Berezin transform and Stratonovich-Weyl...

Rend. Sem. Mat. Univ. Padova 1xx (201x) Rendiconti del Seminario Matematicodella Universita di Padova

c© European Mathematical Society

Berezin transform and Stratonovich-Weyl correspondence

for the multi-dimensional Jacobi group

Benjamin Cahen

Abstract – We study the Berezin transform and the Stratonovich-Weyl correspondenceassociated with a holomorphic representation of the multi-dimensional Jacobi group.

Mathematics Subject Classification (2010). 22E27; 32M10; 46E22.

Keywords. Berezin quantization; Berezin transform; quasi-Hermitian Lie group; uni-tary representation; holomorphic representation; reproducing kernel Hilbert space; multi-dimensional Jacobi group; Stratonovich-Weyl correspondence.

1. Introduction

This paper is part of a program to study Berezin transforms and Stratonovich-Weyl correspondences associated with holomorphic representations. The notion ofStratonovich-Weyl correspondence was introduced in [31] in order to extend theusual Weyl correspondence between functions on R2n and operators on L2(Rn)(see [1], [21]) to the general setting of a Lie group acting on a homogeneous space.Stratonovich-Weyl correspondences were systematically studied by J.M. Gracia-Bondıa, J.C. Varilly and various co-workers, see in particular [23], [20], [18] and[22]. The following definition is taken from [22].

Definition 1.1. Let G be a Lie group and π a unitary representation of G on aHilbert space H. Let M be a homogeneous G-space and µ a (suitably normalized)G-invariant measure on M . Then a Stratonovich-Weyl correspondence for thetriple (G, π,M) is an isomorphism W from a vector space of operators on H to avector space of (generalized) functions on M satisfying the following properties:

(1) W maps the identity operator of H to the constant function 1;

Universite de Lorraine, Site de Metz, UFR-MIM, Departement de mathematiques, BatimentA, Ile du Saulcy, CS 50128, F-57045, Metz cedex 01, France

E-mail: [email protected]

2 B. Cahen

(2) Reality: the function W (A∗) is the complex-conjugate of W (A);

(3) Covariance: we have W (π(g)Aπ(g)−1)(x) = W (A)(g−1 · x);

(4) Unitarity: we have∫M

W (A)(x)W (B)(x) dµ(x) = Tr(AB).

In this context, M is generally a coadjoint orbit of G which is associated with πby the Kirillov-Kostant method of orbits [25]. For instance, consider the case whenG is the (2n+1)-dimensional Heisenberg group Hn. Each non-degenerate coadjointorbit M of G is then diffeomorphic to R2n and is associated with a Schrodingerrepresentation π of Hn on L2(Rn). In this case, the classical Weyl correspondencegives a Stratonovich-Weyl correspondence for the triple (Hn, π,M) [21], [22].

In the case when G is a quasi-Hermitian Lie group and π is a unitary rep-resentation of G (on a Hilbert space H) which is holomorphically induced froma unitary character of a compactly embedded subgroup K of G, we can applyan idea of [20] and we obtain a Stratonovich-Weyl correspondence by modifyingsuitably the Berezin correspondence S [14] (see also [2] and [3]).

More precisely, recall that S is an isomorphism from the Hilbert space of allHilbert-Schmidt operators on H (endowed with the Hilbert-Schmidt norm) onto aspace of square integrable functions on a homogeneous complex domain [32]. Themap S satisfies (1), (2) and (3) of Definition 1.1 but not (4). A Stratonovich-Weyl correspondence W is then obtained by taking the isometric part in the polardecomposition of S, that is, W := (SS∗)−1/2S. Let us mention that B := SS∗ isthen the so-called Berezin transform which have been studied by many authors,see in particular [19], [27], [28], [32], [33].

In [14], we considered the case when the Lie algebra g of G is reductive. Inthis case, we proved that B can be extended to a class of functions which containsS(dπ(X1X2 · · ·Xp)) for X1, X2, . . . , Xp ∈ g and that the restrictions to each simpleideal of g of the mappings X → S(dπ(X)) and X → W (dπ(X)) are proportional(see also [12] and [13] ).

The case when g is not reductive is more delicate. In [16] we investigated thecase of the diamond group and, in [17], we studied B and W in the case of theJacobi group.

The aim of the present paper is to generalize the results of [17] to the case of themulti-dimensional Jacobi group, which is technically more complicated. The multi-dimensional Jacobi group plays a central role in different areas of Mathematics andPhysics and its holomorphic unitary representations were studied intensively, see[26], [9], [10], [4], [6]. In particular, the metaplectic factorization should be usedto reduce the study of the highest weight representations of a quasi-Hermitian Liegroup to that of some generalized multi-dimensional Jacobi group [26]. Then thestudy of the case of the multi-dimensional Jacobi group can be considered as afirst step towards the general case.

Berezin transform... 3

In this paper, we begin by some generalities on the multi-dimensional Jacobigroup (Section 2) and its holomorphic representations (Section 3). Then we intro-duce the Berezin correspondence S, the Berezin transform B and the Stratonovich-Weyl correspondence W (Section 4). In Section 5, we show that, under some tech-nical assumptions, the Berezin transform of S(dπ(X1X2 · · ·Xp)) is well-defined foreach X1, X2, . . . , Xp ∈ g. In Section 6, we identify a class of functions which isstable under B and contains S(dπ(X)) for each X ∈ g. We also give an expressionof W (dπ(X)) in terms of some integrals of Hua’s type (see [24]).

2. The multi-dimensional Jacobi group

The material of this section and of the following section is essentially takenfrom [21], Chapter 4, [26], Chapters VII and XII and [15].

Consider the symplectic form ω on Cn × Cn defined by

ω((z, w), (z′, w′)) =i

2

n∑k=1

(zkw′k − z′kwk).

for z, w, z′, w′ ∈ Cn. The (2n+ 1)-dimensional real Heisenberg group is

H := {((z, z), c) : z ∈ Cn, c ∈ R}

endowed with the multiplication

((z, z), c) · ((z′, z′), c′) = ((z + z′, z + z′), c+ c′ + 12ω((z, z), (z′, z′))).

Then the complexification Hc of H is

Hc := {((z, w), c) : z, w ∈ Cn, c ∈ C}

and the multiplication of Hc is obtained by replacing (z, z) by (z, w) and (z′, z′)by (z′, w′) in the preceding equality. We denote by h and hc the Lie algebras of Hand Hc.

