; -iu

REFERENCEIC/66/18

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL

PHYSICS

CONTINUOUSDEGENERATE REPRESENTATIONS OFNON-COMPACT ROTATION GROUPS

N. LlfAlC

J. NIEDERLEAND

R.• • . . . (

1966PIAZZA OBERDAN

TRIESTE

IC/66/18

INTERNATIONAL ATOMIC ENERGY AGENCY

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

CONTINUOUS DEGENERATE REPRESENTATIONS

OF NON-COMPACT ROTATION GROUPS1"

N. LIMIC*

J. NIEDERLE**

and

R. RACZKA***i

TRIESTE

March 1966

+1 To be submitted to the Journal of Mathematical Physics* On leave of absence from Institute Rudjer BoskoviC, Zagreb.

** On leave of absence from Institute of Physics of the Czechoslovak Academy of Sciences, Prague.*** On leave of absence from Institute of Nuclear Research, Warsaw.

ABSTRACT

Three principal series of continuous most degenerate unitary ir-

reducible representations of an arbitrary non-compact rotation group

SO(p, q) have been derived and their properties discussed in detail. The

corresponding harmonic functions have been constructed.

- 1 -

CONTINUOUS DEGENERATE REPRESENTATIONS OF NON-COMPACT

ROTATION GROUPS

1. INTRODUCTION

In our previous paper [1] the discrete most degenerate represent-

ations of an arbitrary non-compact rotation group SO0(p, q) were derived

and their properties discussed. In the present work we go into the con-

tinuous most degenerate representations of these groups and the cor-

responding harmonic functions.

The harmonic functions for the Lorentz group SO(3,1) were invest-

igated in detail by DOLGINOV and his co-workers [2] . An arbitrary

Lorentz type group was considered by VILENKIN [3] , who derived the ir-

reducible unitary representations of the class one of SO(n, 1). The con-

struction of harmonic functions, which carry one series of the continuous

most degenerate representations of any SOO (p, q) group was given in [4] .

In this paper we present three series of continuous most degenerate re-

presentations of an arbitrary non-compact SOO (p, q) group and we also

construct explicitly a set of corresponding harmonic functions. These

harmonic functions are characterized only by discrete numbers connected

with the representation of the maximal compact subgroup SO(p) x SO(q)

of the group SOJp, q).

In Section 2, two series of continuous representations of the SO0 (p, q)

p ^ q > 1 and corresponding harmonic functions related to the hyper boloids

are constructed. The same problem for the Lorentz type groups SOO (p, 1)

is considered in Section 3. The continuous representations and correspond-

ing harmonic functions related to the cone of an arbitrary SO0 (p, q) group

are investigated in Section 4. Section 5 is devoted to proof of the ir-

reducibility of the derived series of representations. Finally, in Section 6

we discuss some features of the derived representations and harmonic

functions. For instance, it turns out that, except for one series of re-

presentations of the Lorentz type groups, there exist two irreducible re-

- 2 -

presentations of SOo(p, q) differing by parity for any definite eigen-

value of the Casimir operator.

The completeness relations of our harmonic functions, the cor-

responding decomposition of quasi-regular representations and the con-

nection with the Gel'fand-Kostiucenko triplet will be treated in detail in

the next article (Part III).

In the following we shall use the conventional terminology, that is,

we shall speak about representations of the group SOo(p, q) on the Hilbert

space 9L , although we derived only representations of the Lie algebra

R of the considered group on definite vector space & , which is dense

in the Hilbert space <X . However, in Part III of our series of articles

it will be shown that our local representations induce the global irreducible

unitary representations of the group SOo(p,q).

2. CONTINUOUS MOST DEGENERATE REPRESENTATIONS OP SOQ (p, q)

GROUPS ( p ^ q > 1) RELATED TO HYPERBOLOIDS

For the most degenerate representations of SOo(p, q) the ring of in-

variant operators of the corresponding Lie algebra is generated only by

one independent operator. Following the procedure explained in [1] (Sec. 2)

we can represent it as the Laplace-Beltrami operator on the definite vector

space °8 t which is dense in the Hilbert space of functions, the domain of

which are : the following homogeneous spaces of rank one

Then we solve the eigenvalue problem for this invariant operator. It is

obvious that the generalized Fourier images of its eigenfunctions carry

continuous representations of the considered group. The irreducibility

of these representations will be proved in Section 5.

