Poisson transform for hyperboloids Текст научной статьи по специальности «Математика»

POISSON TRANSFORM FOR HYPERBOLOIDS

A.A.ARTEMOV*

Tambov State University, 392622 Tambov, Russia

E-MAIL: ARTEMOV@MAIN.TSU.TAMBOV.RU

§0. Introduction

The Poisson transform, as well as the Fourier transform and spherical functions, is one of the main tools of harmonic analysis on homogeneous spaces (in this connection, see [11]). For applications of it, see for example, [2], [3], [7], [8], [11], [12], [13], [15]. In this paper we consider this transform for hyperboloids SOo(p, q)/SOo(p, q — 1), q ^ 2, in ® n, an important class of semisimple symmetric spaces. Our main interest is an asymptotic behaviour of the Poisson transform at infinity. We write down explicit asymptotic decompositions of Poisson transform for arbitrary C°° functions. For A'-finite functions these decompositions turn out to be absolutely convergent series. For "basic” functions (whose powers form the decompositions) we take both exponents and some other (perhaps more convenient) functions. In this paper we consider the general case p > 1, the case p = 1 (hyperboloids of one sheet) was studied in our paper [1].

§1. Representations of the group SO0(p,q) associated with a cone

In this paragraph we recall some facts about representations of the group G = SOo(p, q) associated with a cone, see [9]. Besides it, we give explicit formulae for eigenvalues of an operator , see (114) and (1.15). They generalize formulae obtained in [9] for eigenvalues of an intertwining operator Aa,e-

Let us consider in IR n, n = p + q, the bilinear form

n

[x,y] = -Xiyi - ... - xpyp + xp+lyp+i + ... + Xnyn = ^ AiXiyi.

t—i

Ai = ... = Ap = -1, Ap+i = ... = An = 1, n = p + q, p ^ 2, q ^ 2. The group G is the connected component of the identity of the group of all linear transformations of M n preserving this form. We shall assume that G acts on ln from the right: x i—► xg, hence we write vectors x in row form.

Consider two accompanying spaces Mp and M 9 consisting of vectors (xi,..., xp) and (xp+i,xn). By ( , ) we denote the standard inner product in both spaces. For x E M n we denote

IX | = ^/x? + ... + x2.

Let S be the intersection of the cone [x, x] = 0 and the cylinder |x| = 1, x £ M n. It is the direct product Si x S2 of two unit spheres Si = S,p-1 C M p and £2 — Sq_1 C 9, so that a vector s 6 5 is the pair

s = (u,v), ueRp, uEM9, (u,w) = 1, (u,v) = 1. (1.1)

Let K denote the maximal compact subgroup of G preserving |x| : K = SO(p) x SO(q). It acts on S transitively, each component on its own sphere, so that S' is a symmetric space of K, the stabilizer Kq of the point s° = (1,0,..., 0,1) is isomorphic to SO(p — 1) x SO(q - 1).

The Euclidean measure ds on S is the product dudv of the Euclidean measures du,dv on the corresponding spheres. It is invariant with respect to K. The volume of S is equal to QpQq, where

= ‘¿7rr/21 T(r/2).

* Partially supported by Goskomvuz RF (grant 95-0-1.7-41).

Let Ai, A2 denote the Laplace-Beltrami operators on spheres Si,S2 respectively. Then the Laplace-Beltrami operator A5 on S is the sum of the operators Ai and A2.

Let R denote the representation of K by rotations in V(S) = C°°(S):

(s) = tp(sk), k G K

It is unitary with respect to the inner product

(V>. V)-Js iJ>{s)<p(s)ds. (1.2)

The representation R decomposes into a direct sum of pairwise non-equivalent irreducible representations pz on spaces Hz = Hfp^ <g) Hm\ where z = (/, m) is a pair of integers called a weight. Here / ranges 7L or N = {0,1, 2,...} for p = 2 or p > 2 respectively, similarly m ranges 7L or N . Therefore the lattice Z of weights z consists of integer points of the plane (p = q = 2), of the upper half-plane (p = 2, q > 2), of the right half-plane (p > 2, <7 = 2), of the first quadrant (p > 2, <7 > 2).

The space Hz is an eigenspace of the operators Ai and A2 and therefore of the operator A5:

AiV? = Aiv?, A2(p=X2ip, As<p = \zip, (<pEHz)

where

Ai = /(2-p —/), A2 = m(2 - q - m), Xz = Xi + A2.

For p > 2, q > 2 the spherical function in Hz with respect to Kq is the product

ip2(s) = V>,(p)(si) №(sn), (1.3)

where

№(t) = cS?(t)/cj1( 1), r > 2,

C^(t) being the Gegenbauer polynomial. If p = 2 or q = 2 then the corresponding factor in (1.3) has to be replaced by (si + isi)1 or (sn-i + isn)m respectively.

