Boundary representations on the Lobachevsky plane Текст научной статьи по специальности «Математика»

BOUNDARY REPRESENTATIONS ON THE LOBACHEVSKY PLANE1

L. I. Grosheva G. R. Derzhavin Tambov State University, Russia

The Lobachevsky plane is the unit disk D — {z E C : zz < 1} (the Poincare model). It is a homogeneous space G/K, where G = SU(1,1), K consists of diagonal matrices (a maximal compact subgroup). Canonical representations R\, A € C, of G in the unitary case (A < 0) were introduced in [3]. In [1], [2] we considered canonical representations in a wider sense: A can be an arbitrary complex number, these representations are not necessary unitary, and they act on spaces wider than in [3]. These canonical representations R\ generate representations L\ of G acting on distributions concentrated at the boundary S of D (boundary representations).

In this paper we study those distributions which are invariant with respect to K. This space have two bases: the first one consists of the delta function concentrated at S and its derivatives in the radial direction:

where p = 1 — zz, and the second one consists of distributions

Ca,0; Ca,1i • ■ ■ j C\,mi • • • j

that are linear combinations of elements of the first basis and are orthogonal pairwise with respect to a bilinear form. This form, an ’’inner product”, is invariant with respect to R\ and L\, it is intimately connected with the Berezin form. They are also eigenfunctions of a differential operator which is the image under R\ of the Casimir element of the Lie algebra of G. Eigenvalues are (A — m)(A — m + 1).

These two bases are expressed each in terms of the other by means of upper triangular matrices with the unit diagonal. We determine explicitly these matrices and matrices of pairwise inner products of elements of these bases. Moreover, we write some generating functions: for instance, for the matrix of inner products of the first basis. In Sections 1-5 we recall some material from [2], [3] and write some other formulas (expressions of some invariant differential operators etc.). Our main results are collected in Section 6.

1 The Lobachevsky plane

Let us realize the Lobachevsky plane as the unit disk D : zz <1 in C. Let S be the unit circle zz — I and D = D U S. The group G = SU(1,1) acts transitively on D and on S by fractional linear transformations

az + b ( a b \

•y i—V . n — ----- n — I ___ I n n — hh — 1

' ~ bz + a' V b a J ’

The stabilizer of the point z = 0 is a maximal compact subgroup K of G consisting of diagonal matrices, so that D = G/K. Introduce on D ’’polar coordinates” p,u, where

p = 1 — zz

Supported by the Russian Foundation for Basic Research (grant No. 05-01-00074a), the Scientific Programs "Universities of Russia” (grant No. ur.04.01.465) and”Devei. Sci. Potent. High. School”, (Templan, No. i.2.02).

and u = eia 6 S, so that z = ru, p = 1 — r2. Let ds(u) denote the measure da on S. The Lebesgue measure dxdy (z = x + iy) on D is (1/2)dpda. The Laplace-Beltrami operator A on D is

A 2 d2

A = pz

dzdz

It is invariant with respect to G. In polar coordinates it is

Ft2 r) n2 f)2

If M is a manifold, then V(M) denotes the Schwartz space of compactly supported infinitely differentiable C-valued functions on M, with a usual topology, and V'(M) denotes the space of distributions on M - of linear continuous functionals on T>(M).

For a differentiable representation of the Lie group G, we retain the same symbol for the corresponding representations of the Lie algebra 0 of G and of the universal enveloping algebra. Let U be a representation of G on functions on D by translations:

(U(g)f){z) = f(z-g),

a quasiregular representation. The Laplace-Beltrami operator A is the image under U of A0, twice the Casimir element.

Let us denote by V{D) the space of restrictions to D of functions from T>(C) with the induced topology, and by V'(D) the space of distributions on C with supports in D. Consider the inner product with respect to the Lebesgue measure on D:

{FJ)d= [ F(z)f(z)dxdy, z = x + iy. (1.1)

Jd

The space T>(D) can be embedded into the space V'(D) by assigning to h 6E T>(D) the functional f^(hJ)D,fEV(D).

2 Elementary representations of G/{±E}

In this section we recall a description of the principal non-unitary series of representations of the group G which are trivial on the center.

A representation Ta, o G C, of G acts on T>(S) by:

{Ta{g)<p)(u) = (p(u ■ g)\bu + a\2a.

The element A0 goes to a scalar operator:

Ta(As)=a(a + l)-E.

The inner product from L2(S,ds)\

I ll\ t n\ ^ - f tit (ft t\/r\( Itl\rl Q i ft l\

JS

is invariant with respect to the pair (Ta,T-a-1)- If cr ^ Z, then Ta is irreducible and Ta is equivalent to T-a-\ (for a € Z there is a ’’partial equivalence”).

