Schr\"odinger model and Stratonovich-Weyl correspondence for Heisenberg motion groups

0.24 MB·

by  Benjamin Cahen

SCHRO¨DINGER MODEL AND STRATONOVICH-WEYL
CORRESPONDENCE FOR HEISENBERG MOTION GROUPS
BENJAMIN CAHEN
7
1
0
2
Abstract. We introduce a Schr¨odinger model for the unitary irreducible representa-
n
a tions of a Heisenberg motion group and we show that the usual Weyl quantization then
J provides a Stratonovich-Weyl correspondence.

6
1

]
T
1. Introduction
R
h. There are different ways to extend the usual Weyl correspondence between functions on
t R2n andoperatorsonL2(Rn)tothegeneralsettingofaLiegroupactingonahomogeneous
a
m space [1], [34], [14], [31]. Here we are concerned with Stratonovich-Weyl correspondences.
[ The notionof Stratonovich-Weyl correspondence was introduced in [51] anditssystematic

study began with the work of J.M. Gracia-Bond`ıa, J.C. V`arilly and their co-workers (see
1
v [33], [29], [26], [32] and also [12]). The following definition is taken from [32], see also [33].
1
6 Definition 1.1. Let G be a Lie group and π be a unitary representation of G on a Hilbert
3
space H. Let M be a homogeneous G-space and let µ be a G-invariant measure on M.
4
0 Then a Stratonovich-Weyl correspondence for the triple (G,π,M) is an isomorphism W
.
1 from a vector space of operators on H to a vector space of functions on M satisfying the
0 following properties:
7
1 (1) the function W(A∗) is the complex-conjugate of W(A);
v: (2) Covariance: we have W(π(g)Aπ(g)−1)(x) = W(A)(g−1 ·x);
i (3) Traciality: we have
X
r
a W(A)(x)W(B)(x)dµ(x) = Tr(AB).
Z
M
Stratonovich-Weyl correspondences were constructed for various Lie group represen-
tations, see [26], [32]. In particular, in [20], Stratonovich-Weyl correspondences for the
holomorphic representations of quasi-Hermitian Lie groups were obtained by taking the
isometric 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 (2n + 1)-dimensional Heisenberg group
acting on R2n ∼= Cn by translations. Each non-degenerate unitary irreducible representa-
tion of G has then two classical realizations: the Schro¨dinger model on L2(Rn) and the
Bargmann-Fock model on the Fock space [30], an intertwining operator between these
2000 Mathematics Subject Classification. 22E45;22E70;22E20;81S10; 81R30.
Key words and phrases. Stratonovich-Weylcorrespondence;Berezinquantization;Berezintransform;
Heisenberg motion group; reproducing kernel Hilbert space; coherent states; Schr¨odinger representation;
Bargmann-Fock representation; Segal-Bargmanntransform.
1
2 BENJAMIN CAHEN
realizations being the Segal-Bargmann transform [30], [27]. In this context, it is well-
known that the usual Weyl correspondence provides a Stratonovich-Weyl correspondence
for the Schro¨dinger realization [6], [54], [49]. It is also known that this Stratonovich-Weyl
correspondence is connected by the Segal-Bargmann transform to the Stratonovich-Weyl
correspondence for the Bargmann-Fock realization which was obtained by polarization
of the Berezin quantization map [44], [43]. In [22], we obtained similar results for the
(2n + 2)-dimensional real diamond group. This group, also called oscillator group, is a
semidirect product of the Heisenberg group by the real line.
Theaimofthepresent paperistoextendthepreceding resultstotheHeisenberg motion
groups. An Heisenberg motion groupis the semidirect product of the (2n+1)-dimensional
Heisenberg group H by a compact subgroup K of the unitary group U(n). Note that
n
Heisenberg motion groups play an important role in the theory of Gelfand pairs, since the
study of a Gelfand pair of the form (K ,N) where K is a compact Lie group acting by
0 0
automorphisms on a nilpotent Lie group N can be reduced to that of the form (K ,H ),
0 n
see [8], [9].
More precisely, we introduce a Schro¨dinger realization for the unitary irreducible repre-
sentations ofa Heisenberg motiongroupandwe prove that we obtaina Stratonovich-Weyl
correspondence by combining the usual Weyl correspondence and the unitary part of the
Berezin calculus for K.
