Berezin transform and Stratonovich–Weyl correspondence for the multi-dimensional Jacobi group

DOI 10.4171/RSMUP/136-17

Berezin transform and Stratonovich–Weyl correspondence for the multi-dimensional Jacobi group

(addendum)

Benjamin Cahen ()

Abstract – We extend the results of [13] to the holomorphic representations of the non- scalar type of the multi-dimensional Jacobi group.

Mathematics SubjectClassification (2010).22E27, 32M10, 46E22.

Keywords.Berezin quantization, Berezin transform; quasi-Hermitian Lie group, multi- dimensional Jacobi group, unitary representation, holomorphic representation of the non-scalar type, Stratonovich–Weyl correspondence.

1. Introduction

We use the notation of [13], Section 2. In particular, we denote byG the multi- dimensional Jacobi group and byKthe subgroup ofG consisting of all elements of the form .0; 0/; c; ^{P 0}_{0 Px}

wherec2RandP 2U.n/.

Recall that the unitary representations ofGconsidered in [13] are holomorphi- cally induced from a unitary character ofK. Here we consider, more generally, the unitary representations ofGwhich are holomorphically induced from a uni- tary representation ofK, see [18], p. 515, and we extend the results of [13] to these representations. The main tool is then the generalized Berezin calculus for a reproducing kernel Hilbert space of vector-valued holomorphic functions, see [2], [17] and [12]. Most of the proofs are similar to those of [13] so we just sketch them briefly.

() Indirizzo dell’A.: Université de Lorraine, Site de Metz, UFR-MIM, Département de mathématiques, Bâtiment A, Ile du Saulcy, CS 50128, F-57045, Metz cedex 01, France E-mail:[email protected]

2. Representations

Let 2Rand let₀be a unitary irreducible representation ofU.n/on a (finite- dimensional) complex vector space ^V. Let be the representation of K on ^V defined by

.0; 0/; c; ^{P 0}₀ Px

De^{i c}0.P /:

We also denote by0andthe corresponding representations of GLn.C/and K^c.

LetMn.C/Dn^C˚h₀˚n be the usual triangular decomposition ofMn.C/. Then0is associated with a dominant integral weight of the form

ƒm1;m2;:::;mnWDiag.a1; a2; : : : ; an/ !

n

X

iD1

miai

wherem1 m2 mnandmi 2Z[15], p. 274. LetmDPn

iD1mi. Then we have0.zIn/Dz^mI^Vfor eachz2C.

Recall that for eachy 2CⁿandY 2Mn.C/such thatY^t DY, we denote a.y; Y /WD .y; 0/; 0; ^{0 Y}_{0 0}

2p^C and

DWD ¹a.y; Y /2p^CWIn YY > 0º Šx CⁿB where^BWD ¹Y 2Mn.C/WY^t DY; In YY > 0ºx .

Now we will apply the general considerations of [18] and [12] to the partic- ular case of the multi-dimensional Jacobi group. Following [18], p. 497, we set K.Z; W /WD..expWexpZ// ¹forZ; W 2DandJ.g; Z/WD..gexpZ//

forg 2G andZ 2 Dand we introduce the Hilbert space^Hof all holomorphic functions on^Dwith values in^Vsuch that

kfk²H WD Z

D

hK.Z; Z/ ¹f .Z/; f .Z/iVd.Z/ <C1 wheredenotes theG-invariant measure on^Ddefined in [13], Section 2.

Letbe the unitary representation ofGon^Hdefined by ..g/f /.Z/DJ.g ¹; Z/ ¹f .g ¹Z/:

In [12], we verified that is obtained by holomorphic induction from, that is, corresponds to the natural action ofG on the square-integrable holomorphic sections of the HilbertG-bundleGVoverG=K ŠD. Moreover,is irreducible sinceis irreducible, see [18], p. 515.

The evaluation mapsK_ZWH ! V; f ! f .Z/ are continuous [18], p. 539.

The vector coherent states of^H are the maps E_Z D K_ZWV ! H defined by hf .Z/; vi^VD hf; EZvi^Hforf 2Handv 2V.

We have the following result, see [18], p. 540.

Proposition2.1. (1) There exists a constantc > 0such that E_ZE_W DcK.Z; W /

for eachZ; W 2D. (2) Forg2GandZ 2D,

EgZ D.g/EZJ.g; Z/:

Now we give explicit formulas forKandJ and we computec(see Lemma 3.1 in [13]).