- 3 -

The homogeneous space X+ and X_ defined in (2,1) can be realized

by the hyperboloid H£ and H^ respectively [1] . The hyperboloid

is determined by the equation

(2, 2)

and imbedded in(a+b)-dimensional Minkowski space M * . By using the

biharmonic co-ordinate system defined in [1] (eqs. (3,4)-(3, 9)) and due

to the properties of the metric tensor g (H£) on the hyperboloid H£

we can write the Laplace-Beltrami operator A(Hp in the form f[l] , (3,10))

& 6 [0, <*> )

where AfS^'^) [A( $*"*)] is the Lap l ace -Be l t r ami opera tor on the sphere S**

fS 5"^ of the compact rotat ion group SO(p) [SO(q)] defined in [1] (eqs. (A, 3),

(A, 4)). If we r e p r e s e n t the eigenfunctions of A(H") a s a product of the

eigenfunctions of A ( S ^ y ), AfS?*') and a function V A ( 8 ) we

m3 mobtain the following differential equation [5] for the latter function:

Here, S.^ (£ ,^,+ p - 2) U iu (tyx + q - 2.)] a re eigenvalues of the

operator A(S'>"' ) [A(S *"')] on the sphere S / " / [ S ^ ' / ] . Therefore

\*/\^ a r e n o n ~ n e g a t i v e integers except the lowest case p = 2[q= 2]

when by definition i , am, [£"., s m ^ , mjni!] integer. A*+ v 2 ' *A « [0,oa) i s o u r a n s a t z for eigenvalues of A(H ) corresponding to

the continuous representations. It is shown in Part III that we obtain in

this way the whole continuous spectrum of A(H£).

The solution of equation (2, 4), regular at the origin, is given by

the function

\/A (Q) y //'''**?i e a

(2,5)

* / ,M i i ) M W - A ^ ) W . i W ^ ^

with N = 2s:

The eigenf unctions of the Laplace-Beltrami operator A(H ) are

then harmonic functions . i (&i vt*5 ) of the form

(2,6)

Y, r 'v/ is given in (2,5) ' ^ ^ and / (co)where Y, r 'v/ is given in (2,5)

are eigenf unctions of AfS** ) and A (S 2"') respectively. They can be

expressed as a product of the usual d-functions of angular momenta and

exponential functions (see [1] , (A, 9), (3,18)). For instance

- 5 -

4 P

I (us) looks like% > • • > ">,

Y <V a < f2, 7)

with the normalization factors

and (2,8)

The harmonic functions Y ($ «' etj'*"'' *!%j constitute an ortho-

gonal set of functions with respect to the measure d^i&.ujo) induced by

our co-ordinate system on the hyperboloid H^

(9, «J,Z) = [pty d<« d&. d& = <ty U- dp fSi). dp (9) , (2, 9)

where ^ ^ =1**1, ' ,„ « „ , . « , »^ c s£ (¥*')<£$ TT <*s$)^ $k).cttk.71dp*

= <J!/'"i$,s£%~ 8.cL8

and the expression for d/u(u) is the same as for dju{u) , but in twiddle

variables.

Let us construct now the carrier space of representations of the

group SO0(p, q). The generalized Fourier transform with respect to?*%

the eigenfunction (2, 6) of a function f(0,u,tf) = P(xt , . . . , xPn ) exp (- X 0l)

where P ^ ^ , . . .Xpt$ ) is an arbitrary polynomial in x1 , and

X1 are expressed in our biharmonic co-ordinates on H^ (3,4)-(3, 9) of

[1] , has the form

- 6 -

A £ * 2 @ ? f A £ €t c r v

(2,10)

All such Fourier transforms determine the Hilbert space <>*•?,<> and

of vectors I -= { L ^ , ^ , 5 ^ 4 , ? 7 ' W ^ w h i c h4f?/aJ

A>±2

, where

even and i u) + •* t , odd respectively. The scalar product in the Hilbert

space < A ? is defined by

where the sum is taken over all integers £ , . . , £ <>i , 4 , . . . , S.;? > ,

m 1 # . . . , Hir%:) , m y , . . . , ni ^ , with i ,^^ + Jt y \ even or odd respectively.