Let us briefly describe our construction. First notice that each Heisenberg motion
group is, in particular, a quasi-Hermitian Lie group and that we can obtain its unitary
irreducible representations as holomorphically induced representations on some general-
ized Fock space by the general method of [46], Chapter XII. Then we can get Schro¨dinger
realizations for these representations by using, as in the case of the Heisenberg group, a
generalized Bargmann-Fock transform. Hence we obtain a Stratonovich-Weyl correspon-
dence for such a Schro¨dinger realization by introducing a generalization of the usual Weyl
correspondence.
Note that, in [45], a Schro¨dinger model and a generalized Segal-Bargmann transform
for the scalar highest weight representations of an Hermitian Lie group of tube type were
introduced andstudied. Let us also mentioned that B. Hall has obtained some generalized
Segal-Bargmann transforms in various situations by means of the heat kernel, see [36] and
references 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 about
the Fock model and the Schro¨dinger model of the unitary irreducible representations of an
Heisenberg groupandaboutthecorrespondingBerezincalculus andWeylcorrespondence.
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 associated
Berezin calculus. We introduce the (generalized) Segal-Bargmann transform and the
Schro¨dinger model inSection6. InSection 7, weshow that theusual Weyl correspondence
also gives a Stratonovich-Weyl correspondence for the Schro¨dinger model. Moreover,
we compare it with the Stratonovich-Weyl correspondence for the Fock model which is
directly obtained by polarization of the Berezin quantization map.
SCHRO¨DINGER MODEL... 3
2. Heisenberg groups
In this section, we review some well-known results about the the Schro¨dinger model
and the Fock model of the unitary irreducible (non-degenerated) representations of the
Heisenberg group. We follow the presentation of [22] in a large extend.
Let G be the Heisenberg group of dimension 2n+1 and g be the Lie algebra of G .
0 0 0
Let {X ,...,X ,Y ,...,Y ,Z˜} be a basis of g in which the only non trivial brackets are
1 n 1 n 0
[X , Y ] = Z˜, 1 ≤ k ≤ n and let {X∗,...,X∗,Y∗,...,Y∗,Z˜∗} be the corresponding dual
k k 1 n 1 n
basis of g∗.
0
For a = (a ,a ,...,a ) ∈ Rn, b = (b ,b ,...,b ) ∈ Rn and c ∈ R, we denote by
1 2 n 1 2 n
[a,b,c] the element exp ( n a X + n b Y + cZ˜) of G . Similarly, for α =
G0 k=1 k k k=1 k k 0
(α1,α2,...,αn) ∈ Rn, β =P(β1,β2,...,βPn) ∈ Rn and γ ∈ R, we denote by (α,β,γ)
the element n α X∗ + n β Y∗ + γZ˜∗ of g∗. The coadjoint action of G is then
k=1 k k k=1 k k 0 0
given by P P
Ad∗([a,b,c])(α,β,γ) = (α+γβ,β −γα,γ).
Now we fix a real number λ > 0 and denote by O the orbit of the element λZ˜∗ of
λ
g∗ under the coadjoint action of G (the case λ < 0 can be treated similarly). By the
0 0
Stone-von Neumann theorem, there exists a unique (up to unitary equivalence) unitary
irreducible representation of G whose restriction to the center of G is the character
0 0
[0,0,c] → eiλc [7], [30]. Note that this representation is associated with the coadjoint
orbit O by the Kirillov-Kostant method of orbits [41], [42]. More precisely, if we choose
λ
the real polarization at λZ˜∗ to be the space spanned by the elements Y for 1 ≤ k ≤ n
k
and Z˜ then we obtain the Schro¨dinger representation σ realized on L2(Rn) as
0
(σ0([a,b,c])f)(x) = eiλ(c−bx+12ab)f(x−a),
see [30] for instance. Here we denote xy := n x y for x = (x ,x ,...,x ) and
k=1 k k 1 2 n
y = (y1,y2,...,yn) in Rn. P
The differential of σ is then given by
0
˜
dσ (X )f(x) = −∂ f(x), dσ (Y )f(x) = −iλx f(x), dσ (Z)f(x) = iλf(x)
0 k k 0 k k 0
where k = 1,2,...,n.
