) = (0 , z = x + iy with x, y ∈ R .Then we have(5.1) S γ ( π γ (exp( − tX )))( z ) = e − it ( a x + a y + a γ ) e − γt ( a + a ) / and, clearly, the function t → S γ ( π γ (exp( − tX )))( z ) is integrable. Then, applying Propo-sition 4.1, we obtain that the measure d ( h E λ ( k e γz k − e γz ) , k e γz k − e γz i γ )associated with − idπ γ ( X ) has density ϕ ( λ ) := 12 π Z R e itλ S γ ( π γ (exp( − tX )))( z ) dt = 1 p πγ ( a + a ) exp (cid:18) − γ ( a + a ) ( λ − a x − a y − a γ ) (cid:19) . which is a Gaussian function. The case X = v , z = 0 (hence e γz = 1) was alreadyconsidered in [23].6. The contraction of SU (1 , to the Heisenberg group In this section we first recall some generalities about the holomorphic discrete seriesof SU (1 ,
1) in the context of the Berezin quantization and its contraction to the non-degenerated unitary irreducible representations of the Heisenberg group which was intro-duced in [8]. We closely follow the exposition of [8], see also [12].Let SU (1 ,
1) denote the group of all matrices g ( a, b ) := (cid:18) a b ¯ b ¯ a (cid:19) BENJAMIN CAHEN where a, b ∈ C satisfy | a | − | b | = 1.Note that SU (1 ,
1) naturally acts on the open unit disk D = ( | z | <
1) by fractionaltransforms g ( a, b ) · z := az + b ¯ bz + ¯ a . For each z ∈ D we denote g z := g ((1 − | z | ) − / , (1 − | z | ) − / z ). Then we have g z · z for each z ∈ D , that is, the map z → g z is a section for the action of SU (1 ,
1) on D .The Lie algebra su (1 ,
1) of SU (1 ,
1) has basis u = 12 (cid:18) − ii (cid:19) ; u = 12 (cid:18) (cid:19) ; u = 12 (cid:18) − i i (cid:19) . We introduce now the holomorphic discrete series ( π n ) of SU (1 , n >
2. Let H n be the Hilbert space of all holomorphic functions f : D → C such that k f k n = Z D | f ( z ) | dµ n ( z ) < ∞ where dµ n ( z ) := n − π (1 − z ¯ z ) n − dx dy , dx dy denoting as usual the Lebesgue measure on C ≃ R .An orthonormal basis of H n is then given by the family f np ( z ) = (cid:0) n + p − p (cid:1) / z p for p ∈ N and the coherent states are e nz ( w ) = (1 − w ¯ z ) − n for z, w ∈ D .Let π n be the representation of SU (1 ,
1) defined on H n by( π n ( g ( a, b )) f )( z ) = ( a − ¯ bz ) − n f ( g ( a, b ) − · z ) . Then the family ( π n ) is the holomorphic discrete series of SU (1 , dπ n is given by ( dπ n ( u ) f )( z ) = n izf ( z ) + 12 i ( z + 1) f ′ ( z )( dπ n ( u ) f )( z ) = n zf ( z ) + 12 ( z − f ′ ( z )( dπ n ( u ) f )( z ) = n if ( z ) + izf ′ ( z ) . For each operator A on H n we denote by S n ( A ) the Berezin symbol of A . Then wehave(6.1) S n ( π n ( g ( a, b )))( z ) = ( a − ¯ az ¯ z − ¯ bz + b ¯ z ) − n (1 − z ¯ z ) n , see [8].Now we introduce the contraction of SU (1 ,
1) to the Heisenberg group at the Lie algebralevel. Let r > C r : h → su (1 ,
1) be the linear map defined by C r ( v ) = ru , C r ( v ) = ru , C r ( v ) = r u . Then we have for each
X, Y ∈ h lim r → C − r ([ C r ( X ) , C r ( Y )] su (1 , ) = [ X, Y ] h . and we say that the family ( C r ) r> is a contraction of su (1 ,
1) to h , see [21], [24], [14].The corresponding group contraction c r : H → SU (1 ,
1) is then given by c r (exp H X ) = exp SU (1 , ( C r ( X )) EREZIN SYMBOLS... 9 and satisfies the following property: for each x, y ∈ H there exists r > r > r < r , the expression c − r ( c r ( x ) c r ( y ) − ) is well-defined and we havelim r → c − r ( c r ( x ) c r ( y ) − ) = xy − , see [8].The following proposition was proved in [8]. For each n > r ( n ) > nr ( n ) = 2 γ . A geometric interpretation of this quite mysterious condition (in terms ofcoadjoint orbits associated with representations) can be found in [8], see also [12]. Proposition 6.1.
