Berezin quantization and holomorphic representations

Berezin Quantization and Holomorphic Representations

BENJAMINCAHEN(*)

ABSTRACT- LetGbe a quasi-Hermitian Lie group and letpbe a unitary highest weight representation ofGrealized in a reproducing kernel Hilbert space of holomorphic functions. We study the Berezin symbol map S and the corre- sponding Stratonovich-Weyl mapWwhich is defined on the space of Hilbert- Schmidt operators acting on the space ofp, generalizing some results that we have already obtained for the holomorphic discrete series representations of a semi-simple Lie group. In particular, we give explicit formulas for the Berezin symbols of the representation operatorsp(g) (forg2G) anddp(X) (forXin the Lie algebra ofG) and we show thatSprovides an adapted Weyl correspondence in the sense of [B. Cahen,Weyl quantization for semidirect products,Differ- ential Geom. Appl. 25 (2007), 177-190]. Moreover, in the case whenGis reductive, we prove thatW can be extended to the operatorsdp(X) and we give the ex- pression ofW(dp(X)). As an example, we study the case whenpis a generic unitary representation of the diamond group.

MATHEMATICS SUBJECT CLASSIFICATION (2010). 22E46; 32M10; 81S10; 46E22;

32M15.

KEYWORDS. Berezin quantization; Berezin transform; quasi-Hermitian Lie group;

coadjoint orbit; unitary representation; holomorphic representation; reprodu- cing kernel Hilbert space; coherent states; Weyl correspondence; Stratonovich- Weyl correspondence.

1. Introduction

In [11] and [12], we introduced the notion of adapted Weyl correspond- ence in order to generalize the usual Weyl quantization [1], [28]. Let us consider a connected Lie groupGwith Lie algebra gand a unitary irre-

(*) Indirizzo dell'A.: Universite de Metz, UFR-MIM, DeÂpartement de matheÂ- matiques, LMMAS, ISGMP-BaÃt. A, Ile du Saulcy 57045, Metz cedex 01, France.

E-mail: cahen@univ-metz.fr

ducible representation ofGon a Hilbert spaceH. Assume thatpis asso- ciated with a coadjoint orbitO gofGby the Kirillov-Kostant method of orbits [34], [35]. In [13], we gave the following definition for the notion of adapted Weyl correspondence (see also [31]).

DEFINITION1.1. An adapted Weyl correspondence is an isomorphism W from a vector spaceAof complex-valued smooth functions on the orbitO (called symbols) onto a vector spaceBof (not necessarily bounded) linear operators onHsatisfying the following properties:

(1) the elements ofBpreserve a fixed dense domainDofH;

(2) the constant function1belongs toA, the identity operator I belongs toBand W(1)ˆI;

(3) A2 Band B2 Bimplies AB2 B;

(4) for each f inAthe complex conjugatef of f belongs toAand the adjoint of W(f)is an extension of W(f);

(5) the elements of D are C¹-vectors for the representation p, the functions X (X~ 2g) defined on O by X(j)~ ˆ hj;Xi are in Aand W(iX)~ vˆdp(X)v for each X2gand each v2 D.

A typical example is the case when G is a connected simply-con- nected nilpotent Lie group. Each coadjoint orbit O of G is then dif- feomorphic to R²ⁿ where nˆ1=2dimO, the unitary irreducible re- presentation of G associated with Ocan be realized in L²(Rⁿ) and the usual Weyl correspondence provides an adapted Weyl correspondence onO[4], [44]. Another construction of adapted Weyl correspondences in the nilpotent case can be found in [40]. Let us also mention that in [6] a Weyl correspondence on a finite dimensional coadjoint orbit of some infinite dimensional Lie group was used to recover the magnetic Weyl calculus.

Besides the nilpotent case, adapted Weyl correspondences have been constructed in various situations, in particular for principal series re- presentations [11], [14], and for unitary representations of semi-direct products of the formV^jK whereK is a semi-simple Lie group acting linearly on a vector space V [13], [21], [22]. Note that adapted Weyl correspondences have different applications in harmonic analysis and deformation theory as the construction of covariant star-products on coadjoint orbits [11] and the study of contractions of Lie group unitary representations [25], [17].

Another way to generalize the usual quantization rules is the notion of Stratonovich-Weyl correspondence [42], [29]. J. M. Gracia-BondõÁa, J. C.

VaÁrilly and various collaborators have studied systematically Stratono- vich-Weyl correspondences [29], [24], [27], [30] (see also [10]).

DEFINITION1.2. [29], [30]Let G be a Lie group andpa unitary re- presentation of G on a Hilbert spaceH. Let M be a homogeneous G-space, for instance a coadjoint orbitOassociated withpas above, and letmbe a (suitably normalized) G-invariant measure on M. Then a Stratonovich- Weyl correspondence for the triple(G;p;M)is an isomorphism W from a vector space of operators onHto a space of (generalized) functions on M satisfying the following properties:

(1) W maps the identity operator ofHto the constant function1;

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

(3) Covariance:we have W(p(g)Ap(g) ¹)(x)ˆW(A)(g ¹x);

(4) Traciality:we have Z

M

W(A)(x)W(B)(x)dm(x)ˆTr(AB):

In fact, each of these two notions of Weyl correspondence has ad- vantages and disadvantages. For physical applications and for the Fourier theory of G, it is convenient to use a Stratonovich-Weyl correspondence instead of an adapted Weyl correspondence [27], [10]. On the other hand, adapted Weyl correspondences have the advantage to connect p and O directly, that is suitable to study contractions.