An operator Aa on T>(S) defined by

(Aa(p)(u) = [ |1 — uv\~2a~2(p(v)ds(v)

Js

intertwines Та with T-cr-i, i.e. Т-а-і(д)Аа = AaTa(g). A sesqui-linear form (A^,tp)s is invariant with respect to the pair (Та,Т&). In particular, for а Є K, this form is an invariant Hermitian form for Ta.

Let us take a basis фт{и) = um, m Є Z, in T>(S). It consists of eigenfunctions of Aa. In particular, for the function гро (it is equal to 1 identically), we have

The operator Aa as well as the factor j(a) is meromorphic in a with simple poles at, a G — (1/2) +N. Here and further N = {0, 1, 2, ...}. Residues of Aa at its poles are some differential operators.

3 Canonical representations

Canonical representations of G are representations R\, A G C, on 'D(D) defined by

(the integral converges absolutely for ReA > — 1 and can be continued analytically to a meromorphic function) intertwines R\ with R-\-2, so that the bilinear form

{Q\F,f)D = {F,Q\J)d respectively.

The multiplication by px+2 converts the sesqui-linear form B\(f,h) into the Berezin form, see [2].

(2.1)

where

j(a) = 2тгГ(-2а - 1)/Г2(-сг).

(2.2)

There is an equality:

j(cr)j(-(J - 1) =

2ir tan СГ7Г

2a + 1

= <i-rff’2^+[2A+4-(2A+5H!,s

+ (A + 2)

so that its radial part (acting on if-invariant functions) is

(3-І)

4 Boundary representations

The canonical representation R\ gives rise to two representations L\ and M\ associated with the boundary S of D. The first one acts on distributions concentrated at S, the second one acts on Taylor coefficients with respect to S. In this paper we consider L\ only.

Let us denote by T,m(D), m G N, the space of distributions ( in V'{D) having the form:

C = <A) (u)S(p) +<pi(u)s'(p) + ■■■ + (4.1)

where 5(p) is the Dirac delta function on the real line, S^(p) its derivative of order k, ipk G V(S). Let £(.D) = UEfc(.D). The representation R\ preserves E(-D) and the filtration So(D) C Si(-D) C .... Denote the restriction of R\ to T,(D) by L\.

Let us assign to the distribution ( of the form (4.1) the column

(<P0,<Pl,---,<Pm, 0,0,...).

Then L\ is a upper triangular matrix with the diagonal T_^_i, T_ ....

Tlio fnrm R, pon Kd f a wnv i-f ic rlo'finor? in f Vi o nQtnral r\n V’ ( T~)\ ■fr»-r

_1_ Iiv IWiill J_/ ^ UUlll L/ V_. U y. AU ikj 1U 11UUU1 m »* Uij Ull t_t YYl ^ -IS j 1W1

Re A > m — 1/2, then we continue it meromorphically. For example,

B\(ipS(p),ip5(p)) = -^-^(A_x-iV>,'>P)s-Let 6(A) denote this inner product with ip = ip = ipQ. We have

—TV \ I o\TVo\ I 1\

6(A) = BX(S, S) = (4.2)

This expression of 6(A) was obtained in [4].

Let us denote by Y,m(D)K and H(D)K the subspaces of Em(D) and £(D) respectively consisting of if-invariant distributions: coefficients ipi in (4.1) have to be constant.

5 Decomposition of boundary representations

From now on we restrict ourselves to the generic case: A ^ (1/2) + N.

First we define some differential operators Wajk on ^(5) - by means of a generating function. Let us consider the following power series in powers of p:

00

(1 - p)ml2F{a + 1, a + 1 + m; 2a + 2;p) = ^ wk(a, —m2)pk. (5.1)

k=0

The coefficients wk(a, —m2) of this series are polynomials in — to2 of degree [k/2] with coefficients rational in a. Keeping in mind that —m2ipm = (d2 / da2)ipm (where u = e*“), we set

d?

Wa>k = wk{a,—j),

notice Wafi = 1. Notice that d2/da2 is nothing but the Laplace-Beltrami operator on S.

The operators Waik play an important role (see [1], [2] for other information). They have simple poles in a at half integer points a satisfying inequalities (—k — 1)/2 ^ a ^ —3/2.

Using Wa£, we define operators : 'D(S) —>■ T,m(D):

The operator (x,m is meromorphic in A, it has simple poles at points A e (1/2) + N for which

A + 3/2 ^ m ^ 2A + 1. It intertwines T_A-i+m with L\.

