Schrödinger model and Stratonovich-Weyl correspondence for Heisenberg motion groups
aa r X i v : . [ m a t h . R T ] F e b SCHR ¨ODINGER MODEL AND STRATONOVICH-WEYLCORRESPONDENCE FOR HEISENBERG MOTION GROUPS
BENJAMIN CAHEN
Abstract.
We introduce a Schr¨odinger model for the unitary irreducible representa-tions of a Heisenberg motion group and we show that the usual Weyl quantization thenprovides a Stratonovich-Weyl correspondence. Introduction
There are different ways to extend the usual Weyl correspondence between functions on R n and operators on L ( R n ) to the general setting of a Lie group acting on a homogeneousspace [1], [34], [14], [31]. Here we are concerned with Stratonovich-Weyl correspondences.The notion of Stratonovich-Weyl correspondence was introduced in [51] and its systematicstudy began with the work of J.M. Gracia-Bond`ıa, J.C. V`arilly and their co-workers (see[33], [29], [26], [32] and also [12]). The following definition is taken from [32], see also [33]. Definition 1.1.
Let G be a Lie group and π be a unitary representation of G on a Hilbertspace H . Let M be a homogeneous G -space and let µ be a G -invariant measure on M .Then a Stratonovich-Weyl correspondence for the triple ( G, π, M ) is an isomorphism W from a vector space of operators on H to a vector space of functions on M satisfying thefollowing properties: (1) the function W ( A ∗ ) is the complex-conjugate of W ( A ) ; (2) Covariance: we have W ( π ( g ) A π ( g ) − )( x ) = W ( A )( g − · x ) ; (3) Traciality: we have Z M W ( A )( x ) W ( B )( x ) dµ ( x ) = Tr( AB ) . Stratonovich-Weyl correspondences were constructed for various Lie group represen-tations, see [26], [32]. In particular, in [20], Stratonovich-Weyl correspondences for theholomorphic representations of quasi-Hermitian Lie groups were obtained by taking theisometric part in the polar decomposition of the Berezin quantization map, see also [29],[16], [17], [3], [4] and [24].The basic example is the case when G is the (2 n + 1)-dimensional Heisenberg groupacting on R n ∼ = C n by translations. Each non-degenerate unitary irreducible representa-tion of G has then two classical realizations: the Schr¨odinger model on L ( R n ) and theBargmann-Fock model on the Fock space [30], an intertwining operator between these Mathematics Subject Classification.
Key words and phrases.
Stratonovich-Weyl correspondence; Berezin quantization; Berezin transform;Heisenberg motion group; reproducing kernel Hilbert space; coherent states; Schr¨odinger representation;Bargmann-Fock representation; Segal-Bargmann transform. realizations being the Segal-Bargmann transform [30], [27]. In this context, it is well-known that the usual Weyl correspondence provides a Stratonovich-Weyl correspondencefor the Schr¨odinger realization [6], [54], [49]. It is also known that this Stratonovich-Weylcorrespondence is connected by the Segal-Bargmann transform to the Stratonovich-Weylcorrespondence for the Bargmann-Fock realization which was obtained by polarizationof the Berezin quantization map [44], [43]. In [22], we obtained similar results for the(2 n + 2)-dimensional real diamond group. This group, also called oscillator group, is asemidirect product of the Heisenberg group by the real line.The aim of the present paper is to extend the preceding results to the Heisenberg motiongroups. An Heisenberg motion group is the semidirect product of the (2 n +1)-dimensionalHeisenberg group H n by a compact subgroup K of the unitary group U ( n ). Note thatHeisenberg motion groups play an important role in the theory of Gelfand pairs, since thestudy of a Gelfand pair of the form ( K , N ) where K is a compact Lie group acting byautomorphisms on a nilpotent Lie group N can be reduced to that of the form ( K , H n ),see [8], [9].More precisely, we introduce a Schr¨odinger realization for the unitary irreducible repre-sentations of a Heisenberg motion group and we prove that we obtain a Stratonovich-Weylcorrespondence by combining the usual Weyl correspondence and the unitary part of theBerezin calculus for K .Let us briefly describe our construction. First notice that each Heisenberg motiongroup is, in particular, a quasi-Hermitian Lie group and that we can obtain its unitaryirreducible representations as holomorphically induced representations on some general-ized Fock space by the general method of [46], Chapter XII. Then we can get Schr¨odingerrealizations for these representations by using, as in the case of the Heisenberg group, ageneralized Bargmann-Fock transform. Hence we obtain a Stratonovich-Weyl correspon-dence for such a Schr¨odinger realization by introducing a generalization of the usual Weylcorrespondence.Note that, in [45], a Schr¨odinger model and a generalized Segal-Bargmann transformfor the scalar highest weight representations of an Hermitian Lie group of tube type wereintroduced and studied. Let us also mentioned that B. Hall has obtained some generalizedSegal-Bargmann transforms