WhenGis a connected semi-simple non-compact real Lie groupGwith finite center andpis a holomorphic discrete series representation ofG, we showed that the Berezin symbol calculus S provides a G-equivariant adapted Weyl correspondence [18]. Moreover,S is an isomorphism from the Hilbert space of all Hilbert-Schmidt operators onH(endowed with the Hilbert-Schmidt norm) onto a space of square-integrable functions onO and, in [20], we obtained a Stratonovich-Weyl correspondenceWby taking the isometric part in the polar decomposition ofS, that is,W:ˆ(SS) ¹⁼²S.

Note that B:ˆSS is the so-called Berezin transform which have been intensively studied, see for instance [26], [37], [38], [43], [45]. The idea of constructing Stratonovich-Weyl correspondences in such a way can be found in [27], see also [2] and [3]. In [20], we proved thatBhenceWcan be extended to a class of functions which containsS(dp(X)) forX2g^cand that the linear formsX!S(dp(X)) andX!W(dp(X)) are proportional. The same results also hold for unitary irreducible representations of a compact semi-simple Lie group, see [16] and [19].

The aim of the present paper is to extend the preceding results to the more general setting of [36], Chapter XII. More precisely, we consider a quasi-Hermitian Lie groupGand a unitary representationpofGwhich is realized in a reproducing kernel Hilbert space of holomorphic functions on some complex domainD. We introduce and study the corresponding Be- rezin calculusSand we compute the Berezin symbols ofp(g) forg2Gand of dp(X) forX2g(Section 4). We show that the coadjoint orbitOasso- ciated withpis diffeomorphic toDand thatSprovides an adapted Weyl correspondence onO(Section 5).

Whengis reductive, we show that the corresponding Berezin transform B can be extended to a space of functions which contains, in particular, S(dp(X)) for eachX2g. This allows us to defineW(dp(X)) forX2g. More precisely, consider the decompositiongˆ Z(g)g₁. . .g_nwhereZ(g) is the center ofgand theg_jare simple ideals. Then we show that for eachj such that g_j is non-compact there exists a constant c_j>0 such that W(dp(X))ˆc_jS(dp(X)) for eachX2g_j(Section 6).

In the general case, it seems difficult to obtain an explicit expression forW(dp(X)). In Section 7 we illustrate the case wheregis solvable by taking G to be the diamond group and p to be a unitary irreducible representation ofGwhich is associated with a generic coadjoint orbit of G. In this case, the Berezin transform has a rather simple expression and we can computeW(dp(X)) for X2g. Then we see in particular that W(dp(X)) and S(dp(X)) are not related in the same way as in the case whereg is reductive.

2. Preliminaries

The material of this section and of the following section is essentially taken from the excellent book of K.-H. Neeb, [36], Chapter VIII and Chapter XII (see also [41], Chapter II and, for the Hermitian case, [32], Chapter VIII and [33], Chapter 6).

Let g be a real quasi-Hermitian Lie algebra, that is, a real Lie al- gebra for which the centralizer in g of the center Z(k) of a maximal compactly embedded subalgebra k coincides with k[36], p. 241. We as- sume that g is not compact. Let g^c be the complexification of g and ZˆX‡iY!Zˆ X‡iY the corresponding involution. We fix a compactly embedded Cartan subalgebra hk, [36], p. 241 and we de- note by h^c the corresponding Cartan subalgebra of g^c. We write D:ˆD(g^c;h^c) for the set of roots ofg^c relative toh^c andg^cˆh^cP

a2D g_a

for the root space decomposition ofg^c. Note thata(h)iRfor eacha2D [36], p. 233. Recall that a root a2D is called compact if a([Z;Z])>0 holds for some element Z2g_a. All other roots are called non-compact [36], p. 235. We write D_k, respectively D_p, for the set of compact, re- spectively non-compact, roots. Note that k^cˆh^c P

a2Dk

g_a [36], p. 235.

Recall also that a subsetD^‡Dis called a positive system if there exists an elementX₀2ihsuch thatD^‡ˆ fa2D : a(X₀)>0ganda(X₀)6ˆ0 for alla2D. A positive system is then said to be adapted if fora2Dk and b2D^‡\Dp we have b(X0)>a(X0), [36], p. 236. Here we fix a positive adapted systemD^‡and we setD^‡_p :ˆD^‡\DpandD^‡_k :ˆD^‡\D_k, see [36], p. 241.

Let G^c be a simply connected complex Lie group with Lie algebra g^c andGG^c, respectively,KG^c, the analytic subgroup corresponding to g, respectively,k. We also setK^c ˆexp (k^c)G^c as in [36], p. 506.

Letp^‡ˆ P

a2D^‡_pg_aandp ˆ P

a2D^‡_pg _a. We denote byP^‡ andP the ana- lytic subgroups ofG^c with Lie algebrasp^‡andp . Then Gis a group of the Harish-Chandra type [36], p. 507, that is, the following properties are satisfied:

(1) g^cˆp^‡k^cp is a direct sum of vector spaces, (p^‡) ˆp and [k^‡;p]p;

(2) The multiplication map P^‡K^cP !G^c, (z;k;y)!zky is a biholo- morphic diffeomorphism onto its open image;

(3) GP^‡K^cP andG\K^cP ˆK.