The Hilbert space 3?^ i^h% / i s a n eigenspace of the parity opera-

tor with the eigenvalue +1(-1) and can be decomposed in the following way

- 7 -

where di h^;^sF •, , £,t ? are finite-dimensional vector spaces

determined by A- ^ ' „ Zi w i t h f i x e d l,f •, a n d I

The series C of the continuous representations of the group

SO(p, q), p > q^ 1 , on the Hilbert space &/,* is induced by the

A,±representation of the corresponding Lie algebra on the vectors X

( 2 . 1 2 )

xAjt H (

where again f(0,w,u) = Pfx1' , . . . , x™ ) exp (- L (**')* )

is a polynomial and x l are expressed in the biharmonic co-ordinates on

the hyperboloid H^ in (3, 4)-(3, 9) of [1]) and L;.. , Bi{ are elements

of the Lie'algebra of the compact and non-compact type respectively

([1] ,(6,2), (6,3)).

The proof of the irreducibility of the derived series C (<? of the

continuous representations is given in Section 5..

The harmonic functions on the hyperboloid H^ can be obtained by

exchanging p, ^ rp and q, i <q respectively and vice versa only in the

function V T (6) contained in the harmonic function

1 ^ ^ ^ ^P A +

on the hyperboloid H ? . The series C^

of the continuous most degenerate irreducible unitary representations of

-8-

-oxP*q), P ^ q > l on &^,p are then constructed by the same procedure

as described above.

3. CONTINUOUS MOST DEGENERATE REPRESENTATIONS OF SO(p, 1)

GROUPS RELATED TO THE HYPERBOLOIDS

The spaces X+ and X_ (2,1) can be realized now by hyperboloids

H., and Up respectively [1] . The biharmonic co-ordinates on H ^

and Hj, are introduced again a s in [1] (Sec. 3). We consider the Lorentz-

type groups separately because the range of 6 on the hyperboloid H^

is (-oo, co ) and therefore the solution of the eigenvalue problem of A(H )

is different from the corresponding one in the previous case. On the

hyperboloid H^ the range of 9 is from zero to infinity since we re-

strict ourselves to the upper sheet of the hyperboloid H £ . Of course,

the upper sheet of H» is a transitive manifold only under the proper

SO0(p, 1) group, i .e . , under the group of transformations g= (g;* ), for

which gyy is positive.

The Laplace-Beltrami operator A(Hy) has the form ([1] , (5, 2)):

6

where AfS'""' ) is the Laplace-Beltrami operator for the SO(p) group.

Representing the eigenfunctions of A(H^) as a product of eigenfunctions

of A(Sfi~') and a function V {$) , we obtain the following dif-

(&ferential equation for V. (&) •W

3,2)

where if,-, (JL r.^ + p-2) and A1 + (^-~-) a re eigenvalues of A(S'""')

and A(H'J) respectively. Both independent solutions of eq. (3, 2) are

regular at the origin and can be taken as orthogonal functions V ($}

- 9 -

in the form

<*?/ +iA)

and

A

V (e) -%

w i t h

— . ± ,> 2 >

X =

and A t [0,ve) } ( non-negative integer except the case p - 2

when & =. in , m integer. The eigenfunctions of the Laplace-Beltrami

operator A(H ) are harmonic functions orthogonal with respect to the

measure dp (vt8) = I - ^ i ' du d9 . Their form is

where / fa) are eigenfunctions of A(S^* ) expressed in

equation (2, 7) and V, (9) is given in (3, 3).

The construction of the carrier space of the representation of the

group SO0(p,l) is analogous to the previous case. Thus the generalized

Fourier transform with respect to the eigenfunction (3, 4) of a function

ild^lo } = P(x" ,...,V*} exp f- i^xtf), where PCx' , . . . , x'**) is an

arbitrary polynomial and x1- are expressed in biharmonic co-ordinates

on H^ : (3,4)-(3,9) of [1] , has the form

-10-

*"<>•• >

(3.5)

All such Fourier transforms determine the Hilbert space < *,-/

of vectors *. — i ^X _ ' J jof^^l*[«^Jffor which A fi =

= H \«X J , ^ 1 ° and for which w + i ^ i is even or odd respectively.