in various situations by means of the heat kernel, see [36] andreferences therein. Then one can hope for futher generalizations of our results to quasi-Hermitian Lie groups.This paper is organized as follows. In Section 2, we review some well-known facts aboutthe Fock model and the Schr¨odinger model of the unitary irreducible representations of anHeisenberg group and about the corresponding Berezin calculus and Weyl correspondence.In Section 3, we introduce the Heisenberg motion groups and, in Section 4 and Section 5,we describe their unitary irreducible representations in the Fock model and the associatedBerezin calculus. We introduce the (generalized) Segal-Bargmann transform and theSchr¨odinger model in Section 6. In Section 7, we show that the usual Weyl correspondencealso gives a Stratonovich-Weyl correspondence for the Schr¨odinger model. Moreover,we compare it with the Stratonovich-Weyl correspondence for the Fock model which isdirectly obtained by polarization of the Berezin quantization map. CHR ¨ODINGER MODEL... 3 Heisenberg groups
In this section, we review some well-known results about the the Schr¨odinger modeland the Fock model of the unitary irreducible (non-degenerated) representations of theHeisenberg group. We follow the presentation of [22] in a large extend.Let G be the Heisenberg group of dimension 2 n + 1 and g be the Lie algebra of G .Let { X , . . . , X n , Y , . . . , Y n , ˜ Z } be a basis of g in which the only non trivial brackets are[ X k , Y k ] = ˜ Z , 1 ≤ k ≤ n and let { X ∗ , . . . , X ∗ n , Y ∗ , . . . , Y ∗ n , ˜ Z ∗ } be the corresponding dualbasis of g ∗ .For a = ( a , a , . . . , a n ) ∈ R n , b = ( b , b , . . . , b n ) ∈ R n and c ∈ R , we denote by[ a, b, c ] the element exp G ( P nk =1 a k X k + P nk =1 b k Y k + c ˜ Z ) of G . Similarly, for α =( α , α , . . . , α n ) ∈ R n , β = ( β , β , . . . , β n ) ∈ R n and γ ∈ R , we denote by ( α, β, γ )the element P nk =1 α k X ∗ k + P nk =1 β k Y ∗ k + γ ˜ Z ∗ of g ∗ . The coadjoint action of G is thengiven by Ad ∗ ([ a, b, c ]) ( α, β, γ ) = ( α + γβ, β − γα, γ ) . Now we fix a real number λ > O λ the orbit of the element λ ˜ Z ∗ of g ∗ under the coadjoint action of G (the case λ < G whose restriction to the center of G is the character[0 , , c ] → e iλc [7], [30]. Note that this representation is associated with the coadjointorbit O λ by the Kirillov-Kostant method of orbits [41], [42]. More precisely, if we choosethe real polarization at λ ˜ Z ∗ to be the space spanned by the elements Y k for 1 ≤ k ≤ n and ˜ Z then we obtain the Schr¨odinger representation σ realized on L ( R n ) as( σ ([ a, b, c ]) f )( x ) = e iλ ( c − bx + ab ) f ( x − a ) , see [30] for instance. Here we denote xy := P nk =1 x k y k for x = ( x , x , . . . , x n ) and y = ( y , y , . . . , y n ) in R n .The differential of σ is then given by dσ ( X k ) f ( x ) = − ∂ k f ( x ) , dσ ( Y k ) f ( x ) = − iλx k f ( x ) , dσ ( ˜ Z ) f ( x ) = iλf ( x )where k = 1 , , . . . , n .On the other hand, if we consider the complex polarization at λ ˜ Z ∗ to be the spacespanned by the elements X k + iY k for 1 ≤ k ≤ n and ˜ Z then the method of orbits leadsto the Bargmann-Fock representation π defined as follows [13].Let F be the Hilbert space of holomorphic functions F on C n such that k F k F := Z C n | F ( z ) | e −| z | / λ dµ λ ( z ) < + ∞ where dµ λ ( z ) := (2 πλ ) − n dx dy . Here z = x + iy with x and y in R n .Let us consider the action of G on C n defined by g · z := z + λ ( b − ia ) for g = [ a, b, c ] ∈ G and z ∈ C n . Then π is the representation of G on F given by π ( g ) F ( z ) = α ( g − , z ) F ( g − · z )where the map α is defined by α ( g, z ) := exp (cid:0) − icλ + (1 / b + ai )( − z + λ ( − b + ai )) (cid:1) for g = [ a, b, c ] ∈ G and z ∈ C n . BENJAMIN CAHEN
The differential of π is then given by dπ ( X k ) F ( z ) = 12 iz k F ( z ) + λi ∂F∂z k dπ ( Y k ) F ( z ) = 12 z k F ( z ) − λ ∂F∂z k dπ ( ˜ Z ) F ( z ) = iλF ( z ) . As in [35], Section 6 or [27], Section 1.3, we can verify by using the previous formulasfor dπ and dσ that the Segal-Bargmann transform B : L ( R n ) → F defined by B ( f )( z ) = ( λ/π ) n/ Z R n e (1 / λ ) z + ixz − ( λ/ x f ( x ) dx is a (unitary) intertwining operator between σ and π . The inverse Segal-Bargmanntransform B − = B ∗ is then given by B − ( F )( x ) = ( λ/π ) n/ Z C n e (1 / λ )¯ z − ix ¯ z − ( λ/ x F ( z ) e −| z | / λ dµ λ ( z ) . For z ∈ C n , consider the coherent state e z ( w ) = exp(¯ zw/ λ ). Then we have thereproducing property F ( z ) = h F, e z i F for each F ∈ F where h· , ·i F denotes the scalarproduct on F .Now, we introduce the Berezin quantization map and we review some of its properties.Let C be the space of all operators (not necessarily bounded) A on F whose domaincontains e z for each z ∈ C n . Then the Berezin symbol of A ∈ C is the function S ( A )defined on C n by S ( A )( z ) := h A e z , e z i F h e z , e z i F . We have the following result, see for instance [22].