Moreover, there exists an open connectedK-invariant subsetDp^‡such thatGK^cP ˆexp (D)K^cP , [36], p. 497. We denote byz: P^‡K^cP !P^‡, k: P^‡K^cP !K^candh: P^‡K^cP !P the projections ontoP^‡-,K^c- and P -component. ForZ2p^‡andg2G^cwithgexpZ2P^‡K^cP , we define the elementgZofp^‡bygZ:ˆlogz(gexpZ). Note that we haveD ˆG0.

We also denote byg!gthe involutive anti-automorphism ofG^cwhich is obtained by exponentiatingX!X. We denote byp_p^‡,p_k^c andpp the projections of g^c onto p^‡, k^c and p associated with the direct decom- positiong^c ˆp^‡k^cp .

The G-invariant measure on D is dm(Z):ˆx₀(k(expZexpZ))dm_L(Z) where x₀ is the character on K^c defined by x₀(k)ˆDet_p^‡(Adk) and dm_L(Z) is a Lebesgue measure on D[36], p. 538 (in fact, this result is proved in [36] under the assumption that p^‡ is abelian but by adapting the arguments of [16] we see that the same result holds in the general case).

3. Representations

We keep the notation of Section 2. We fix a unitary characterx ofK.

We also denote by x the extension of x to K^c. We set K_x(Z;W)ˆ x(k(expWexpZ)) ¹ forZ;W2 Dand J_x(g;Z)ˆx(k(gexpZ)) forg2G andZ2 D. We consider the Hilbert spaceH_xof holomorphic functions on Dsuch that

kfk²_x:ˆ

Z

D

jf(Z)j²Kx(Z;Z) ¹dm(Z)5‡ 1

In the rest of the paper, we assume thatHx6ˆ(0). ThenHxcontains the polynomials [36], p. 546. Moreover, the formula

p_x(g)f(Z)ˆJ_x(g ¹;Z) ¹f(g ¹Z)

defines a unitary representation ofGonHxwhich is a highest weight re- presentation with highest weightl:ˆdxj_h^c[36], p. 540.

Note also that Hx is a reproducing kernel Hilbert space. More pre- cisely, there exists a constant c_x>0 such that if we set e_Z(W):ˆ

c_xK_x(W;Z) then we have the reproducing property f(Z)ˆ hf;e_Zi_x for each f 2 Hx and each Z2 D [36], p. 540. Here h;i_x denotes the inner product onHx.

Applying the reproducing property to the constant functionf(Z)ˆ1 we see thatc_x is given by

c_x¹ˆ Z

D

K_x(Z;Z) ¹dm(Z):

The following lemma will be need later.

LEMMA 3.1. Let E₁;E₂;. . .;E_m be a basis of p^‡ such that E_j2g_aj where aj2D^‡_p for jˆ1;2;. . .;m. Then we have l([E_j;Ej])50 for each jˆ1;2;. . .;m.

PROOF. For Z2p^‡, we write ZˆP^m

jˆ1z_jE_j and we setp_j(Z)ˆz_j. We also setjZj ˆ

P^m

jˆ1jz_jj² ₁₌₂

.

First, we fixjˆ1;2;. . .m. We note that ifkˆexpHwithH2hthen we havep_j(Ad(k)Z)ˆe^aj(H)p_j(Z). Then, performing the change of variables

Z!kZˆAd(k)Zin the integral h1;p_ji_x ˆ

Z

D

z_jKx(Z;Z) ¹dm(Z);

we geth1;p_ji_xˆe ^aj(H)h1;p_ji_xfor eachH2hhenceh1;p_ji_xˆ0.

Consequently, we can choose an orthonormal basis (fp)_p0 ofHx such that f0(Z)ˆ1=k1k_xˆc¹⁼²x and f1(Z)ˆ kp_jk ¹p_j(Z). It is well-known that the reproducing kernel ofHxis also given bye_Z(W)ˆP

l0f_l(Z)f_l(W). Then we have

eZ(Z)ˆcxx(k(expZexpZ)) ¹cx‡ kp_jk ²jz_jj² (3:1)

for eachjˆ1;2;. . .;m.

On the other hand, by adapting the arguments of [15], Section 4, we get k(expZexpZ)ˆexp (p_k^c([Z;Z])‡O(jZj³)) forZclose to 0. Then we have

c_xx(k(expZexpZ)) ¹ˆc_x(1 p_k^c([Z;Z])‡O(jZj³)):

(3:2)

Thus, combining (3.1) and (3.2) and applying toZˆz_jE_jforz_jclose to 0, we obtain

l([E_j;E_j])jz_jj²‡O(jz_jj³)c_x¹kp_jk ²jz_jj²:

This implies thatl([E_j;E_j])50 for eachjˆ1;2;. . .;m. p Now, we give an explicit expression for the derived representationdp_x. IfLis a Lie group andX is an element of the Lie algebra ofL then we denote byX^‡the right invariant vector field onLgenerated byX, that is, X^‡(h)ˆ d

dt(exptX)hj_tˆ0 forh2L.

By differentiating the multiplication map from P^‡K^cP onto P^‡K^cP , we can easily prove the following result.