The scalar product in the Hilbert space **•/»/ is defined as

(3,6)where the sum is taken over all integers' i , . . , i ,-,, -. , m, , , . , m . -,,

i ^ i 1 "*/and of + j£ , , is even or odd respectively. The Hilbert space

/l +

( 36 ' ) is again the eigenspace of the parity operator with the eigen-

value + 1 (-1). The structure of the Hilbert space d££, has the form

where ^?,(, £((} ai*e the finite-dimensional vector spaces containing

yA,et,.-ttn} with fixedK A * V . - , ^ J witniixea

A +

The series C '^ of the continuous representations of the group

, 1) on the Hilbert space 3t'^ is induced by the representation of

-11-

the corresponding Lie algebra on the vectors XA,±

//>>± = / / yA''*'">fm / A .

(3,8)

where f(0,<y) is a function as in (3, 5) and L.. , B,t are the represent-

ations of the generators of the compact and non-compact one-parameter

subgroups of the group SOjp, 1) respectively.

The Laplace-Beltrami operator AfH^) on the hyperboloid H^

has the form [1] fSec. 5)

Since A(H^) has again the continuous spectrum of the form -/)*-

and eigenvalues of A f S ^ ) are - i . Uth. + p - 2 ) , the eigenfunctions

of A(H?) can be expressed as

Y

where now

*>*

(3,1 0)

K =rp{,-A<'-'

-12-

(3,11)

) Ceo) * is given in (2,6) and Ae[O,oct)t / being a non-negative

integer except for the case p = 2 when by definition $.± = n ^ , m^-integer.

The series C of the continuous most degenerate irreducible

unitary representations of the group SOft(p,H on the Hilbert space <#V*

are easily obtained from those constructed in Section 2 by omitting depend-

ence on all twiddle variables.

As will be proved in Section 5,, we construct two series cC'J

and C^L of irreducible unitary representations related to the hyper-

boloids H, and H^ respectively.

4. CONTINUOUS MOST DEGENERATE REPRESENTATIONS OF SO(p, q)

GROUPS RELATED TO THE CONE

In this section we derive the continuous most degenerate represent-

ations of an arbitrary SO0(p, q) group on the Hilbert spaced (X) of

functions the domain X of which is the following homogeneous space of

rank one under the action of SOo(p, q) [6] ; ^

so, (MA - /

Here, T p ^ ' 2 is the group of translations in the p+q- 2-dimensional

Minkowski space MF"1'V'1 .

The homogeneous space X can be realized by the cone C defined as

fr*/** . . . + (x*)x- (x"")1 (K***f = V .

Following the general procedure described in Section 2 of [1] we

have to introduce first the biharmonic co-ordinate system on the cone

Ca . Then we would try to find the metric tensor g (C ) on the cone

c£ and construct the Laplace-Beltrami operator. However, it turns out

that the metric tensor is singular on the cone and hence the Laplace-

Beltrami operator does not exist. Therefore, we have to construct the

second order Casimir operator Qz directly from the algebra. Calculating

-13-

the Cartan metric tensor from the Lie algebra R {[1] , (6,1)) of the group

, q) we easily find that the Casimir operator has the form

K} (4,2)

The biharmonic co-ordinate system on the cone is introduced as0

xk = r . x ' k , k= J<2,»,¥> ,o ,j> . .. (4 s\

x s r . x *~ p+7tp+ ?)•• >/>i't

where x'* , x1^ have the same structure as in formulae (3, 5) -(3, 7) of

[1] . We represent now the Lie algebra R of the group SC (p,q) with

respect to the parametrization (4, 3) by the operators L r and B^ :

where i, j = 1, 2, . . . , p . The analogous expressions hold for L t..,

i* j = P+l . P+2-, . . . , p+ q.