Proposition 2.1. (1)
Each A ∈ C is determined by S ( A ) ; (2) For each A ∈ C and each z ∈ C n , we have S ( A ∗ )( z ) = S ( A )( z ) ; (3) For each z ∈ C n , we have S ( I F )( z ) = 1 . Here I F denotes the identity operatorof F ; (4) For each A ∈ C , g ∈ G and z ∈ C n , we have π ( g ) − Aπ ( g ) ∈ C and S ( A )( g · z ) = S ( π ( g ) − Aπ ( g ))( z );(5) The map S is a bounded operator from L ( F ) (endowed with the Hilbert-Schmidtnorm) to L ( C n , µ λ ) which is one-to-one and has dense range.Proof. For (1) and (2), see [10] and [25]. Note that (4) follows from the following property:For each g ∈ G and each z ∈ C n , we have π ( g ) e z = α ( g, z ) e g · z , see [20]. Finally, (5) is aparticular case of [52], Proposition 1.19. (cid:3) Recall that the Berezin transform is then the operator B on L ( C n , µ λ ) defined by B = S ( S ) ∗ . Thus we have the integral formula B ( F )( z ) = Z C n F ( w ) e | z − w | / λ dµ λ ( w ) , see [10], [11], [52], [48] for instance. Recall also that we have B = exp( λ ∆ /
2) where∆ = 4 P nk =1 ∂ /∂z k ∂ ¯ z k , see [52], [43]. CHR ¨ODINGER MODEL... 5
Note that Berezin transforms have been studied, in the general setting, by many au-thors, see in particular [52], [47], [28], [48] and [56].Note also that S allows us to connect π to O λ as shown by the following proposition.Here we denote by g c the complexification of g . Proposition 2.2. [22]
Let Φ λ be the map defined by Φ λ ( z ) := n X k =1 (Re z k X ∗ k + Im z k Y ∗ k ) + λ ˜ Z ∗ . Then (1)
For each X ∈ g c and each z ∈ C n , we have S ( dπ ( X ))( z ) = i h Φ λ ( z ) , X i . (2) For each g ∈ G and each z ∈ C n , we have Φ λ ( g · z ) = Ad ∗ ( g ) Φ λ ( z ) . (3) The map Φ λ is a diffeomorphism from C n onto O λ . Now we aim to transfer S to operators on L ( R n ). To this goal, we define S ( A ) := S ( B AB − ) for A operator on L ( R n ). Of course, the properties of S give rise to similarproperties of S . In particular, S is a bounded operator from L ( L ( R n )) to L ( C n , µ λ )and S is G -covariant with respect to σ .Moreover, denoting by I B the (unitary) map from L ( L ( R n )) onto L ( F ) defined by I B ( A ) = B AB − , we have S = S I B then S ( S ) ∗ = ( S I B )( S I B ) ∗ = S I B I ∗ B ( S ) ∗ = S ( S ) ∗ = B . This shows that the Berezin transform corresponding to S is the same as the Berezintransform corresponding to S . Then we can write the polar decompositions of S and S as S = ( B ) / U and S = ( B ) / U where the maps U : L ( F ) → L ( C n , µ λ ) and U : L ( L ( R n )) → L ( C n , µ λ ) are unitary.Moreover, as in the proof of [17], Proposition 3.1, we can verify that U is a Stratonovich-Weyl correspondence for ( G , π , C n ) and that U is a Stratonovich-Weyl correspondencefor ( G , σ , C n ). Note that G -covariance of U and U immediately follows from G -covariance of S and S . Note also that we have U = U I B .Now, we show how to use the usual Weyl correspondence in order to get anotherStratonovich-Weyl correspondence for σ . The Weyl correspondence on R n is defined asfollows. For each f in the Schwartz space S ( R n ), let W ( f ) be the operator on L ( R n )defined by W ( f ) φ ( p ) = (2 π ) − n Z R n e isq f ( p + (1 / s, q ) φ ( p + s ) ds dq. The Weyl calculus can be extended to much larger classes of symbols (see for instance[38]). In particular, if f ( p, q ) = u ( p ) q α where u ∈ C ∞ ( R n ) then we have, see [53], W ( f ) φ ( p ) = (cid:18) i ∂∂s (cid:19) α ( u ( p + (1 / s ) φ ( p + s )) (cid:12)(cid:12)(cid:12) s =0 . From this, we can deduce the following proposition. Consider the action of G on R n given by g · ( p, q ) := ( p + a, q + λb ) where g = [ a, b, c ]. BENJAMIN CAHEN
Proposition 2.3. [22]
Let Ψ λ be the map defined by Ψ λ ( p, q ) := n X k =1 ( q k X ∗ k − λp k Y ∗ k ) + λ ˜ Z ∗ . Then (1)
For each X ∈ g c and each ( p, q ) ∈ R n , we have W − ( dσ ( X ))( p, q ) = i h Ψ λ ( p, q ) , X i . (2) For each g ∈ G and each ( p, q ) ∈ R n , we have Ψ λ ( g · ( p, q )) = Ad ∗ ( g ) Ψ λ ( p, q ) . (3) The map Ψ λ is a diffeomorphism from R n onto O λ . (4) For each ( p, q ) ∈ R n , we have Φ λ ( q − λpi ) = Ψ λ ( p, q ) . Assume that R n is equipped with the G -invariant measure ˜ µ := (2 π ) − n dpdq . Thenone has the following result. Proposition 2.4. [30] , [22] The map W − is a Stratonovich-Weyl correspondence for ( G , σ , R n ) . The following proposition asserts that if we identify R n with C n by the map j : ( p, q ) → q − λpi then the unitary part in the polar decomposition of S coincides with the inverseof the Weyl transform, see [43] and [48]. Proposition 2.5.
Let J be the map from L ( C n , µ λ ) onto L ( R n ) defined by J ( F ) = F ◦ j .Then we have U = ( W J ) − . Finally, note that we can obtain Stratonovich-Weyl correspondences for ( G , σ , O λ )and ( G , π , O λ ) by transferring W − and U by using Φ λ and Ψ λ . More precisely, let ν λ be the G -invariant measure on O λ defined by ν λ := (Φ − λ ) ∗ ( µ λ ) = (Ψ − λ ) ∗ (˜ µ ). Then themaps τ Φ λ : F → F ◦ Φ − λ from L ( C n , µ λ ) onto L ( O λ , ν λ ) and τ Ψ λ : F → F ◦ Ψ − λ from L ( R n ) onto L ( O λ , ν λ ) are unitary and we have τ Φ λ = τ Ψ λ J . Hence we can assert thefollowing proposition. Proposition 2.6.