LEMMA3.2. Let X2g^cand gˆz k y where z2P^‡;k2K^cand y2P . We have

(1) dz_g(X^‡(g))ˆ(Ad(z)p_p^‡(Ad(z ¹)X))^‡(z).

(2) dkg(X^‡(g))ˆ(p_k^c(Ad(z ¹)X))^‡(k).

(3) dh_g(X^‡(g))ˆ(Ad(k ¹)pp (Ad(z ¹)X))^‡(y).

From this, we deduce the following proposition (see also [36], p. 515).

PROPOSITION3.3. For X 2g^c and f 2 Hx, we have dpx(X)f(Z)ˆdx(p_k^c(Ad((expZ) ¹)X))f(Z) (df)_Z adZ

1 e ^adZp_p^‡(e ^adZX) :

In particular, ifgis reductive then

(1) If X 2p^‡then dp_x(X)f(Z)ˆ (df)_Z(X).

(2) If X 2k^c then dpx(X)f(Z)ˆdx(X)f(Z)‡(df)_Z([Z;X]).

(3) If X 2p then dp_x(X)f(Z)ˆ dx([Z;X])f(Z) 1

2(df)_Z([Z;[Z;X]]).

4. Berezin symbols

In this section, we introduce the Berezin calculus onD[8], [9] and we give explicit expressions for the Berezin symbols of p_x(g),g2G and of dpx(X),X2g^c following the same lines as in [18].

Consider an operator (not necessarily bounded)AonHxwhose domain containse_Z for eachZ2 D. Then the Berezin symbol ofAis the function S_x(A) defined onDby

Sx(A)(Z):ˆhA e_Z;e_Zi_x heZ;eZi_x :

We can verify that an operator is determined by its Berezin symbol and that if an operatorAhas adjointAthen we haveSx(A)ˆSx(A) [8], [23].

Moreover, we have the following equivariance property.

LEMMA 4.1. (1) For each g2G and each Z2 D, we have p_x(g)e_Zˆ J_x(g;Z) ¹e_gZ.

(2) Let A be an operator on H_x whose domain contains the coherent states eZfor each Z2 D. Then, for each g2G, the domain ofpx(g ¹)Apx(g) also contains eZfor each Z2 Dand we have

S_x(p_x(g) ¹Ap_x(g))(Z)ˆS_x(A)(gZ) (4:1)

for each g2G and Z2 D.

PROOF. (1) Forg2GandZ;W2 D, we have

he_W;p_x(g)e_Zi_xˆ hp_x(g ¹)e_W;e_Zi_xˆ(p_x(g ¹)e_W)(Z)

ˆJx(g;Z) ¹eW(gZ)ˆJx(g;Z) ¹heW;egZi_x: Then we havepx(g)eZˆJx(g;Z) ¹egZ:

(2) This is an immediate consequence of (1). p

PROPOSITION4.2. (1)For g2G and Z2 D, we have Sx(px(g))(Z)ˆx k(expZg ¹expZ) ¹k(expZexpZ)

: (2) For X2g^c and Z2 D, we have

Sx(dpx(X))(Z)ˆdx p_k^c(Ad(z(expZexpZ) ¹expZ)X) :

In particular, ifp^‡is abelian and X2k^c, we have

Sx(dpx(X))(Z)ˆdx X‡p_k^c[logh(expZexpZ);[X;Z]]

:

PROOF. (1) We have

hpx(g)e_Z;e_Zi_xˆ(px(g)e_Z)(Z)ˆJ(g;Z) ¹e_gZ(Z)

ˆc_xx(k(gexpZ)) ¹x(k(exp (gZ)expZ)) ¹: Now, we write

gexpZˆexp (gZ)k(gexpZ)h(gexpZ) Then we have

expZgexpZˆh(gexpZ)k(gexpZ)exp (gZ)expZ:

Thus, sincegˆg ¹, we get

k(expZg ¹expZ)ˆk(gexpZ)k(exp (gZ)expZ) This gives

hpx(g)eZ;eZi_xˆcxx(k(expZg ¹expZ)) ¹ hence the result.

(2) The result easily follows from the computation of the derivative of the functionSx(px(exptX))(Z) attˆ0 by means of Lemma 3.2. p

5. Adapted Weyl correspondence

In this section, we show that the Berezin calculusA!Sx(A) gives an adapted Weyl correspondence on the coadjoint orbit of G associated withp_x.

Recall thatxis a unitary character ofK. Then the linear extension of dx2ktok^c(also denoted bydx) is zero ong_afor eacha2Dc. Moreover we havedx(h)iR. Consider the linear formjong^cdefined byjˆ idxonk^c andjˆ0 onp. Then we havej(g)Rand the restrictionj₀ofjtogis an element ofg. We denote byO(j₀) the orbit ofj₀ in g for the coadjoint action ofG. Then we have the following result.

PROPOSITION5.1.

(1) For each X 2g^c and each Z2 D, we have S(dp_x(X))(Z)ˆihc_x(Z);Xi wherec_x(Z):ˆAd exp ( Z)z(expZexpZ)

j₀:

(2) For each g2G and each Z2 D, we havec(gZ)ˆAd(g)c(Z).

(3) The mapcis a diffeomorphism fromDontoO(j₀).

