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Vol. 13 (1978) REPORTS ON MATHEMATHICAL PHYSICS No. 2

ELEMENTARY REPRESENTATIONS AND INTERTWINING OPERATORS FOR THE GROUP SU*(4)

V. K. DOBREV and V. B. PETKOVA

Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, Sofia, Bulgaria

(Received April 14, 1977)

Global realizations of all elementary induced representations (EIR) of the group SU*(4), which is the double covering group of SO t (5, l), are given. The Knapp-Stein intertwining operators are constructed and their harmonic analysis carried out. The invariant subspaces of the reducible EIR are introduced and the differential intertwining operators between partially equivalent EIR are defined. Invariant sesquilinear forms on pairs of invariant subspaces are constructed. Differential identities between invariant sesquilinear forms on pairs of irreducible components of the reducible representations are derived. The results will be applied elsewhere to the nonperturbative analysis of Euclidean conformal invariant quantum field theory with fields of arbitrary spin.

Introduction

The group SU*(4) is the double covering group of SO’(5, 1) (the identity component

of 0(5, 1)). These groups and their representation theory are studied because the group

0’ (5, 1) (including reflection of the first five axes) is the Euclidean conformal group

for 4 space-time dimensions. In a previous work [4] the so-called type I (or symmetric

tensor) representations of 0 + (iV, 1) were studied in detail. The reader can find there also

a rather comprehensive review of results and methods useful in considering the representa-

tion theory of semi-simple Lie groups (especially of split-rank one).

The aim of this paper is to give a global construction of all elementary representa-

tions of SU”(4) (and thus of all one- and two-valued representations of SOt(5, 1)) and

to construct and study the intertwining operators between (partially) equivalent repre-

sentations. We follow the general fram.ework of ref. [4] and whenever possible cast the

results in such a form that comparison with the case SO + (5, 1) (as considered in [4]) is

straightforward. The present paper will make possible the extension of the applications

(see [S]) to conformal invariant quantum field theory, e.g. we shall be able to consider

fields of arbitrary spin.

We define the group SU*(4) as in [lo], where also one can find some results about

234 V. K. DOBREV and V. B. PETKOVA

the group structure. We define an appropriate Cartan involution in the Lie algebra su*(4) and construct the Iwasawa decompositions for su*(4) and the Iwasawa and Bruhat decompositions for SU*(4) following the general scheme for semisimple groups (cf. [7, 13, 2, 181) and give the relation between these two decompositions. We also display the isomorphism between su*(4) and so(5, 1).

We define the unitary irreducible representations (UIR) of M = SU(2) x SU(2) (double covering of SO(4)) as in [l]. We display two standard realizations of the elementary induced representations (ETR) following [14, 181 and construct the infinitesimal generators for them.

It is a well known result (cf. [12, 161) that almost all EIR are irreducible. We construct explicitly the Knapp-Stein intertwining operators [14] and carry out their harmonic analysis.

We also construct invariant sesquilinear forms on pairs of EIR of two types (the second involving the integral kernel of the Knapp-Stein intertwining operators). We obtain the result (which could also be derived from more general considerations) that the complementary series of unitary representations occur only for symmetric tensor representations of M (or SO(4)). For that reason Grensing [IO] has constructed the Knapp- Stein intertwining operators only for such representations.

We study in detail the exceptional EIR. We construct the invariant subspaces of the reducible representations and define explicitly the differential intertwining operators some of which were introduced for type I representations of SO t (5, 1) in [4]. Such operators for arbitrary representations of SO t (N, 1) were also studied in [6]. We discuss their results in Subsection 5.B. After the present paper was completed we received a paper by Zhelo- benko [22]. The author has shown that for any real reductive group the intertwining operators between reducible representations (besides the Knapp-Stein operators), cor- respond to the positive noncompact roots of the root system of gc relative to lj” (lj’ is the complexification of the Cartan subalgebra lj). The list of these intertwining operators for all simple real algebras of split rank 1 is given. The operators are defined infinitesimally in a realization of the elementary representations with a representation space consisting of complex-valued functions on the group.