Let Ez be the projection operator in V(S) (or in L2(S)) onto Hz:

(e2v) (s°k) = (R(k)<p, ipz)/(rpz, ipz),

Let V£(S), £■ = 0,1, denote the subspace of functions ip G 'D(S) of parity e\ ip(—s) = (—l)ey?(s). Let <7 (E C . The representation Tat€ of G acts on T>£(S) in the following way:

(r»,.(»)*>)(«) =v(i^j) M’-

It is continuous and indefinitely differentiable. The restriction of TaiC to K is the representation R£ of K

by rotations in Ve(S). It is the direct sum of representations pz with z from the lattice ZE : I + m = e.

Here and further the sign = denotes the congruence modulo 2.

Denote

<x* = 2 — n — a.

The form (1.2) is invariant with respect to the pair T^, 7V»iC, so that

= (V’.IWGT1)*'). (1-4)

Let us denote by Lq the following element of the Lie algebra 9 of G:

0 . • M

0 0 . . 0

0 . . 0 }

Since Lq centralizes Kq, the operator Tai£(L0) preserves the family of spherical functions xJjz. Namely, it carries ipz to a linear combination of four ’’neighboring” functions:

4

Ta,e(L0)rpz = ^ 7i(z) fkfaz) V’z+e., (1.5)

»=1

where 7i(z) are some positive functions, e, are the following four vectors on the plane: e\ = (1,1), =

(1, —1), 63 = (—1,1), e4 = (—1, —1), and /?,(<r; z) are the following functions of z G 1R 2:

Pi(<r\ z) = a - I - m, /32{cr\ z) = <r-l + m + q — 2,

/?3(<x; z) = cr-\-l — m + p — 2, /?4 (cr; z) = a + l + m + n — 4.

The line Pi(<r\ z) = 0 on the plane z is called a barrier for Ta>£ if it meets Z£. If the line /?,■ = 0 is barrier, then Vff'C'i denote the sum of subspaces Hz, z G Z£, for which /?,(cr;z) > 0.

The subspaces V0)e,i are invariant with respect to G in the representation TCt£. Any invariant subspace is the sum of intersections of spaces Va%cti- Therefore, if a is not integer, then Ta)C is irreducible.

Let A0 be the following element of the universal enveloping algebra of g (it differs from the Casimir element by a factor only):

Ag = 'y ^ A» Aj {^Eij A,'XjEji^ ,

i<j

Eij being the standard matrix basis. The representation T0t£ carries it to a scalar operator:

T0'E(Ab) = a* aE. (1.6)

We shall use the following notation for a character of M *:

xX'£ = |x|Asgnex = x+ + (-l)exA , (A GC ,£ = 0,1).

Define an operator A0t£ on V£(S):

(^,cV>)(*) = / ( “ [s,s\)a ,£(p(s)ds. (1.7)

J s

The integral converges absolutely for Rea- < 3 — n and can be extended to other a by analyticity to a meromorphic function. The operator A0y£ is a continuous operator in V£(S).

The operator Aa<£ intertwines the representations Ta<£ and Ta-1£:

Tff*t£Aa)e = Aat£T(jt£.

On every subspace Hz it is a scalar operator:

Aate (p = a(<7,e\z)(p, <p£Hz, (1.8)

where

T(3 — n — cr)r ( — cr')

«(*,£;*) = 2*+"Tri(-ir—--------------------7—i-------------------------------------J-. (1.9)

By (1.8) and (1.9) we have

Ao*,tAo,t = j(cr,e)E, (1-10)

where

7(<r, e) = 2V-*r(a + l)r(<r* + l)r (-^ - a) Y - <r*) •

• [(-ircos^ - COS (cr + |)tt] . [(-l)e+PCOS^ - COs((7 + |)tt] .

Thus if cr is not integer, then the representations Tffi£ and Ta»i£ are equivalent. In reducible case there is a partial equivalence.

The operator Aa^t interacts with the form (1.2) as follows:

(Aa>eil>t<p) = (^,Aw,E(py (1.11)

Let us extend the representation T0yt to the space Ve(S) of distributions on 5 of parity e - by formula

(1.4): now in (1.4) ip is a distribution in Ve(S), y is a function in V£(S), and (ip,<p) means the value of a distribution xp at a function ip. Indeed, it is an extension by attaching to a function xp E 'D£(S) the functional ip i—► (ip,ip) in VE(S) by means of (12). Similarly we can extend the operator Aa<t to V£(S) -by (1.11).