PROOF. Statement (1) is an immediate consequence of (2) of Proposi- tion 4.2, and Statement (2) follows from Statement (1) and Lemma 4.1. In order to prove Statement (3), we first show that for Z2 D, the equality c_x(Z)ˆj₀impliesZˆ0.

Assume that c_x(Z)ˆj₀. Then we have Ad(z(expZexpZ))j₀ˆ Ad(expZ)j₀. This implies that

hj₀;Ad(z(expZexpZ) ¹)Zi ˆ hj₀;Ad(exp ( Z))Zi:

Thus, taking into account thatj₀j_pˆ0, we find thathj₀;[Z;Z]i ˆ0. Ap- plying Lemma 3.1, we getZˆ0.

Now, suppose thatc_x(Z)ˆc_x(Z⁰) for someZ;Z⁰2 D. Chooseg;g⁰2G such that g0ˆZ and g⁰0ˆZ⁰. By (2), we have Ad(g)j₀ˆAd(g⁰)j₀. Then we have Ad(g ¹g⁰)j₀ˆj₀and by (2) again we getc_x((g ¹g⁰)0)ˆj₀. By using the assertion already proved, we obtain that (g ¹g⁰)0ˆ0. Thus we haveg ¹g⁰2K^cP \GˆK. HenceZˆg0ˆg⁰0ˆZ⁰. This proves thatc_xis injective. By using (2) we see thatc_xis also surjective hence bi- jective. It remains to show thatc_xis regular. Using (2) again, it is sufficient to verify thatc_xis regular atZˆ0. But, using Lemma 3.2, we easily obtain that

h(dc_x)₀(V);Xi ˆ hj₀;[X;V V]i

for eachV 2p^‡ and eachX2g^c. Assuming that (dc_x)₀(V)ˆ0 for some V 2p^‡and takingXˆVin the preceding equality, we gethj₀;[V;V]i ˆ0

henceV ˆ0 by Lemma 3.1. p

6. Stratonovich-Weyl correspondence

We retain the notation from the previous sections. We denote byL2(Hx) the space of all Hilbert-Schmidt operators onH_x. It is well-known that the mapS_xis a bounded operator fromL₂(H_x) intoL²(D;m) which is one-to-one and has dense range [39], [43].

The Berezin transform is the operator onL²(D) defined byBx:ˆSxS_x or, equivalently, by the integral formula

BxF(Z)ˆ Z

D

F(W) jhe_Z;e_Wij²_x

heZ;eZi_xheW;eWi_x dm(W) (6:1)

(see [8], [43], [45] for instance). As a consequence of the equivariance property forS_x,B_xcommute withr(g) for eachg2G, whereris the left- regular representation ofGonL²(D;m_x).

Now, we introduce the polar decomposition ofSx:Sxˆ(SxS_x)¹⁼²Wx ˆ B¹⁼²x Wx where Wx:ˆBx¹⁼²Sx is a unitary operator from L2(Hx) onto L²(D;m). We immediately obtain the following proposition which is analo- gous to Theorem 3 of [27] and Proposition 3.1 of [20]. Note that, by (2) of Proposition 5.1, the measure m₀:ˆ(c_x¹)(m) is a G-invariant measure on O(j₀).

PROPOSITION6.1. 1)The map Wx:L2(Hx)!L²(D;m_x)is a Stratono- vich-Weyl correspondence for the triple(G;px;D), that is, we have

(1) W(A)ˆW(A);

(2) W(p(g)Ap(g) ¹)(Z)ˆW(A)(g ¹Z);

(3) Wxis unitary.

2) Similarly, the map W_x:L₂(H_x)!L²(O(j₀);m₀) defined by Wx(A)ˆWx(A)c ¹ is a Stratonovich-Weyl correspondence for the triple(G;px;O(j₀)).

Note that here we have relaxed the requirement that W_x maps the identity operatorIto the constant function 1 (see Definition 1.2) since it is not adapted to the present setting whereIis not Hilbert-Schmidt. How- ever, this requirement should be hold in some generalize sense, see [29].

In the rest of this section, we assume thatgis reductive. Thenp^‡and p are abelian, [p^‡;p ]k^c [36], p. 241 andDis bounded [36], p. 504. We will extend the Berezin transform to a class of functions which contains Sx(dpx(X)) forX2g^c in order to define and studyWx(dpx(X)).

We introduce some additional notation. Let (Ea)_a2D^‡_p be a basis forp^‡ such thatE_a2g_aanda([E_a;E_a])ˆ2 for eacha2D^‡_p. Leta₁;a₂;. . .;a_nbe an enumeration of D^‡_p. We write ZˆPⁿ

kˆ1z_kEak for the decomposition of Z2p^‡ in the basis (E_ak). If f is a holomorphic function on D, then we denote by@_kf the partial derivative off with respect toz_k. We say that a function f(Z) on D is a polynomial of degree q in the variable Z if f Pⁿ

kˆ1zkEak

is a polynomial of degree q in the variables z1;z2;. . .;zn. For Z;W2 D, we setl_Z(W):ˆlogh(expZexpW)2p .

LEMMA6.2.

(1) For Z;W 2 Dand V2p^‡, we have d

dte_Z(W‡tV)

tˆ0ˆ e_Z(W)dx([l_Z(W);V]):

and, for Z;W 2 Dand V2p^‡, we have d

dtlZ(W‡tV)

tˆ0ˆ1

2[lZ(W);[lZ(W);V]]:

(2) The function (@_k1@_k2 @_kqe_Z)(W) is of the form e_Z(W)P(l_Z(W)) where P is a polynomial of degreeq.

