Finite dimensional analysis and polynomial quantization on a hyperboloid of one sheet Текст научной статьи по специальности «Математика»

FINITE DIMENSIONAL ANALYSIS AND POLYNOMIAL QUANTIZATION ON A HYPERBOLOID OF ONE SHEET

V.F.Molchanov*, N.B.Volotova

Tambov State University, 392622 Tambov, Russia e-mail: molchanov@main.tsu.tambov.ru

This paper has two goals. The first one is to study ’’finite dimensional analysis” on the hyperboloid X of one sheet in M3,i.e. the decomposition of representations of the group G = SL(2, E) ( or its factor-group SOo(l, 2)) acting on spaces An of restrictions to X of polynomials of degree ^ 21. These representations (equipped a G-invariant bilinear form) can be regarded as a finite dimensional analogue of the so-called canonical representations, see, for example, [4], [1].

We solve this problem with using a machinery of ’’usual”, infinite dimensional, harmonic analysis (see, for example, [5]): //-invariants, Poisson and Fourier transforms, spherical functions, a Plancherel formula.

The decomposition in question appears also when we tensor a finite dimensional irreducible representation 7r/ of G by its contragradient 7r/. Although the formula for the decomposition of 7T/<g>7r/ is well-known (7r/<g>7T/ = 7To —}— 7Ti —{-.. . + 7T2/), explicit constructions and formulae given in this paper are interesting both themselves (for example, a differential formula for the Poisson transform, etc.) and as a ruling example for generalizations.

The second goal is to construct a polynomial quantization (symbol calculus) for the hyperboloid X,

i.e. quantization on the space S(X) of polynomials on X. For almost all values of a parameter a E C we introduce in S(A’) a structure of an associative algebra and establish that the corresponding principle is true (for ’’the Planck’s constant” h one has to take —1/2a). The space S'(A') is formed by covariant symbols of operators corresponding to elements of the universal enveloping algebra of the Lie algebra 9 of G in a representation Ta. We define also contravariant symbols for the same operators. It gives us two transforms: O and B. The first one, the transform O, is a map on a space of operators (A = OD means that operators D and A have one and the same polynomial F as the covariant and the contravariant symbol, respectively). The second one, the Berezin transform B, is a transform of 5(^), it carries contravariant symbols to covariant symbols. We give full and explicit descriptions of these transforms O and B. For example, for B, we write the spectral decomposition, an explicit expresión in terms of the Laplacian, an explicit ’’deformation” decompositon.

Some fragments of this work were published in [7], [8].

The theory developed in the present paper can be generalized to para-Hermitian symmetric spaces in a very natural way. It will be published elsewhere.

§1. Representations of the group SL(2,M)

In this Section we recall some material about representations of the group G = SL(2,IR), see, for example, [3]. The group G consists of real 2x2 matrices

«=(“ £),<*«-/*r = 1. (11)

* Partially supported by RFBR (grant 96-15-96249) and Goskomvuz RF (grant 95-0-1.7-41).

Let g be the Lie algebra of G, and il the universal enveloping algebra of g. Take the following basis of g:

L‘ = (>/02 -1/2)’ = (0 o)-L- = (°, 0)' <12>

The centre of 11 is generated by the element

Ai = L; + i(£+L- + £-L+)

For A £ C, e = 0,1, we denote

tx'£ = |*|Asgne* = tl +

a character of E*. We shall also use the following symbols for ’’generalized powers”:

a= a(a — 1)... (a — n+1), = a(a + 1)... (a + n — 1). (1.3)

For a G C, £ = 0,1, let VayC denote the space of complex valued C°° functions f(x) on E such that the function

f(t) = (1.4)

is also of class C°°. For the topology on Vff£, see, for example, [3]. The representation Ta>£ of the group G acts on Va e by

(7V,«(9)/)(<)=/(i)(/3i + i)2<’'' (1.5)

where

< = <.,= “i±l (1.6)

^+s

Permuting in (1.5) a with 6 and /? with 7, we obtain the contragradient representation Tay.

faiS(g)f)(t) = + a)2o'e

where

i=fg=6-^±i,g=(6, 7), (1.7)

y 'yt + a' y \(3 ot)' K }

The representations Ta<e and Tai£ are equivalent, the equivalence is given by the map / 1—► /, see (1.4). A differentiable representation of G gives rise to a representation of g and a representation of U. We retain for them the symbol of the representation of G.