Now consider the group S := Sp(n,C)∩SU(n, n) ' Sp(n,R) [26], p. 501, [21],p. 175. Then S consists of all matrices

h =

(P QQ P

), P,Q ∈Mn(C), PP ? −QQ? = In, PQt = QP t

and Sc = Sp(n,C).The group S acts on H by

h · ((z, z), c) = (h(z, z), c) = (Pz +Qz, Qz + P z, c)

where the elements of Cn and Cn×Cn are considered as column vectors. Then wecan form the semi-direct product G := H oS called the multi-dimensional Jacobi

4 B. Cahen

group. The elements of G can be written as ((z, z), c, h) where z ∈ Cn, c ∈ R andh ∈ S. The multiplication of G is thus given by

((z, z), c, h) · ((z′, z′), c′, h′) = ((z, z) + h(z′, z′), c+ c′ + 12ω((z, z), h(z′, z′)), hh′).

The complexification Gc of G is then the semi-direct product Gc = HcoSp(n,C)whose elements can be written as ((z, w), c, h) where z, w ∈ Cn, c ∈ C, h ∈Sp(n,C) and the multiplication of Gc is obtained by replacing z and z′ by w andw′ in the preceding formula.

We denote by s, sc, g and gc the Lie algebras of S, Sc, G and Gc. The Liebrackets 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 θ denotes conjugation over the real form g of gc. For X ∈ gc, we setX∗ = −θ(X). We can easily verify that if X =

((z, w), c,

(A BC −At

))∈ gc then we

haveX∗ = ((−w,−z),−c,

(At −C−B −A

)).

Also, we denote by g → g∗ the involutive anti-automorphism of Gc which isobtained by exponentiating X → X∗ to Gc.

Let K be the subgroup of G consisting of all elements((0, 0), c,

(P 00 P

))where

c ∈ R and P ∈ U(n). Then the Lie algebra k of K is a maximal compactlyembedded subalgebra of g and the subalgebra t of k consisting of all elementsof the form ((0, 0), c, A) where A is diagonal is a compactly embedded Cartansubalgebra of g [26], p. 250. Following [26], p. 532, we set

p+ =

{((y, 0), 0,

(0 Y0 0

)): y ∈ Cn, Y ∈Mn(C), Y t = Y

}and

p− =

{((0, v), 0,

(0 0V 0

)): v ∈ Cn, V ∈Mn(C), V t = V

}.

Then we have the decomposition gc = p+ ⊕ kc ⊕ p−.Henceforth we denote by a(y, Y ) the element

((y, 0), 0,

(0 Y0 0

))of p+. Also, we

denote by pp+ , pkc and pp− the projections of gc onto p+, kc and p− associatedwith the above direct decomposition.

Let P+ and P− be the analytic subgroups of Gc with Lie algebras p+ and p−.Then we have

P+ =

{((y, 0), 0,

(In Y0 In

)): y ∈ Cn, Y ∈Mn(C), Y t = Y

}and

P− =

{((0, v), 0,

(In 0V In

)): v ∈ Cn, V ∈Mn(C), V t = V

}.

In particular, we see that G is a group of the Harish-Chandra type [26], p. 507(see also [30]), 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+KcP− → Gc, (z, k, y)→ zky is a biholomorphicdiffeomorphism onto its open image;

(3) G ⊂ P+KcP− and G ∩KcP− = K.

We can easily verify that g =((z0, w0), c0,

(A BC D

))∈ Gc has a P+KcP−-

decomposition

g =

((y, 0), 0,

(In Y0 In

))·(

(0, 0), c,

(P 00 (P t)−1

))·(

(0, v), 0,

(In 0V In

))if and only if Det(D) 6= 0 and, in this case, we have y = z0−BD−1w0, Y = BD−1,v = D−1w0, V = D−1C, P = A − BD−1C = (Dt)−1 and c = c0 − (1/4)i(z0 −BD−1w0)tw0.

We denote by ζ : P+KcP− → P+, κ : P+KcP− → Kc and η : P+KcP− →P− the projections onto P+-, Kc- and P−-components.

We can introduce an action (defined almost everywhere) of Gc on p+ as follows.For Z ∈ p+ and g ∈ Gc with g expZ ∈ P+KcP−, we define the element g ·Z of p+

by g·Z := log ζ(g expZ). From the above formula for the P+KcP−-decomposition,we deduce that the action of g =

((z0, w0), c0,

(A BC D

))∈ Gc on a(y, Y ) ∈ p+ is

given by g · a(y, Y ) = a(y′, Y ′) where Y ′ := (AY +B)(CY +D)−1 and

y′ := z0 +Ay − (AY +B)(CY +D)−1(w0 + Cy).

This implies that

D := G · 0 = {a(y, Y ) ∈ p+ : In − Y Y > 0} ∼= Cn × B.

where B := {Y ∈Mn(C) : Y t = Y, In − Y Y > 0}.Now we introduce a useful section Z → gZ for the action of G on D. Let

Z = a(y, Y ) ∈ D. Define gZ :=((z0, z0), 0,

( P QQ P

))∈ G as follows. We set

z0 = (In − Y Y )−1(y + Y y), P = (In − Y Y )−1/2, Q = (In − Y Y )−1/2Y.

Then one has gZ · 0 = Z.From the above formula for the action of G on D, we can deduce the G-invariant

measure µ on D. Let µL be the Lebesgue measure on D ' Cn × B. Thus, weeasily obtain that dµ(Z) = Det(1−Y Y )−(n+2) dµL(y, Y ), see for instance [5]. Thisresult can be also deduced from the general formula for the invariant measure, see[26], p. 538.

In the rest of the paper, we fix the normalization of the Lebesgue measureas follows. For y ∈ Cn, write y = (a1 + ib1, a2 + ib2, . . . , an + ibn) with aj , bj ∈R for j = 1, 2, . . . n. Then we take the measure Lebesgue on Cn to be dy :=da1db1da2db2 · · · dandbn. Similarly, writing Y ∈ B as Y = (ykl), we denote by dYthe Lebesgue measure on B defined by dY :=

∏kl dykl. Thus we set dµL(y, Y ) :=

dy dY .

6 B. Cahen

Now we aim to compute the adjoint and coadjoint actions of Gc. First, wecompute the adjoint action of Gc as follows. Let g = (v0, c0, h0) ∈ Gc wherev0 ∈ C2n, c0 ∈ C and h0 ∈ Sc = Sp(n,C) and X = (w, c, U) ∈ gc where w ∈ C2n,c ∈ C and U ∈ sc. We set exp(tX) = (w(t), c(t), exp(tU)). Then, since thederivatives of w(t) and c(t) at t = 0 are w and c, we find that

Ad(g)X =d

dt(g exp(tX)g−1)|t=0

=(h0w − (Ad(h0)U)v0, c+ ω(v0, h0w)− 1

2ω(v0, (Ad(h0)U)v0),Ad(h0)U).