T? IS it3st = * *st

where s = 1, 2, . . . , p , t = p + l , p+2, ; . . , p+q and x I J , x|f are

defined in (4, 3). The operators L ,y are r-independent and have the

same form as in previous cases. The corresponding representation of

the Casimir operator Q^ has the form

Q2 s -1 f r*l2 + f^./j.rjL ] . (4, 6)

-14_

The left-invariant measure on the cone is given by

where d/Li(u) is defined by (2,8).

Fromhere we again pursue our general procedure, i, e. , we first

solve the eigenvalue problem for the Casimir operator Q^ (4, 6). The

eigenfunctions of the operator Q are the solutions of the differential

equation

where we put the ansatz A + (2—3 ) , A t (-*-=, <*=>) for the spectrum

of the operator Qz . It will be shown in Part III of our work that we

do not lose any part of the spectrum of Q2 in this way. Hence the

general form of the eigenfunction of the operator Q^ has the form

r*• «?(w,w), where °i - - ~\ + i A and (f> (U,LJ) is a function which

can be chosen in an arbitrary way. It is convenient for our purpose to

restrict eigenfunctions of the Casimir operator Q^ to be the harmonic

functions of the following form

I (r, U>,UJJ — r . 1 (tu) . 1 (t*j) f

where ot = - ^—3 + i / \ and the functions / ^ ^

are defined in (2, 7) for p > 1, and for p = 1 this function is equal to one.

The generalized Fourier transform with respect to the eigen-

function (4, 9) of a function f(r,u,S) = Pf*" >..., x'*' ) exp (- L (x1}2 ) ,1=/

where Pfx'' , . . . , x'9*2 ) is an arbitrary polynomial in xx , and x1 are

defined in our hiharmonic co-ordinates (4,3) on the cone c£ , is given

-15-

(4,10)

For p ^ q > 1 as in previous cases, all such generalized Fourier

transforms determine the Hilbert space $ JZ » of the vectors

where Jt +1. even and i + ^r> odd respectively. The Hilbert

space 3tpt<i can be decomposed in the form

where ^ ^ , / 7 are the iinite-dimensional vector-spaces

determined by X ^ „ « % with fixed i and

The scalar product in the Hilbert space 3C i~ is defined by

where the sum is taken over all integers Si , . . i <» > , i2 , . . £

m , m , . . , m g / z j , with 2.,^ + i ^ , even or odd respectively.

The representation of the series C J" on the Hilbert space ^?',^

is defined by the representation of the algebra on the vectors X. '^

-16-

^

with f(r,wfu) as in (4,10).

For q = 1 the corresponding representations are constructed in

a way completely analogous to that used in Section 3.

5. IRREDUCIBILITY

A) The representations related to the hyperboloid for p > q > 2. The

maximal compact subalgebra (consisting of the generators Lt- •) with any

generator Bit of a one-parameter non-compact subgroup generates

the whole algebra. Therefore the problem of the irreducibility of the

representation of the algebra can be solved considering only the set of

generators L^ together with one of the generators Bst ,

For the proof we take the generator B . and represent it by ^

the definition (2,12) on the vectors (2, 9) which determine the Hilbert

space <5 ' 0 3t'^ . Calculating the explicit form of the operator Bpi

from the Expression (6, 3) of [1] we easily find that it can map an arbitrary

vector f. 5- <= 3C*'~.e r only to such vectors f, r, for which

Jt\ = Jt.k , +1 and jP- ^ = Jtt7 . + 1. Hence we have at least two in-

variant subspaces ^ , ^ with respect to the representation of the algebra

for the same eigenvalue of the Casimir operator. In the following we

shall show that any of them is irreducible, i. e., they do not contain the

invariant subspaces with respect to the representation of the algebra.

To show that there is no invariant subspace in the vector space

determined by the vectors (2, 9) with respect to the algebra R it is suf-

ficient to find vectors X.A'if%?'fal e dtAlt t T e

-17-

such that every B has non-vanishing components in

four neighbouring subspaces <#\ - - <**/ £ +/ • Choosing X. ' ?%!'&*)_

, where m2 - m2 - .. . = m^"1 = rn$/£} = 0 ' anc*

, {mil t i 2 , S.x,,.,, SL, , 4 , £r i . have the minimal possible values,

we calculate from (6, 3) of [1] and (2,12):

B X o,

2

2.