(3) For each X₁;X₂;. . .;X_q2g^c, the operator dp_x(X₁X₂ X_q)is a sum of terms of the form P(Z)@k1@k2 @kqwhere P is a polynomial in Z of degree2q.

(4) Each holomorphic differential operator on D with polynomial coefficients has Berezin symbol. In particular, for each X₁;X₂;. . .;X_q2g^c, S_x(dp_x(X₁X₂ X_q))is well-defined and is a sum of terms of the form P(Z)Q(l_Z(Z))where P is a polynomial of degree 2q and Q is a polynomial of degreeq.

PROOF. We verify (1) by a routine computation using Lemma 3.2. (2) is deduced from (1) by induction onqand (3) follows from (2) and Proposition 3.3. Finally, (4) is a direct consequence of (2) and (3). p Letg₁;g₂;. . .;g_rbe a maximal orthogonal subset ofD^‡_p as in [36], p. 503.

We set qx:ˆ Max1sr(l‡l0)([E_gr;Eg_r]) where l0ˆdx₀j_h^c. The follow- ing proposition is analogous to Proposition 4.1 of [20].

PROPOSITION 6.3. If qqx then for each X1;X2;. . .;Xq2g^c, the Berezin transform of Sx(dpx(X1X2 Xq))is well-defined.

PROOF. We imitate the proof of Proposition 4.1 of [20]. We fixZ2 D and we choose g2G such that g0ˆZ. By performing the change of variablesW!gWin the integral (6.1) we get

(B_xF)(Z)ˆ Z

D

F(gW)he_W;e_Wi_x¹dm(W)

ˆ Z

D

F(gW)c_x(x:x₀)(k(expWexpW))dm_L(W) (6:2)

Note that if FˆS_x(dp_x(X₁X₂ X_q)) then we have F(gW)ˆ Sx(dpx(Y1Y2 Yq))(W) whereYk:ˆAd(g_Z¹)Xkforkˆ1;2;. . .;q. Now we verify that the conditionqqximplies that the function

W!S_x(dp_x(Y₁Y₂ Y_q))(W)(x:x₀)(k(expWexpW))

is bounded hence integrable on the bounded domainD. By Lemma 6.2, the functionS_x(dp_x(Y₁Y₂ Y_q)) is a sum of terms of the formP(W)Q(l_W(W)) wherePis a polynomial andQis a polynomial of degreeq. By [36], p. 504, each W2 D can be written as WˆAd(k) P^r

kˆ1tsEg_s

where k2K and ts2] 1;1[ forsˆ1;2;. . .r. From the orthogonality of the rootsg_s, we get

logh(expWexpW))ˆAd(k) X^r

sˆ1

t_s 1 t²_s E_gs

! (6:3)

and

k(expWexpW)ˆkexp X^r

sˆ1

log 1

1 t²_s [E_gs;Eg_s]

! k ¹: (6:4)

Then we have

(x:x₀)(k(expWexpW))ˆY^r

sˆ1

(1 t²_s) ^{(l‡l0)([Egs;Egs])}: Thus we see that the conditionqqximplies that the functions

W!P(W)Q(lW(W))(x:x₀)(k(expWexpW))

are bounded hence the result. p

In particular, ifqx1 then the Berezin transform ofSx(dpx(X)) is well- defined for X2g^c. In the rest of this section, we studyBxSx(dpx(X)) in order to defineWx(dpx(X)).

Sincegis reductive, we have the direct decompositiongˆg₀g₁ g_N where g₀ˆ Z(g), the ideals g₁;. . .;g_m are non-compact and the ideals g_m‡1;. . .;g_Nare compact.

Let jˆm‡1;. . .;N. Note that g_jk. Then, since x is a unitary

character ofK, we havel(X)ˆdx(X)ˆ0 for eachX2g_j. Moreover, for X2g_jandZ2p^‡, we have [X;Z]2g_j\p^‡k^c\p^‡ˆ(0). Therefore, by using Proposition 3.3 we see thatdp_x(X)ˆ0 for eachX2g_j.

The following proposition is the generalization of Proposition 5.2 in [20].

PROPOSITION6.4. The Berezin transform maps Sx(dpx(g^c))onto itself and for each jˆ0;1;. . .;m there exists a constant c_j>0 such that B_xS_x(dp_x(X))ˆc_jS_x(dp_x(X))for each X 2g^c_j.

PROOF. First we consider the linear formbxdefined ong^c by (6:5) bx(X):ˆBxSx(dpx(X))(0)ˆ

Z

D

Sx(dpx(X))(Z)x(k(expZexpZ))dm(Z):

Changing variablesZ!k ¹Zin this integral we see thatb_x(Ad(k)X)ˆ bx(X) for each k2K and each X2g^c. In particular, for X2g_a and kˆexpH with H2h we get e^a(H)bx(X)ˆbx(X). Then we see that b_x(X)ˆ0 for eachX2g_a,a2D. Moreover, for each H2h^c, we have b_x(H)ˆc_x¹l(H)‡

Z

D

l([logh(expZexpZ);[H;Z]])x(k(expZexpZ))dm(Z):