On the other hand, let us denote by ξ = (u, d, ϕ), where u ∈ C2n, d ∈ C andϕ ∈ (sc)

∗, the element of (gc)∗ defined by

〈ξ, (w, c, U)〉 = ω(u,w) + dc+ 〈ϕ,U〉.

Moreover, for u, v ∈ C2n, we denote by v × u the element of (sc)∗ defined by〈v × u, U〉 := ω(u, Uv) for U ∈ sc.

Let ξ = (u, d, ϕ) ∈ (gc)∗ and g = (v0, c0, h0) ∈ Gc. Then, by using the relation〈Ad∗(g)ξ,X〉 = 〈ξ,Ad(g−1)X〉 for X ∈ gc, we obtain

Ad∗(g)ξ =(h0u− dv0, d,Ad∗(h0)ϕ+ v0 × (h0u− d

2v0))

By restriction, we also get the formula for the coadjoint action of G. The followinglemma will be needed later.

Lemma 2.1. [15] The elements ξ0 of g∗ fixed by K are the elements of the form(0, d, ϕλ) where d, λ ∈ R and ϕλ ∈ s∗ is defined by 〈ϕλ,

(A BC D

)〉 = iλTr(A).

3. Holomorphic representations

The holomorphic representations of the multi-dimensional Jacobi group werestudied by many authors, see in particular [26], [9], [10], [4], [5] and [6]. We followhere the general presentation of [26], Chapter XII (see also [14]).

Let χ be a unitary character of K. The extension of χ to Kc is also denotedby χ. We set Kχ(Z,W ) := χ(κ(expW ∗ expZ))−1 for Z, W ∈ D and Jχ(g, Z) :=χ(κ(g expZ)) for g ∈ G and Z ∈ D. We consider the Hilbert space Hχ of allholomorphic functions f on D such that

‖f‖2χ :=

∫D|f(Z)|2Kχ(Z,Z)−1cχdµ(Z) < +∞

where the constant cχ is defined by

c−1χ =

∫DKχ(Z,Z)−1 dµ(Z).


We shall see that, under some hypothesis on χ, cχ is well-defined and Hχ 6= (0).In that case, Hχ contains the polynomials [26], p. 546. Moreover, the formula

πχ(g)f(Z) = Jχ(g−1, Z) f(g−1 · Z)

defines a unitary representation of G on Hχ which is a highest weight representa-tion [26], p. 540.

The space Hχ is a reproducing kernel Hilbert space. More precisely, if weset eZ(W ) := Kχ(W,Z) then we have we have the reproducing property f(Z) =〈f, eZ〉χ for each f ∈ Hχ and each Z ∈ D [26], p. 540. Here 〈·, ·〉χ denotes theinner product on Hχ.

Here we fix χ as follows. Let γ ∈ R and m ∈ Z. Then, for each k =((0, 0), c,

(P 00 P

))∈ K, we set χ(k) := eiγc(DetP )m.

We need the following lemma.

Lemma 3.1. [24] Let λ ∈ R. The integral

Jn(λ) :=

∫B

Det(In − Y Y )λdY

is convergent if λ > −1 and in this case we have

Jn(λ) = πn(n+1)

2Γ(2λ+ 3)Γ(2λ+ 5) · · ·Γ(2λ+ 2n− 1)

(λ+ 1)(λ+ 2) · · · (λ+ n)Γ(2λ+ n+ 2)Γ(2λ+ n+ 3) · · ·Γ(2λ+ 2n).

Then we have the following result.

Proposition 3.2. (1) Let Z = a(y, Y ) ∈ D and W = a(v, V ) ∈ D. We set

E(y, v, Y, V ) := 2yt(In − V Y )−1v + yt(In − V Y )−1V y + vtY (In − V Y )−1v.

Then we have

Kχ(Z,W ) = Det(In − Y V )m exp(γ

4E(y, v, Y, V )

).

(2) We have Hχ 6= (0) if and only if γ > 0 and m + n + 1/2 < 0. In this case,we also have c−1

χ = (2π)nγ−nJn(−m− n− 1/2).

(3) For each g =((z0, z0), c0,

( P QQ P

))∈ G and each Z = a(y, Y ) ∈ D, we have

J(g, Z) = eiγc0 Det(QY + P )−m exp(γ

4

(zt0z0 + 2zt0Py + ytP tQy

− (z0 + Qy)t(PY +Q)(QY + P )−1(z0 + Qy))

8 B. Cahen

Proof. We can verify (1) and (3) by computations based on the formula for κgiven in Section 2. To prove (2), recall that, by [26], Theorem XII.5.6, we haveHχ 6= (0) if and only if

Iχ :=

∫DKχ(Z,Z)−1 dµ(Z) <∞.

Then we have to study the convergence of Iχ. By taking into account the expres-sion of µ given in Section 2, we get

Iχ =

∫D

exp(−γ

4E(y, y, Y, Y )

)Det(In − Y Y )−m−n−2dµL(y, Y )

and, by making the change of variables y → (In − Y Y )1/2y whose Jacobian isDet(In − Y Y ), we find that

Iχ =

∫Cn×B

Det(In − Y Y )−m−n−1 exp(−γ

4(2yty + ytY y + ytY y)

)dy dY.

But by [21], p. 258, we have

Iχ =

(2π

γ

)n ∫B

Det(In − Y Y )−m−n−3/2 dY

for γ > 0. The result then follows from Lemma 3.1

Note that we can deduce from (3) of Proposition 3.2 an explicit but rathercomplicated expression for πχ(g). Now we consider the derived representationdπχ.

Here we use the following notation. If L is a Lie group and X is an element ofthe Lie algebra of L then we denote by X+ the right invariant vector field on Lgenerated by X, that is, X+(h) = d

dt (exp tX)h|t=0 for h ∈ L.By differentiating the multiplication map from P+×Kc×P− onto P+KcP−,

we can easily prove the following result.

Lemma 3.3. Let X ∈ gc and g = z k y where z ∈ P+, k ∈ Kc and y ∈ P−.We have

(1) dζg(X+(g)) = (Ad(z) pp+(Ad(z−1)X))+(z).

(2) dκg(X+(g)) = (pkc(Ad(z−1)X))+(k).

(3) dηg(X+(g)) = (Ad(k−1) pp−(Ad(z−1)X))+(y).

From this, we easily deduce the following proposition (see also [26], p. 515).

Proposition 3.4. For X ∈ gc , f ∈ Hχ and Z ∈ D, we have

dπχ(X)f(Z) = dχ(pkc(e− adZ X)) f(Z)− (df)Z

(pp+(e− adZ X)

).

In particular, we have


(1) If X ∈ p+ then dπχ(X)f(Z) = −(df)Z(X).