We also prove irreducibility of two series of representations partially equivalent to reducible representations. We derive some differential identities between invariant sesquilinear forms on pairs of irreducible components of exceptional representations.

We note that the formulae for the infinitesimal generators of the representations of M (and SO(4)) and G (and SOt(5, 1) (and some results derived by infinitesimal methods) look similar to analogous formulae for SOt (3, 1) (see [17]) and S00(4, 2) (see [2O]) respectively. This happens because the representations of SO t (3, 1) are also realized in the space of polynomials of two complex (Pauli) spinors and the Lie algebras of SO(4) and SO? (3, 1) are connected through the Weyl unitary trick. However, the results for the representations can not be compared directly because the corresponding groups differ

significantly (Cf. [ 151).

ELEMENTARY REPRESENTATIONS AND INTERTWINING OPERATORS 235

1. Group stmcture l.A. The group SU*(4). First we introduce some notation. Let

qk = -iok (k=1,2,3), &El= i y 3 1 1 (1.1)

where ok are the Pauli matrices

ul=[Yf J, u*=[P -J, u3=[i _;I. It is well known that the matrices q,, (p = 1,2, 3,4) are a representation of the quaternion

units satisfying

949p = 9p94 = 4r, qjqk = &jklql- djkq4. (1.2)

An arbitrary (real) quaternion is parametrized by 4 real numbers uP and is written

as a e ePqP = alql+a2q2+a3q3+a4q4 E Q (l-3)

(where Q stands for the quaternion field). Then by definition U*(4) is a matrix group, whose elements are described by

(1.4)

The 16-parameter group U*(4) is thus isomorphic to GL(2, Q). Now we can define the matrix group G:

G = SU*(4) E {g E U*(4), detg = l}. (1.5)

In the standard definition of SU*(4) [13] the matrices differ from (1.5) by a real orthogonal transformation. The present definition was introduced in [lo] without pointing out the connection to the quaternions.

By a straightforward computation we obtain

detg = a262+/?2y2-2(a*/?)(y* 9-2(a.y)(/?* S)+~(C(.~)(~.~)+~E~~~~~B~Y~B~,

(1.6a)

where

u2 f aPaP = ~a + = ~+a- = deta, “p = cr,/% = +(@++Bd(+), (1.6b)

&4123 = 1, CJ+ E a,q,+,

q$ is the hermitian conjugated matrix (q$ = q4, qj+ = - qj). Thus condition detg = 1 gives one constraint and the group W*(4) is 15-dimensional (like the group O(5, 1)).

It is useful to have different expressions for detg, namely

detg = u~~~+B~~~-(?~+~B+-(c(Y+~~_+)+, (1.6~)


(where (gy_+~?B_+)+ = pj+yp+) and three more expressions obtained from (1.6~) by

simultaneous cyclic permutation of the matrices in the last two terms.

l.B. The Lie algebra of SU*(4). It is easily seen that the Lie algebra of G is [ll]

g={Z=[I i]IaO,b,,,c,,d,ER, trZ=O)=su*(4). (1.7)

Noting that trZ = tr_a+ tr_d = 2(a4+dJ we shall give an explicit expression for the

basis of the Sdimensional real vector space g

00 = - 1 0 & [ qk" 1 Yk = - 1 2 0 0’ 2

[ 1 0 qk’

[ 4r 1 -q;:o

’ (1.8a)

(1.8b)

It is easy to check that the basis elements (1.8a) span the maximal compact subalgebra f of g, while the elements (1.8b) span a vector space p [ll].

Next we define the Cartan involution 0

82 z -z+, ‘ZEg. (1.9)

With this definition we can state

PROPOSITION 1.1. The Cartan decomposition of g is given by

g = mp. (1.10)

Proof: Having in mind the properties of f and p already mentioned it remains only to prove that

ZEf*ez=Z, zEp=+ez= -2 (1.11)

which is straightforward application of (1.8). Next we single out important subalgebras of g. Let a be a subspace of p which is maximal

subject to the condition [Z, 2’1 = 0 if Z, 2’ E a. It turns out that a is one-dimensional subalgebra of g and we shall take D of (1.8b) as its generator. The centralizer m of a in f iS spanned by the matrices & and Yk. It is isomorphic to su(2)@su(2).