Proposition2.2. (1) LetZDa.y; Y /2DandW Da.v; V /2D. We set E.y; v; Y; V /WD2y^t.In V Y /x ¹vNCy^t.In V Y /x ¹V yx C Nv^tY .In V Y /x ¹v:N Then,

K.Z; W /Dexp

4E.y; v; Y; V /

0.In YV /:x (2) ^H6D.0/if and only if > 0andmCnC1=2 < 0. In this case,

cDDim.V/ 2

n

Jn. m n 3=2/ ¹:

(3) For eachgD .z0;zN0/; c0; ^{P Q}Qx Px

2Gand eachZ Da.y; Y /2D, we have J.g; Z/De^{i c0}exp

4.z₀^tzN₀C2zN₀^tP yCy^tP^tQyx

.Nz0C xQy/^t.P Y CQ/.Qyx C xP / ¹.zN0C xQy/

0..QYx C xP /^t/ ¹:

Proof. (1) and (3) are simple calculations. To prove (2), we use the formula .In Y Y /x ¹ DDet.In Y Y /x ¹C.In Y Y /x ^t

for eachY 2B, whereC.A/denotes the cofactor matrix of a matrixA. From this formula, we deduce

Tr0.In Y Y /x ¹ DDet.In Y Y /x ^mTr0.C.In Y Y /x ^t/

and then we can prove (2) by following the same lines as in the proof of Proposi-

tion 3.2 of [13], using Theorem XII.5.6 of [18].

Proposition 3.4 of [13] can be generalized as follows.

Proposition2.3. ForX 2g^c,f 2HandZ2D,

d .X /f .Z/Dd.pk^c.e ^adZX // f .Z/ .df /Z.p_pC.e ^adZX //:

In particular,

(1) ifX 2p^C, thend .X /f .Z/D .df /Z.X /;

(2) ifX 2k^c, thend .X /f .Z/Dd.X /f .Z/C.df /Z.ŒZ; X /; (3) ifX 2p , then

d .X /f .Z/D.dıpk^c/

ŒZ; X C 1

2ŒZ; ŒZ; X 

f .Z/

.dfZıp_pC/

ŒZ; X C1

2ŒZ; ŒZ; X 

:

Let.Ek/be a basis ofp^C. Then, for eachf 2Hand eachk, we denote .@kf /.Z/D d

dtf .ZCtEk/jtD0: From the preceding proposition we deduce the following result.

Proposition2.4. For eachX1; X2; : : : ; Xq 2g^c,d .X1X2 Xq/is a sum of terms of the formP .Z/@_k1@_k2 @_kr wherer q andP .Z/is a polynomial of degree2q.

3. Berezin calculus

First we introduce the Berezin quantization map associated with0, see [3], [4], [1], [5], and [19].

Let'Q0be the linear form onMn.C/defined by'Q0 D i ƒm1;m2;:::;mnonh₀and Q

'0 D0onn^˙. We denote by'0 the restriction of'Q0tou.n/. Then the orbito.'0/ of'0 under the coadjoint action ofU.n/is then said to be associated with0.

Note that the stabilizer of'₀for the coadjoint action ofU.n/contains the torus H₀WD ¹Diag.i a₁; i a₂; : : : ; i a_n/Wa_j 2Rº:

We say that such an element'0 isregular if the stabilizer of'0 is equal toH0, see [5]. Then we can verify that'₀ is regular if and only if one hasm₁ > m₂ >

> m_n. In the rest of the paper, we assume that'₀is regular.

Note also that a complex structure ono.'₀/is then defined by the diffeomor- phismo.'₀/' U.n/=H₀ ' GLn.C/=H₀^cN whereN is the analytic subgroup of GLn.C/with Lie algebran .

Without loss of generality, we can assume that^Vis the space of holomorphic functions on (a dense open set of)o.'0/as in [5]. For' 2 o.'0/there exists a unique functione' 2 V(called a coherent state) such thata.'/ D ha; e'iV for eacha2V. The Berezin calculus ono.'0/associates with each operatorB on^V the complex-valued functions.B/ono.'0/defined by

s.B/.'/D hBe_'; e_'iV

he'; e'i^V

which is called the symbol ofB. Then we have the following proposition, see [14], [1] and [5].

Proposition3.1. (1) The mapB !s.B/is injective.

(2) For each operatorB on^V, we haves.B/Ds.B/. (3) For each'2o.'₀/,k 2U.n/andB 2End.V/, we have

s.B/.Ad.k/'/Ds.0.k/ ¹B0.k//.'/:

(4) For eachA2u.n/and'2o.'0/, we haves.d0.A//.'/Dih'; Ai. We also need the following result, see [10] and [12].