In §5 we shall need the following operator A^j}, r G N , in +r (•£):

(Aa,l<p)(s) = J^(-[s,s\Y ,£ <u,u>r <p(s)ds, (1.12)

see (1.1), so that for r = 0 it is (1.7). The operator A^j} intertwines the representation Re+r of K with itself (but for r > 0 it is not intertwining operator for G). Therefore the spaces Hz are eigenspaces of it:

<P = ar{(r,e]z) <p, <p€HZy zeZe+r. (1.13)

To write explicit expressions of ar, we need to make some preparations. We shall use the ’’generalized powers”:

= a(a — l)...(a — m + 1), = a(a + l)...(a + m — 1).

Introduce the following polynomials Srj(l) in / of degree r:

r(*0

S

.(() — ___________________[(r-k)

r;U ^2k-i(2j-k)\(k-j)\

(the summation is taken over k satisfying inequalities j ^ k ^ 2j, k ^ r). Notice that these polynomials satisfy the following recurrence relation:

Srj = Sr-i,j-i + (/ + 1 + 2j — r)Sr-ij.

Theorem 1.1. The eigenvalues ar(a,£-,z) of the operator A^j} are given by the formulae:

n (rr c- — 9CT+n -tt-T /_ 1 \m________________I (3 — n — O')_________________

r(- fA»(o-- r,z))T(- %/34(<r- r,z))T(l+ I)

4

‘(¿) |a_1a,F(1 + ^3(°'~r;z)'1 + \/34(<r-r;z),l+ |;a2) (1.14)

_ 2ff+n-r7rf/jxm__________________r(3 - n - a)---------------

r(- \Pi{cr -r]z))Y(- ±/?2(<T-r;*)) i

.\^2jS (I)__________________r(^T1 -v + r-j)____________________ (1 15)

2. ^ Wr( _ i_H(7 _ r + 2j. 2))r( _ jA(<r _ r + 2j. z)) V-K)

j= o

(formula (1.14) is written for p > 2, for p = 2 one has to replace I by l/IJ.

Proof. To prove (114), we use the method from [9]. For definiteness, assume p > 2, q > 2. Putting in (1.13) ip = xpz and taking it at s = s° (recall xpz(s°) = 1), we obtain by (1.12):

ar(a,£]z) =

= i)p_iiVi J J (si-sny,,esritp(iP\si)xp^\sTl)-(l-s2i)IL^(l-sl)!L^1dsidsn.

Denote n = (p — 2)/2, v — (q — 2)/2 and compute separately the integrals

d± = J j (x - y)^.xrxp\p\x)(l - x2)(p_3)/2V'^)(y)(1 ~ y2)(q~3^'2dxdy.

Let us pass to Fourier transforms. The Fourier transform of the function x+ is (see [5] p.196):

T(t) = iT(X + l){e‘A,r/2/;A"1 - e_iA,r/2CA_1}. (1.16)

Denote for brevity p = (p — 2)/2, u — (q — 2)/2. The Fourier transforms of the functions (1 —

ar2)^—3)/2-0/P)C^) an(l (1 — y2)+~3^2are expressed by means of Bessel functions (see [6] 7.321)

and are equal to

A(t) = 1/2) + (-lycn.Wlil). (117)

B(t) = imT^r(v + 1/2 )[«;•' + (-l)mCl/,+m(|<|). (1.18)

respectively. Therefore

1 r°°

d+ = —^p_1fii_1(-Or / T(t)A^(-t)B(t)dt.

¿TT J-00

We can rewrite it as follows:

1 . J . |* 1 roo

d+=-Qp^a,^(-) |^ j l-rT(t)A(-at)B(t)dt.

Introducing here (1.16), (1.17) and (1.18) we obtain

d+ = (27r)/i+t/+12r+,+m+1r(A + 1) [e^(-l)' - e"irL(-l)m+r •

. j . r 1 r 00

w) L=l“"'‘ I t-X-’‘-''-r-'iJM^V^m(t)dt.

The last integral is computed with the help of formula [6] 6.574(1) and we finally obtain

j ol — A—r n/2/_i\m_______________________________________________________________1)_#

+ _ { 1 r((A — I + 711 + g + r)/2)r((A — l — m + r + 2)/2)r(/ + p/2)

i d V\ »_,/-A + / + m- r-A + /--m-g + 2- r p 3^^

Ada) L=ia V 2 ’ 2 ’ 2;a /'

Clearly d_ = (—l),+m+r<i+, so that for z 6 Z£+r we have ar(cr, e; z) = d+ + (—1 )£d- = 2d+ with A = a*. It proves (114).

Now we make the differentiation in (1.14). We need some formulae from differential calculus (see, for example, [6] 0.433(1), 0.432(1), 0.431(1)):

(¿y = (ir. i=°

(¿)r=E dn^ <»•»>

;=o

where

««=(L20) dri = ¥j\= 2Jj!(r - 2j)!