To construct the Iwasawa decomposition [13] of g we use the restricted root system of g relative to a. Let a* be the space of linear functionals over a. Each functional is de- termined by its value on D. We define for A E a*, I # 0

ga = {Z Egl P, Zl = WW), (1.12a)

A = {2Ea*l A We obtain

n = @+,A->,

and the corresponding subalgebras

ii = &+,

# 0, ga # {OH, (1.12b)

A*(D) = +l, (1.13a)

u = 9A_ (1.13b)


are spanned by

0 0 , c, = u*--s, =

[ 1 41: 0 (1.13c)

respectively. Obviously fi = &t. Using (1.8) and (1.13) we also obtain the decomposition (valid

generally for semisimple Lie algebras) :

0 = fi@?t~a~trt. (1.14)

It is easily seen that the map

J: m@fi-+_,

J(Z+.Z’) = Z-tZ’+8Z’ (for ZEm,Z’Efi)

is bijective and that g = f@a@n (1.15)

which is the Iwasawa decomposition of g. The algebra g is isomorphic to so(5, I), the isomorphism given explicitly by

xjk = &jkl(xi+x), X4& = -x& = Xk-Yk,

x,5 = -&, = sp, (1.16a)

X,,, = -X,,, = U,, X,, = -X0, = D. (1.16b)

Indeed,

[XAB, &DI = B~cXBD+~)~~~XAC-I~)ADXBC-~~~BCXAD (1.17)

where A, B, C, D = 0, 1, . . . ,5, qll = . . . = qs5 = -qoO = 1, TAB = 0 for A # B. Note that the subalgebra f is isomorphic to SO(~) (and the elements X,, span an algebra

isomorphic to so(4)).

l.C. Structure of the group SU*(4) and decompositions. The maximal compact subgroup K of G is (cf. [l I])

K E U*(4)nU(4) = SU*(4)nSU(4),

i somorphic to Spin(S). If g E K

s_r’+_B_s’ = 0, a2+/92 = 1, y2+82 = 1.

It follows that a2 = fJ2, /3” = y2.

We list other important subgroups starting with the abelian noncompact [IO]:

(1.18)

(1.19a)

(1.19b)

subgroups

(1.20)

238 V. IL DOBREV and V. B. PETKOVA

N= nb~GJ nb=

fcr= (&EG, i, = [; ;I, ..R4).

Let M be the centralizer of A in K. Then

uo M= mEKI m= o z, [ 1

(1.21)

(1.22)

(1.23)

The matrices u, v E SU(2), so M = SU(2) x SU(2). Let M’ be the normalizer of A in K. The Weyl group of the pair (G, A) (the restricted Weyl group) W s M’/M has two elements: W = (1, In). Let 0 E 9. Then we have (as a matrix 0 E K)

M’={m’EKl either m’EM, or m’=com, mEM}. (1.24)

The subgroups N and i+? are conjugated under the Weyl transformation $ = wNm_‘. We shall make the following choice for the representative cc) of the nontrivial element 9:

0 -1 0=

[ 1 1 0' (1.25)

We also note that the (maximal abelian) Cartan subgroup H of G is given by

H=AHM (1.26a)

where HM is a diagonal subgroup of M, i.e. if g E HM then

g = [; o,], u(‘)= [id” ;+], #')ER. (1.26b)

It is easy to check that f, a, n, ii, m are the Lie algebras of K, A, N, fi, M respectively. Next we shall construct the Iwasawa decomposition [13] of G. It is well known that

every element g E G may be represented uniquely in the factorized form (k, E K, nb, EN, ‘iI E A)

g = k&,ar. (1.27a)

For the factors in (1.27a) we have:

PROPOSITION 1.2. Let kI = [;I iI] and g = [F 51. Then we have

1 1

?!I = r/pz+gz ~‘r-!(+co, ,s, = JB2+62 p,

The proof consists of a straightforward matrix multiplication.

(1.27b)


The order of the factors in the Iwasawa decomposition is a matter of convenience (however the expressions (1.27b) depend on it).