The map W := τ Ψ λ W − is a Stratonovich-Weyl correspondence for ( G , σ , O λ ) , the map W := τ Φ λ U is a Stratonovich-Weyl correspondence for ( G , π , O λ ) and we have W = W I B . Generalities on Heisenberg motion groups
In order to introduce the Heisenberg motion groups, it is convenient to write the ele-ments of the Heisenberg group G and its multiplication law as follows. For each z ∈ C n , c ∈ R , we denote here by ( z, ¯ z, c ) the element G which is denoted by [Re z, Im z, c ] inSection 2. Moreover, for each z, w ∈ C n , we denote zw := P nk =1 z k w k and we considerthe symplectic form ω on C n defined by ω (( z, w ) , ( z ′ , w ′ )) = i zw ′ − z ′ w ) . for z, w, z ′ , w ′ ∈ C n . Then the multiplication of G is given by(3.1) (( z, ¯ z ) , c ) · (( z ′ , ¯ z ′ ) , c ′ ) = (( z + z ′ , ¯ z + ¯ z ′ ) , c + c ′ + ω (( z, ¯ z ) , ( z ′ , ¯ z ′ ))) , the complexification G c of G is G c = { (( z, w ) , c ) : z, w ∈ C n , c ∈ C } and the mul-tiplication of G c is obtained by replacing ( z, ¯ z ) by ( z, w ) and ( z ′ , ¯ z ′ ) by ( z ′ , w ′ ) in Eq.3.1. CHR ¨ODINGER MODEL... 7
Now, let K be a closed subgroup of U ( n ). Then K acts on G by k · (( z, ¯ z ) , c ) =(( kz, ¯ kz ) , c ) and we can form the semidirect product G := G ⋊ K which is called aHeisenberg motion group. The elements of G can be written as (( z, ¯ z ) , c, k ) where z ∈ C n , c ∈ R and k ∈ K . The multiplication of G is then given by(( z, ¯ z ) , c, k ) · (( z ′ , ¯ z ′ ) , c ′ , k ′ ) = (( z, ¯ z ) + ( kz ′ , ¯ kz ′ ) , c + c ′ + ω (( z, ¯ z ) , ( kz ′ , ¯ kz ′ )) , kk ′ ) . We denote by K c the complexification of K and we consider the action of K c on C n × C n given by k · ( z, w ) = ( kz, ( k t ) − w ) (here, the subscript t denotes transposition). The group G c is then the semidirect product G c = G c ⋊ K c . The elements of G c can be written as(( z, w ) , c, k ) where z, w ∈ C n , c ∈ C and k ∈ K c and the multiplication law of G c is givenby(( z, w ) , c, k ) · (( z ′ , w ′ ) , c ′ , k ′ ) = (( z, w ) + k · ( z ′ , w ′ ) , c + c ′ + ω (( z, w ) , k · ( z ′ , w ′ )) , kk ′ ) . We denote by k , k c , g and g c the Lie algebras of K , K c , G and G c . The derived actionof k c on C n × C n is then A · ( z, w ) := ( Az, − A t w ) and the Lie brackets of g c are given by[(( z, w ) , c, A ) , (( z ′ , w ′ ) , c ′ , A ′ )] = ( A · ( z ′ , w ′ ) − A ′ · ( z, w ) , ω (( z, w ) , ( z ′ , w ′ )) , [ A, A ′ ]) . Let ˜ K be the subgroup of G defined by ˜ K := { ((0 , , c, k ) : c ∈ R , k ∈ K } . Also,let h be a Cartan subalgebra of k . Then the Lie algebra ˜ k of ˜ K is a maximal compactlyembedded subalgebra of g and the subalgebra h of g consisting of all elements of the form((0 , , c, A ) where c ∈ R and A ∈ h is a compactly embedded Cartan subalgebra of g [46], p. 250.Following [46], Chapter XII.1, we set p + = { (( z, , ,
0) : z ∈ C n } and p − = { ((0 , w ) , ,
0) : w ∈ C n } and we denote by P + and P − the corresponding analytic sub-groups of G c , that is, P + = { (( z, , , I n ) : z ∈ C n } and P − = { ((0 , w ) , , I n ) : w ∈ C n } .Note that G is a group of the Harish-Chandra type [46], p. 507 (see also [50] and [37],Chapter VIII), that is, the following properties are satisfied:(1) g c = p + ⊕ ˜ k c ⊕ p − is a direct sum of vector spaces, ( p + ) ∗ = p − and [˜ k c , p ± ] ⊂ p ± ;(2) The multiplication map P + ˜ K c P − → G c , ( z, k, y ) → zky is a biholomorphic dif-feomorphism onto its open image;(3) G ⊂ P + ˜ K c P − and G ∩ ˜ K c P − = ˜ K .We denote by p p + , p ˜ k c and p p − the projections of g c onto p + , ˜ k c and p − associated withthe above direct decomposition.We can easily verify that each g = (( z , w ) , c , k ) ∈ G c has a P + ˜ K c P − -decompositiongiven by g = (( z , , , I n ) · ((0 , , c, k ) · ((0 , w ) , , I n )where c = c − i z w . We denote by ζ : P + ˜ K c P − → P + , κ : P + ˜ K c P − → K c and η : P + ˜ K c P − → P − the projections onto P + -, ˜ K c - and P − -components.We can introduce an action (defined almost everywhere) of G on p + as follows. For Z ∈ p + and g ∈ G c , we define g · Z ∈ p + by g · Z := log ζ ( g exp Z ). From the aboveformula for the P + ˜ K c P − -decomposition, we deduce that if g = (( z , w ) , c , k ) ∈ G and Z = (( z, , , ∈ p + then we have g · Z = log ζ ( g exp Z ) = (( z + kz, , , D := G · p + ≃ C n here.A useful section Z → g Z for the action of G on D can be obtained by using Proposition4.5 of [21]. Here we get g Z = (( z, ¯ z ) , , I n ) for each Z = (( z, , , z ∈ C n . BENJAMIN CAHEN