On the other hand, if we consider the complex polarization at λZ˜∗ to be the space
spanned by the elements X +iY for 1 ≤ k ≤ n and Z˜ then the method of orbits leads
k k
to the Bargmann-Fock representation π defined as follows [13].
0
Let F be the Hilbert space of holomorphic functions F on Cn such that
0
kFk2 := |F(z)|2e−|z|2/2λdµ (z) < +∞
F0 ZCn λ
where dµ (z) := (2πλ)−ndxdy. Here z = x+iy with x and y in Rn.
λ
LetusconsidertheactionofG onCn definedbyg·z := z+λ(b−ia) forg = [a,b,c] ∈ G
0 0
and z ∈ Cn. Then π is the representation of G on F given by
0 0 0
π (g)F(z) = α(g−1,z)F(g−1 ·z)
0
where the map α is defined by
α(g,z) := exp −icλ+(1/4)(b+ai)(−2z +λ(−b+ai))
for g = [a,b,c] ∈ G and z ∈ C(cid:0)n. (cid:1)
0
4 BENJAMIN CAHEN
The differential of π is then given by
0
1 ∂F
dπ (X )F(z) = iz F(z)+λi
0 k k
 2 ∂z
k

 1 ∂F

 dπ0(Yk)F(z) =2zkF(z)−λ∂z
k
 dπ0(Z˜)F(z) =iλF(z).



As in [35], Section 6 or [27], Section 1.3, we can verify by using the previous formulas
for dπ and dσ that the Segal-Bargmann transform B : L2(Rn) → F defined by
0 0 0 0
B (f)(z) = (λ/π)n/4 e(1/4λ)z2+ixz−(λ/2)x2 f(x)dx
0
ZRn
is a (unitary) intertwining operator between σ and π . The inverse Segal-Bargmann
0 0
transform B−1 = B∗ is then given by
0 0
B−1(F)(x) = (λ/π)n/4 e(1/4λ)z¯2−ixz¯−(λ/2)x2 F(z)e−|z|2/2λdµ (z).
0 ZCn λ
For z ∈ Cn, consider the coherent state e (w) = exp(z¯w/2λ). Then we have the
z
reproducing property F(z) = hF,e i for each F ∈ F where h·,·i denotes the scalar
z F0 0 F0
product on F .
0
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 domain
0 0
contains e for each z ∈ Cn. Then the Berezin symbol of A ∈ C is the function S0(A)
z 0
defined on Cn by
hAe , e i
S0(A)(z) := z z F0.
he , e i
z z F0
We have the following result, see for instance [22].
Proposition 2.1. (1) Each A ∈ C is determined by S0(A);
0
(2) For each A ∈ C and each z ∈ Cn, we have S0(A∗)(z) = S0(A)(z);
0
(3) For each z ∈ Cn, we have S0(I )(z) = 1. Here I denotes the identity operator
F0 F0
of F ;
0
(4) For each A ∈ C , g ∈ G and z ∈ Cn, we have π (g)−1Aπ (g) ∈ C and
0 0 0 0 0
S0(A)(g ·z) = S0(π (g)−1Aπ (g))(z);
0 0
(5) The map S0 is a bounded operator from L (F ) (endowed with the Hilbert-Schmidt
2 0
norm) to L2(Cn,µ ) which is one-to-one and has dense range.
λ
Proof. For(1) and(2), see [10] and [25]. Note that(4) follows fromthefollowing property:
For each g ∈ G and each z ∈ Cn, we have π (g)e = α(g,z)e , see [20]. Finally, (5) is a
0 0 z g·z
(cid:3)
particular case of [52], Proposition 1.19.
Recall that the Berezin transform is then the operator B0 on L2(Cn,µ ) defined by
λ
B0 = S0(S0)∗. Thus we have the integral formula
B0(F)(z) = F(w)e|z−w|2/2λdµ (w),
λ
ZCn
see [10], [11], [52], [48] for instance. Recall also that we have B0 = exp(λ∆/2) where
∆ = 4 n ∂2/∂z ∂z¯ , see [52], [43].
k=1 k k
P
SCHRO¨DINGER 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 S0 allows us to connect π to O as shown by the following proposition.