Next we proceed with the construction of the (Gel’fand-Naimark)- Bruhat decomposi-

tion [2, 71, It is we11 known that up to a submanifold of lower dimension every element

of G can be written in a unique way as a product

g = n+,nb,aBmB

where nXs Eh7, nb,oN, asEA, mr,EM.

PROPOSITION 1.3. For g = [ I

“f

y!? and S # 0 we have

(1.28a)

In the case ,S = 0 (such elements form the II-dimensional manifold wNAM) we have

(1.29a)

(1.29b)

Proof: Straightforward matrix multiplication.

Remark. Several special cases of (1.28b) ( namely (1.32) and (2.21) below) are displayed in [lo].

It was mentioned in the introduction that the simply connected group SU*(4) is a double covering of SOt(5, 1) [1 11. The covering is easily displayed using the Iwasawa or Bruhat decompositions. Indeed, let G = KAN, SO t (5,f) = K,A,N,. Then A0 and AJ0 are simply connected and isomorphic to A and N (respectively), while K z Spin(S) is the double

covering of K0 = SO(5). Analogously if I?NAM and i,,k, AoMo are (locally isomorphic)

dense submanifolds of G and SOr(5, 1) respectively, then 15 and rJb are isomorphic, while A4 = SU(2) x SU(2) is the double covering of MO = SO(4).

The group SO’ (5,l) is the conformal group of the 4dimensional Euclidean space R4 and so we shall consider G = SU*(4) as the quantum~mechani&l Euclidean conformal group. First we note that

G/NAM z KfiU z S4 (1.30)

240 V. K. DOBREV and V. B. PEJTKOVA

(where S4 is the unit sphere in P) can be identified with the one-point compactification of R4. The group G acts in a natural way on the homogeneous space S4. This action will be displayed by using the decomposition (1.28) and the isomorphism between R4 and

the abelian subgroup I?. We define as in [4] the transformation g: x H x’ of R4 by

g-rnx = nx,n-la-lm-l. (1.31) Thus we obtain

(a) g = k,, translations of R4,

x' =x--x,; (1.32a)

(b) g = m, rotations of R4,

X' = A-lx, (1.32b)

where (cf. [l]) A E SO(4), A,; = $r(qiuqp+);

(c) g = a, dilatations of P,

x’ = x/la] ; (1.32c)

(d) g = &,, special conformai transformations of R4; for 1 -2bx+b2x2 # 0

x’ = (x-x2b)/(l-2bx+b2x2); (1.32d)

for I -2bx+b2x2 = 0, x’ $ R4, x' = co E S4 (co is the point added to R4 to compactify

it).

l.D. Relationship between the Iwasawa and the Bruhat decompositions.

PROWSITION 1.4. Let us have Cfor !j # 0) g = ktnb,at = t+&.,aBmB . Then we can express the 2 x 2 matrices (1.27b) in the Iwasawa factors by those of the Bruhat decomposition (1.28b) :

1 1 (XI = --.

I/1+x; &J, 01 = - - J1+x; XBWB,

(1.33a)

bt = &[&+(1-kx;)$]uB, iad = laA/(1+x3. The inverse formulae of (1.33a) are

(1.33b)

In the case cj = 0 we express (1.27b) by (1.29b):

(1.34a)

ELEMENTARY REPRESENTATIONS AND INTEFCTWININ G OPERATORS 241

The inverse of (1.34a) are obvious. We note only

bB’ = -jwyf * (1.34b)

Proof: Formulae (1.33a) are obtained by substituting in (1.27b) the inverse formulae of (1.28b). The remaining equations are verified in a similar fashion.

We mention two special cases. First the Iwasawa decomposition of ii,:

-

lx n, =

[ I 0 1 = kdw(4,

where

k,= i& 1 x [ 1 -x+ 1’

la(x)1 = 1/(1+x*);

then the Bruhat decomposition of k:

where

k tLf

= _r,s [ 1 = &uVWa(k)m(k),

x(k) = -$ &?+, b+(k) = -@+,

In the case 2 = ,S = 0 instead of (1.36a) we have

(1.35)

(a’ = 62 # 0) (1.36a)

la(k)1 = l/a*,

k=[; &.w[; _;].,., (B”=y”=l>. (1.36b)

It is also interesting to note that almost every element of K can be decomposed in the form

k=k,m (a*=d*#O). (1.37)

PROPOSITION 1.5. Let k be as in (1.36a), then in (1.37)

x = -$ _Bs+, a0 rn=j&;j [ 1

. (1.38)

Proof It suffices to apply the Bruhat decomposition of k,

k, = &nbWax,

where (1.39a)

b(x) = -Lx, 1+x*

la,1 = 1+x*. (1.39b)

Note. Elements k, may be considered as forming the factor space K/M.