(2) If X ∈ kc then dπχ(X)f(Z) = dχ(X)f(Z) + (df)Z([Z,X]).

(3) If X ∈ p− then

dπχ(X)f(Z) = (dχ ◦ pkc)(−[Z,X] +

1

2[Z, [Z,X]]

)f(Z)

− (dfZ ◦ pp+)

(−[Z,X] +

1

2[Z, [Z,X]]

).

Now we need to introduce some notation. As usual, we write Z ∈ D as Z =a(y, Y ) where y = (yj)1≤j≤n ∈ Cn and Y = (ykl)1≤k,l≤n ∈ B. Define

I := {1, 2, . . . , n} ∪ {(k, l) : 1 ≤ k, l ≤ n}

and consider i ∈ I. Then we define ∂i as follows. If i ∈ {1, 2, . . . , n} then ∂i is thepartial derivative with respect to yi and if i = (k, l) then ∂i is the partial derivativewith respect to ykl. Moreover, we say that P (Z) is a polynomial of degree ≤ q ifP (a(y, Y )) is a polynomial of degree ≤ q in the variables yj and ykl.

From the preceding proposition we deduce the following result.

Proposition 3.5. For each X1, X2, . . . , Xq ∈ gc the operator dπχ(X1X2 · · ·Xq)is a sum of terms of the form P (Z)∂i1∂i2 · · · ∂ir where r ≤ q, i1, i2, . . . , ir ∈ I andP (Z) is a polynomial of degree ≤ 2q.

Proof. By Proposition 3.4 we see that, for each X ∈ gc, dπχ(X) is of the formP 0(Z) +

∑i P

i(Z)∂i where P 0(Z), P i(Z) are polynomials of degree ≤ 2. Theresult then follows by induction on q.

4. Generalities on the Stratonovich-Weyl correspondence

In this section, we review some general facts about the Berezin correspondence,the Berezin transform and the Stratonovich-Weyl correspondence.

First at all, recall that the Berezin correspondence on D is defined as follows.Consider an operator (not necessarily bounded) A on Hχ whose domain containseZ for each Z ∈ D. Then the Berezin symbol of A is the function Sχ(A) definedon D by

Sχ(A)(Z) :=〈AeZ , eZ〉χ〈eZ , eZ〉χ

.

We can verify that each operator is determined by its Berezin symbol and thatif an operator A has adjoint A∗ then we have Sχ(A∗) = Sχ(A) [7], [8]. Moreover,for each operator A on Hχ whose domain contains the coherent states eZ for eachZ ∈ D and each g ∈ G, the domain of πχ(g−1)Aπχ(g) also contains eZ for eachZ ∈ D and we have

Sχ(πχ(g)−1Aπχ(g))(Z) = Sχ(A)(g · Z),

10 B. Cahen

that is, Sχ is G-equivariant, see [14]. We have also the following result.

Proposition 4.1. [14]

(1) For g ∈ G and Z ∈ D, we have

Sχ(πχ(g))(Z) = χ(κ(expZ∗g−1 expZ)−1κ(expZ∗ expZ)

).

(2) For X ∈ gc and Z ∈ D, we have

Sχ(dπχ(X))(Z) = dχ(pkc(Ad(ζ(expZ∗ expZ)−1 expZ∗)X

).

Let ξ be the linear form on gc defined by ξ = −idχ on kc and ξ = 0 on p±.Then we have ξ(g) ⊂ R and the restriction ξχ of ξ to g is an element of g∗. Inthe notation of Section 2 we have ξχ = (0, γ,−mϕ0) where ϕ0 ∈ s∗ is defined by

〈ϕ0,( P QQ P

)〉 = iTr(P ).

We denote by O(ξχ) the orbit of ξχ in g∗ for the coadjoint action of G. Thisorbit is said to be associated with πχ by the Kostant-Kirillov method of orbits,see [25], [14]. Moreover, we have the following result.

Proposition 4.2. [14]

(1) For each Z ∈ D, let Ψχ(Z) := Ad∗(exp(−Z∗) ζ(expZ∗ expZ)

)ξχ. Then, for

each X ∈ gc and each Z ∈ D, we have

S(dπχ(X))(Z) = i〈Ψχ(Z), X〉.

(2) For each g ∈ G and each Z ∈ D, we have Ψχ(g · Z) = Ad∗(g) Ψχ(Z).

(3) The map Ψχ is a diffeomorphism from D onto O(ξχ).

In order to make the expression of Ψχ more explicit, we introduce the follow-ing notation. For ϕ ∈ s∗, let α(ϕ) the unique element of s such that 〈ϕ,X〉 =Tr(α(ϕ)X) for each X ∈ s. In particular, one has α(ϕ0) = 1

2

(iIn 00 −iIn

). Moreover,

for u = (x, x) ∈ C2n and u = (y, y) ∈ C2n we have

θ(v × u) =1

2

(−iyxt iyxt

−iyxt iyxt

).

Note also that θ intertwines Ad∗ and Ad. Then we have the following result.

Proposition 4.3. [15] The map ψχ : D → O(ξχ) is given by

ψχ(a(y, Y )) =(−d(y1, y1), γ, ϕ(y, Y )

)where y1 = (In − Y Y )−1(y + Y y) and

ϕ(y, Y ) := −mAd∗(

(In − Y Y )−1/2 (In − Y Y )−1/2Y(In − Y Y )−1/2Y (In − Y Y )−1/2

)ϕ0−

γ

2(y1, y1)×(y1, y1).


Moreover, we have

α(ϕ(y, Y )) = −γ4

(−iy1y

t1 iy1y

t1

−iy1yt1 iy1y

t1

)− m

2i

(A(Y ) B(Y )

−B(Y ) −A(Y )

).

where

A(Y ) :=(In + Y Y )(In − Y Y )−1/2(In − Y Y )−1/2;

B(Y ) :=− 2Y (In − Y Y )−1/2(In − Y Y )−1/2.

Now we recall briefly the construction of the Stratonovich-Weyl correspondence[20], [13], [14]. Denote by L2(Hχ) the space of all Hilbert-Schmidt operators onHχ and by µχ the G-invariant measure on D defined by dµχ(Z) = cχdµ(Z). Thenthe map Sχ is a bounded operator from L2(Hχ) into L2(D, µχ) which is one-to-oneand has dense range [29], [32]. Moreover, the Berezin transform is the operator onL2(D, µχ) defined by Bχ := SχS

∗χ. We can easily verify that we have the following

integral formula for Bχ

(4.1) BχF (Z) =

∫DF (W )

|〈eZ , eW 〉|2χ〈eZ , eZ〉χ〈eW , eW 〉χ

dµχ(W )

(see [7], [32], [33] for instance).