1

(5,1)

jA,*-4,

where , i = iff,j ,

[1] , N(je,I) B N(i |

a r e defined by (6, 7) or (6, 8) of

, where N ;^ > and Nrs , are

-18-

defined by the expressions (3,21) of [1] , and

' Zi --tiO

for odd* p and q f

for even p and odd q .

(5,2)

for even p and q .

The only coefficients in the expression (5,1) which can vanish for

a n d . If p(q)non-negative integers £lfA* ^\v\ S ' r e A~

is odd then the coefficient A_(je^ ) (A-W^, )) is zero for Jt^ = 0

ft Hi ~ 0) *n accordance with the fact that we have the representation.

If p(q) is even,A_(^ ) = 0 (&-ftf/z) = o) for k^ = 1 ( ^ = l). This

does not mean that the representation is reducible because the mapping

3P A>± - ^ -> 2P ^ / always exists and the operator

is skew-symmetric on the vector space determined by the vectors

(2, 9).

Thus we proved that the second order Casimir operator is not suf-

ficient to specify the irreducible representations. The complete specific-

ation of the irreducible representations is achieved by the commutative

invariant algebra generated by the second order Casimir operator and

an operator P , eigenvalues of which are ±1 . Let us show that the

operator P is the representation of the parity operator px*= - x , k =

1, 2, . . . p+ q. From the explicit form of the harmonic functions we

easily calculate the representation of the parity operator on any harmonic

function "-19-

(5,3)

V.

Then the representation of the parity operator on the space ^pt7 is

defined by the expression

p yAi ir-i f%p ir't V%? / Y A> *'"' %' *'"' (%} VI \ ^

It follows that the vectors X '" of the space 36*'* are common eigen-

vectors of the operator A(H£) with the eigenvalue A + (•*-— —-) and

the operator P with the eigenvalue ± 1 respectively.

Completely analogous proof holds for the series C '" of the re-

presentations related to the hyperboloid H I .

B) The representations related to the hyperboloid for q = 2 . The

proof of the irreducibility of the continuous representations of the seriesA+ A,*-

C ' and C does not differ from the previous one as the vector

B , 9 * " ' W , where

with (mxl , SLZ , . . . , i.fi . having minimal possible values, has

essentially the same structure as in the previous case.

C) The representations related to the hyperboloid for q = 1. The proofA

of the irreducibility of the continuous representations of the series CjiP

can be obtained by specifying the one which is derived in A) above. Omitting

completely the indices S. , . . . , £. ,^, •> , m , . . . m ,-f/ -1 in A) we obtain

1(5 ,5)

-20-

where A ± ( l ^ j ) are defined as before, N( i^ ]

in (3, 21) of [1] and

) = N f e j is determined

(5,6)

Ji "• 1 ~ 2i

A,tA,tFor the continuous representations of the series C uy we choose,

as in the case of discrete representations [1] , the vector X ' ^

w h e r e

have the minimal possible values.

a n d

l /

(5,7)

- 2 1 -

where A+f^) are defined as before, N(lrfi i ) = N,, is defined in (3, 21)

of [1] and

p2i i* -4

Analyzing the coefficients in the expressions (5, 5) and (5, 7) one

can verify again that there is no invariant subspace, i, e., the represent-A,± A

ations of the series C ^ and Cif) are irreducible. Hence again the

irreducible representations of the series C '" on the Hilbert space

3Cpj are characterized by the eigenvalue of both the Casimir operator

A(H ) and the parity operator P .

D) The representations related to the cone. By the same argument

as in the previous section we establish first the existence of at least two

invariant subspaces ""^a and </*/>,£ with respect to the representation

of the group for the fixed value of the Casimir operator Q^ . Then we

prove their irreducibility as before. Thus, for instance, for q > 2 .