Now we compute the adjoint and coadjoint actions of G c . Let g = ( v , c , k ) ∈ G c where v ∈ C n , c ∈ C , k ∈ K c and X = ( w, c, A ) ∈ g c where w ∈ C n , c ∈ C and A ∈ k c . We can easily verify thatAd( g ) X = ddt ( g exp( tX ) g − ) | t =0 = (cid:0) k w − (Ad( k ) A ) · v , c + ω ( v , k w ) − ω ( v , (Ad( k ) A ) · v ) , Ad( k ) A (cid:1) . Now, let us denote by ξ = ( u, d, φ ), where u ∈ C n , d ∈ C and φ ∈ ( k c ) ∗ , the element of( g c ) ∗ defined by h ξ, ( w, c, A ) i = ω ( u, w ) + dc + h φ, A i . Also, for u, v ∈ C n , we denote by v × u the element of ( k c ) ∗ defined by h v × u, A i := ω ( u, A · v ) for A ∈ k c . Then, from the above formula for the adjoint action, we deducethat for each ξ = ( u, d, φ ) ∈ ( g c ) ∗ and g = ( v , c , k ) ∈ G c we haveAd ∗ ( g ) ξ = (cid:0) k u − dv , d, Ad ∗ ( k ) φ + v × ( k u − d v ) (cid:1) By restriction, we also get the analogous formula for the coadjoint action of G . Fromthis, we see that if a coadjoint orbit of G contains a point ( u, d, φ ) with d = 0 then it alsocontains a point of the form (0 , d, φ ). Such an orbit is called generic .4. Fock model for Heisenberg motion groups
In this section, we introduce the Fock model of the unitary irreducible representationsof G by using the general method of [46], Chapter XII that we describe here briefly.Let ρ be a unitary irreducible representation of K on a (finite-dimensional) Hilbert space V and λ ∈ R . Let ˜ ρ be the representation of ˜ K on V defined by ˜ ρ ((0 , , c, k ) = e iλc ρ ( k )for each c ∈ R and k ∈ K .For each Z, W ∈ D , let K ( Z, W ) := ˜ ρ ( κ (exp W ∗ exp Z )) − and for each g ∈ G , Z ∈ D ,let J ( g, Z ) := ˜ ρ ( κ ( g exp Z )), [46], Chapter XII.1. Consider the Hilbert space ˜ F of allholomorphic functions on D with values in V such that k f k F := Z D h K ( Z, Z ) − f ( Z ) , f ( Z ) i V dµ ( Z ) < + ∞ where µ denotes an invariant G -measure on D . Then the equation˜ π ( g ) f ( Z ) = J ( g − , Z ) − f ( g − · Z )defines a unitary representation of G on ˜ F . This representation can be also obtained byholomorphic induction from ˜ ρ , that is, it corresponds to the natural action of G on thesquare-integrable holomorphic sections of the Hilbert G -bundle G × ˜ ρ V over G/K ∼ = D [22]. Note also that ˜ π is irreducible since ˜ ρ is irreducible, [46], p. 515.Here we can easily compute K and J . For each Z = (( z, , , , W = (( w, , , ∈ D ,we have K ( Z, W ) = e λz ¯ w/ I V and for each g = (( z , ¯ z ) , c , k ) ∈ G and Z = (( z, , , , ∈D , we have J ( g, Z ) = exp (cid:0) iλc + λ ¯ z ( kz ) + λ | z | (cid:1) ρ ( k ) . Moreover, µ can be taken to be the G -invariant measure on D ≃ C n defined by dµ ( Z ) := λ n (2 π ) − n dx dy . Here Z = (( z, , ,
0) and z = x + iy with x and y in R n . From nowon, we identify Z = (( z, , , ∈ D with z ∈ C n and each function on D with thecorresponding function on C n . CHR ¨ODINGER MODEL... 9
Consequently, the Hilbert product on ˜ F is given by h f, g i ˜ F = Z C n h f ( z ) , g ( z ) i V e − λ | z | / (cid:18) λ π (cid:19) n dx dy and we get the following formula for ˜ π :(˜ π ( g ) f )( z ) = exp (cid:0) iλc + λ ¯ z z − λ | z | (cid:1) ρ ( k ) f ( k − ( z − z ))where g = (( z , ¯ z ) , c , k ) ∈ G and z ∈ C n .In fact, in order to use the results of Section 2, it is convenient to replace ˜ π by anequivalent representation π whose restriction to G is precisely π . To this aim, weconsider the Fock space F of all holomorphic functions f : C n → V such that k f k F := Z C n k f ( z ) k V e −| z | / λ dµ λ ( z ) < + ∞ . Let J : ˜ F → F be the unitary operator defined by J ( f )( z ) = f ( iλ − z ) and set π ( g ) := J ˜ π ( g ) J − for each g ∈ G . Then we have( π ( g ) f )( z ) = exp (cid:0) iλc + i ¯ z z − λ | z | (cid:1) ρ ( k ) f ( k − ( z + iλz ))where g = (( z , ¯ z ) , c , k ) ∈ G and z ∈ C n .We can easily compute the differential of π : Proposition 4.1.
Let X = (( a, ¯ a ) , c, A ) ∈ g . Then, for each f ∈ F and each z ∈ C n , wehave ( dπ ( X ) f )( z ) = dρ ( A ) f ( z ) + i ( λc + ¯ az ) f ( z ) + df z ( − Az + iλa ) . Clearly, one has F = F ⊗ V . For f ∈ F and v ∈ V , we denote by f ⊗ v the function z → f ( z ) v . Moreover, if A is an operator of F and A is an operator of V then wedenote by A ⊗ A the operator of F defined by ( A ⊗ A )( f ⊗ v ) = A f ⊗ A v .Let τ be the left-regular representation of K on F , that is, ( τ ( k ) f )( z ) = f ( k − z ).Then we have(4.1) π (( z , ¯ z ) , c , k ) = π (( z , ¯ z ) , c ) τ ( k ) ⊗ ρ ( k )for each z ∈ C n , c ∈ R and k ∈ K . Note that this is precisely Formula (3.18) in [8].5. Stratonovich-Weyl correspondence via Berezin quantization