Applying (1.19) to the product a1 F{a,b\c\a2) where F is the hypergeometric function, we obtain

(¿)ra'F(a,6;c;a2)L=i =

r(e)r(l-a)r(l-6)^0|. r(c — a — 6 — j) WA rfi-t)

T(c — a)T(c — 6) 4^ r(l-a-i)r(l-6-j) V W ’

i *

The last sum is precisely Srj(l)- □

§2. Eigenfunctions of the Laplace-Beltrami operator

The hyperboloid X : [x, x\ = 1 in № n is a homogeneous space G/H, the stabilizer H of the point a:° = (0, 1) is isomorphic SOo(p,q — 1). Let us introduce on X the ’’polar” coordinates t,s (/£l,

s E S) as follows: x = (sh< • u,cht ■ v), s = (u,v), see (1.1). In these coordinates the operator Laplace-Beltrami A is:

a=-£ - i(p - i>cthi+<* - i)thíi

Notice that

— (u, v) = s (t —*■ +00), (2.1)

so that S can be regarded as a boundary of X (in sense of Karpelevich).

Let U denote the representation of G on C°°(X) by translations:

(tf(fiO/)(*) = f(xg).

It is continuous and indefinitely differentiable. It generates representations of Lie algebra of G and its universal enveloping algebra which we denote by the same symbol U. The element A0 is mapped just to A:

U (Afl) = A. (2.2)

For a E C , £ = 0, 1, denote by 7ia e the subspace of C°°(X) consisting of functions f(x) satisfying

A/ = <r*crf, f{-x) = (-l)e/(*). (2-3)

It is closed. Clearly 7{a¡e = 'Ha>Let Ua,e denote the restriction of U to Ti0i£.

Let us separate the variables t and s in (2.3): set f(t,s) = R(t)(p(s). Then the function ip on S has to belong to a subspace Hz, z = (l,m), see §1, and the function R(t) on the real line must satisfy the equation

d2R r. „. , -i dR r 1(1 + p — 2) m(m -f q — 2) 'i _ , _ *\

“dF"'í(p-1)cthí + (?-1)thíldr + {-^-------------------------------tfi-----}R = ^R- <2-4)

Since f(—t,s) = f(—t,u,v) = f(t,—u,v) = = (—\)lf(t,s), the function R has parity /:

R(-t) = (-1 )lR(t). (2.5)

and ip belongs to V£(S), so that z E Ze.

For simplicity, we assume p > 2 (then / E N ) in the main part of this section, and in the end of it we

indicate the differences for p = 2.

Let us make the change of the variable and of the function in (2.4):

th 2t = y, R= (chí)c7(th¿),/r,

then for F we obtain the hypergeometric equation with parameters (—a + / + m)/2, (—a 4■ I — m — q + 2)/2, / + p/2. Notice that the first two parameters are precisely — (l/2)/?i(cr; z) and —(l/2)/?2(cr; 2:). Thus the function

R(*^i) = (cMHthO'f g;th2«) = (2.6)

= (chi)" (tht)‘ i/31((T1z),-ÍA(o',2);/+|;th2i),

is a solution of (2.4) satisfying (2.5). It is invariant with respect to mi—>2 — q — m and a i—► a* (the last statement follows from [4] 2.1(23)).

Theorem 2.1. For any function f £ 7iaiC there exist functions tpz from Hz,z E Zz, such that

/(*>*) = (2-7)

This series converges with respect to the topology of the space C°°(X). In particular, we can differentiate equation (2.7) with respect to t and apply the operators A\ and A2 as many times as desired.

The theorem is proved similarly to the corresponding theorem in [1].

Let us describe invariant subspaces in 7ia e. Let (* = 1,2, 3,4) denote the subspace in 'Ha^

consisting of the functions / for which the functions <pz from (2.7) are contained in Va,£,i, see §1.

Theorem 2.2. (see [15], [13]). The subspaces and i — 1,2, are closed G-invariant subspaces in 7i<7,e, and any such a subspace in 7iai£ is a sum of intersecions of subspaces above. In particular,

if a is not integer, then Ua,e. is irreducible.

For proof, it suffices to observe how the operator U(Lq) acts on functions h(a,z\x) =

R(cr, z\ t)ipz(s), z £ Ze. For brevity, denote them hz. Namely, as we shall see in §4,

4

U(L0)ht =Y, M".*) (2-8)

1 — 1

where

* } Uw. * = 3>4>

and 6i(z) are some non-zero numbers. □

Let us decompose R(a, z;i) in series in powers of 77 = (ch£)-2. Firstly we have:

R(a,z]t) = g(a,z)(chtyW(a,z-,r)) + g(a*,z)(cht)a W(<r*,z\r)), (2.10)

where

(T + l + m —a + I — m — q + 2 4 — n \ /«„„x

W{a,z-,rj) = (thi) ------------------,--------------------------------(2.11)

r(<r + (n — 2)/2)T(/ + p/2)