Let ρ be the left-regular representation of G on L2(D, µχ). As a consequenceof the equivariance property for Sχ, we see that Bχ commute with ρ(g) for eachg ∈ G.

Consider the polar decomposition of Sχ: Sχ = (SχS∗χ)1/2Wχ = B

1/2χ Wχ where

Wχ := B−1/2χ Sχ is a unitary operator from L2(Hχ) onto L2(D, µχ). Note that, by

(2) of Proposition 4.2, the measure µ0 := (Ψ−1χ )∗(µχ) is a G-invariant measure on

O(ξχ). The following proposition is then immediate.

Proposition 4.4. 1) The map Wχ : L2(Hχ) → L2(D, µχ) is a Stratonovich-Weyl correspondence for the triple (G, πχ,D), that is, we have

(1) Wχ(A∗) = Wχ(A);

(2) Wχ(πχ(g)Aπχ(g)−1)(Z) = Wχ(A)(g−1 · Z);

(3) Wχ is unitary.

2) Similarly, the mapWχ : L2(Hχ)→ L2(O(ξχ), µ0) defined byWχ(A) = Wχ(A)◦Ψ−1χ is a Stratonovich-Weyl correspondence for the triple (G, πχ,O(ξχ)).

Note that we have relaxed here (1) of Definition 1.1 which is not adapted to thepresent setting since I is not Hilbert-Schmidt. However, this requirement shouldbe hold in some generalize sense, see for instance [22].

12 B. Cahen

5. Extension of the Berezin transform

The aim of this section is to extend the Berezin transform to a class of func-tions which contains Sχ(dπχ(X)) for each X ∈ gc, in order to define and studyWχ(dπχ(X)). This question was already investigated in [14] in the case of a re-ductive Lie group and in [17] in the case of the Jacobi group.

For Z, W ∈ D, we set lZ(W ) := log η(expZ∗ expW ) ∈ p−.

Lemma 5.1. (1) For each Z, W ∈ D and V ∈ p+, we have

d

dteZ(W+tV )

∣∣t=0

= −eZ(W ) (dχ◦pkc)(

[lZ(W ), V ] +1

2[lZ(W ), [lZ(W ), V ]]

).

(2) For each Z, W ∈ D and V ∈ p+, we have

d

dtlZ(W + tV )

∣∣t=0

= pp−

([lZ(W ), V ] +

1

2[lZ(W ), [lZ(W ), V ]]

).

(3) For each i1, i2, . . . , iq ∈ I and Z ∈ D, the function (∂i1∂i2 · · · ∂iqeZ)(W ) isof the form eZ(W )Q(lZ(W )) where Q is a polynomial on p− of degree ≤ 2q.

(4) For each X1, X2, . . . , Xq ∈ gc, the function Sχ(dπχ(X1X2 · · ·Xq))(Z) is asum of terms of the form P (Z)Q(lZ(Z)) where P and Q are polynomials ofdegree ≤ 2q.

Proof. The proof of this lemma is similar to those of Lemma 6.2 of [14] andLemma 5.2 of [17]. Note that the proof of (1) is essentially based on Lemma 3.3,that (3) is a consequence of (1) and (2) and, finally, that (4) follows from (3) andProposition 3.5.

We can then establish the main result of this section.

Proposition 5.2. If q < 14 (−m − 2n) then for each X1, X2, . . . , Xq ∈ gc, the

Berezin transform of Sχ(dπχ(X1X2 · · ·Xq)) is well-defined.

Proof. First, we fix Z ∈ D and we make the change of variables W → gZ ·W inEquation 4.1. Then we obtain

(BχF )(Z) =

∫DF (gZ ·W )〈eW , eW 〉−1

χ dµχ(W ).

We take F = Sχ(dπχ(X1X2 · · ·Xq) and we set Yk := Ad(g−1Z )Xk for k =

1, 2, . . . , q. Then, by G-invariance of Sχ, we have

F (gZ ·W ) = Sχ(dπχ(Y1Y2 · · ·Yq))(W )

for each W ∈ D. Recall that, by the preceding lemma, the function

Sχ(dπχ(Y1Y2 · · ·Yq))(W )


is a sum of terms of the form P (W )Q(lW (W )) where P and Q are polynomialsof degree ≤ 2q. Then we have to prove that, for each q < 1

4 (−m − 2n) and eachpolynomials P and Q of degree ≤ 2q, the integral

I :=

∫DP (W )Q(lW (W )) dµ(W )

is convergent.First, we note that if W = a(y, Y ) then

lW (W ) =

((0,−(In − Y Y )−1(y + Y y)

), 0,

(0 0

−(In − Y Y )−1Y 0

)).

Thus we have

I = cχ

∫DP (y, Y )Q(−(In − Y Y )−1(y + Y y),−(In − Y Y )−1Y )

× exp(−γ

4(2yt(In − Y Y )−1y + yt(In − Y Y )−1Y y + ytY (In − Y Y )−1y)

)×Det(In − Y Y )−m−n−2 dµL(y, Y ).

As in the proof of Proposition 3.2, we make the change of variables y →(In − Y Y )1/2y and we find that

I = cχ

∫DP ((In − Y Y )1/2y, Y )Q(−(In − Y Y )−1/2(y + Y y),−(In − Y Y )−1Y )

× exp(−γ

4(2yty + ytY y + ytY y)

)Det(In − Y Y )−m−n−1 dµL(y, Y ).

Now we make the following remarks.(1) Since P is a polynomial of degree ≤ 2q and B is bounded, there exists a

constant C0 > 0 such that

|P ((In − Y Y )1/2y, Y )| ≤ C0

∑r≤2q

|y|r

for each (y, Y ) ∈ Cn × B.(2) By using the classical formula for the inverse of a matrix, for each Y ∈ B

we have(In − Y Y )−1 = Det(In − Y Y )−1C(In − Y Y )t

where C(A) denotes the cofactor matrix of a matrix A. From this we deduce thatthere exists a constant C ′0 > 0 such that

|Q(−(In − Y Y )−1(In − Y Y )1/2(y + Y y),− (In − Y Y )−1Y )|

≤ C ′0 Det(In − Y Y )−2q∑r≤2q

|y|r

14 B. Cahen

for each (y, Y ) ∈ Cn × B.(3) For each (y, Y ) ∈ Cn × B, we have

2yty + ytY y + ytY y = 2(yty + Re(ytY y)) ≥ 2(1− ‖Y ‖)|y|2.

Here ‖ · ‖ denotes the operator norm corresponding to the Hermitian norm on Cn.By using these remarks, we can reduce the study of the convergence of I to

that of the integral

I ′ :=

∫D

Det(In − Y Y )−2q−m−n−1|y|4qe−γ2 |y|

2(1−‖Y ‖) dµL(y, Y ).