In particular, we have

T'AL>) = tjt-« (1.8)

T,,c(L+) = t2ji+2<rt (1.9)

(1.10)

For the contragradient representation one has to multiply by —1 the right hand side of (1.8) and to permute the right hand sides of (1.9) and (1.10).

The element A0 goes to a scalar operator:

T„,e(A9) = i,,c(All) = a(CT + l)£ (1.11)

The bilinear form _

/OO

F(t)f(t)dt (1.12)

-OO

is invariant with respect to the pair (Ta^,T-a-iiC), so that

(T«,,(9)F,f) = (F,T-,-U(g-')f) (1-13)

The operator with the kernel (t — s)~2a~2'e intertwines Ta<£ and (for points a where this

operator has poles, one has to take the first Laurent coefficient). If T0y£ is irreducible, then it is equivalent to T_a_i e. It will be more convenient to us to deal with other intertwining operator:

ds

/OO -OO

which intertwines Tffi£ and T_a_i>e:

-T— a— l,c (^)-^<r,e = (fiO> 9 E G,

and also TCt£ and T_a_iiC. The composition of these operators there and back is a scalar operator:

A. — a — \l£-A.ff}£ —U)q((T,£S)E (1.14)

where

tJ°^’^ = 2^TTtg^2<T ~

The operator Aa>£ is self-dual with respect to the form (1.12):

(.Aff>eFJ) = (FiAa,ef). (1.15)

Let us extend the representation Ta<£ to the dual space Va E (linear continuous functionals, distributions) by formula (1.13) where F E T>a £ and / E V-a-it£ (V0iE is embedded in £ by assigning to

F E Va>£ the functional / 1—► (F, /)). Similarly we extend the operator A„i£ - by means of (1.15).

Formulae (1.8) - (110) for the representation Ta e on Vae generate a representation Ta (notice that it does not depend on e) of g given by the same formulae. We shall assume that this representation acts on one of the spaces: C°° functions (defined on E or on intervals); the space P of polynomials; the space Z of distributions, concentrated at zero. The two latter spaces are Verma moduli.

The action of g on P in the basis tk is given by:

Ta(Li)tk = (k - a)tk, (1.16)

Ta(L+)tk = (2<r - k)tk+1, (1.17)

Ta(L-)tk = ktk~1, (1.18)

The space Z consists of linear combinations of the delta function 6(x) and its derivatives. We have

T,(L{)6^ = (-<r - 1 -

T„(i+)6<*> = —k(2<r + k +

T„(L-)6{k> = 6(i+1l The operator € carries the basis tk in P to the basis in Z (with factors):

/OO

(1 - ts)-2ff~2'£sk ds = U)k{a, £)6(-k\t), (1.19)

-OO

where

a,k(a, e) = (-l)k (2<T + [)[ib+i]tg((2cr - c)*/2)

(for k = 0 formula (1.19) is equivalent to (1.14) and for k > 0 is obtained by the differentiation). And, inversely, the operator A-a-i,c carries the basis 6(k) to the basis tk (with factors):

A-a-ijSW = (2cr)Wtk (1.20)

The representations Ta<£ are irreducible except the case 2<r E 7L,£ = 2a (here and further the sign = means the congruence modulo 2). Denote N = {0, 1,2,... }. For 2a E N, £ = 2a, the representation Ta>£ has an invariant finite dimensional subspace Va, consisting of polynomials in t of degree ^ 2a, so that

dimV^r = 2a + 1. In this case let na denote the restriction T„i£ to Va. On the basis tk of Va it acts by

formulae (1.16) - (1.18) - with replacing Ta by 7rff.

The operator Affi£ vanishes on V„, and its derivative

i d

Aa = — ATi£ (1.21)

OT r=«7,e=2<7

gives rise to an equivalence of Va and the finite dimensional quotient space for T_a_ iiC. Moreover, we have on Va:

' 7r2

A—1JtA0 = (1.22)

Any irreducible finite dimensional representation of G (and of g) is equivalent to one of TTa. We shall say that the basis eo,e\,... , a in the representation space is standard, if the action of Li,L+,L_ in this basis is the same as (1.16) - (1.18) in the basis 1,tla in V„. The standard basis is unique up to the factor. The vector e0 is minimal: L_eo = 0, and the vector e2a is maximal. L+e2a = 0. A vector em is obtained from the minimal vector eo as follows:

Cm = (2<r)W L+e°'

Let 2a E N. Then the module Z with respect to T-a-1 or T-a-i has a g-invariant submodule spanned by with k ^ 2a + 1. The operator A-a-ii£ vanishes on it (see (120)) and maps the quotient module Z-a-1 generated by 6^k\0 ^ k ^ 2a, onto Va. The operator Aa carries the basis tk in Va to the standard

basis [(2(7)^')]“1 in Z-a..\ with the factor (cf. (1.22)):

A tk =_______-______ ___!__________k)(t)

2(7+1 (2 a)(k) [)

For any irreducible subfactor of T0i£ there exists a unique up to the factor G-invariant bilinear form. In particular, such a form Ba on V„ is defined by

= 1)‘(2*) (1.23)

Let us else write a bilinear form Ba on Va which is G-invariant with respect to the pair (71^, 7^):

B„(ttin = {-1)*(2") (1.24)

This form can be written as follows:

B.ULh) = -^VL{4.h,h) (1.25)

§2. Tensor product 7n <g> vi

Let 21 E N. The representation Ri = 7T/ <S> tt/ of G acts on the space Wi = Vi ® Vi which consists of polynomials f(^,r]) in two variables £ and 77 of degree ^ 21 with respect to each of them by

(ßZ + <5)(7i7 + a)

see (1.6) and (1.7). The polynomial

N = l-tri (2.1)

has the following property:

i-«=

(M + 5)(t»7 + <*)

Therefore, the polynomial

$ = AT2' = (1 -^)2' (2.2)

is fixed under Ri:

R,(g)<t> = * (2.3)

It is easy to check that for any k = 0,1,... , 21 the polynomial

ujt = N2,~kr]k, (2.4)

is annihilated by L_ and is an eigenvector for L\ with the eigenvalue — k. Therefore, it is a minimal

(k)

vector and generates an irreducible subspace Wf of Wi where 7rk acts. Since the sum of dimensions is

precisely the dimension of Wf. 1 + 3 + 5 + ... + (4/ + 1) = (2/ + l)2, we obtain the decomposition:

Rl - TTO + TTl + • • • + TT2/)

and, respectively,

Wi = W,(0) + + ... + (2.5)

where

WW _ N2l~kwjck/l

A standard basis in Wj^ is N21 Fktm where FkiTn are functions defined by (4.4) below. In particular, the minimal and maximal vectors are uk, see (2.4), and

vk = N2l-k£k.

Take on Wi the bilinear invariant form Bi which is the ’’tensor square” of Bl, see (1.24), (1.25), namely, for pure tensors:

Bi(<p®xl>,<p\ <g> rpi) = Bl((p,rpi)B,(xp,ipl)

so that

.€"Y) = (-i)‘+m(2/)

2/\~V2/N _1

m

and Bi is equal to zero for other pairs of basis elements £kT]m. The form Bi is invariant with respect to R,:

Bi(Ri(g)fuRi(g)h) = BiVij2)

The subspaces Wjk^ are pairwise orthogonal with respect to Bi.

Denote

m=Bi(uk,vk) (2.6)

A computation with using the binomial formula for generalized powers (see, for example, [6], Sect. I, No. 35)

(a + 6)l"l = £('’f)0BW"--’l (2.7)

gives

j= o

tk*{2l-k)\{2l + k + l)\

W‘ - (-1) -------(2k + 1)!(2/)!2----- (2 8)

§3. Hyperboloid of one sheet

In this section we show that the tensor product Ri = 7T/ <g) 7rj is equivalent to the representation of G by translations on a space of polynomials on the hyperboloid of one sheet in M3 (see Theorem 3.1).

Let us introduce in M3 the bilinear form

[x,y] = -xiyi + x2y2 + x3y3 (3.1)

Let X and Xq denote the hyperboloid [x, x] = 1 and the cone [x, x] = 0, x ^ 0, respectively. Let us realize K3 as the space of matrices

_ 1 f 1 - X3 X\ + x2

~~ 2 \xi + x2 1 + x3

Then detx = (1/4)( 1 — [x,x]). The group G acts on these matrices as follows:

xh^g~1xg (3.2)

On X, it acts transitively. The stabilizer of the basic point

x° = (0,0,1)= ' 0 0

0 l,

is the diagonal subgroup H of G. Under (3.2), where g is given by (1.1), the point x° goes to the point

x = (ay + /36, «7 — /36, a6 -f fi-y)

The action (3.2) gives a right action of G on vectors x E M3 by means of matrices from SOo(l, 2). So we get a homomorphism of G onto SOo(l,2) (with the kernel ±E).