!l a' * T((<7 +1 — m + p)/2)r((cr + 1 + m + n - 2)/2)

(2-12)

This formulae are obtained from (2.6) with the help of the transform [4] 2.10(1). Expanding for / > 0 the first factor (thi)* = (1 — rj)1^2 in the right hand side of (2.11) in the binomial series, we can presented W as the sum of a series of powers of 77:

W(a,z-,Tj) = ^2wr((T,z)r}r, (2.13)

r=0

Further, expand W in a series of powers of 1 — £ with £ = th< and denote this sum by V(a, z;£):

OO

V(<rtz\Z) = ^ur(o-,2)(1 -0r, (2.14)

r=0

(Unfortunately, for p > 1 the function V is not expressed in terms of the hypergeometric function of (1 — 0/2). F°r t > 0 we have

V(<rtz]£)=W(ffiz\rj)i 77 =l-f2, (2.15)

so that we can rewrite (2.10) for t > 0 in the following way

R(a, *; <) = g(v, *)(cht)’ V(a, z;f) + g(a‘, ^(chi)"' V(a’ ,z;0, (2.16)

The coefficients vr and wr of series (2.13) and (2.14) are linked by formulae

r — 1

/ \ 1

W

’ j=o

[r/2] / _ . \

Vr(<r,z)= ^(-i)*2r-2^r k jwr-k{cr,z), (2.18)

where crj are defined by (1.20).

Let us write vr and wr explicitly. Assume

Recr > r - (n - 2)/2, (219)

so that Re(<r* — a) < —2r and we have from (2.16):

£=1

Remembering (2.6), we obtain:

Comparing this with formulae (1.14) and (1.9), we obtain:

*(*.*) = (2.20)

\r/ a(<r*,*)

By analyticity in cr we can take off the restriction (2.19), so that (2.20) holds for all cr for which the right hand side of (2.20) makes sense.

Substituting into (2.20) explicit expressions (1.15), (1.9) we obtain

vr((T,z) = (-^’'^¿2J5rj(01(1/2)/?l(CT'':)1(j)1(1/2)^IT’z)lO>. (2.21)

r-U (a + (n-4)/2)0)

Similarly

wT

(,}_(‘ '''-uvjw wh«-wU)

Notice that formulae (2.21) and (2.22) are connected by means of (2.17) and (2.18).

Lemma 2.3. The coefficients vr(a, z) and wr(a, z) are polynomials in Ai = /(2—p — /), A2 = m(2 — q — m) of degree r in Ai and A2 separately.

The lemma follows from that (2.11) is invariant with respect to mi—>•2 — q — m and /1-+ 2 — p — I (the latter statement follows from [4] 2.1(23)).

Let us normalize these polynomials so that the highest coefficient with respect to \2 (i.e. the coefficient

of A2) is equal to 1. We denote these normalized polynomials by Sr(cr; Ai, \2) and wr(a; Ai, A2), so that

vr(*,z) = (-l)r 2rrl^-p^(r) vr{<T; At, A2), (2.23)

Wr(<T,z) = (~iy 22rr!((r I n-4)(r) (^i *1 » *2) (2.24)

Let us write some first polynomials vr, wr :

0 0 1

V0 = W0 = 1,

00 \ > /

Vi = Wl = -Ai 4- A2 + (T{(T + q - 2),

v2 = [ — Ai + A2 + <t(<t 4- q — 2)] [ — Ai + A2 + (<r — l)(cr + <7 — 3)+p — l]+ 2(2cr -+- n — 4)Ai,

w2 = [ — Ai + A2 + cr (o’ + q — 2)] [ — Ai -+- A2 + (c — 2)(cr + q — 4)] + 2(2<r -f- n — 4)Ai,

Let us consider the case p = 2. Then / ranges TL and we have to replace / by |/| in formulae (2.6), (2.12), (2.14), (2.22). In formula (2.9) it is necessary to make an exception for / = 0, in which case we have

x.(a z\-{ foi*7’2) A (**1*). * = !» 2,

I 6i(z), pi_2(tr,z) (3i-2(a*,z), ¿=3,4, formulae (2.11), (2.21), (2.22) remain valid.

(3.2)

§3. H-INVARIANTS

We refer to [10]. Let us show elements invariant under H in representations of G described in §1. These invariants belong to V (S) or to its subfactors.

Theorem 3.1. The space of H-invariant elements from Ve(S) in the representation T0i£ is one-dimensional except the case q = 2, a = —m — 1, mEN , m = £ + 1, in which case this space is two-dimensional. For the generic case, a basis is the distribution

or its first Laurent coefficient when it has a pole. In the exceptional case a basis consists, for example, of s“m_1 and <5(m)(sn)sgn(s„_i).