Let h = [ a , a , a ] ∈ H and let g n := c r ( n ) ( h ) = exp SU (1 , ( C r ( n ) ( a v + a v + a v )) . Then (1)
For each z ∈ C , we have lim n → + ∞ S n ( π n ( g n )) (cid:18) z √ γn (cid:19) = s γ ( π γ ( h ))( z ) . (2) For each p, q ∈ N , we have lim n → + ∞ h π n ( g n ) f np , f nq i n = h π γ ( h ) f γp , f γq i γ . (3) For each n > , let B n : H γ → H n be the unitary operator defined by B n ( f γp ) = f np for each p ∈ N . For each f ∈ H γ , we have lim n → + ∞ k ( B − n π n ( g n ) B n ) f − π γ ( h ) f k γ = 0 . Proof.
Here we just detail the proof of (1) since it is of some interest for our purpose.Let a ∈ R , β ∈ C and let R ∈ R ∪ i R such that R = − a + | β | . Then we have(6.2) exp (cid:18) ai β ¯ β − ai (cid:19) = g (cid:18) cosh R + sinh RR ai, sinh
RR β (cid:19) . From this we deduce that if we denote R ( n ) = r ( n )( a + a − r ( n ) a ) / then we have g n = g ( α n , β n ) with α n = cosh R ( n ) − ir ( n ) a sinh R ( n )2 R ( n ) ; β n = r ( n )( a − ia ) sinh R ( n )2 R ( n ) . Also, by Eq. 6.1, we have S n ( π n ( g n )) (cid:18) z √ γn (cid:19) = (cid:18) α n − ¯ α n | z | γn − ¯ β n z √ γn + β n ¯ z √ γn (cid:19) − n (cid:18) − | z | γn (cid:19) n . Then log (cid:18) S n ( π n ( g n )) (cid:18) z √ γn (cid:19)(cid:19) ∼ − n ( α n −
1) + n ¯ β n z √ γn − nβ n ¯ z √ γn . Consequently, since n ( α n −
1) = n (cid:18) cosh R ( n ) − − ir ( n ) a sinh R ( n )2 R ( n ) (cid:19) has limit γ ( a + a ) − iγa and nβ n / √ γn has limit a − ia when n → ∞ , we see thatlim n → + ∞ S n ( π n ( g n )) (cid:18) z √ γn (cid:19) = exp (cid:0) − γ ( a + a ) + iγa + ( a + ia ) z − ( a − ia )¯ z (cid:1) . The result hence follows by Eq. 5.1. (cid:3)
For each z ∈ C let f γz := k e γz k − γ e γz ∈ H γ and for each z ∈ D let f nz := k e nz k − n e nz ∈ H n .Let X = a v + a v + a v ∈ h with ( a , a ) = (0 , ϕ f γz the density of the spectral measure µ f γz correspond-ing to − idπ γ ( X ) and by ϕ f nz the density of the spectral measure µ f nz corresponding to − idπ n ( C r ( n ) ( X )), see Section 4 and Section 5. Then we have the following contractionresult for these densities. Proposition 6.2.