Proposition3.2. LetZ2D. There exists a unique elementkZinK^csuch that k_Z DkZ andk_Z² D.expZexpZ/ ¹. Eachg2G such thatg0DZ is then of the formgDexp. Z/ .expZexpZ/k_Z¹hwhereh2K. Consequently, the mapZ !gZ WDexp. Z/ .expZexpZ/k_Z¹ is a section for the action ofG on^D.

More explicitly, for eachZ Da.y; Y /2D, we have kZ D

.0; 0/; i

8E.y; y; Y; Y /;

.In YY /x ¹⁼² 0 0 .In Y Y /x ¹⁼²

and

gZ D

. w; w/;N i

8.y^t.In Y Y /x ¹Y yx yN^tY .In Y Y /x ¹y/; M.Y /N wherewWD .In Y Y /x ¹.yNC xY y/and

M.Y /WD

.In YY /x ¹⁼² Y .In Y Y /x ¹⁼² Y .Ix _n YY /x ¹⁼² .I_n Y Y /x ¹⁼²

:

Now, following [17], [2], [12], we define the pre-symbolS0.A/of an operatorA by

S₀.A/.Z/Dc¹.k_Z¹/E_ZAE_Z.k_Z¹/

and the Berezin symbolS.A/ofAis defined as the complex-valued function on Do.'0/given by

S.A/.Z; '/Ds.S0.A/.Z//.'/:

For eachg 2 G andZ 2 D, let k.g; Z/WD g_Z¹g ¹ggZ 2 K. Then we can write

k.g; Z/D

.0; 0/; c.g; Z/;

P .g; Z/ 0 0 P .g; Z/

; wherec.g; Z/2RandP .g; Z/2U.n/.

We have the following properties ofS, see [12].

Proposition3.3. (1) Each operatorAis determined byS.A/. (2) For each operatorA, we haveS.A/DS.A/.

(3) We haveS.IH/D1.

(4) For each operatorA,g2G,Z 2Dand'2o.'0/, we have S.A/.gZ; '/DS..g/ ¹A.g//.Z;Ad.P .g; Z//'/:

Now, we give some formulas for the Berezin pre-symbol of.g/for g 2 G and for the Berezin symbol ofd .X /forX 2 g^c. For' 2u.n/, we denote by '^s the linear form onsdefined by

D

'^s;_{P Q}

Qx Px

E

D h'; Pi and by'^e the linear form ongdefined by

D '^e;

.z;z/; c;N _{P Q}

Qx Px

E

D h'; Pi Cc:

We also denote by'^s and'^ethe extensions of'^s and'^etos^candg^c.

Proposition3.4([12]). (1) Forg2GandZ2D, we have S0..g//.Z/D k_Z¹.expZg ¹expZ/ ¹.k_Z¹/

: (2) For eachX 2g,Z 2Dand'2o.'0/, we have

S.d .X //.Z; '/DihAd.g_Z/'^e; Xi:

Recall that0 2 g is said to beregularif the stabilizerG.0/of0 for the coadjoint action is connected and if the Hermitian form.Z; W /! h0; ŒZ; Wi is not isotropic [12].

Lemma 3.5. The linear form'₀^e is regular if and only if we havem_j > 0for eachj ormj < 0for eachj.

Proof. On the one hand, by using the formula for the coadjoint action ofG given in [13], Section 2, we can verify thatG.'₀^e/consists of all matrices of the form .0; 0/; c; ^{P 0}_{0 Px}

wherec2RandP 2U.n/is such that Ad.P /'₀ D'₀. Since'₀ is assumed to be regular as an element ofu.n/, we getP 2H₀. Hence G.'₀^e/ŠRH₀is connected.

On the other hand, for eachZ Da.y; Y /2D, we have h'₀^e; ŒZ; Zi D h'0; YYxi i

2jyj²

D X

j

mjjYjj² i 2jyj²

whereY1; Y2; : : : ; Yndenote the columns ofY. The result hence follows.

Let us assume that'₀^e is regular and denote by^O.'₀^e/the orbit of'₀^e for the coadjoint action ofG. Then we have the following proposition, see [12].

Proposition3.6. The map‰WDo.'0/!gdefined by

‰.Z; '/DAd.gZ/'^e is a diffeomorphism form^Do.'₀/onto^O.'₀^e/such that

‰.gZ; '/DAd.g/ ‰.Z;Ad.P .g; Z//.'//

for eachg2G,Z2Dand' 2o.'0/.

More precisely, with the notation of[13], we have

‰.Z; '/D

.w; w/; ;N Ad.M.Y //'^s

2.w; w/N .w; w/N whereZ Da.y; Y /2DandwWD .In Y Y /x ¹.yNC xY y/.

4. Berezin transform and Stratonovich–Weyl correspondence

Here we introduce the Berezin transform associated withS and the corresponding Stratonovich–Weyl correspondence, following [12].