We set

I(Y ) :=

∫Cn|y|4qe−

γ2 |y|

2(1−‖Y ‖) dy

and, passing to spherical coordinates, we see that there exists some constantsC, C ′ > 0 such that, for each Y ∈ B, we have

I(Y ) = C

∫ +∞

0

x4q+2n−1e−γ2 (1−‖Y ‖)x2

dx = C ′(1− ‖Y ‖)−2q−n.

Then we have to study the integral

I ′′ :=

∫B

Det(In − Y Y )−2q−m−n−1(1− ‖Y ‖)−2q−n dY.

Now denote by λs(Y Y ) the maximum of the eigenvalues of Y Y and recall that‖Y ‖2 = λs(Y Y ). Then we have

Det(In − Y Y ) ≤ 1− λs(Y Y ) = 1− ‖Y ‖2 ≤ 2(1− ‖Y ‖)

for each Y ∈ B. Thus we obtain

Det(In − Y Y )−2q−m−n−1(1− ‖Y ‖)−2q−n ≤ 22q+n Det(In − Y Y )−4q−m−2n−1

for each Y ∈ B. But by Lemma 3.1, we see that Jn(−4q −m− 2n− 1) hence I ′′

converges if q < 14 (−m− 2n). This ends the proof.

6. Stratonovich-Weyl symbols of derived representation operators

Here we assume that −m > 2n+4. Then, by Proposition 5.2, Bχ(Sχ(dπχ(X)))is well-defined for each X ∈ gc. We aim to define also Wχ(dπχ(X)) for X ∈ gc.To this goal, we first introduce a space of functions on D which is stable under Bχand contains Sχ(dπχ(X)) for each X ∈ gc.


Recall that, by Proposition 4.2 we have Sχ(dπχ(X))(Z) = iξ(Ad(g−1Z )X) for

each X ∈ gc and Z ∈ D. This leads us to introduce the vector space S generatedby the functions Z → φ0(Ad(g−1

Z )X) where X ∈ gc and φ0 is an element of (gc)∗

which is Ad∗(K)-invariant. Such elements φ0 were determined in [15], see Lemma2.1 above. The following proposition is analogous to Proposition 6.2 of [17].

Proposition 6.1. Let φ : D × gc → C be a function such that(i) For each Z ∈ D, the map X → φ(Z,X) is linear;(ii) For each X ∈ gc, g ∈ G and Z ∈ D, we have φ(g ·Z,X) = φ(Z,Ad(g−1)X).Then

(1) The element φ0 of (gc)∗ defined by φ0(X) := φ(0, X) is fixed by K;

(2) For each X ∈ gc and Z ∈ D, we have

φ(Z,X) = φ0(Ad(g−1Z )X).

We also have

φ(Z,X) =φ0

(Ad(ζ(expZ∗ expZ)−1 expZ∗)X

)=(φ0 ◦ pkc)


).

(3) For each X ∈ gc, the function ψ : D×gc → C given by ψ(·, X) = Bχ(φ(·, X))is well-defined and satisfies (i) and (ii).

(4) The vector space S is generated by all the functions Z → φ(Z,X) for φ asabove and X ∈ gc. Moreover, S is stable under Bχ.

Proof. (1) By (ii), for each k ∈ K and X ∈ gc, we have

(Ad∗(k)φ0)(X) = φ0(Ad(k−1)X) = φ(0,Ad(k−1)X) = φ(k·0, X) = φ(0, X) = φ0(X).

Then φ0 is fixed by K.(2) The first assertion follows from (ii). To prove the second assertion, recall

that by [15], there exists kZ ∈ K such that gZ = exp(−Z∗)ζ(expZ∗ expZ)k−1Z .

Then we have

φ(Z,X) = φ0

(Ad(kZζ(expZ∗ expZ)−1 expZ∗)X

)= φ0


)and, noting that φ0|p± = 0 by Lemma 2.1, we can conclude that

φ(Z,X) = (φ0 ◦ pkc)(Ad(ζ(expZ∗ expZ)−1 expZ∗)X

).

(3) By using the same arguments as in the proof of Proposition 5.2, we canverify that, for each X ∈ gc, the Berezin transform of φ(·, X) is well-defined. Thesecond assertion follows from the fact that Bχ commutes to the ρ(g), g ∈ G.

(4) This follows from the preceding statements.

16 B. Cahen

Now we need the following lemmas.

Lemma 6.2. For each Y ∈ B, we have

I1(Y ) : =

∫Cn

yty exp(−γ

4

(2yty + ytY y + ytY y

))dy

=2

γ

(2π

γ

)nDet(In − Y Y )−1/2 Tr

((In − Y Y )−1

)I2(Y ) : =

∫Cn

ytY y exp(−γ

4


))dy

=2

γ

(2π

γ

)nDet(In − Y Y )−1/2

(n− Tr

((In − Y Y )−1

)).

Proof. For s ∈ [0, 1] and Y ∈ B, let us introduce

Js(Y ) =

∫Cn

exp(−γ

4(2yty + sytY y + sytY y)

)dy.

By [21], p. 258, we have

Js(Y ) =

(2π

γ

)nDet(In − s2Y Y )−1/2.

Then, by computing the derivative of Js(Y ) at s = 1, we get∫Cn

(ytY y+ytY y) exp(−γ

4


))dy

= − 4

γ

(2π

γ


((In − Y Y )−1Y Y

).

Thus we have

(6.1) I2(Y ) + I2(Y ) = − 4

γ

(2π

γ


(−In + (In − Y Y )−1

).

On the other hand, by integrating by parts, we get

J1(Y ) = −∫Cn

yk∂

∂k

(exp

(−γ

4


)))dy

=γ

4

∫Cn

yk(2yk + 2etkY y) exp(−γ

4


))dy

for each k = 1, 2, . . . , k. By summing up over k, we obtain

(6.2) nJ1(Y ) =γ

2(I1(Y ) + I2(Y )).


This last equation implies that I2(Y ) is real since J1(Y ) and I1(Y ) are real. Conse-quently, Equation 6.1 gives the desired value for I2(Y ) hence Equation 6.2 providesthe desired value for I1(Y ).

The following lemma gives a useful expression for Kχ(Z,Z) which will be usedin the proof of Proposition 6.4 .

Lemma 6.3. For each Z = a(y, Y ), let z0 := (In − Y Y )−1(y + Y y). Then wehave

Kχ(Z,Z) = exp(γ

4

(2zt0z0 − zt0Y z0 − zt0Y z0

))Det(In − Y Y )m.