As we have already mentioned, the representations L •• (4,4) of the

generators of the compact one-parameter subgroups have the same form

on 36 LZ as in Section 2. Moreover, since the operators L,y are

reduced by the subspaces <# *„ ; i,b ,7ff..exactly to the same operators

as in Section 2, the subspaces ^p~a ••£;?,,%* a r e irreducible with

respect to the maximal compact subgroup. Now we proceed as before,A & £

i .e . , we consider the operator B*, .„ and the vector X. ' f^/'j/W =

where mi = ... . m - s = ... - m = ow h e r e m i = . . . . m

ani Jnij], jrii^j , S.z, i 2 , . . . , S. ,p , ^, JC r? , ^ have the minimal possible

values. Using the definitions (4, 5) and (4,13) we easily compute the

following expression:

- 2 2 -

yA'e'e

(5,9)

•+ i 2-A + t-4. + iL-t^i • AJ*J. Aji).

2

where A±(JL)t N(i,i) are defined as in the expression (5,1) and <f) ' '

as in expression (5, 2).

Using the same analysis as previously, we can check that there

are no invariant subspaces of the vector spaces determined by the vectors

(4,1 0) with respect to the representation of the algebra.

The representations related to the cone for q = 1 and q = 2. The

corresponding proofs of irreducibility we obtain as previously.

The unitarity of the representations of the group SO0 (p, q) on the

Hilbert spaces ^ p'Z related to all three homogeneous spaces will be

proved in Part III of our work,

6. SUMMARY

Three most degenerate principal series of the continuous irreducible

unitary one-valued representations of an arbitrary non-compact rotation

group SO0 (p, q) have been constructed. These series are related to

three homogeneous spaces of rank one under the action of SOO (p, q), i. e.,

to the hyperboloid H £ and H* and to the cone C £ .

-23-

Generally, the most degenerate continuous irreducible unitary re-

presentations of SO0(p, q) are characterized by two numbers" A and p .

The former determines the eigenvalue of the second order Casimir opera-

tor and the latter is the eigenvalue of the parity operator.

In particular cases the situation is as follows:

i) SO#(p, q), p > q > 1. The constructed representations are determined

by both A and p . A is real from the range (0, °o ) and (-<»,o©)

for representations related to the hyperboloids and to the cone re-

spectively; p has the value + 1 .

ii) SO0(p, 1). Two series of representations of the Lorentz type group,

namely, those related to the hyperboloid H£ and to the cone Cj* ,

are also characterized by both A and p , whereas the represent-

ations of the series related to the upper sheet of the hyperboloid Hp

are characterized only by the number A , The range of A j . s

again (0,°») and (-c*:>,e») for representations related to the hyper-

boloids and to the cone respectively; p is equal to + 1 .

The harmonic functions of the derived three series of continuous re-

presentations have been explicitly constructed. They are labelled by the

numbers /[ , p, from the corresponding above-mentioned ranges and by

a set of integers SL%, , . . , $.,..f. , m^, . . ., ny^, _,e eigenvalues of th

commuting operators defined in (7, 8) of [1] .

ni - » which determine the eigenvalues of the maximal set of compactLvi J

ACKNOWLEDGMENTS

The authors are grateful to Professors Abdus Salam and P. Budini

and the IAEA for the hospitality extended to them at the International Centre

for Theoretical Physics, Trieste.

It is also a pleasure to thank Professor K. Maurin and Dr. J. Fischer

for interesting discussions.

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REFERENCES AND NOTES

[1] R. RACZKA, N. LI MIC and J. NIEDERLE, ICTP preprint IC/66/2,

Tries te , (Submitted to J. Math. Phys.) .

[2] A . Z . DOLGINOV, Soviet Phys. - JETP 3, 589 (1956),

A. Z.. DOLGINOV.and J. N. TOPTYGIN, Soviet Phys. - JETP 10,

1022 (1960),

A. Z. DOLGINOV and A. N. MOSKALEV, Soviet Phys.- JETP 10,

1202 (1960).

[3] N.Ya. VILENKIN, Trudy Moskov. Mat. Obsc. (Russian) VZt 185

(1963).

[4] J. FISCHER, J. NIEDERLE and R. RACZKA, to be published in

J. Math. Phys. (1966).

J . NIEDERLE and R. RACZKA, ICTP preprint IC/65/89, Tr ies te .

[5] Here and elsewhere we keep the notation from [1] . Let us keep in

mind that the brackets a re defined as follows ;

[6] The authors a re grateful to Dr. O. NACHTMAN for a valuable

discussion on the group of motion on the cone.

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ii

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