Now, as in [36], p. 552, we consider the decompositionD1D2 Dm

corresponding to the direct decomposition gˆg₀g₁ g_N. For jˆ1;2;. . .;m, we also seth_j:ˆh\g_j,l_j ˆlj_hj, etc. Then for eachH2h_j we have

bx(H)ˆc_x¹lj(H)‡ Z

D_j

lj([logh(expZ_j expZj);[H;Zj]])x_j(k(expZ_jexpZj))dm_j(Zj);

with obvious notation. Therefore, by [20], Proposition 5.1, there exists a constantc_j>0 such thatb_x(H)ˆc_jl_j(H) for eachH2h_j. This implies that Bx(Sx(dpx(X)))(0)ˆcjSx(dpx(X))(0) for eachX2g_j. Finally, replacingXby Ad(g ¹)Xand applying equivariance (see Lemma 4.1) we obtain the desired

result. p

As a consequence of this proposition, we can define Bx¹⁼² on the space of functions fSx(dpx(X)):X2g^cg and then Wx on the space fdpx(X):X2g^cg. More precisely, for each jˆ0;1;. . .;m we have Wx(dpx(X))ˆc_j ¹⁼²Sx(dpx(X)) for each X2g^c_j. Note that c0ˆ1 but the explicit computation of the constants c_j for j>0 is not easy in general, see Section 6 of [20] for the case GˆSU(p;q).

7. Example: the diamond group

In this section we show by considering the case of the diamond group that, in general, one cannot expect to get results similar to those obtained in Section 6 forgreductive.

We fix an integern1. The diamond (real) Lie algebragis the semi- direct product ofRwith the (2n‡1)-dimensional Heisenberg Lie algebra.

More precisely, g has basisfH;Z;~ X₁;. . .;X_n;Y₁;. . .;Y_ng, the non-trivial brackets being given by

[H;Xk]ˆ Yk; [H;Yk]ˆXk; [Xk;Yk]ˆZ:~

LetGbe the connected and simply connected (real) Lie group with Lie algebrag. ThenGis a non-exponential solvable Lie group. Eachg2Gcan be written uniquely as

gˆexptHexpX

k

x_kX_k‡X

k

y_kY_k‡sZ~

wheret;s;x_kandy_kare real numbers. Then we denotegˆ(t;s;(x_k);(y_k)).

The group law ofGis given by

(t;s;(x_k);(y_k)):(t⁰;s⁰;(x⁰_k);(y⁰_k))ˆ(t⁰⁰;s⁰⁰;(x⁰⁰_k);(y⁰⁰_k)) where

t⁰⁰ˆt‡t⁰ s⁰⁰ˆs‡s⁰‡1

2 X

k

cost⁰(x_ky⁰_k x⁰_ky_k) sint⁰(x_kx⁰_k‡y_ky⁰_k) x⁰⁰_kˆx⁰_k‡x_kcost⁰ y_ksint⁰

y⁰⁰_kˆy⁰_k‡x_ksint⁰‡y_kcost⁰:

LetfH;Z~;X₁;. . .;X_n;Y₁;. . .;Y_ngbe the dual basis ofg. The coadjoint action of gˆ(t;s;(xk);(yk))2G on jˆ[d;c;(ak);(bk)]:ˆdH‡cZ~‡

P

k akX_k‡P

k bkYis given by Ad(g)jˆ[d⁰;c⁰;(a⁰_k);(b⁰_k)] wherec⁰ˆcand d⁰ˆd‡c

2 X

k

x²_k‡y²_k

‡X

k

a_ky_k b_kx_k a⁰_kˆa_kcost‡b_ksint c(x_ksint y_kcost) b⁰_kˆ a_ksint‡b_kcost c(x_kcost‡y_ksint):

From this, it follows that if a coadjoint orbit ofGcontains a point of the form [d;0;(a_k);(b_k)] then it is trivial or diffeomorphic toRS¹. Moreover, if a coadjoint orbit contains a point [d;c;(ak);(bk)] withc6ˆ0 then it has dimension 2nand contains a unique point of the form [d;c;(0);(0)] with c6ˆ0. Such an orbit is called generic.

In the rest of this section, we fix j₀ˆ[d₀;c₀;(0);(0)]2g such that c₀6ˆ0 and we consider the generic coadjoint orbitO(j₀) ofj₀. We assume thatc0>0 (the case wherec050 can be treated similarly).

The subalgebra hˆspanfH;Zg~ is a compactly embedded Cartan subalgebra ofg. Here we havekˆh. We choose the positive root to bea defined by a(H)ˆ i. Then we see that p^‡ has basis (E_j)_1jn where E_jˆ1=2(X_j iY_j). We writeZˆP

j z_jE_j for the decomposition ofZ2p^‡ in this basis.

TheP^‡K^cP -decomposition ofgˆ(t;s;(xk);(yk))2Gis given by gˆexpX

k

a_k(X_k iY_k) exp (uH‡vZ) exp~ X

k

b_k(X_k‡iY_k) where uˆt, vˆs (i=4)P

k (x²_k‡y²_k), a_kˆ1=2(x_k‡iy_k)e ^it and b_k ˆ1=2(x_k iy_k).

From this, we deduce that ifZˆP

j zjEj then we have k(expZexpZ)ˆexpi

2jZj²Z~ where jZj ˆ P

j jzjj²₁₌₂

. Moreover the action of (t;s;(xk);(yk))2G on ZˆP

j zjEj2 Dis given bygZˆP

j (zj‡xj‡iyj)e ^itEj. Note that here we haveD ˆp^‡.