The action of G on functions / on X by translations will be denoted by U:

U(g)f(x) = f(g~1xg) (3.3)

Introduce on X coordinates £,77:

x = N-'iZ + r),^- 77, 1 + £t/)

(for N, see (2.1)), so that in the matrix realization we have:

, _ 1 ( -rf -v N \ f 1

These coordinates are defined on X except x3 = —1. The action (3.2) is given by fractional linear transformations (separately in each variable £ and rj):

= 9, V * ^ V ~ T) ~ 9

see (1.6) and (1.7), so that

U(g№,v) = f(lv)

The basic point x° has coordinates £ = 0,77 = 0. An element g E G, see (1.1), carries x° to the point x with coordinates

f = 7/6, V = P/<*> (3-4)

so that N = \/a6.

Write in £,77 the G-invariant measure dx, the Laplace-Beltrami operator A and the Poisson bracket {<P, V>}:

dx = dx(£,r]) = N 2d£drj (3-5)

A = £/<A») = "2& (36)

Ш] = М2(?г?і_дгдф\ . .

\ді дт) дті К 4

A polynomial / on М3 is called harmonic (with respect to (3.1)), if

д2 д2 д2 \ , 4 ~дЩ + Щ + Щ)г-° (3'8)

Let S and Ті denote the spaces of all polynomials and of harmonic polynomials, respectively. Let

Tik denote the subspace of V. consisting of homogeneous polynomials of degree k. We shall denote

the restriction of 7ik to X by Hk(X) - and similarly for other spaces. For harmonic polynomials this restriction is an one-to-one map. Besides it, we have S(A’) = 7i(X). The space ?ik{X) is invariant and irreducible with respect to (3.3), the corresponding representation is equivalent to 7Г*. Equation (3.8) gives for polynomials from lik{X) the condition (cf. (3.6) and (1.11)):

AF = k(k + 1)F (3.9)

Denote

Ak='Ho{X) + 'Hi(X) + ...Hk(X) (3.10)

(the space of the restrictions to X of polynomials, or of harmonic polynomials, of degree ^ k).

Remember Ri from §2. Let us divide all polynomials /(£,77) from Wi by Ф, see (2.2):

£/ = ф-1/ (3.11)

We obtain the space Ф”1 Wi of some rational functions in £, 77.

It follows from (2.3) that C intertwines Ri and the restriction of U to Ф-1И^. In particular, (2.3) gives that the function 1 (identically equal to 1) is fixed under U.

Theorem 3.1. Ф-1И^ = A2i-

Proof. By virtue (3.10) and (2.5), it suffices to indicate in Ф~1Wi at least one element from Tik(X) for

each k = 0,1,... ,21. Such an element is, for example, the minimal vector

(see (2.4)), because (x\ — x2) belongs to Нк(Х). □

§4. Finite dimensional harmonic analysis on hyperboloid

s

In this Section we give explicit constructions and formulae for the decomposition of the representation U of G on A21 (and therefore of the tensor product Ri = ni <g> 7Г/ from §2).

Let us transfer the bilinear form Bi from Wi to A21 by means of C, see (3.11), and retain the symbol,

so that

Bt( ф-^^ф^ГУ) = (-D*+mQ') (^j

and Bt is equal to zero for other pairs of basis elements Ф~1^*'’т;т, k,m ^ 21.

Let us indicate intertwining operators between Пк and U on A2i (k ^ 21).

Recall that H is the diagonal subgroup of G. The subspace of Я-invariants for ттк is non-trivial if and

only if k G N. Then it is one-dimensional and is spanned by

h(t) = i *.

The corresponding Poisson kernel is defined as follows:

Pk(x\t) = Pk{Z,T)\t) = (tt k(g~l)0k){t)

where x — g xx°g and £,77 are obtained from g by (3.4). Here it is explicitly:

Pk(Z,Wt) =

N

(4.1)

or

Pk(x,t) = [x,y\k,

where у = ((t2 + l)/2, (t2 — l)/2), i), a point of the cone Xq.