The operator A0>£ transfers da£ to 0a*i£ with a factor:

(3-1)

where

j(a,e) = 21-<77TIVtr((7+ l)r( ~2~ - -a) (-l)£cos(<r+ |)tt-cos^

By (3.1) we have another expression for the factor 7 from (1.10):

l(°’i e) = ](<?, e)j{a*, e). (3.3)

§4. POISSON TRANSFORM According to [11] the Poisson transform Pa<£ associated with the //-invariant Qa>£ is defined as follows

(Pol£p) (x) = J (r<T(g~1)9<j,cS) (s)<p(s)ds = [x, s]a’e<p(s)ds, (4.1)

where x = x°g. It is a linear continuous operator from V£(S) to C°°(X). (The continuity is proved

similarly to [14] pp. 113-114). It intertwines Ta* >£ with U:

U(g) Pff,£ = Pa,e Ta.i£(g), geG. (4.2)

Hence (see (1.6) and (2.2)):

A o Pa e = cr* aPa £.

The function (P0tetp){x) has parity e:

(p„,f ¥>)(-*) = (-1 )c(p<,>ey?)(x),

so that the image of Pa<£ is contained in Tia,£- The Poisson transform interacts with the operator Aa,e as follows

Po,e Affi£ = j{ct,e)Pa.t£. (4.3)

As a function of a, Pai£ is a meromorphic function with poles at points a = — 1— e — 2k, k EN . So

firstly we consider a ^ —I — e — 2k.

Theorem 4.2. Let y E V£(S). The decomposition (2.7) for the function Pa,eW E have the form:

(P<T,c<p)(i,s)= ^2 X(<r,z) (Ez<p)(s), (4.4)

z£Z‘

where Ez are projection operators from §i, the numbers x{v, z) ~ let us call them the Poisson coefficients - are given for p > 2 by the formula:

X(<r,z) = (-1) 22 a 7r/ + p/2)r((cr -l-m + 2)/2)T((<r - I + m + q)/2)- ^

for p = 2 one has to replace I by |/|. The radial factor from (4.4), i.e. the function

V(<r,z;t) = x(<r,z) R(<r,z;t), (4.6)

is expressed in the terms of W, V, see (2.11), (2.15), as follows

}V((T,z\t) = (-iy^(cht)aa(a*}z)W(<r,z,Tj) + (cht)a*j((r>£)W((r*,z]r?)} = (4.7)

= (-l)£{(chi)<7a(o-*,2)K((r,2;0 + (cht)a’ j((T, e)V(a*, 2;£)}, (4-8)

where a and j are given by formulae (1.9) and (3.2) respectively.

Proof. Expand the function Po,cV G 'Ha.e into the series (2.7). Elements k G K preserve the coordinate t. Therefore, by (4.2) with g = k, we obtain that for every weight z G Z£ the map <p t—► ipz of the space T>£(S) into the space Hz commutes with rotations k G K. Since the representations pz (see §1) are irreducible and pairwise inequivalent, the map <p <pz differs from Ez by the factor only - we denote its by x{a, z)> - so that <pz = x(&, z) Ez<p, and we obtain the decomposition (4.4).

Now let us take ip G Hz, z G Z£. Then = Ez(p, and series (4.4) is reduced to one term only:

(Po,e<p) (*,«) = <p(s), (4.9)

where the function ^(cr, z\t) is defined by (4.6). Introduce (2.10) in (4.6), we obtain:

W(<ryz]t) = x{v,z){(cht)ag((T,z)W((r,z]r)) + (ch<)* g(<r*, z)W{cr*, 2; 77)}. (4.10)

Assume that Reo- > (2 — n)/2. Then Recr* < (2 — n)/2, and by (4.10) and (2.11) we have

V(<r,z\t) ~ (cht)ax{(r,z)g{(T,z) (t -> +00). (4.11)

On the other hand, the left hand side of (4.9) behaves as

(ch*)ff(-l)e J {-[s,s\Y'£ip(s)ds= (chty (~\y a(a*-, z)<p(s)

when t —> +00 (we used (2.1), (1.7), (1.8)), so that by (4.9) we have:

'¡'(o-, z\t) ~ (ch/)<7(-l)£a(cr*; z) (t —► +00). (412)

Comparing (4.11) and (4.12) we obtain

(-l)*a(<rV) = x(v,z)g((r,z). (4.13)

Now by analyticity in a we may take off the restriction Recr > (2 — n)/2. Substituting expressions (1.9) of a and (2.12) of g into (4.13) we obtain (4.5).

Let us apply (4.3) to <p G HZ) z G Z£ and use (1.8) and (4.10). Comparing coefficients, we obtain

a(<r, z)x(<7, z) = j(a,e)x{cr* ,z). (4.14)

From (4.13) and (4.14) we find that

X(<r,z)g(a*,z) = (-1 )£j((J,e).