0 λ
Here we denote by gc the complexification of g .
0 0
Proposition 2.2. [22] Let Φ be the map defined by
λ
n
Φ (z) := (Rez X∗ +Imz Y∗)+λZ˜∗.
λ k k k k
X
k=1
Then
(1) For each X ∈ gc and each z ∈ Cn, we have
0
S0(dπ (X))(z) = ihΦ (z),Xi.
0 λ
(2) For each g ∈ G and each z ∈ Cn, we have Φ (g ·z) = Ad∗(g)Φ (z).
0 λ λ
(3) The map Φ is a diffeomorphism from Cn onto O .
λ λ
Now we aim to transfer S0 to operators on L2(Rn). To this goal, we define S1(A) :=
S0(B AB−1) for A operator on L2(Rn). Of course, the properties of S0 give rise to similar
0 0
properties of S1. In particular, S1 is a bounded operator from L (L2(Rn)) to L2(Cn,µ )
2 λ
and S1 is G -covariant with respect to σ .
0 0
Moreover, denoting by I the (unitary) map from L (L2(Rn)) onto L (F ) defined by
B0 2 2 0
I (A) = B AB−1, we have S1 = S0I then
B0 0 0 B0
S1(S1)∗ = (S0I )(S0I )∗ = S0I I∗ (S0)∗ = S0(S0)∗ = B0.
B0 B0 B0 B0
This shows that the Berezin transform corresponding to S1 is the same as the Berezin
transform corresponding to S0. Then we can write the polar decompositions of S0 and
S1 as S0 = (B0)1/2U0 and S1 = (B0)1/2U1 where the maps U0 : L (F ) → L2(Cn,µ ) and
2 0 λ
U1 : L (L2(Rn)) → L2(Cn,µ ) are unitary.
2 λ
Moreover, asintheproofof[17],Proposition3.1,wecanverifythatU0 isaStratonovich-
Weyl correspondence for (G ,π ,Cn) and that U1 is a Stratonovich-Weyl correspondence
0 0
for (G ,σ ,Cn). Note that G -covariance of U0 and U1 immediately follows from G -
0 0 0 0
covariance of S0 and S1. Note also that we have U1 = U0I .
B0
Now, we show how to use the usual Weyl correspondence in order to get another
Stratonovich-Weyl correspondence for σ . The Weyl correspondence on R2n is defined as
0
follows. For each f in the Schwartz space S(R2n), let W (f) be the operator on L2(Rn)
0
defined by
W (f)φ(p) = (2π)−n eisqf(p+(1/2)s,q)φ(p+s)dsdq.
0
ZR2n
The Weyl calculus can be extended to much larger classes of symbols (see for instance
[38]). In particular, if f(p,q) = u(p)qα where u ∈ C∞(Rn) then we have
α
∂
W (f)φ(p) = i (u(p+(1/2)s)φ(p+s)) ,
0
(cid:18) ∂s(cid:19) (cid:12)s=0
(cid:12)
see [53]. (cid:12)
From this, we can deduce the following proposition. Consider the action of G on R2n
0
given by g ·(p,q) := (p+a,q +λb) where g = [a,b,c].
6 BENJAMIN CAHEN
Proposition 2.3. [22] Let Ψ be the map defined by
λ
n
Ψ (p,q) := (q X∗ −λp Y∗)+λZ˜∗.
λ k k k k
X
k=1
Then
(1) For each X ∈ gc and each (p,q) ∈ R2n, we have
0
W−1(dσ (X))(p,q) = ihΨ (p,q),Xi.
0 0 λ
(2) For each g ∈ G and each (p,q) ∈ R2n, we have Ψ (g ·(p,q)) = Ad∗(g)Ψ (p,q).
0 λ λ
(3) The map Ψ is a diffeomorphism from R2n onto O .
λ λ
(4) For each (p,q) ∈ R2n, we have Φ (q −λpi) = Ψ (p,q).
λ λ
Assume that R2n is equipped with the G -invariant measure µ˜ := (2π)−ndpdq. Then
0
one has the following result.