The Poisson transform Vk '■ Vk —*■ A21 (recall к ^ 2/) is defined by

(!Pk<p)(£,v) = Bk(ick(g~1)Ok,<p) (4-2)

where В к is the bilinear form (1.23) on V*. It is an intertwining operator:

Vk^k(g) = U(g)Vk

So that, by (3.10), its image is "Hi\X). It transfers the basis tm in Vk to a standard basis in 7ik{X). Let Fk,m denote this standard basis multiplied by (— \)k:

Ft,m = (-1 )kVktm (4.3)

By (4.1), (4.2) and (1.23) we have

FkM = ЛГ-*(“)"* ± (*) (m*_y)<~V-' (4-4)

(in fact, the summation is taken over 0 ^ j ^ m for m ^ к and over m — к ^ jк for m ^ к). In

particular, the minimal and maximal vectors are

Notice that

Fk,m(£, V) — Fk,2m— k(Vt 0

Since 7ik{X) is irreducible, values of B\ on the basis Fk>m differ from values of В к on the basis tm by

the factor only, which in virtue of (2.6) is equal to /i/t, see (2.8), so that by (1.23) we have

В|(А,т,Л.М-т)='(-1Г(“) Wk (4.6)

Returning to the Poisson kernel (4.1), we can now write it as follows:

=Е(-1)‘+т(2г!)‘т^.«-™«.л) (4.7)

m=0 ' '

so that for t fixed, it is an element of 7ik(X).

Now define the Fourier transform Tk- A21 —► Vk as follows:

(FkF)(t) = B,(Pk(-,t),F).

Introducing here (4.7), we obtain

2k /9k\

W/Cft.at-m,/5’) (4.8)

m=0 \m/

The Fourier transform T intertwines U and nk:

TTfc (g)?k = FkU(g), and is conjugated to the Poisson transform:

Bl(F,Vk<p) = Bk(rkF,<p), (4.9)

where ip 6 14, F € An) and the composition of these two transforms is a scalar operator:

TkPk — HikE

(these properties are obtained from (4.8) and properties of the Poisson transform).

Therefore, if F G 'Hk(X), then

B,(F, F) = ^Bk(FkF,FtF) and, for an arbitrary F 6 A21, we have

21

B:(F,F) = J2^pii(^F,TkF).

k=0

It can be regarded as an analogue of a Plancherel formula, fifl being an analogue of a Plancherel measure. The image of the //-invariant 0k under the Poisson transform Vk is called the spherical function:

*k=Vk9k (4.10)

By (4.3) we have *i!k = (—1 )kFk)k so that by (4.4)

_i k

= (-1 )*AT* (“) (*) (il?)* j

=(-i)‘RT'^(-)

where Pk is the Legendre polynomial. The spherical function is //-invariant.

From (4.10) and (4.9) we have

B{*k,F)= Bk(0ktFkF)

A shifted spherical function [/(#-1)'I'k is an analogue of the Bergman kernel:

U(g-l)<Sk(u,v)= (4-11)

m=0 \m/

where £,77 correspond to g by (3.4).

For any //-invariant function Q E A21 we can define an operator in A21 ~ the convolution F —*■ Q ★ F with Q:

(Q*F)(tv) = B,(U(g-')Q,F)

where £, 77 correspond to g by (3.4).

In particular, the convolution with 4>-1 is the identity operator, so that $~1 plays the role of the delta function. The shifted function U(g~1)^~l is the following function of two pairs variables (the Berezin kernel):

Ki(£,rj\u,v) =

or, in terms of points of X,

(1 - W7)(l -ivY21

.(1 -£rç)0 - uv)

Thus, this kernel Ki has the following reproducing property:

Bl(Kl(x,),F(-)) = F(x),FeA2i

For the spherical function the convolution with /i^1 is the projection in A21 onto 7ik(X) (it follows from (4.11) and (4.6)).

Therefore, we have the decomposition:

21

= <412) Jfc=0

(it could be obtained also by a direct computation). Formula (4.12) also can be regarded as a Plancherel formula.