Together with (4.13) it proves (4.7) and (4.8). □

Formula (4.6) together with (4.5) can be proved by a direct computation. Integral representations obtained in this way will be used in §5. Namely, by (4.9) and (4.1) we have

tf(<r,2;/) = (chi)‘7^p-i^,-i J J (~£x + y)*7'6ipi(x)ipm(y) ■

•(1-z2) 2 (1 - y2) 2 dxdy, (4-15)

where £ = tht. As in §1, using Fourier transform, we obtain

*(<r,z;t) = J°° [T(i) + (-l)‘T(-i)]^(i()B(-i)di,

where T, A, B are given by (1.16), (1.17), (1.18). So

= (chi)£72/i+t/+27r^+l/+1i/+m+1 r(cr + 1)-

•(e^ -(-l)£c-“)r" J°° t-°~'—'-lJl.+i№J*+m(.t)dt. (4.16)

The last integral is computed with the help of formula [6] 6.574(1), and we obtain (4.6) with \ given by

(4.5).

Now let us prove (2.8). According to (4.2) the operator U(Lq) acts on the functions = Pa,e^z in the same way as the operator Ta*|C(Lo) acts on the functions rf}z (see (15)), i.e.

4

z+ei• (4-17)

1=1

By (4.9) and (4.6) we have 'I'* = xi*7, Z)hz- Substituting this into (4.17) we obtain

A

* Nx(*,* + e,-)

■h

z+ei >

U(Lo)hz ='}T'Yi(z)pi(a*,z)------------

fri X{<r,z)

which is (2.8).

§5. Asymptotic behavior of the Poisson transform

First we expand the Poisson transform into series of powers of 1 — tht for the A'-finite functions <p G T^(S), i.e. for linear combinations of functions from Hz. In this case the expansion is given by absolutely convergent series.

For a G C , r G N , define the differential operators L„ir, MGyT on 5 as follows. For r > 0 we set

L,,r = «,(**; A,, Aj), M,,r = wr(<r';Ai,A2),

where ur, wr are the polynomials from §2, see (2.23), (2.24), Ai, A2 are the Laplace-Beltrami operators on S\,Si, see § 1; and for r = 0 we set LVir = 1, Ma<r = 1.

Theorem 5.1. Let a be generic: a ^ (n/2) + Z , a £ — 1 — e — 2N . For any K-finite function ip G 'Dg(S) its Poisson transform \P0)Etp) (t, s) has the following expansion in series of powers of 1 — tht:

OO

(p<7,e<p) (<»*) = (chOa Xr(a’(Air-+r,e-r<p) («)(1 - thty + r=0

OO

+(chi)ff* ^t/r(cr,£)^L<7|r^(s)(l - tht)r, (5.1)

r=0

(V) * f r

where A\ 1 is the operator from §i,

Ir(<T,£) = (-l)C+r

\ I

= (->№, e)^V|)W

= (—l)r21 a r —j,7T^n 4^2T(o- + l)r^~—— <7 — rj ^cos^q- + — cos^e -f f)^] ‘

Both series in (5.1) converge absolutely.

Proof. It suffices to consider the case when <p E Hz, z E Ze. Then we have (4.9) and (4.8). Expand both functions V in (4.8) into series (2.14) of powers of 1 — £. Then

'H(cr,z-,t) =

oo oo

= (chi)" £(-!)'<cr-, z)vr(<r, *)(1 - i)r + (chi)"' ^(-l)£i(ff, £)^(<T*, Z)( 1 - 0r.

r=0 r=0

Now applying (2.20) to the first series and (2.23) to the second series, we obtain

OO

^(cr, 2; t) = (ch*)ff xr(<r, e)ar((T* + r, e + r; z)(l - £)r +

r=0

+(ch t)°* 'Yyr(a, e)Sr(<7*; Ai, A2)(l ~Or»

r=0

whence (5.1) follows at once. The absolute convergence follows from the absolute convergence of the hypergeometric series (2.14) □

Equation (4.3) gives two relations for operators occurring in (5.1):

L„,rA,:t = j(<r, e)^f^A(;lrt+r, (5.2)

yr\<7) £)

A(;)+riC+rA„,, = j(c, e)»d^L'.,r, (5.3)

Denote by ur(a) the coefficient in (5.2) (it does not depend on e):

u,r{a) = j(a, = 2> + n - 2)M (<r + £)M (5.4)

Notice that these two formulae (5.2), (5.3) are connected by means of (1.10), namely, multiplying (5.2) by Aa*i£ from the right and using (1.10), (3.3) and replacing a by a*, we obtain (5.3).