Proposition 2.4. [30], [22] The map W−1 is a Stratonovich-Weyl correspondence for
0
(G ,σ ,R2n).
0 0
Thefollowing propositionasserts thatifwe identify R2n withCn by themapj : (p,q) →
q−λpi then the unitary part in the polar decomposition of S1 coincides with the inverse
of the Weyl transform, see [43] and [48].
Proposition 2.5. Let J be the map from L2(Cn,µ ) onto L2(R2n) definedby J(F) = F◦j.
λ
Then we have U1 = (W J)−1.
0
Finally, note that we can obtain Stratonovich-Weyl correspondences for (G ,σ ,O )
0 0 λ
and (G ,π ,O ) by transferring W−1 and U0 by using Φ and Ψ . More precisely, let ν
0 0 λ 0 λ λ λ
be the G -invariant measure on O defined by ν := (Φ−1)∗(µ ) = (Ψ−1)∗(µ˜). Then the
0 λ λ λ λ λ
maps τ : F → F ◦ Φ−1 from L2(Cn,µ ) onto L2(O ,ν ) and τ : F → F ◦Ψ−1 from
Φλ λ λ λ λ Ψλ λ
L2(R2n) onto L2(O ,ν ) are unitary and we have τ = τ J. Hence we can assert the
λ λ Φλ Ψλ
following proposition.
Proposition 2.6. The map W := τ W−1 is a Stratonovich-Weyl correspondence for
1 Ψλ 0
(G ,σ ,O ), the map W := τ U0 is a Stratonovich-Weyl correspondence for (G ,π ,O )
0 0 λ 2 Φλ 0 0 λ
and we have W = W I .
1 2 B0
3. 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 ∈ Cn,
0
c ∈ R, we denote here by (z,z¯,c) the element G which is denoted by [Rez,Imz,c] in
0
Section 2. Moreover, for each z, w ∈ Cn, we denote zw := n z w and we consider
k=1 k k
the symplectic form ω on C2n defined by P
i
ω((z,w),(z′,w′)) = (zw′ −z′w).
2
for z,w,z′,w′ ∈ Cn. Then the multiplication of G is given by
0
(3.1) ((z,z¯),c)·((z′,z¯′),c′) = ((z +z′,z¯+z¯′),c+c′ + 1ω((z,z¯),(z′,z¯′))),
2
the complexification Gc of G is Gc = {((z,w),c) : z,w ∈ Cn,c ∈ C} and the mul-
0 0 0
tiplication of Gc is obtained by replacing (z,z¯) by (z,w) and (z′,z¯′) by (z′,w′) in Eq.
0
3.1.
SCHRO¨DINGER MODEL... 7
Now, let K be a closed subgroup of U(n). Then K acts on G by k · ((z,z¯),c) =
0
((kz,k¯z),c) and we can form the semidirect product G := G ⋊ K which is called a
0
Heisenberg motion group. The elements of G can be written as ((z,z¯),c,k) where z ∈ Cn,
c ∈ R and k ∈ K. The multiplication of G is then given by
((z,z¯),c,k)·((z′,z¯′),c′,k′) = ((z,z¯)+(kz′,k¯z′),c+c′ + 1ω((z,z¯),(kz′,k¯z′)),kk′).
2
Wedenoteby Kc thecomplexification ofK andwe consider theactionofKc onCn×Cn
given by k·(z,w) = (kz,(kt)−1w) (here, the subscript t denotes transposition). The group
Gc is then the semidirect product Gc = Gc ⋊Kc. The elements of Gc can be written as
0
((z,w),c,k) where z, w ∈ Cn, c ∈ C and k ∈ Kc and the multiplication law of Gc is given
by
((z,w),c,k)·((z′,w′),c′,k′) = ((z,w)+k ·(z′,w′),c+c′ + 1ω((z,w),k·(z′,w′)),kk′).
2
We denote by k, kc, g and gc the Lie algebras of K, Kc, G and Gc. The derived action
of kc on Cn ×Cn is then A·(z,w) := (Az,−Atw) and the Lie brackets of gc 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˜ := {((0,0),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 compactly
0
embedded subalgebra of g and the subalgebra h of g consisting of all elements of the form
((0,0),c,A) where c ∈ R and A ∈ h is a compactly embedded Cartan subalgebra of g
0
[46], p. 250.