In this section, we introduce the Berezin quantization map associated with π and thecorresponding Stratonovich-Weyl correspondence. We consider first the Berezin quanti-zation map associated with ρ [5], [15], [55].Let us fix a positive root system of k relative to h and denote by Λ ∈ ( h c ) ∗ the highestweight of ρ and by k c = n + ⊕ h c ⊕ n − the corresponding triangular decomposition of k c .Let ˜ ϕ be the element of ( k c ) ∗ defined by ˜ ϕ = − i Λ on h and by ˜ ϕ = 0 on n ± . Wedenote by ϕ the restriction of ˜ ϕ to k . Then the orbit o ( ϕ ) of ϕ under the coadjointaction of K is said to be associated with ρ [14], [55].Here we assume that ϕ is regular in the sense that the stabilizer of ϕ for the coadjointaction of K is precisely the connected subgroup H of K with Lie algebra h [15].Note that a complex structure on o ( ϕ ) is then defined by the diffeomorphism o ( ϕ ) ≃ K/H ≃ K c /H c N − where H is the connected subgroup of K with Lie algebra h and N − is the analytic subgroup of K c with Lie algebra n − . Without loss of generality, we can assume that V is a space of holomorphic sections of acomplex line bundle over o ( ϕ ) as in [15]. Let ϕ ∈ o ( ϕ ). For each ˆ ϕ = 0 in the fiber over ϕ , there exists a unique function e ˆ ϕ ∈ V (a coherent state) such that a ( ϕ ) = h a, e ˆ ϕ i V e ˆ ϕ for each a ∈ V .The Berezin calculus on o ( ϕ ) associates with each operator B on V the complex-valuedfunction s ( B ) on o ( ϕ ) defined by s ( B )( ϕ ) = h Be ˆ ϕ , e ˆ ϕ i V h e ˆ ϕ , e ˆ ϕ i V which is called the symbol of B . This definition makes sense since the right side of theequation does not depend on ˆ ϕ in the fiber over ϕ but only on ϕ . We denote by Sy ( o ( ϕ ))the space of all such symbols. Then we have the following proposition, see [25], [5] and[15]. Proposition 5.1. (1)
The map B → s ( B ) is injective. (2) For each operator B on V , we have s ( B ∗ ) = s ( B ) . (3) For each ϕ ∈ o ( ϕ ) , k ∈ K and B ∈ End( V ) , we have s ( B )(Ad ∗ ( k ) ϕ ) = s ( ρ ( k ) − Bρ ( k ))( ϕ ) . (4) For each A ∈ k and ϕ ∈ o ( ϕ ) , we have s ( dρ ( A ))( ϕ ) = i h ϕ, A i . In our papers [18], [19] and [23], we developped a general method for constructing aBerezin quantization map associated with a unitary representation of a quasi-HermitianLie group which is holomorphically induced from a unitary irreducible representation ofa maximal compactly embedded subgroup. This construction goes as follows.The evaluation maps K z : F →
V, f → f ( z ) are continuous [46], p. 539. The vectorcoherent states of F are the maps E z = K ∗ z : V → F defined by h f ( z ) , v i V = h f, E z v i F for f ∈ F and v ∈ V . Here we have that E z v = e z ⊗ v , that is, we have ( E z v )( w ) = e ¯ zw/ λ v .Let F s be the subspace of F generated by the functions e z ⊗ v for z ∈ C n and v ∈ V .Then F s is a dense subspace of F . Let C be the space consisting of all operators A on F such that the domain of A contains F s and the domain of A ∗ also contains F s . Then,following an idea of [40] and [2], we first introduce the pre-symbol S ( A ) of A ∈ C by S ( A )( z ) = ( E ∗ z E z ) − / E ∗ z AE z ( E ∗ z E z ) − / = e − z ¯ z/ λ E ∗ z AE z . The Berezin symbol S ( A ) of A is thus defined as the complex-valued function on C n × o ( ϕ ) given by S ( A )( z, ϕ ) = s ( S ( A )( z ))( ϕ ) . By applying Proposition 4.4 of [23] we can see that S has the following properties. Proposition 5.2. (1)
Each A ∈ C is determined by S ( A ) . (2) For each A ∈ C , we have S ( A ∗ ) = S ( A ) . (3) We have S ( I F ) = 1 . (4) For each A ∈ C , g = (( z , ¯ z ) , c, k ) ∈ G , z ∈ C n and ϕ ∈ o ( ϕ ) , we have S ( A )( g · z, ϕ ) = S ( π ( g ) − Aπ ( g ))( z, Ad ∗ ( k − ) ϕ ) . Moreover, we can decompose S according to the decomposition F = F ⊗ V . Let f be a complex-valued function on C n and f be a complex-valued function on o ( ϕ ). Thenwe denote by f ⊗ f the function on C n × o ( ϕ ) defined by ( f ⊗ f )( z, ϕ ) = f ( z ) f ( ϕ ). CHR ¨ODINGER MODEL... 11
Proposition 5.3.
Let A ∈ C and let A be an operator on V . Then A ⊗ A ∈ C andwe have S ( A ⊗ A ) = S ( A ) ⊗ s ( A ) . From this, we deduce the following result. We denote by ϕ the restriction to g of theextension of ˜ ϕ ∈ ( k c ) ∗ to g c which vanishes on p ± . We also denote by O ( ϕ ) the orbit of ϕ for the coadjoint action of G . Proposition 5.4. [23](1)