For each λ ∈ R and each z ∈ C , we have lim n → + ∞ ϕ f nz/ √ γn = ϕ f γz .Proof. By Eq. 4.2, we have ϕ f nz/ √ γn ( λ ) = 12 π Z R S n ( π n (exp( tC r ( n ) ( X ))( z/ p γn ) e − itλ dt, so, taking into account (1) of Proposition 6.1, we see that in order to get the resultwe have just to verify that the dominated convergence theorem can be applied. Tothis end, we first note that z/ √ γn ∈ D for n large enough. Then the expression S n ( π n (exp( tC r ( n ) ( X ))( z/ √ γn ) is well-defined for n large enough and we have S n ( π n (exp( tC r ( n ) ( X ))))( z/ p γn ) = S n ( π n ( g z/ √ γn ) − π n (exp( tC r ( n ) ( X ))) π n ( g z/ √ γn )))(0)= S n ( π n (exp( t Ad( g z/ √ γn ) − C r ( n ) ( X ))))(0)by (2) of Proposition 2.2.Now, let us denote by ( b nij ) ≤ i,j ≤ the matrix of Ad( g z/ √ γn ) − : su (1 , → su (1 ,
1) inthe basis ( u i ) ≤ i ≤ and introduce c n := a b n + a b n + r ( n ) a b n c n := a b n + a b n + r ( n ) a b n c n := a b n + a b n + r ( n ) a b n . Then we haveAd( g z/ √ γn ) − ( C r ( n ) ( X )) = Ad( g z/ √ γn ) − ( r ( n ) a u + r ( n ) a u + r ( n ) a u )= r ( n )( c n u + c n u + c n u ) . Note that the sequences ( c n ), ( c n ) and ( c n ) are convergent; we denote by c , c and c the corresponding limits. Since we have( c n ) + ( c n ) − ( c n ) = a + a − r ( n ) a , we get c + c − c = a + a > . We denote d n := (( c n ) + ( c n ) − ( c n ) ) / for n large enough. Then, by Eq. 6.1 andEq. 6.2, we obtain S n ( π n (exp( t Ad( g z/ √ γn ) − C r ( n ) ( X ))))(0) = (cid:0) cosh( r ( n ) d n t ) − ic n d − n sinh( r ( n ) d n t ) (cid:1) − n . EREZIN SYMBOLS... 11
Hence | S n ( π n (exp( t Ad( g z/ √ γn ) − C r ( n ) ( X ))))(0) | ≤ (cosh( r ( n ) d n t )) − n ≤ (cid:0) ( r ( n ) d n t ) (cid:1) − n ≤ (cid:16) γ n d n t (cid:17) − n . Finally, since there exists
C > d n ≥ C for each n large enough, we concludethat there exists C ′ > | S n ( π n (exp( t Ad( g z/ √ γn ) − C r ( n ) ( X ))))(0) | ≤ (cid:18) C ′ t n (cid:19) − n ≤ (1 + C ′ t ) − for each n sufficiently large. The result follows. (cid:3) The case X = v , z = 0 considered in [23] corresponds to the limitlim n → + ∞ π Z R e − itλ (cid:0) cosh r ( n ) t (cid:1) − n dt = 12 π Z R e − itλ e − γt / dt = 1 √ πγ e − λ /γ . The contraction of SU (2) to the Heisenberg group The contraction of the unitary irreductible representations of SU (2) to the unitaryirreducible representations of H was investigated in [27] and [7]. Some applications toFourier multipliers can be found in [15]. This contraction is quite analogous to that of theprevious section but a little bit more complicated since the unitary irreductible represen-tations of SU (2) are finite-dimensional while the (non-degenerated) unitary irreduciblerepresentations of H are not.Let us denote the elements of SU (2) as g ( a, b ) := (cid:18) a b − ¯ b ¯ a (cid:19) , a, b ∈ C , | a | + | b | = 1 . The Lie algebra su (2) of SU (2) has basis u ′ = 12 (cid:18) ii (cid:19) ; u ′ = 12 (cid:18) −