Since we have [H;E_j]ˆ iE_j for eachjˆ1;2;. . .;n, we immediately see that Tr_p^‡ad(uH‡vZ)~ ˆ inu for each u;v2C. The character x₀ is thenx₀(exp (uH‡vZ))~ ˆe ^inu. In particular we havex₀(k(expZexpZ))ˆ1 for each Z2p^‡. The G-invariant measure on D ˆp^‡ is then the Le- besgue measure m_L. Here we normalizem_L as follows. We take dm_L(Z)ˆ (2p) ⁿcⁿ₀dx₁dy₁ dx_ndy_n whereZˆP

j (x_j‡iy_j)E_j,x_j;y_j2R.

We consider the character x of K defined by x(exp (uH‡vZ))~ ˆ e^i(d0u‡c0v)foru;v2R. Then we havedxj_k ˆij₀j_k. We can easily verify that

eZ(W)ˆx ¹(k(expZexpW))ˆe^c20hW;Zi with the notationhW;Zi:ˆP

j z_jw_jforZˆP

j z_jE_jandWˆP

j w_jE_jinp^‡. We also have

J(g;Z)ˆexpi(d0t‡c0s) exp c₀ 2

X

k

zk(xk iyk)‡c₀ 4

X

k

(x²_k‡y²_k)

!

for Z2p^‡ and gˆ(t;s;(x_k);(y_k))2G. Note that we have c_xˆ1 here.

Therefore,p_xis given by

px(g)f(Z)ˆexpi(d0t‡c0s) exp c0

4 X

k

(x²_k‡y²_k) expc0

2 X

k

zk(xk iyk)e^it fX

k

(zke^it xk iyk)Ek

forZ2p^‡,gˆ(t;s;(x_k);(y_k))2Gandf 2 H_x. Note that the norm ofH_xis defined by

kfk²_xˆ Z

p^‡

jf(Z)j²e ^c20jZj2dm_L(Z):

We can verify that p_x can be also obtained from O(j₀) by using the general method of [5], taking the positive polarization span_CfH;Z;~ X_j‡iY_j;1jng at j₀, see [7], p. 189-190.

By differentiatingpx we get

S_x(dp_x(H))(Z)ˆid₀‡1 2ic₀jZj² Sx(dpx(Z))(Z)~ ˆic₀

S_x(dp_x(X_k))(Z)ˆ1

2c₀(z_k z_k) S_x(dp_x(Y_k))(Z)ˆ i

2c₀(z_k‡z_k):

This gives c(Z)ˆ

d0‡1

2c0jZj²

H‡c0Z~‡X

k

c0z_k z_k

2i X_k X

k

c0z_k‡z_k 2 Y_k:

Note that c(Z)ˆAd(g)j₀ where gˆ(t;0;Re(e^itZ);Im(e^itZ)). Note also that the Berezin symbol ofpx(g) is

S_x(p_x(g))(Z)ˆexpi(d₀t‡c₀s) exp c₀

2jZj²

exp c₀

4 X

k

(x²_k‡y²_k)

exp c₀

2e^itX

k

z_k(x_k iy_k)

exp c₀

2e^itjZj²

exp c₀

2 X

k

(x_k‡iy_k)z_k

forgˆ(t;s;(x_k);(y_k))2GandZ2p^‡. The Berezin transform is given by

Bx(F)(Z)ˆ Z

p^‡

e^c20(hW;Zi‡hZ;Wi jZj² jWj²)F(W)dm_L(W)

ˆ Z

p^‡

e ^c20jWj2F(Z W)dm_L(W):

In particular, we consider the function F_g;U defined by F_g;U(Z)ˆ e^igRehZ;Ui forg2CandU2p^‡ such thatjUj ˆ1. Then we have

Bx(F_g;U)(Z)ˆF_g;U(Z) Z

p^‡

e ^c20jWj²e ^igRehW;Widm_L(W)ˆe ^g2=2c0F_g;U(Z):

On the other hand, let us introduce the LaplacianL:ˆ4Pⁿ

kˆ1

@²

@z_k@z_k. Clearly L(F_g;U)ˆ g²F_g;U. ThenB_x(F_g;U)ˆexp (L=2c₀)(F_g;U) for eachg;Uand we have recovered the well-known formula B_xˆexp (L=2c₀), see the in- troduction of [43] for instance. We are now in position to compute Wx(dpx(X)) forX2g. Indeed, one has

Wx(dpx(X))ˆB_x¹⁼²Sx(dpx(X))ˆexp ( L=4c0)Sx(dpx(X)):

Then we immediately obtainWx(dpx(X))ˆSx(dpx(X)) forXˆX1;. . .;Xn; Y1;. . .;Yn;Z~and

Wx(dpx(H))ˆSx(dpx(H)) 1

4c0 L(Sx(dpx(H)))ˆSx(dpx(H)) 1 2in:

In particular, we cannot find a basis (X_j) of g for which there exists a family (c_j) of real numbers such thatWx(dpx(X_j))ˆc_jSx(dpx(X_j)) for each j, as this is the case wheng is reductive, see Section 6.

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Manoscritto pervenuto in redazione il 19 Marzo 2012.