Let us take an arbitrary function y? G V£ (S) (not necessarily K-finite) and decompose it in a series of

its Fourier components:

Ezip-

z£Zr

By the continuity of the Poisson transform we obtain

z£Zc

Apply to each term here formulae (4.9) and (4.8), we represent Po,e<P as a sum:

(p<t,£¥>)(M) = (chOa(P(7+ir¥?)(i.s) + (ch0ff (P<7~e^)(^>s), where £ = thi and the operators P^e are defined in the following way:

zeZr

(P„>)(i.») = E (-l)c>(o-.c)V(<r*,*;€)(^*v)(*)-

z£Zr

For arbitrary function <p we can state only that the (5.1) has to be understood as an asymptotic expansion.

Theorem 5.2. Let a be generic: a £ (n/2) + Z , cr £ -1 — e — N . For arbitrary function <p G ^(5) its Poisson transform has the following asymptotic expansion for t —► +oo:

OO

(/^)(*>5) ~ (chOff 5Zxr(<T,e)^(;.)+r e+r^(5)(l - tht)r +

r=0

OO

+(cht)° ^yr(o-,e)^Lcr(rv?)(s)(l - tht)r. (5.5)

r=0

The asymptotic equality (5.5) is understood as an asymptotic expansion of both functions (^Pae ¥>)(?.*) which means the following - for example, for (p^ip) : for every TV G N there exists a constant C such

that

N

(p££v)(ts)-J2xr(<r,£)(A[r-+r,£+r(p){s){l-0’

r=0

for all £ G [0,£o], 5 G S, £o is some number from (0,1).

The proof of Theorem 5.2 goes similarly to the case p = 1, see [1]. Here we restrict ourselves to indicating some steps of the proof, cf [1].

Define the function 4>(cr, z;£) of £ by the equation:

'H(<r,z]i) = (chty$(a,z\Z),

where z G Z£, £ = tht and the function 'P is given by (4.6) (or (4.7), (4.8)). Let us take in (4.9) <£> = xp2 and s = s° then by definition (4.1) of the Poisson transform we obtain for 4> an integral representation, which is (4.15) where the factor (chi)^ has to be omitted. This integral converges absolutely for Recr > —1/2 and can be extended on the <r-plane as a meromorphic function. The function $ has parity e + m = I. Denote

X(<r,z]Z) = a(<r*,z)V(cr,z;t)y

where a and V are given by formulae (1.9) and (2.14) respectively.

Then we express X in terms of $:

X(<T,z,Z) = (-l)e (/¿((7, z)$(<7, z\£) + n(<T,z)$(<T,z]tf), (5.6)

where z = (/, m), / = 2 — p — /,

sin (/?3(cr, z)tt/2) ■ sin [pA(a)z)'K/2sj

p((T,z) =:

sin(cr -f (n — 2)/2)7t • sin(/ + p/2)îr

for even p one has to evaluate a undetermined form in (5.6). (Formula (5.6) is obtained from (2.11) by means of [4] 2.10(1)).

For Re<r > 0 the function z;£) in (5.6) is estimated by means of integral representation (4.15), the function <I>(cr, ?;£) in (5.6) is estimated by means of integral representation (4.16), where one has to replace / by / and a by a*. Thus, for Recr > 0 the function X is estimated by some constant independent of 2.

Furhter, we have the following recursions for X:

Orri _L Q _ A

X(a\ I, m;£) = X(a\ /, m - 2;f) +------------—------X(<r+ 1; I, m — l;f),

(T + 1

■^X(a ;/,m;0 = ——^-----------------------------x<rX(a -I1 ,m]€) +<rX(<r -1,1+ 1 ,m;£)-

¿1 + p — I

Applying these formulae, we can estimate X successively for all a (first for Recr > —1, next for —2 < Recr ^ —1 etc.). This gives us desired estimates to prove (5.5). □

We can include a G — 1 — e — 2N (where Oat£ has poles) dividing 0a£ by a suitable function of <r, say, r((<7 + 1 + e)/2). Then we have to divide by the same function both Pa£ and xr(a, e), yr(<T,e) in Theorems 5.1 and 5.2.

Similarly we consider the expansions of the Poisson transform in series (absolutely convergent for K-finite functions and asymptotic for arbitrary functions) of powers of 77 = (ch/)-2 and £ = —e~2t. Then we replace operators A%)+rft+r and Lffir by operators 2~r Bat£J and 2~rM0yT for 77 and operators C0,r and Ka<r for C, where

1 r_1 B°'e'r = _ r’+ 1)B| 4r-'+r)-i,,+r-i.

c°.r=p(-vr-krr+kQA(X+k>

K*,r = ¿(_1)r'‘ (l) (ff + k + " - 2)Ir'‘1

with ur,crj defined by formulae (5.4), (1.20). These pairs of operators are linked by relations similar to (5.2),(5.3), for example,

xr(<r*,e) yr(<r, e)

Ma<rAai£ =

Ba<£irAa'£ = j((r, e)Vr^ * ^ Ma. ,r.

Xr[(T, €)

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