The same Poisson transform Vk can be obtained by using the representation T-k-i on distributions (recall that its factor-representation on Z-k-1 is equivalent to 7Tjt, see §1), then the Poisson transform is written in a differential form. Namely, //-invariant is <5^^(i), we take the bilinear form (1.12), so that:

rOO

i'Pk<p)(Z,v) = ck / T-k-iig'1)^^)^)^

/00

6^k\t)Tic(g)ip(t)dt

■OO

where Ck = [(26)^')] 1 = k\/(2k)\ and 77 are connected with g by (3.4). In particular, for basis function tm we have

k

(Vktm)(tr,) = Ck(-iy{7t

(at -f 7)m(/3t + <5)

2k — m

t=0

§5. Polynomial quantization on the hyperboloid

Let us apply to the hyperboloid X the scheme of quantization from [4], [1]. For an algebra of operators we take the algebra of operators Ta(X), X 6 11, acting on functions of £, see §1. For a supercomplete systeme we take the kernel of the intertwining operator A-a-1,0 namely,

Let us call the function F of £, rj, defined by

nt,rl) = ^ilT4xmt:,’i), (5.1)

the covariant symbol of the operator T0(X). It does not depend on e.

The covariant symbol of the identity operator is 1, see §3. The covariant symbols of operators corresponding to the basis (1.2) of g are — ax3, cr(xi +£2), —^(^l — ^2), respectively. The covariant symbol of

the operator Ta(Lr_) = (d/d£,)r, see (1.13), is the minimal vector FTy0 in 7ir(X) up to the factor, namely,

cov. symb. Ta{Lr_) = (-l)r(2<r)rFr,0 (5.2)

Lemma 5.1. For any X E ii of degree k the covariant symbol of the operator Ta(X) is a polynomial in Ak with coefficients depending on a polynomially.

The lemma follows from (1.8) - (110) immediately.

The operator D(= Ta(X)) is reconstructed by its covariant symbol F in the following way (cf. [4], [1]). First we rewrite formula (1.14) as follows:

=c J

where c = l/uo(a, e) and dx(u,v) is given by (3.5). Here and further integrals are taken over M2 or E. Then by (5.1)

(D<p)(t) = cj F(£,v)||^y>(u)dx(tz,v) (5.3)

so that the kernel K(£,u) of D is

*({,«) = c/f(f,.dv

J Ф(u, v) (1 — UV)2

Inversely, the covariant symbol is expressed by means of the kernel (we use (1.19) with k = 0):

F(Cv) = ^ J I<(t,u)<b(u,ri)du

The correspondence D —*■ F commutes with g: if F is the covariant symbol of the operator D =

Ta(X), X Є it, then U(L)F, where L Є g, is the covariant symbol of the operator

T0(bdLX) = [Ta(L),D\

Theorem 5.2. The set of the covariant symbols of all operators T„(X),X Є U, is the space S'(A'), if 2a £ N, and the space Ага, if 2а Є N.

The theorem follows from (5.2) and the g-equivariance of D —► F.

The multiplication of operators gives rise to a multiplication * of covariant symbols: if D — D\D2)

then F = F\ * F2. This multiplication * can be written as follows:

Fi*F2= (5.4)

(it includes (5.1) as a particular case), or, using (5.3) for (5.4),

(Fi * F2)(£, 77) = J Fi(i,v)F2{u,r])e(i,T]]u,v)dx(u,v) (5.5)

where

„/л ч Ф(І,у)Ф(и,г])

B(£, 77; u, v) = c————-------

the Berezin kernel. In terms of points x Є X, the Berezin kernel is

2а,є

m \ f[x,y] + lYa'e B(x,y) = cl-------------1

It is invariant with respect to translations:

В(д~1хд,д~1уд) = B(x,y) (5.6)

Thus, we have

Theorem 5.3. The set of covariant symbols indicated in Theorem 5.2 (S(X) for 2a £ N, and A20 for 2(7 6 NJ is an associative algebra with the multiplication *.