11 0 (cid:19) ; u ′ = 12 (cid:18) i − i (cid:19) . For each integer m >
0, we denote by F m the space of all complex polynomials of degree ≤ m endowed with the Hilbertian norm k f k m = Z C | f ( z ) | m +1 π (1 + z ¯ z ) − m − dx dy. Then F m is a Hilbert space of dimension m +1 (its Hilbert product is denoted by h· , ·i m )and an orthonormal basis of F m is the family f mp ( z ) = (cid:0) mp (cid:1) / z p for p = 0 , , . . . , m .For each w, z ∈ C , let e mz ( w ) = (1 + w ¯ z ) m . Then we have the reproducing property h f, e mz i m = f ( z ) for each f ∈ F m and z ∈ C .We define the representation ρ n of SU (2) on F m by( ρ m ( g ( a, b )) f )( z ) = ( a + ¯ bz ) m f (cid:18) ¯ az − b ¯ bz + a (cid:19) . Then ρ m is a unitary irreductible representation of SU (2) whose differential is given by ( dρ m ( u ′ ) f )( z ) = − m izf ( z ) + 12 i ( z − f ′ ( z )( dρ m ( u ′ ) f )( z ) = − m zf ( z ) + 12 ( z + 1) f ′ ( z )( dρ m ( u ′ ) f )( z ) = m if ( z ) − izf ′ ( z ) . For each operator A on F m , we denote by S m ( A ) the Berezin symbol of A . Then wecan verify that, see [7],(7.1) S m ( ρ m ( g ( a, b )))( z ) = ( a + ¯ az ¯ z + ¯ bz − b ¯ z ) m (1 + z ¯ z ) − m . For each r >
0, let C ′ r : h → su (2) be the linear map defined by C ′ r ( v ) = ru ′ , C ′ r ( v ) = ru ′ , C ′ r ( v ) = r u ′ . Then we can verify that ( C ′ r ) is a contraction of su (2) to h the corresponding contractionof SU (2) to H being given by c ′ r (exp H X ) = exp SU (2) ( C ′ r ( X ))for each X ∈ h , see [27].For each integer m >
0, let r ( m ) > mr ( m ) = 2 γ . Also, let B ′ m : F m →F m ⊂ H γ be the unitary operator defined by B ′ m ( f mp ) = f γp . Then we have the followingresult which is analogous to Proposition 6.1 Proposition 7.1. [7] Let h = [ a , a , a ] ∈ H such that ( a , a ) = (0 ,
0) and for each m >
0, let g m := c ′ r ( n ) ( h ) = exp SU (2) ( C ′ r ( m ) ( a v + a v + a v )) . Then(1) For each z ∈ C , we havelim m → + ∞ S m ( ρ m ( g m )) (cid:18) z √ γm (cid:19) = s γ ( π γ ( h ))( z ) . (2) For each p, q ∈ N , we havelim n → + ∞ h ρ m ( g m ) f np , f nq i m = h π γ ( h ) f γp , f γq i γ . (3) For each polynomial f ∈ H γ , we havelim m → + ∞ k ( B ′ m ρ m ( g m ) B ′− m ) f − π γ ( h ) f k γ = 0 . (This makes sense since we have then f ∈ F m for each m large enough).For each z ∈ C let f mz := k e mz k − m e mz ∈ F m and for each X = a v + a v + a v ∈ h with( a , a ) = (0 , µ f mz the spectral measure d ( h E λ f mz , f mz i m ) correspondingto − idρ m ( C r ( m ) ( X )) (see Section 4). Proposition 7.2.
The sequence µ f mz/ √ γm converges to µ f γz in S ′ ( R ) . EREZIN SYMBOLS... 13
Proof.