Let g = (( z , ¯ z ) , c , k ) ∈ G . For each z ∈ C n and ϕ ∈ o ( ϕ ) , we have S ( π ( g ))( z, ϕ ) = exp (cid:0) iλc + i ¯ z z − λ | z | − λ | z | + λ ¯ zk − ( z + iλz ) (cid:1) s ( ρ ( k ))( ϕ ) . (2) For each X = (( a, ¯ a ) , c, A ) ∈ g , z ∈ C n and ϕ ∈ o ( ϕ ) , we have S ( dπ ( X ))( z, ϕ ) = iλc + i az + a ¯ z ) − λ ¯ z ( Az ) + s ( dρ ( A ))( ϕ ) . (3) For each X = (( a, ¯ a ) , c, A ) ∈ g , z ∈ C n and ϕ ∈ o ( ϕ ) , we have S ( dπ ( X ))( z, ϕ ) = i h Φ( z, ϕ ) , X i where the map Φ : C n × o ( ϕ ) → g ∗ is defined by Φ( z, ϕ ) = (cid:0) i ( − z, ¯ z ) , λ, ϕ − λ ( z, ¯ z ) × ( z, ¯ z ) (cid:1) . Moreover Φ is a diffeomorphism from C n × o ( ϕ ) onto O ( ϕ ) . Consider now the Berezin transforms B := SS ∗ , B := S ( S ) ∗ , b := ss ∗ and thecorresponding maps U := B − / S , U := ( B ) − / S and w := b − / s . We fix a K -invariant measure ν on o ( ϕ ) and we endow C n × o ( ϕ ) with the measure µ λ ⊗ ν . Also,we consider the action of G on C n × o ( ϕ ) given by g · ( z, ϕ ) := ( g · z, Ad ∗ ( k ) ϕ )for g = (( z , ¯ z ) , c , k ) ∈ G . Then we have the following results. Proposition 5.5. [23]
The map U is a Stratonovich-Weyl correspondence for ( G, π, C n × o ( ϕ )) . Proposition 5.6. [23]
For each f ∈ L ( C n × o ( ϕ ) , µ λ ⊗ ν ) , we have B ( f )( z, ψ ) = Z C n × o ( ϕ ) k B ( z, w, ψ, ϕ ) f ( w, ϕ ) dµ λ ( w ) dν ( ϕ ) where k B ( z, w, ψ, ϕ ) := e −| z − w | / λ |h e ˆ ψ , e ˆ ϕ i V | h e ˆ ϕ , e ˆ ϕ i V h e ˆ ψ , e ˆ ψ i V . In particular, for each f ∈ L ( C n ) and f ∈ Sy ( o ( ϕ )) , we have B ( f ⊗ f ) = B ( f ) ⊗ b ( f ) . Moreover for each A operator on F and A operator on V , we have U ( A ⊗ A ) = U ( A ) ⊗ w ( A ) . Note that it is well-known that if ∆ := 4 P nk =1 ( ∂ z k ∂ ¯ z k ) is the Laplace operator then wehave B = exp( λ ∆ / U = exp( − λ ∆ / S . Hence, by applyingProposition 5.4 and Proposition 5.6, we obtain the following result. Proposition 5.7. [23]
For each X = (( a, ¯ a ) , c, A ) ∈ g , z ∈ C n and ϕ ∈ o ( ϕ ) , we have U ( dπ ( X ))( z, ϕ ) = icλ + w ( dρ ( A ))( ϕ ) + 12 Tr( A ) + i az + a ¯ z ) − λ ¯ z ( Az ) . Schr¨odinger model for Heisenberg motion groups
Here we introduce the Schr¨odinger representations of G by using a Segal-Bargmanntransform which is obtained by a slight modification of B . More precisely, let us definethe map B from L ( R n , V ) ∼ = L ( R n ) ⊗ V to F ∼ = F ⊗ V by B := B ⊗ I V or, equivalently,by the integral formula B ( f )( z ) = ( λ/π ) n/ Z R n e (1 / λ ) z + ixz − ( λ/ x f ( x ) dx for each f ∈ L ( R n , V ).Now, by analogy with the case of the Heisenberg group, we define the Schr¨odingerrepresentation σ of G on L ( R n , V ) by σ ( g ) := B − π ( g ) B . Similarly, recalling that τ is therepresentation of K on F given by ( τ ( k ) F )( z ) = F ( k − z ), we define the representation˜ τ of K on L ( R n , V ) by ˜ τ := B − τ ( k ) B . Then we have the following proposition. Proposition 6.1.
Let g ∈ G , k ∈ K and g = ( g , k ) ∈ G . Then we have σ ( g ) = σ ( g )˜ τ ( k ) ⊗ ρ ( k ) .Proof. Let f ∈ L ( R n ) and v ∈ V . Then by Eq. 4.1 we have σ ( g )( f ⊗ v ) = ( B − ⊗ I V )( π ( g ) τ ( k ) ⊗ ρ ( k ))( B ⊗ I V )( f ⊗ v )= ( B − π ( g ) τ ( k ) B ) f ⊗ ρ ( k ) v = σ ( g )( B − τ ( k ) B ) f ⊗ ρ ( k ) v, hence the result. (cid:3) The following proposition gives an explicit expression for dσ ( X ) when X is of the form((0 , , , A ) where A ∈ k . Proposition 6.2. (1)
For each A = ( a kl ) ∈ k , we have d ˜ τ ( A ) = 12 λ X k,l a kl ∂ ∂x k ∂x l + 12 X k,l a kl (cid:18) x k ∂∂x l − x l ∂∂x k (cid:19) − λ x ( Ax ) + 12 Tr( A ) . (2) For each X = ((0 , , , A ) with A ∈ k , we have dσ ( X ) = d ˜ τ ( A ) ⊗ I V + I F ⊗ dρ ( A ) where d ˜ τ ( A ) is as in (1).Proof. In order to prove the first statement, first note that for each A ∈ k and F ∈ F we have ( dτ ( A ) F )( z ) = − ( dF ) z ( Az ) = − X k ∂F ∂z k ( z )( e k ( Az )) . To simplify the notation we denote by k B ( z, x ) the kernel of B , that is, k B ( z, x ) := (cid:18) λπ (cid:19) n/ e (1 / λ ) z + ixz − ( λ/ x . Then, for each f ∈ S ( R n ) we have( dτ ( A ) B f )( z ) = − Z R n (cid:18) λ z ( Az ) + ix ( Az ) (cid:19) k B ( z, x ) f ( x ) dx. CHR ¨ODINGER MODEL... 13
Thus writing z ( Az ) = P k,l a kl z k z l and integrating by parts, we get Z R n z ( Az ) k B ( z, x ) f ( x ) dx = − (cid:18) λπ (cid:19) n/ X k,l a kl Z R n e (1 / λ ) z + ixz ∂ ∂x k ∂x l ( e − ( λ/ x f ( x )) dx and, similarly, Z R n ix ( Az ) k B ( z, x ) f ( x ) dx = − (cid:18) λπ (cid:19) n/ X k,l a kl Z R n e (1 / λ ) z + ixz ∂∂x l ( e − ( λ/ x x k f ( x )) dx. The first statement hence follows. The second statement is an immediate consequence ofProposition 6.1 . (cid:3)
Note that σ is completely determined by the fact that σ ( g , I n ) = σ ( g ) ⊗ I V and byProposition 6.2.7. Stratonovich-Weyl correspondence via Weyl calculus