We fix aK-invariant measureono.'0/normalized as in [5], Section 2. Then the measure Q WD ˝ on ^Do.'₀/is invariant under the action of G on Do.'₀/ given by g.Z; '/ WD .g Z;Ad.P .g; Z// ¹'/ and the measure O.'₀^e/WD.‰ ¹/./Q is aG-invariant measure on^O.'₀^e/.

We denote by^L2.H/the space of Hilbert-Schmidt operators on ^Hendowed with the Hilbert-Schmidt norm. We also endow End.V/with the Hilbert-Schmidt norm. We denote byL².Do.'0//(respectivelyL².D/; L².o.'0//; L².O.'^e₀///

the space of functions on ^Do.'0/ (respectively ^D, o.'0/, ^O.'₀^e/) which are square-integrable with respect to the measureQ (respectively,,O.'₀^e/). Then we have the following result, see for instance [6].

Proposition4.1. The Berezin transformbWDssis given by b.a/. /D

Z

o.'0/

a.'/ jhe ; e'i^Vj² he'; e'iVhe ; e iV

d.'/

for eacha 2L².o.'0//

Similarly, we have the following proposition.

Proposition 4.2. The Berezin transformB WD S S is a bounded operator ofL².Do.'0//and that, for eachf 2 L².Do.'0//, we have the following integral formula

B.f /.Z; /D Z

Do.'0/

k.Z; W; ; '/ f .W; '/ d.W /d.'/

where

k.Z; W; ; '/WDjh..g_Z¹gW// ¹e ; e'i^Vj² he_'; e_'iVhe ; e iV

:

Consider the left-regular representationofGonL².Do.'₀//defined by . .g/.f //.Z; '/Df .g ¹.Z; '//:

Clearly, is unitary. Moreover, sinceS isG-equivariant, we immediately verify that for each f 2 L².D o.'0// and each g 2 G, we have B. .g/f / D .g/.B.f //.

Now, we introduce the polar decomposition ofSWL₂.H/!L².Do.'₀//. We can writeS D.S S/¹⁼²W DB¹⁼²W whereW WDB ¹⁼²S is a unitary operator from^L2.H/toL².Do.'0//. Then we have the following proposition, see [12].

The main point is thatW isG-equivariant sinceS (henceB) isG-equivariant.

Proposition4.3. (1) WWL₂.H/ ! L².Do.'₀//is a Stratonovich–Weyl correspondence for the triple.G; ;Do.'₀//.

(2) The map^Wfrom^L2.H/toL².O.'₀^e//defined by^W.f / D W .f ı‰/is a Stratonovich–Weyl correspondence for the triple.G; ;O.'₀^e//.

5. Extension of the Berezin transform

Here we generalize Proposition 5.2 of [13], that is, we extendB to a class of functions which containsS.d .X //forX 2 g^c, in particular in order to define W .d .X //.

ForZ; W 2 D, we setlZ.W / WD log.expZexpW / 2 p . We need the following lemma which is the direct generalization of Lemma 5.1 of [13].

Lemma 5.1. (1) For eachZ; W 2D,V 2p^Candv2V, d

dt .EZv/.W CtV /ˇ ˇ_tD0 D c.dıp_k^c/

Œl_Z.W /; V C1

2Œl_Z.W /; Œl_Z.W /; V  

.E_Zv/.W /:

(2) ForZ; W 2DandV 2p^C, d

dt lZ.W CtV /ˇ

ˇ_tD0Dpp

ŒlZ.W /; V C1

2ŒlZ.W /; ŒlZ.W /; V   :

(3) The function.@_k1@_k2 @_kqEZv/.W / is of the form Q.lZ.W //.EZv/.W / whereQis a polynomial of degree2qwith values inEnd.V/.

(4) For eachX1; X2; : : : ; Xq 2g^c, the functionS0.d ..X1X2 Xq//is a sum of terms of the form .kZ/ ¹P .Z/Q.lZ.Z//.kZ/ where P and Q are polynomials of degree2qwith values inEnd.V/.

By combining the arguments of the proof of Proposition 6.5 in [8] with those of the proof of Proposition 5.2 in [13], we then obtain the following result. Recall thatmWDP

imi.

Proposition5.2. Ifq < ¹₄. m 2n/then for eachX₁; X₂; : : : ; X_q 2 g^c, the Berezin transform ofS.d .X₁X₂ X_q//is well-defined.

We have then generalized Proposition 5.2 of [13]. However, it seems difficult to obtain here an explicit expression forW .d .X //,X2g, as in [13], Section 6.

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Manoscritto pervenuto in redazione l’8 agosto 2015.