Following [46], Chapter XII.1, we set p+ = {((z,0),0,0) : z ∈ Cn} and p− =
{((0,w),0,0) : w ∈ Cn} and we denote by P+ and P− the corresponding analytic sub-
groups of Gc, that is, P+ = {((z,0),0,I ) : z ∈ Cn} and P− = {((0,w),0,I ) : w ∈ Cn}.
n 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) gc = p+ ⊕˜kc ⊕p− is a direct sum of vector spaces, (p+)∗ = p− and [˜kc,p±] ⊂ p±;
(2) The multiplication map P+K˜cP− → Gc, (z,k,y) → zky is a biholomorphic dif-
feomorphism onto its open image;
(3) G ⊂ P+K˜cP− and G∩K˜cP− = K˜.
We denote by pp+, p˜kc and pp− the projections of gc onto p+, ˜kc and p− associated with
the above direct decomposition.
We can easily verify that each g = ((z ,w ),c ,k) ∈ Gc has a P+K˜cP−-decomposition
0 0 0
given by
g = ((z ,0),0,I )·((0,0),c,k)·((0,w ),0,I )
0 n 0 n
where c = c − iz w . We denote by ζ : P+K˜cP− → P+, κ : P+K˜cP− → Kc and
0 4 0 0
η : P+K˜cP− → 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 ∈ Gc, we define g · Z ∈ p+ by g · Z := logζ(gexpZ). From the above
formula for the P+K˜cP−-decomposition, we deduce that if g = ((z ,w ),c ,k) ∈ G and
0 0 0
Z = ((z,0),0,0) ∈ p+ then we have g·Z = logζ(gexpZ) = ((z +kz,0),0,0). Note that
0
D := G·0 = p+ ≃ Cn here.
A useful section Z → g for the action of G on D can be obtained by using Proposition
Z
4.5 of [21]. Here we get g = ((z,z¯),0,I ) for each Z = ((z,0),0,0), z ∈ Cn.
Z n
8 BENJAMIN CAHEN
Now we compute the adjoint and coadjoint actions of Gc. Let g = (v ,c ,k ) ∈ Gc
0 0 0
where v ∈ C2n, c ∈ C, k ∈ Kc and X = (w,c,A) ∈ gc where w ∈ C2n, c ∈ C and
0 0 0
A ∈ kc. We can easily verify that
d
Ad(g)X = (gexp(tX)g−1)|
t=0
dt
= k w −(Ad(k )A)·v ,c+ω(v ,k w)− 1ω(v ,(Ad(k )A)·v ),Ad(k )A .
0 0 0 0 0 2 0 0 0 0
(cid:0) (cid:1)
Now, let us denote by ξ = (u,d,φ), where u ∈ C2n, d ∈ C and φ ∈ (kc)∗, the element of
(gc)∗ defined by
hξ,(w,c,A)i = ω(u,w)+dc+hφ,Ai.
Also, for u,v ∈ C2n, we denote by v × u the element of (kc)∗ defined by hv × u,Ai :=
ω(u,A· v) for A ∈ kc. Then, from the above formula for the adjoint action, we deduce
that for each ξ = (u,d,φ) ∈ (gc)∗ and g = (v ,c ,k ) ∈ Gc we have
0 0 0
Ad∗(g)ξ = k u−dv ,d,Ad∗(k )φ+v ×(k u− dv )
0 0 0 0 0 2 0
(cid:0) (cid:1)
By restriction, we also get the analogous formula for the coadjoint action of G. From
this, we see that if a coadjoint orbit of G contains a point (u,d,φ) with d 6= 0 then it also
contains a point of the form (0,d,φ ). Such an orbit is called generic.
0
4. Fock model for Heisenberg motion groups
In this section, we introduce the Fock model of the unitary irreducible representations
of G by using the general method of [46], Chapter XII that we describe here briefly.
LetρbeaunitaryirreduciblerepresentationofK ona(finite-dimensional)Hilbertspace
V and λ ∈ R. Let ρ˜ be the representation of K˜ on V defined by ρ˜((0,0),c,k) = eiλcρ(k)
for each c ∈ R and k ∈ K.