LE. The Haar measure. We shall summarize here some results of ref. [4] concerning the invariant measure on G and its expression in terms of the invariant measures of its

242 V. K. DOBRBV and V. B. PETKOVA

subgroups in the Iwasawa decomposition. Since the group G is unimodular, its Haar measure is both left and right invariant; the measures of the non-unimodular factors are chosen to be left invariant.

Let dk be the Haar measure on K normalized by

s dk = 1. (1.4(-Q K

Set further

da E %, dna E db,db,db,db, = db. (1.41)

Then the Haar measure on G has the form

dg = d(kn,u) = dkdbda. (1.42)

We shall also need the expression for dk in terms of the factors k, and m of (1.37); it is

dk = d(k,m) = 6 7cz (Eq’ dm9

(1.43)

where dx E dx, dxz dx3dx4 = dii,

and dm is the normalized Haar measure on M:

s dm = 1. M

2, EIementary induced representations

2.A. The unitary irreducible representations of M. The unitary irreducible representations of it4 (= SU(2) x SU(2)) are characterized by two positive integers, say ml and mz. Following [l] we realize these representations in the space V’ (I 3 (ml, mz)) of homogeneous polynomials of two complex two-vectors, i.e.,

I/’ = {p): c2x c2 4 Cl y(Aw+, pz) = Amyhm’p(W+, Z), A, p EC}, (2.1)

where

Wl [I E c2, 21

w= z= w2 [I 22

E c2,

and we use the hermitean conjugate two-vector w+ (rather than w) for convenience, Ob- viously dimV’ = (m, + l)(m2 + 1).

The UIR D’ is given by the formula [l]:

[o’(m)v](w+,z) 3 ~)(w+u,w+z), WO

m = o e, . [ 1 (2.2)

(From now on we shall not mark transposition of complex two-vectors, e.g. instead of w+ Z 5 we shall write 5.)


We note for future reference

ml!m2!fp(E4, v’z) = (Kg)“’ (~oiz)12cw, Zl)

a where E = q2,- = aZ form

It follows from this equation that the sesquilinear

(2.3)

defines a D’-invariant scalar product in V’. Every representation of A4 is a representation of SO(4) and we shall use sometimes

the standard labelling of the representations of SO(4)

Ill, l,] E [*,;*I ) *,:*q. (2.4)

The numbers Ii, I2 are simultaneously integer or half-integer. In the latter case the representations of SO(4) are double-valued (spinor) representations.

In order to find the infinitesimal generators of the representation D' we use (2.2') and obtain

&VP, z) = f@&p(%z), (2.5a)

Y&-K z) = fZ%k &(“, z). (2.5b)

The second order Casimir operator of the representation D’ is

Q(I) 3 -f(x,‘+Yf) = ~[ml(~l+l)+m2(~2+~11. (2-G

In our considerations very important role along with D’ will play its so-called “mirror image” representation Df, acting in the same space V’ by the formula

[Dt(m)q](iG, z) E [D’(m”)pll(W, z) = &-b, u’z) (2.7a)

where mm is the Weyl conjugated matrix of m:

(2.7b)

The representations D’ and Df, are not equivalent except for ml = m2. The mirror

image 0; is equivalent to the representation DT, where r” = (m2, m,) for 1 = (ml, md- This is seen by the following

244 V. K. DOBKBV and V. B. PETKOVA

PROPOSITION 2.1. The intertwining operator of 0’ and 0; is given by

B: VL V’, (2.8a)

(Bv)(z, 2) = &x, Ez), pl E v’; (2.8b) its inverse

B-l: V’-+ V 7

is given by the same formula (2.8b) for ~1 E V’.