Proof. The result follows from Proposition 3.2 by a routine computation. Alter-natively, by [14], Lemma 4.1, we have

〈eZ ,eZ〉χ = 〈egZ ·0, egZ ·0〉χ = 〈J(gZ , 0)π(gZ)e0, J(gZ , 0)π(gZ)e0〉χ= |J(gZ , 0)|2 = |χ(κ(gZ))|2

and, by taking into account the expressions of χ and gZ , we then recover thedesired formula for Kχ(Z,Z).

Let us introduce the following integral of Hua’s type:

Kn(λ) :=

∫B

Tr((In − Y Y )−1) Det(In − Y Y )λdY.

Since the maximum of the eigenvalues of (In−Y Y )−1 is (1−λs(Y Y ))−1, we have

Tr((In − Y Y )−1

)≤ n(1− λs(Y Y ))−1 ≤ nDet(In − Y Y )−1

and then we see that Kn(λ) converges for λ > 2 since Jn(λ) converges for λ > 1,see Lemma 3.1.

Also, we denote by φ1 and φ2 the elements of S defined by φ10 = (0, 1, 0) and

φ20 = (0, 0, ϕ0). We are now in position to establish the following proposition.

Proposition 6.4. Let

µn :=cχnγ

(2π

γ

)nKn(−m− n− 3

2) ; νn := −1 +

2cχn

(2π

γ

)nKn(−m− n− 3

2).

Let φ ∈ S defined by φ0 = (0, d, λϕ0) with d, λ ∈ C. Let ψ ∈ S such that ψ(·, X) =Bχ(φ(·, X)) for each X ∈ gc. Then we have ψ0 = (0, d, dµn + λνn).

Proof. We will use the formula

ψ0(X) =

∫Dφ0(Ad(g−1

Z )X))Kχ(Z,Z)−1cχdµ(Z)

18 B. Cahen

in order to compute the Berezin transforms ψ1(·, X) and ψ2(·, X) of φ1(·, X) andφ2(·, X).

We write ψ10 = (0, d1, λ1ϕ0) with d1, λ1 ∈ C. Let H1 := ((0, 0), 1, 0). Then we

have Ad(g−1Z )H1 = H1 hence φ1

0(Ad(g−1Z )H1) = 1 for each Z ∈ D. This gives

ψ10(H1) =

∫DKχ(Z,Z)−1cχdµ(Z) = 1.

On the other hand, we also have ψ10(H1) = d1. Then we find d1 = 1.

Now, let H2 :=((0, 0), 0,

(In 00 −In

)). Then, for each Z ∈ D, we have

Ad(g−1Z )H2 =

(?,i

2zt0z0,

((In−Y Y )−1(In+Y Y ) ?

? ?

))where, as usual, z0 = (In − Y Y )−1(y + Y y). Consequently, we have

φ10(Ad(g−1

Z )H2) =i

2zt0z0.

Thus, by Lemma 6.3, we get

ψ10(H2) =

icχ2

∫Dzt0z0 exp

(−γ

4

(2zt0z0 − zt0Y z0 − zt0Y z0

))×Det(In − Y Y )−m−n−2 dy dY

and we make the change of variables y = z0−Y z0 whose Jacobian is Det(In−Y Y ).Hence, by using Lemma 6.2, we obtain

ψ10(H2) =

icχγ

(2π

γ

)nKn

(−m− n− 3

2

).

On the other hand, it is clear that ψ10(H2) = iλ1n. Finally, we find that

λ1 =cχnγ

(2π

γ

)nKn

(−m− n− 3

2

)= µn.

Similarly, we write ψ20 = (0, d2, λ2ϕ0). Since we have φ2

0(Ad(g−1Z )H1) = 0 for

each Z ∈ D, we first obtain d2 = ψ20(H1) = 0. Moreover, for each Z = a(y, Y ) ∈ D,

we also have

φ20(Ad(g−1

Z )H2) =iTr(In − Y Y )−1(In + Y Y )

=i(−n+ 2 Tr

((In − Y Y )−1

)).

Then, changing variables y → (In − Y Y )1/2y, we get

ψ20(Ad(g−1

Z )H2) =− in+ 2icχ

∫B×Cn

exp(−γ

4


))× Tr

((In − Y Y )−1

)Det(In − Y Y )−m−n−1 dy dY.


Thus, by using [21], p. 248, we obtain

ψ20(Ad(g−1

Z )H2) = −in+ 2icχ

(2π

γ

)nKn

(−m− n− 3

2

).

Also, we have ψ20(Ad(g−1

Z )H2) = iλ2n. This gives

λ2 = −1 +2cχn

(2π

γ

)nKn

(−m− n− 3

2

)= νn.

This finishes the proof.

Recall that cχ can be expressed in terms of the Hua’s integral Jn(−m−n− 32 )

which can be explicitely computed, see Proposition 3.2 and Lemma 3.1. However,it seems difficult to compute Kn(−m− n− 3

2 ) similarly.Now we give the matrix of Bχ in a suitable basis of S. First, we consider the

basis of gc consisting of the elements:

Xi = ((ei, 0), 0, 0) ; Yj = ((0, ej), 0, 0) ; Fij =((0, 0), 0,

(0 Eij0 0

));

Gij =((0, 0), 0,

(0 0Eij 0

)); H1 = ((0, 0), 1, 0) ; Aij =

((0, 0), 0,

(Eij 00 −Eji

))

for i, j = 1, 2, . . . , n, Eij denoting the n × n complex matrix whose ij-th entry is1 and all of whose other entries are 0.

Note that φ2(·, Xi) = φ2(·, Yj) = φ2(·, H1) = 0. Then, from the precedingproposition, we easily deduce the following result.

Corollary 6.5. The functions φ1(·, Xi), φ1(·, Yj), φ1(·, H1), φ1(·, Fij), φ1(·, Gij),

φ1(·, Aij), φ2(·, Fij), φ2(·, Gij) and φ2(·, Aij) form a basis for S in which Bχ hasmatrix I2n+1 O O

O I3n2 OO µnI3n2 νnI3n2

.

Recall that for each X ∈ gc, we have Sχ(dπχ(X)) ∈ S. Consequently, we see

that Wχ(dπχ(X)) = B−1/2χ (Sχ(dπχ(X))) is well-defined. Moreover, we have the

following proposition.

Proposition 6.6. For each X ∈ SpanC{H1, Xi, Yj , 1 ≤ i, j ≤ n}, we haveWχ(dπχ(X)) = Sχ(dπχ(X)). For each X ∈ SpanC{Fij , Gij , Aij , 1 ≤ i, j ≤ n}, wehave

Wχ(dπχ(X)) = Sχ(dπχ(X)) + i(1− ν−1/2n )

(γµn

1− νn+m

)φ2(·, X).