For each Z,W ∈ D, let K(Z,W) := ρ˜(κ(expW∗expZ))−1 and for each g ∈ G, Z ∈ D,
let J(g,Z) := ρ˜(κ(gexpZ)), [46], Chapter XII.1. Consider the Hilbert space F˜ of all
holomorphic functions on D with values in V such that
kfk2 := hK(Z,Z)−1f(Z),f(Z)i dµ(Z) < +∞
F˜ Z V
D
where µ denotes an invariant G-measure on D. Then the equation
π˜(g)f(Z) = J(g−1,Z)−1f(g−1·Z)
defines a unitary representation of G on F˜. This representation can be also obtained by
holomorphic induction from ρ˜, that is, it corresponds to the natural action of G on the
∼
square-integrable holomorphic sections of the Hilbert G-bundle G × V over G/K = D
ρ˜
[22]. Note also that π˜ is irreducible since ρ˜ is irreducible, [46], p. 515.
HerewecaneasilycomputeK andJ. ForeachZ = ((z,0),0,0), W = ((w,0),0,0) ∈ D,
we have K(Z,W) = eλzw¯/2I and for each g = ((z ,z¯ ),c ,k) ∈ G and Z = ((z,0),0,0),∈
V 0 0 0
D, we have
J(g,Z) = exp iλc + λz¯ (kz)+ λ|z |2 ρ(k).
0 2 0 4 0
Moreover, µcanbetakentobethe(cid:0)G-invariantmeasureon(cid:1)D ≃ Cn definedbydµ(Z) :=
λn(2π)−ndxdy. Here Z = ((z,0),0,0) and z = x + iy with x and y in Rn. From now
on, we identify Z = ((z,0),0,0) ∈ D with z ∈ Cn and each function on D with the
corresponding function on Cn.
SCHRO¨DINGER MODEL... 9
Consequently, the Hilbert product on F˜ is given by
n
λ
hf,gi = hf(z),g(z)i e−λ|z|2/2 dxdy
F˜ ZCn V (cid:18)2π(cid:19)
and we get the following formula for π˜:
(π˜(g)f)(z) = exp iλc + λz¯ z − λ|z |2 ρ(k)f(k−1(z −z ))
0 2 0 4 0 0
where g = ((z ,z¯ ),c ,k) ∈ G an(cid:0)d z ∈ Cn. (cid:1)
0 0 0
In fact, in order to use the results of Section 2, it is convenient to replace π˜ by an
equivalent representation π whose restriction to G is precisely π . To this aim, we
0 0
consider the Fock space F of all holomorphic functions f : Cn → V such that
kfk2 := kf(z)k2 e−|z|2/2λdµ (z) < +∞.
F V λ
ZCn
Let J : F˜ → F be the unitary operator defined by J(f)(z) = f(iλ−1z) and set
π(g) := Jπ˜(g)J−1 for each g ∈ G. Then we have
(π(g)f)(z) = exp iλc + 1iz¯ z − λ|z |2 ρ(k)f(k−1(z +iλz ))
0 2 0 4 0 0
where g = ((z ,z¯ ),c ,k) ∈ G a(cid:0)nd z ∈ Cn. (cid:1)
0 0 0
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 ∈ Cn, we
have
(dπ(X)f)(z) = dρ(A)f(z)+i(λc+ 1a¯z)f(z)+df (−Az +iλa).
2 z
Clearly, one has F = F ⊗V. For f ∈ F and v ∈ V, we denote by f ⊗v the function
0 0 0 0
z → f (z)v. Moreover, if A is an operator of F and A is an operator of V then we
0 0 0 1
denote by A ⊗A the operator of F defined by (A ⊗A )(f ⊗v) = A f ⊗A v.
0 1 0 1 0 0 0 1
Let τ be the left-regular representation of K on F , that is, (τ(k)f )(z) = f (k−1z).
0 0 0
Then we have
(4.1) π((z ,z¯ ),c ,k) = π ((z ,z¯ ),c )τ(k)⊗ρ(k)
0 0 0 0 0 0 0
for each z ∈ Cn, c ∈ R and k ∈ K. Note that this is precisely Formula (3.18) in [8].