Proof: The only thing left is to check using the definitions:

BD7 = D:B or D!+ LO. =: B-ID’ (2.8~)

Remark. We note that if m t-, LI E SO(4) as in (1.32), then m“’ H I,&, E SO(4) (Is E O(4), for x = (z, x.,) c R4, IJ = (-x,x4)), i.e. the mirror image transformation of SO(4) (cf. [21]). That is the reason that we called the Weyl conjugated representation the mirror image representation.

2.B. Definition of the induced representations. We shall give two equivalent realizations of the elementary induced representations (induced by the parabolic subgroup MNA) (cf. [14, 41).

Let V’ be the finite-dimensional Hilbert space (2.1) of the UIR D’ of M. Let c E: C,

x E [I, c] = (ml, m,; c) 5 [II, I,; c], (2.9)

and DX is the finite-dimensional representation of MA given by

D”(ma) 3 lal-2-cD’(m). (2.10)

Then we define

c,= (f: GxC2xC2 -+Cl fEC’YG), f(g)EV’,fkman,~,z) = [D”(m-‘a-‘)f](g, i?, z) = lal”‘f(g, Wu”, wz)). (2.11)

The representation Zx of G induced by the finite-dimensional representation Dx of

MNA (N being represented trivially) is defined by

[=(g)fl(g’) = fk-'A g,g' EG, f EC,. (2.12)

Each Zx is called an elementary induced representation of G. The K-invariant scalar product in 6X is defined by setting

<fl> f2>% = $- ~Wf,(k),f,Oh (2.13a) * K

(factor x2/3! is introduced for convenience). Using the factorization of dk (1.43), CO- variance properties of f and M-covariance of < , ) we obtain

(fly fz>ax = s

Rd

(1Fx2)4 <f,(kJ, f2bW. (2.13b)


The representation QY is continuous with respect to the topology defined by (2.13). Another realization of the elementary induced representations is the noncompact

picture (there is also the so called compact picture). Let

c, = R4 x C2 x C2 -P CJ f E Cm(R4), f(x) E V’,

-j%,+)} (2.14)

where Hk, is a homogeneous polynomial of degree k, ml, m, in the first, second and third

variable, respectively;

ox E (g, -x4)/x2 for x = (x,x4). (2.14b)

The representation TX is defined by

[Tx(g)f](x, tj, z) = [P(mu)f](xs, w, z) = l~l-~-‘f(x,, Ku, v+z), (2.15)

xs is defined through the Bruhat decomposition g-%zX = n,,n-la-‘m-‘. For the various subgroups of G we obtain from (2.15) and (1.32)

(a) lrunslations

[TX(;;x,)fJ(x, w, z) =f(x-x’, W, z), (2.16a)

(c) dilatations

[T”(m)f](x, W, z) =f(A-lx, ti;u, o+z), (2.16b)

[T*(alfl(xt W, 4 =f(bl-‘x, 12, z)-&, I4

(2.1643

(d) We$ inversion (then for special conformal transformations we use nb = &&b~)

LW~lfl(x, w, 3 = -,,F+, f wx (2.16d)

In (2.16d) we have used

w-l& = &~~~~~a(x, CO)-‘m(x, CO)” (2.16e)

where la(x, o)l = x2, m(x, w) E ___ -1 $0 1 1 I/F 0 x+ *

We note that the asymptotic behaviour (for x + co) of the functions of C, guarantees the smoothness of the right-hand side of (2.16d) for x -+ 0.

The operator which realizes the equivalence between Zx and TX is

A,: %.,-PC,, (&f)(x, w, z) = f&9 w, z), (2.17)

(&‘f)(g, W, z) E la12+Efx, WU+, wz), g = ii,nfzm.


The G-invariant scalar product in C, is defined by setting (for c = icr, (r E R)

(fi,f,)C, = 5 <fi(x),fi(x)>dx. RI

(2.18)

The representation TX is continuous with respect to the topology defined by (2.18). The operator A0 is isometric. Indeed using (1.35) and (2.13~) we obtain

= (bG’fl)(k), cG!fz)t~x)~ (1 $)4 S = (&$I, Ai;‘fi)nx. (2.19)

The spaces cx and C, are not complete as normed linear spaces. Their completion with respect to their scalar products will be denoted by 8, and H,, respectively. (Note however that they are complete with respect to some Frechet space topologies-cf. [5a].)