A monomial

ck m

h,m^r, (5.7)

is the covariant symbol for the operator

3=0

(it can be checked by applying of D to <I>). The kernel K(£,u) of this operator is

*«.«) = E (' “ m) v^TJ)tk+i6im+i)(t ~ “)

Now define contravariant symbols. In accordance with a general scheme (see, for example, [1]), a function F(£, 77) is the contravariant symbol for the following operator A acting on functions <£>(£):

= cJ F(u, *0^’ V^v(u)dx(u’v) (5-9)

(notice the difference from (5.3) in the one argument only), so that the kernel L(£,u) of the operator A

J 4>(ti, l>) (1 — UV)2

Inversely,

F(i,ri)=-¥^J L(t,W«,ri)dt

where we denote

$*«,>)) = f>_„-1|e(i, 7,)= 1

For polynomials F(£, tj) from S'(A') corresponding operators A are differential operators. In particular, the monomial (5.7) is the contravariant symbol for the operator

r_m /.. ~A 1 / a \ m+3

E

j=0

(it is proved by a direct computation with using (1.19)), so that the kernel of this operator is

3=0

Thus we have the following diagram

*«■«) = E f “•m) (_2J2)(^/+i^(« - 0

D—A

where arrows \ are co and arrows f are contra. Let us consider their compositions: O = contraoco, 3 = coo contra (the latter is called the Berezin transform).

Theorem 5.4. Let A = 0(D) (i.e. there exists a polynomial F E 5(^) which is the covariant and contravariant symbol for the operators D and A respectively). Then A is the transpose of D with respect to d£ with replacing <x by —a — 1:

, (511)

To prove, one has to compare (5.8) and (5.10). On the language of kernels, (5.11) means the permutation of the arguments and the substitution a —> —a — 1:

L(i,n) = K(u,0

For B we have two theorems. The first one follows from (5.9) and (5.3).

Theorem 5.5. Let F\ — BF (i.e. F and F\ from S"^) are respectively the contravariant and covariant symbols of an operator A). Then F\ is obtained from F by means of an integral operator with the Berezin kernel:

Fi(Z, K v)dx(u>v)

In fact, for polynomials F the Berezin transform is reduced to a differential operator.

Theorem 5.6. Let F\ = BF (as in Theorem 5.5). Then Fi is obtained from F by the following differential operator

T(—2cr + r)r(—2cr — r — 1)

B(A) =

r(—2cr)r(—2<r — 1)

r(r+l)=A

Proof. It follows from Theorem 5.5 and the invariance of B, see (5.6), that the co- and contra-correspondences commute with g.

Therefore, it is sufficient to consider some polynomial F from 7ik(X). For example, take the minimal one: F = Fkto, see (4.5). Then, by (5.10), the operator A is

A =

1 (A.

(—2<T — 2)(*) \di

In its turn, A has, by (5.1), the covariant symbol

Fx = bk(a)F (5.12)

where

^*(g) = (—2(r _ 2)(>) ' (2{r)t*)(~1)* (5-13)

r(—2cr + A:)r(—2<r — k — 1)

(5.14)

T(—2cr)r(—2<r — 1)

Due to (3.9), equality (5.12) with (5.14) means F\ = B(A)F □

Theorem 5.6 is a spectral decomposition of the Berezin transform: on every 'Hk(X) it is the multiplication by bk(cr).

Let cr —► —oo. Then by [2] 1.18(4), we have

1_±A

Therefore, by (5.5) and (3.6) we obtain

n n 1 *r2^F\ dF2

Fi*F2 = F,F,--N

It gives us that on the algebra S(A') the correspondence principle is true: if cr —>• —oo, then

Fx*F2 —► F\F2, —2a(F\ * F2 — F2 * Fi) —> {Fi, F2},

where {Fi,^} is the Poisson bracket, see (3.7). So, for the Planck constant we take h = —1/2a.

Moreover, we can write explicitly an asymptotic decomposition of B when a —► —oo. It is much more convenient to use not powers of h but ’’generalized powers”, see (1.3), of —2a — 2.

Then the decomposition turns out to be given by a series which terminates on each 7ik(X).

Theorem 5.7. There is the following decomposition of the Berezin transform:

---------------------m!--------------------------(-2a - 2)(~> (515)

m=0 v '

Proof. The eigenvalue (5.13) of B can be written as

t (/* + 2)W 6* = -^)-

where p. = —2cr — 2. By (2.7) we have

k

(/i + 2)W = ((/*-*+!) + (* + !))[*> = (^) (/j - k + l)(*-mJ(fc + 1)H

so that

m—o x ' '

_ 1 (k + m)\ 1

m! (k — m)\ /i(m)

The fraction (A; + m)!/(A: — m)! can be written as njLi(^ ~J^)- ^ut ^ is just the eigenvalue on 7ik{X) of the nominator of the first fraction in (5.15). □

REFERENCES

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