For distributions £a,m(<^), we have the following ’’orthogonality relations”:

^(6,m(^),6,rW) = 0, r^m, (5.3)

where

,, , m! r(2A — 2m + 2)T(A + 1)T(A + 2}

“(A,™) = r(2A - m + 2)r2(A - m +1) (5'4>

_m! A^(A + 1)(TO+1)

47r (2A — m + l)(m)

Here and further we use the following notation:

= o(a — 1)... (a — m + 1), = a(a + 1)... (a + m — 1). (5.5)

The ’’old basis” tpS^{p) can be expressed in terms of the ’’new basis”

<p8{k)(p) = (5-6)

m=0

Let Vx,m be the image of £x,m- This space is contained in Em(D), it is an eigenspace for Aa

with the eigenvalue (A — m)(A — m + 1). _

The boundary representation Lx is diagonalizable (recall A ^ \ + N) which means that S(D) decomposes into the direct sum of VXim, m € N, the restriction of Lx to V\^m is equivalent to T-X—l+m (by £a ,m) ■

6 The space of if-invariant distributions

Consider the space £(D), see Section 4. In this space we take a subspace of distributions

invariant with respect to K. We have two bases in the space E(D)K : the first one consists of the delta function 8(p) and its derivatives:

and the second one consists of distributions

Ca,0> Ca,1j ■ ■ ■ > Ca,rm ■ ■ ■ >

where

Ca ,m = o)-

0

Therefore, Cx.m is eigendistribution for Aa:

0

Aa Ca,rn = (A - m)(A - m + l)CA,m-The intersection V\>m fl T,(D)K is one dimensional, it is spanned just by C\,m-

Theorem 6.1 The second basis is orthogonal with respect to the form B\, the orthogonality relations are

= j8(A,m), (6.1)

= o, (6.2)

where

TOi(A(m))4

p(X, m) = 6(A) • , wo -------------s---(6.3)

’ v ' (2A)(2m)(2A +1 -m)M v '

we use the notation (5.5), the factor 6(A) is given by (4-2).

Proof. Let us set ip = ip = %pQ in (5.2) and (5.3). In virtue of (2.1) we have

(-4—A—l+m^o, V>o)s = j(-.a - 1 + rn) ■ 2tt.

It gives (6.1) and (6.2) with

B{ A, m) = 27T7'(—A + m — l)a(A, m), for a(A,m), see (5.4). Substituting (2.2) and (5.4), we obtain

^ , r(2A-2m + l)r(2A-2m + 2)r(A + l)r(A + 2)

«A, m) = -™!--------------------r(2A — m + 2)T4{A — m + 1)-' (6'4)

Selecting here the factor 6(A), see (4.2), we get (6.3). □

Theorem 6.2 The elements of bases in question are expressed in terms of each other as follows:

= Vf D-.WlHtu1-*1)' , ,Bri

Proof. Let us compute Waiki'ipo)- Since {d? / da2){ipo) — 0, we have to take m = 0 in (5.1). We obtain

w ( i \ — 1_ (a + 1)2(CT + 2)2 ... (cr + k)2

iA'M - fc! {2<7 -f 2)(2<t + 3}... (2u + k + 1)

- 1 r> + i + 1>r<V + J> (6.8)

Since

ft! r2(u + 1)T(2ct + ft + 1) Ca ,m = ^X^i^Po) =

m j

“ (m — rj!

r=0

formula (6.8) gives

s=0

c = W i^I™]r ^ ~ m +5 + ~2m + 2)

' ' ' Vsyr2(A-m+l)r(2A-2m + s + 2) '

r-

It can be rewritten as (6.5).

Putting ip = ipo in (5-6) and using (6.8), we obtain

jM/p) _ VV-i)"*-»- (m) . r2(-A + m)r(-2A + 2rj_^ _ d [P)~fr'0[ ’ \r) r2(—A + r)r(—2A + rn + r) ^

It can be rewritten as (6.6) or (6.7). □

Theorem 6.3 There is the following expansion:

, OO k

exp (ujA8[p) = YJl^F(-\ + k,-\ + k--2\ + 2k--u) - Ca,*, ^ k-0

where F is the Gauss hypergeometric function.