Along with TX we shall use its conjugate representation T: and its Weyl conjugate representation Txm. Here for x = [I, c]

i3 [L-c], x., = IL, -4, (2.9’)

where I,,, denotes that the inducing representation of A4 is 0:.

2.C. Injinitesimal generators and Casimir operators of the elementary representations. We shall write the infinitesimal generators in the noncompact picture for the subgroups

fi, M, A, N. Using (2.16) we have

(a) translations

T&x, W, z) = -&- W&)fI(x, W, z)xt=o = -&-f&-x’, W, zL=o, P

Tp = -VP f -?__. ax,* (2.2Oa)

(b) rotations: for m = 1+$&q, 0

0 l+fa:qk 1 , s;,s;l < 1, xkf@, Ii’, 2) = as; - --!- (T”(m)f)(x, s, z)~.=P,=o - -

= ~Wqk~+~x~v~-tX4VL-&kIIXivjf(X, w, 2)~

where 3 = ((r,, &, 6,);

(2.2Ob’)

Yk = a -+*qk-- az -_tXkV,+~x,V,-&kijX1Vj; (2.20b”)

(c) dilatations:

Df(x, W, z) = -& (T”(a)f)(x, w, z)+l = (-2-c-xv)f(x, k, 2). (2.2&)


(d) To find the infinitesimal generators of special conformal transformations we first calculate (see [2]) using (1.28) (cf. (1.32))

n;‘&, = ~X,n;;ia’-lm’-l, (2.21)

x’ = x-x2b

0 , Ia’1 = o, m’ =

, e c 1 o = l-2bx+b2x2 (a # 0 since b + 0).

Thus we have

CJ(x, w, z) = -$- (T%)f)(x, w, Z)b=O

where *f(

x-bx2 ,wI,ZI

11 = (c;+Q-(~, w, z),

0 b=O

c; = 2(2+c)x,+ (2x,x,- &x2)V,,

c: = 212X,-x”~q.q;& -X”z5&q&.

Using homogeneity of the representation functions we obtain from (2.21):

c: = -x, wql,-&+z’qt+ ,

( 1

(2.20d)

(2.21)

(2.22)

In accord with the general fact of operator (or Schur) irreducibility of elementary representations the Casimir operators are multiples of the unit operator; in particular,

is given by 6,(x) = D”+CT+4D-2(X;+Y:) (2.23)

%(X) = ~[ml(mI+2)+m,(m2+2)]+c2-4. (2.24)

3. Properties of the elementary induced representations

3.A. Irreducibility of the elementary induced representations. We shall review some general properties of the elementary induced representations.

As already mentioned every EIR of G on C, is operator irreducible in the sense of Schur’s lemma. Nevertheless, as we shall see, some of EIR are topologically reducible in the sense that there exist nontrivial closed invariant subspaces in some of C,.


It will be shown in Section 5 that the representations labelled by

Xi;” = (21-n+ 1) n- 1; --I--Y- 1) E [n-l- 1, I; -z-v- I], (3.la)

1= o,;, 1, . . . . v = 1,2, *..; It = 1,2, . ..) 21+1,

contain finite-dimensional invariant subspaces. For fixed 1, v and 12 there are three more reducible representations obtained from

T;,n (= Ts,,) by a chain of intertwining maps, which establish pairwise partial equi-

valence. They are labelled as follows:

& = (Fz-1,21--n+l;/+v+l) E [I+l-n,I;1+v+l] = XC”, (3.lb)

x;m = (2z+v-n+1,v+n-1;-z-1) E [n-z-l,Z+v; -Z-l], (3.lc)

x;; = (v+n-1,2z+v-n+l;Z+l) E [l+l--n,I+v;I+l] = &. (3. Id)

Since the Casimir operators of G are multiples of the identity for each elementary representation, they must have the same value for the representations (3.1). Indeed the Casimir (2.24) has the value