Let z ∈ C . To simplify the notation, let F ( t ) := S γ ( π γ (exp( tX )))( z ) and for each m ∈ N , let F m ( t ) := S m ( ρ m (exp( tC ′ r ( m ) ( X ))))( z/ √ γm ). By (1) of Proposition 7.1, wehave lim m → + ∞ F m ( t ) = F ( t ) for each t ∈ R .Moreover, by the Cauchy-Schwarz inequality, we have | F m ( t ) | = |h ρ m (exp( tC ′ r ( m ) ( X ))) f mz , f mz i m |≤ k ρ m (exp( tC ′ r ( m ) ( X ))) f mz k m k f mz k m ≤ ρ m is unitary.This implies that ( F m ) converges to F in S ′ ( R ). Indeed, for each φ ∈ S ( R ), thedominated convergence theorem shows that lim m → + ∞ R R F m φ = R R F φ since | F m φ | ≤| φ | ∈ L ( R ) for each m ≥ F − : S ′ ( R ) → S ′ ( R ) is continuous, the result follows from (1) of Propo-sition 4.1. (cid:3) We conclude with the following example. We consider the case where X = v , z = 0hence f mz = 1.First, let us compute the spectral measure µ := d ( h E λ , i m ) corresponding to theoperator A := − idρ m ( u ′ ) on F m .By Proposition 4.1, we have just to compute the inverse Fourier transform of the func-tion S m ( ρ m (exp( − tu ′ )))(0) = (cid:0) cos (cid:0) t (cid:1)(cid:1) m = (cid:18) (cid:19) m m X k =0 (cid:18) mk (cid:19) e it ( m − k ) . But we can easily verify that for each λ ∈ R , we have F ( δ λ ) = e − iλt . Consequently, weget µ = (cid:18) (cid:19) m m X k =0 (cid:18) mk (cid:19) δ k − m . In fact, we can also find this result directly but this is a little bit more longer. This canbe done as follows. First, we remark that a basis of F m consisting of eigenvectors of A is F mk = ( z − k ( z + 1) m − k , k = 0 , , . . . m, the eigenvalue associated with F mk being λ k := k − m .Then, denoting by P k the orthogonal projection operator of F m on the line generatedby F mk , we have µ = m X k =0 h P k (1) , i m δ λ k . It remains to compute h P k (1) , i m . One has P k (1) = k F mk k − m h , F mk i m F mk hence h P k (1) , i m = k F mk k − m |h , F mk i m | . Now, on the one hand, we have h , F mk i m = ( − k since h , z q i m = 0 for each q > m = m X k =0 (cid:18) mk (cid:19) ( − k F mk hence h , F mk i m = (cid:18) (cid:19) m (cid:18) mk (cid:19) ( − k k F mk k m which gives k F mk k m = 2 m (cid:0) mk (cid:1) − and finally h P k (1) , i m = (cid:0) (cid:1) m (cid:0) mk (cid:1) as required.Similarly, we can verify that the spectral measure corresponding to the operator − idρ m ( C r ( m ) ( v )) = − ir ( m ) dρ m ( u ′ )is µ m := (cid:18) (cid:19) m m X k =0 (cid:18) mk (cid:19) δ r ( m )( k − m . Consequently, applying Proposition 7.2 and taking the results of Section 5 into account,we obtain the following result.
Proposition 7.3.