Proof By (6.7) we have

(-1)“-* {(-A + t)[”-‘]}i

expl“^rm = £jfcA*2> (_2A + 2*)l—*l

k—0 77i—h

OO jc

------ IT

2

fc=0 s=0 v

The latter series is just the Gauss hypergeometric function F(—A + k, —A + fc; -2A + 2k; —u). □ Theorem 6.4 The pairwise inner products of elements of the first basis are given by:

„ , ,H, „ flT(A + 1)T(A + 2)F(2A — m - r + 1}

>(?)) =----------r2(A - m + 1)T2(A — r + 1)----- (M)

(A(mU(rH2

= • (2A)(™+r) ^6'10^

= (_2A)[J]-- (6-n)

Proof. Let m ^ r. We express distributions $(m)(p) and S^(p) in terms of C\tk, see (6.6), then use the orthogonality relations (6.1) and (6.2). We get

fp c(m.) r \ cfr)/ \\ i \™+r (m\ fr\ ^ A + m)r( 2A + 2s)

(BAa> W(p)) = }J-D {s){s)rH-\ + s)r(-2\ + m + s)

T2(—A + r)T(—2A + 2s)

5 — 0

2/................

/3(A, m).

T2(—A + s)r(—2A + r + s)

Substituting here expressions (6.4) of f3{\,m) and making transformations with Gammas, we obtain

For brevity, denote 2A = t. The sum in (6.12) can be written as F(t — m — r + l)T(i — 2r + 1)

r(i + 2)

• ^s\(™Vr)(t-2s + l)(t-m-s)(r-s\t-r-sf-s\t + l)(s\ (6.13)

s=0 / V /

The sum in (6.13) is a polynomial in t of degree 2r + 1. It turns out that it has a simple expression:

^(t-2s + l)(t-m-s)^-sHt-r-s)^-s\t + l)^ = (t + l)(2r+1) (6.14)

r(‘ + 2)«6.15)

T(t -2 r +1;

The combinatorial identity (6.14) can be proved, for example, as follows. The polynomial in the right hand side of (6.14) has numbers -1,0,1,..., 2r — 1 as its roots. Therefore, one has to check that the polynomial in the left hand side of (6.14) vanishes at these numbers.

Substituting (6.15) in (6.13), we obtain that (6.13) is equal to T(t — m — r + 1). It proves (6.9) and hence (6.10) and (6.11). □

Denote

cmr(A) = 7^-Bi(i<”‘>(p),iW(p)).

0(A)

By (6.11) we have

|(_A)H(-A)W\2 <w(A) = (-1)^ 1 (_2A)[ro+r] J • (6-16)

Let C(A) be the matrix (cmr (A)), m, r = 0,1,.... Consider the following generating function of two variables u,v for the matrix C(A):

°2_ um vr

$(A;^,u) = ^ cmr(A) • — ■ —. (6.17)

m,r=0

Theorem 6.5 The function 3>(A;u,v) can be expressed in terms of the Gauss hypergeometric

function whose argument is a polynomial in u, v of order two:

4>(A; u,v) = F(—A, — A; — 2A; — u — v — uv). (6.18)

Proof. By (6.16) and (6.17) we have

(-A)!ml(-A)[m](_A)W(_A)M ( ,u,v) - 2^ a)[m+r]m!rj

m,r=0 v 7

By [1] 5.7(8), it is a hypergeometric function of two variables:

$(A; u, v) = Fs(—\, —A, —A, —A, —2A; —u, —v).

Fortunately, this function with given parameters can be reduced to the Gauss hypergeometric function - by [1] 5.10 (4). It just gives (6.18). □

Formula (6.18) can be rewritten as follows:

Bx(exp (u-^)S{p),exp {vJ-)5(p)) = bW ' F{-\, -A; -2A; -u - v

uv

Let us write down the matrix C(A) and matrices of passage from one basis to other. Let the basis {Ca,™} is expressed in terms of the basis {<^(p)} by means a matrix M(A). We have

/

C{ A) =

1 A A(A-l)2 A(A —1)(A—2)2

2 2(2A—1) 4(2A—1)

A A3 A3(A—1) A3(A—1)(A—2)2

2 2(2A—1) 4(2A—1) 4(2A—1)(2A—3)

A(A-l)2 A3(A—1) A3(A-1)3 A3(A—1)3(A—2)

2(2A—1) 4(2A—1) 4(2A—1)(2A—3) 8(2A—1)(2A—3)

A(A—1)(A—2)2 A3(A—1)(A—2)2 A3(A—1)3(A—2) A3(A—1)3(A—2)3

4(2A—1) 4(2A—1))(2A—3) 8(2A—1)(2A—3) 8(2A—1)(2A—3)(2A—i

\

M( A) =

( 1 -0 0 0

V-

/1

A2(A-1) 2(2A—1)

-(A-l)

1

0

A(A-l)2 2(2A—1)

A-l

1

0

A2(A—1)(A—2) 4(2A—3)

3(A—1)2(A—2) 2(2A-3)

— 5(A 2)

A(A—1)(A—2)2 4(2A—X)

3(A—1)(A—2)2 2(2A—3)

| (A — 2)

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