0 0
5. Stratonovich-Weyl correspondence via Berezin quantization
In this section, we introduce the Berezin quantization map associated with π and the
corresponding 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 ∈ (hc)∗ the highest
0 0
weight of ρ and by kc = n+ ⊕hc ⊕n− the corresponding triangular decomposition of kc.
0
Let ϕ˜ be the element of (kc)∗ defined by ϕ˜ = −iΛ on h and by ϕ˜ = 0 on n±. We
0 0 0 0
denote by ϕ the restriction of ϕ˜ to k. Then the orbit o(ϕ ) of ϕ under the coadjoint
0 0 0 0
action 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 coadjoint
0 0
action of K is precisely the connected subgroup H of K with Lie algebra h [15].
0 0
Note that a complex structure on o(ϕ ) is then defined by the diffeomorphism o(ϕ ) ≃
0 0
K/H ≃ Kc/HcN− where H is the connected subgroup of K with Lie algebra h and
0 0 0 0
N− is the analytic subgroup of Kc with Lie algebra n−.
10 BENJAMIN CAHEN
Without loss of generality, we can assume that V is a space of holomorphic sections of a
complex line bundle over o(ϕ ) as in [15]. Let ϕ ∈ o(ϕ ). For each ϕˆ 6= 0 in the fiber over
0 0
ϕ, there exists a unique function e ∈ V (a coherent state) such that a(ϕ) = ha,e i e
ϕˆ ϕˆ V ϕˆ
for each a ∈ V.
TheBerezin calculus ono(ϕ ) associates witheach operatorB onV the complex-valued
0
function s(B) on o(ϕ ) defined by
0
hBe ,e i
ϕˆ ϕˆ V
s(B)(ϕ) =
he ,e i
ϕˆ ϕˆ V
which is called the symbol of B. This definition makes sense since the right side of the
equation does not depend on ϕˆ in the fiber over ϕ but only on ϕ. We denote by Sy(o(ϕ ))
0
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
0
s(B)(Ad∗(k)ϕ) = s(ρ(k)−1Bρ(k))(ϕ).
(4) For each A ∈ k and ϕ ∈ o(ϕ ), we have s(dρ(A))(ϕ) = ihϕ,Ai.
0
In our papers [18], [19] and [23], we developped a general method for constructing a
Berezin quantization map associated with a unitary representation of a quasi-Hermitian
Lie group which is holomorphically induced from a unitary irreducible representation of
a maximal compactly embedded subgroup. This construction goes as follows.
The evaluation maps K : H → V,f → f(z) are continuous [46], p. 539. The vector
z
coherent states of F are the maps E = K∗ : V → F defined by hf(z),vi = hf,E vi for
z z V z F
f ∈ F and v ∈ V. Here we have that E v = e ⊗v, that is, we have (E v)(w) = eλz¯w/2v.
z z z
Let Fs be the subspace of F generated by the functions e ⊗v for z ∈ Cn and v ∈ V.
z
Then Fs 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 Fs and the domain of A∗ also contains Fs. Then,
following an idea of [40] and [2], we first introduce the pre-symbol S (A) of A ∈ C by
0
S (A)(z) = (E∗E )−1/2E∗AE (E∗E )−1/2 = e−λzz¯/2E∗AE .
0 z z z z z z z z
The Berezin symbol S(A) of A is thus defined as the complex-valued function on Cn×
o(ϕ ) given by
0
S(A)(z,ϕ) = s(S (A)(z))(ϕ).
0
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 ) = 1.
F
(4) For each A ∈ C, g = ((z ,z¯ ),c,k) ∈ G, z ∈ Cn and ϕ ∈ o(ϕ ), we have
0 0 0
S(A)(g ·z,ϕ) = S(π(g)−1Aπ(g))(z,Ad∗(k−1)ϕ).
Moreover, we can decompose S according to the decomposition F = F ⊗ V. Let f
0 0
be a complex-valued function on Cn and f be a complex-valued function on o(ϕ ). Then
1 0
we denote by f ⊗f the function on Cn ×o(ϕ ) defined by (f ⊗f )(z,ϕ) = f (z)f (ϕ).
0 1 0 0 1 0 1