For each φ ∈ S ( R ) , we have lim m → + ∞ (cid:18) (cid:19) m m X k =0 (cid:18) mk (cid:19) φ r γm ( k − m ) ! = Z R √ πγ e − λ /γ φ ( λ ) dλ. References [1] D. Arnal,
The ∗ -exponential, Quantum theories and geometry (Les Treilles, 1987), 23-51, Math. Phys.Stud. 10, Kluwer Acad. Publ., Dordrecht, 1988.[2] D. Arnal, M. Cahen and S. Gutt,
Representations of compact Lie groups and quantization by defor-mation,
Acad. R. Belg. Bull. Cl. Sc. 3e s´erie LXXIV, 45 (1988) 123-141.[3] M. B. Bekka and P. de la Harpe,
Irreducibility of unitary group representations and reproducing kernelsHilbert spaces,
Expo. Math. 21, 2 (2003) 115-149.[4] F. A. Berezin,
Covariant and contravariant symbols of operators , Math. USSR Izv. 6, 5 (1972), 1117-1151.[5] F. A. Berezin,
Quantization , Math. USSR Izv. 8, 5 (1974), 1109-1165.[6] F. A. Berezin,
Quantization in complex symmetric domains,
Math. USSR Izv. 9, 2 (1975), 341-379.[7] B. Cahen,
Contraction de SU(2) vers le groupe de Heisenberg et calcul de Berezin , Beitr¨age AlgebraGeom. 44, 2 (2003), 581-203.[8] B. Cahen,
Contraction de SU (1 , vers le groupe de Heisenberg , Mathematical works, Part XV.Luxembourg: Universit´e du Luxembourg, S´eminaire de Math´ematique. (2004) 19-43 .[9] B. Cahen, Contractions of SU (1 , n ) and SU ( n + 1) via Berezin quantization , J. Anal. Math. 97 (2005)83-102.[10] B. Cahen, Berezin quantization on generalized flag manifolds , Math. Scand. 105 (2009), 66-84.[11] B. Cahen,
Contractions of semisimple Lie groups via Berezin quantization , Illinois J. Math. 53, 1(2009), 265-288.[12] B. Cahen,
Contractions of Discrete Series via Berezin quantization , J. Lie Theory 19 (2009), 291-310.[13] M. Cahen, S. Gutt and J. Rawnsley,
Quantization on K¨ahler manifolds I , Geometric interpretationof Berezin quantization, J. Geom. Phys. 7 (1990) 45-62.[14] A. H. Dooley, Contractions of Lie groups and applications to analysis, In Topics in Modern HarmonicAnalysis, Proc. Semin., Torino and Milano 1982, Vol. I , Ist. di Alta Mat, Rome, 1983, pp. 483-515.[15] A. H. Dooley and S. K. Gupta,
The Contraction of S p − to H p − , Monatsh. Math. 128 (1999),237-253.[16] A. H. Dooley and J. W. Rice, Contractions of rotation groups and their representations , Math. Proc.Camb. Phil. Soc. 94 (1983) 509-517.[17] A. H. Dooley and J. W. Rice,
On contractions of semisimple Lie groups , Trans. Am. Math. Soc. 289(1985), 185-202.[18] J. Faraut and A. Koranyi,
Function Spaces and Reproducing Kernels on Bounded Symmetric Do-mains , J. Funct. Anal. 88 (1990) 64-89.
EREZIN SYMBOLS... 15 [19] B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989.[20] C. Fronsdal,
Some ideas about quantization , Rep. Math. Phys. 15 (1979), 111-145.[21] E. In¨on¨u , E. P. Wigner, On the contraction of groups and their representations, Proc. Nat. Acad.Sci. USA 39 (1953) 510-524.[22] A. W. Knapp, Representation theory of semi simple groups. An overview based on examples, Prince-ton Math. Series t. 36 , 1986.[23] S. Luo,
Discrete Series of SU (1 , and Gaussian Deformation , Lett. Math. Phys. 42, 1 (1997), 1-10.[24] J. Mickelsson and J. Niederle, Contractions of Representations of de Sitter Groups , Commun. math.Phys. 27 (1972), 167-180.[25] C. C. Moore,
Ergodicity of flows on homogeneous spaces , Amer. J. Math. 88 (1966), 154-178.[26] K-H. Neeb, Holomorphy and Convexity in Lie Theory, de Gruyter Expositions in Mathematics, Vol.28, Walter de Gruyter, Berlin, New-York 2000.[27] F. Ricci,
A Contraction of SU (2) to the Heisenberg Group , Monatsh. Math. 101 (1986), 211-225.[28] S. C. Scull, Spectra in representations of semisimple Lie groups , Proc. Amer. Math. Soc. 41, 1 (1973),287-293.[29] S. C. Scull,
Positive operators and automorphism groups of bounded symmetric domains , Rep. Math.Phys. 10, 1 (1976), 1-7.[30] M. E. Taylor, Noncommutative Harmonic Analysis, Mathematical surveys and monographs no 22,Amer. Math. Soc. 1986.[31] N. J. Wildberger,
On the Fourier transform of a compact semisimple Lie group , J. Austral. Math.Soc. A 56 (1994) 64-116.
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