20 B. Cahen

Proof. For each X ∈ gc we have

Sχ(dπχ(X)) = dχ(Ad(g−1Z )X) = iγφ1(·, X)− imφ2(·, X).

Now, by using the preceding corollary, we see that the matrix of B−1/2χ with respect

to the above basis of S isI2n+1 O OO I3n2 O

O −µnν−1/2n

1+ν1/2n

I3 ν−1/2n I3n2

.

This implies that for X ∈ {H1, Xi, Yj , 1 ≤ i, j ≤ n}, we have Wχ(dπχ(X)) =Sχ(dπχ(X)) and, for X ∈ {Fij , Gij , Aij , 1 ≤ i, j ≤ n}, we have

Wχ(dπχ(X)) = iγ

(φ1(·, X)− µnν

−1/2n

1 + ν1/2n

φ2(·, X)

)− imν−1/2

n φ2(·, X).

Hence the result follows.

References

[1] S. T. Ali and M. Englis, Quantization methods: a guide for physicists and analysts,Rev. Math. Phys., 17 (2005), pp. 391–490.

[2] J. Arazy and H. Upmeier, Weyl Calculus for Complex and Real Symmetric Domains,Harmonic analysis on complex homogeneous domains and Lie groups (Rome, 2001).Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 13, no3-4 (2002), pp. 165–181.

[3] J. Arazy and H. Upmeier, Invariant symbolic calculi and eigenvalues of invariantoperators on symmeric domains, Function spaces, interpolation theory and relatedtopics (Lund, 2000) pp. 151–211, de Gruyter, Berlin, 2002.

[4] S. Berceanu, A holomorphic representation of the Jacobi algebra, Rev. Math. Phys.,18 (2006), pp. 163–199.

[5] S. Berceanu, A holomorphic representation of the multidimensional Jacobi algebra,arXiv:math/0604381v2.

[6] S. Berceanu and A. Gheorghe, On the geometry of Siegel-Jacobi domains, Int. J.Geom. Methods Mod. Phys., 8 (2011), pp. 1783–1798.

[7] F. A. Berezin, Quantization, Math. USSR Izv., 8 (1974), pp. 1109–1165.

[8] F. A. Berezin, Quantization in complex symmetric domains, Math. USSR Izv., 9(1975), pp. 341–379.

[9] R. Berndt and S. Bocherer, Jacobi forms and discrete series representations of theJacobi group, Math. Z., 204 (1990), pp. 13–44.

[10] R. Berndt and R. Schmidt, Elements of the representation theory of the Jacobi group,Progress in Mathematics 163, Birkhauser Verlag, Basel, 1998.

[11] B. Cahen, Berezin quantization for discrete series, Beitrage Algebra Geom., 51(2010), pp. 301–311.


[12] B. Cahen, Stratonovich-Weyl correspondence for compact semisimple Lie groups,Rend. Circ. Mat. Palermo, 59 (2010), pp. 331–354.

[13] B. Cahen, Stratonovich-Weyl correspondence for discrete series representations,Arch. Math. (Brno), 47 (2011), pp. 41–58.

[14] B. Cahen, Berezin Quantization and Holomorphic Representations, Rend. Sem. Mat.Univ. Padova, 129 (2013), pp. 277–297.

[15] B. Cahen, Global Parametrization of Scalar Holomorphic Coadjoint Orbits of aQuasi-Hermitian Lie Group, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Mathe-matica, 52 (2013), pp. 35–48.

[16] B. Cahen, Stratonovich-Weyl correspondence for the diamond group, Riv. Mat. Univ.Parma, 4 (2013), pp. 197–213.

[17] B. Cahen, Stratonovich-Weyl correspondence for the Jacobi group, Comm. Math.,22 (2014), pp. 31–48.

[18] J. F. Carinena, J. M. Gracia-Bondıa and J. C. Varilly, Relativistic quantum kine-matics in the Moyal representation, J. Phys. A: Math. Gen., 23 (1990), pp. 901–933.

[19] M. Davidson, G. Olafsson and G. Zhang, Laplace and Segal-Bargmann transformson Hermitian symmetric spaces and orthogonal polynomials, J. Funct. Anal., 204(2003), pp. 157–195.

[20] H. Figueroa, J. M. Gracia-Bondıa and J. C. Varilly, Moyal quantization with compactsymmetry groups and noncommutative analysis, J. Math. Phys., 31 (1990), pp. 2664–2671.

[21] B. Folland, Harmonic Analysis in Phase Space, Princeton Univ. Press, 1989.

[22] J. M. Gracia-Bondıa, Generalized Moyal quantization on homogeneous symplecticspaces, Deformation theory and quantum groups with applications to mathemati-cal physics (Amherst, MA, 1990), 93-114, Contemp. Math., 134, Amer. Math. Soc.,Providence, RI, 1992.

[23] J. M. Gracia-Bondıa and J. C. Varilly, The Moyal Representation for Spin, Ann.Physics, 190 (1989), pp. 107–148.

[24] L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Clas-sical Domains, Translations of Mathematical Monographs Vol. 6, American Mathe-matical Society, Providence, Rhode Island, 1963.

[25] A. A. Kirillov, Lectures on the Orbit Method, Graduate Studies in Mathematics Vol.64, American Mathematical Society, Providence, Rhode Island, 2004.

[26] K-H. Neeb, Holomorphy and Convexity in Lie Theory, de Gruyter Expositions inMathematics, Vol. 28, Walter de Gruyter, Berlin, New-York 2000.

[27] T. Nomura, Berezin Transforms and Group representations, J. Lie Theory, 8 (1998),pp. 433–440.

[28] B. Ørsted and G. Zhang, Weyl Quantization and Tensor Products of Fock andBergman Spaces, Indiana Univ. Math. J., 43, 2 (1994), pp. 551–583.

[29] J. Peetre and G. Zhang, A weighted Plancherel formula III. The case of a hyperbolicmatrix ball, Collect. Math., 43 (1992), pp. 273–301.

[30] I. Satake, Algebraic structures of symmetric domains, Iwanami Sho-ten, Tokyo andPrinceton Univ. Press, Princeton, NJ, 1971.

22 B. Cahen

[31] R. L. Stratonovich, On distributions in representation space, Soviet Physics. JETP,4 (1957), pp. 891–898.

[32] A. Unterberger and H. Upmeier, Berezin transform and invariant differential oper-ators, Commun. Math. Phys., 164, 3 (1994), pp. 563–597.

[33] G. Zhang, Berezin transform on compact Hermitian symmetric spaces, ManuscriptaMath., 97 (1998), pp. 371–388.

Received submission date; revised revision date

Berezin transform and Stratonovich-Weyl...

Documents

Transcript of Berezin transform and Stratonovich-Weyl...