In this section we first introduce a slight modification of the usual Weyl correspondencein the spirit of our previous works, see for instance [14].Recall that the Berezin calculus s associates with each operator B on V a complex-valued function s ( B ) on o ( ϕ ) which is called the symbol of B and that the space of allsuch symbols is denoted by Sy ( o ( ϕ )), see Section 5. Then the unitary part w of s is anisomorphism from End( V ) onto Sy ( o ( ϕ )).Now we say that a complex-valued smooth function f : ( p, q, ϕ ) → f ( p, q, ϕ ) is a symbolon R n × o ( ϕ ) if for each ( p, q ) ∈ R n the function f ( p, q, · ) : ϕ → f ( p, q, ϕ ) is an elementof Sy ( o ( ϕ )). In that case, we denote ˆ f ( p, q ) := w − ( f ( p, q, · )). A symbol f on R n × o ( ϕ )is called an S -symbol if the function ˆ f belongs to the Schwartz space S ( R n , End( V )) ofrapidly decreasing smooth functions on R n with values in End( V ). For each S -symbolon R n × o ( ϕ ), we define the operator W ( f ) on the Hilbert space L ( R n , V ) by W ( f ) φ ( p ) = (2 π ) − n Z R n e isq ˆ f ( p + (1 / s, q ) φ ( p + s ) ds dq. Of course, W can be extended to much larger classes of symbols as the usual Weylcalculus, see Section 2. As an immediate consequence of the definition of W , we have thefollowing proposition. Proposition 7.1. (1) W is a unitary operator from L ( R n , V ) onto L ( L ( R n , V )) ; (2) For each f ∈ S ( R n ) and f ∈ Sy ( o ( ϕ )) , we have W ( f ⊗ f ) = W ( f ) ⊗ w − ( f ) . In order to compare W and U , it is convenient to transfer U to operators on L ( R n , V )in the spirit of Proposition 2.5. First, for any operator A on L ( R n , V ), we define S ( A ) := S ( BAB − ). Clearly, one has S S ∗ = SS ∗ = B . Then the unitary part U of S is givenby U ( A ) := U ( BAB − ) for any operator A on L ( R n , V ). Moreover, we have(7.1) U = B − / S = (cid:0) ( B ) − / ⊗ b − / (cid:1) (cid:0) S ⊗ s (cid:1) = ( B ) − / S ⊗ b − / s = U ⊗ w with obvious notation. Hence we are in position to extend Proposition 2.5 to Heisenbergmotion groups. Proposition 7.2.
We have U = ( J − ⊗ I Sy ( o ( ϕ )) ) W − . Proof.
By Proposition 7.1, Proposition 2.5 and Eq. 7.1, we have( J − ⊗ I Sy ( o ( ϕ )) ) W − = ( J − ⊗ I Sy ( o ( ϕ )) )( W − ⊗ w ) = ( J − W − ) ⊗ w = U ⊗ w = U . This is the desired result. (cid:3)
Now consider the action of G on R n × o ( ϕ ) given by g · ( p, q, ϕ ) := ( j − ( g · j ( p, q )) , Ad ∗ ( k ) ϕ )for g = (( z , ¯ z ) , c , k ) ∈ G . Then we have the following result. Proposition 7.3. (1) W − is a Stratonovich-Weyl correspondence for ( G, σ, R n × o ( ϕ )) . (2) For each X = (( a, ¯ a ) , c, A ) ∈ g , p, q ∈ R n and ϕ ∈ o ( ϕ ) , we have W − ( dσ ( X ))( p, q, ϕ ) = iλc + 12 Tr( A ) + i (cid:16) ¯ aj ( p, q ) + aj ( p, q ) (cid:17) − λ j ( p, q )( Aj ( p, q )) + w ( dρ ( A ))( ϕ ) . Proof. (1) For each g = (( z , ¯ z ) , c , k ) ∈ G let us denote by L g the operator of L ( C n × o ( ϕ ) , µ λ ⊗ ν ) defined by ( L g F )( z, ϕ ) = F ( g · z, Ad ∗ ( k ) ϕ ) . Then the covariance property for U can be rewritten as L g U ( A ) = U ( π ( g ) − Aπ ( g ))for each g ∈ G and A ∈ L ( F ). This gives the following covariance property for U : L g U ( A ) = U ( σ ( g ) − Aσ ( g ))for each g ∈ G and A ∈ L ( L ( R n , V )). But by Proposition 7.2 we have U = ( J − ⊗ I Sy ( o ( ϕ )) ) W − . Thus we get( J ⊗ I Sy ( o ( ϕ )) ) L g ( J − ⊗ I Sy ( o ( ϕ )) ) W − ( A ) = W − ( σ ( g ) − Aσ ( g ))for each g ∈ G and A ∈ L ( L ( R n , V )).Now let ( ˜ L g f )( p, q, ϕ ) := f ( j − ( g · j ( p, q )) , Ad ∗ ( k ) ϕ )for each g = (( z , ¯ z ) , c , k ) ∈ G and ( p, q, ϕ ) ∈ R n × o ( ϕ ). Since it is clear that for each g ∈ G we have ˜ L g = ( J ⊗ I Sy ( o ( ϕ )) ) L g ( J − ⊗ I Sy ( o ( ϕ )) ) , we see that ˜ L g W − ( A ) = W − ( σ ( g ) − Aσ ( g ))for each g ∈ G and A ∈ L ( L ( R n , V )). Hence W − is G -covariant. The other propertiesof a Stratonovich-Weyl correspondence can be easily verified.(2) For each X ∈ g c , we have U ( dπ ( X )) = U ( dσ ( X )) = (( J − ⊗ I Sy ( o ( ϕ )) ) W − ( dσ ( X ))hence the result follows from Proposition 5.7. (cid:3) CHR ¨ODINGER MODEL... 15
Finally, we can obtain Stratonovich-Weyl correspondences for (
G, π, O ( ϕ )) and for( G, σ, O ( ϕ )) by transferring U and W − by means of Φ. LetΨ := Φ ◦ ( j ⊗
1) : R n × o ( ϕ ) → O ( ϕ )and let ˜ ν be the G -invariant measure on O ( ϕ ) defined by ˜ ν := (Φ − ) ∗ ( µ λ ⊗ ν ) =(Ψ − ) ∗ (˜ µ ⊗ ν ). Consider also the unitary maps τ Φ : F → F ◦ Φ − from L ( C n × o ( ϕ ) , µ λ ⊗ ν )onto L ( O ( ϕ ) , ˜ ν ) and τ Ψ : F → F ◦ Ψ − from L ( R n × o ( ϕ ) , ˜ µ ⊗ ν ) onto L ( O ( ϕ ) , ˜ ν ).Then we have the following proposition. Proposition 7.4.
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