(&(x) = (I+~--~+l)~+2(Z~+21+vn)-3 (3.2)

for all the representations (3.1). The same value of the Casimir operator have two additional representations (for

fixed I, v, n):

x;z m (21+v+l,v-1; -I+n-1) E [-I-l,Z+v; -I-l+n], (3.3a)

2;;; z (v-1,22+v+1;5--12+1) E [I+l,I+v;l+l-?zn] = ii;.‘. (3.3b)

These representations are partially equivalent to (3.1), but are irreducible (see Sec. 5). All nonexceptional elementary representations different from (3.1) are topologically

irreducible [12, 131. The importance of the elementary representations comes from the fact that every

IR of G (in a certain sense) is equivalent either to anelementary IR or to an irreducible component of an elementary reducible representation. For more details see [4], Sec. 3.A (see also [19, 31).

We shall use some facts about characters of EIR as given in [4] (cf. also [IS]). For infinitely differentiable functions p of compact support on G the operator

(3.4)

is known to be trace class. Its trace

O,(P) = TrzYv) is called the character of the EIR.

We introduce an auxiliary function

(3.5)

13.6)

ELEMENTARY REPRESENTATIONS AND INTERTWINING OPERATORS

with covariance condition

F,(m’mam’-‘) = F,(ma), m’ E M’.

The character 0, is expressed in terms of FP according to

0,(v) = { Ial-‘[trD’(m)]Fp(ma)dmda. MA

The characters of conjugate or Weyl conjugate representations coincide:

%JV) = t&(V),

e,@(v) = 0, (v).

249

(3.7)

(3.8)

(3.9a)

(3.9b)

Indeed, (3.10a) is proved by change of variables and use of (3.7) (cf. [4]), while for (3.10b) we need only to show that

trDt = trDP (3.10)

which is easily done in appropriate basis of V’ and VT

3.B. Invariant sesquilinear forms on pairs of EZR. Suppose we have an invariant sesquilinear form L on the spaces C,, x C,, given by cfi E C,,, i = 1,2)

Lcf, A = ~<f,(Mz(x)>~x. (3.11)

In writing (3.11) we have a restriction on x1 and xz, namely if x1 = [lx, cl], x2 = [l,, c2]

then either 1, = II or I, = II,. Checking invariance of (3.1 l), i.e. establishing

L( 7Yg)fi 9 wdf2) = w-1 JJ (3.12) for g = o we obtain

c1+c2 = 0, (3.13a)

r, = Ii. (3.13b)

(Invariance for g = Z E Eis trivial and follows from (3.13) for g = a E A and g = m E M.) Denoting for x = [I, c]

*x E [I, -C] (3.14) we conclude :

PROPOSITION 3.1. There exist an invariant sesquilinear form L on the product Ce, x C, given by (3.11) (JI E G,, f2 E C,).

Remark. In the case of unitary characters of A (c = ia, CT E R) we have *x = [l, -C] = [l, c] = x and so (3.15) defines an invariant scalar product in C, since (cf. (2.18))

(For nonunitary characters of A (2.13) is only K-invariant.) This scalar product gives rise to the so-called principal series of unitary irreducible representations-cf. [2, 14, 41.


4. Intertwining operators

4.A. Definition and construction of the intertwining operators. By a well known theorem (see [19], Vol. I, Sec. 4.5) a character determines every unitary irreducible representation uniquely (up to equivalence). Since, according to (3.10) the unitary principal series representations x = [1, c], xrn and 2 have the same character, they must be unitarily equivalent. Therefore there should be operators intertwining these representations.

First we introduce the equivalence maps between the representations Z6 and ZX

and between Zx and Zxm:

B,: cxo --) a,, (4. la)

(B,f)(g, z;, z) = (Bfk))(k z) = fk, Z&P ~9 (4.lb) (cf. (2.8b)),

B * (I; -+ cxm XCU’ (4. lc)

and the action of B, is given by (4.lb) for f E a,.

Note that BxBgm = id&, , BxmB; = idaX_ (4. Id)

(where by idx is denoted the unit operator in the space X). The intertwining properties of B, and BxW

TxBx = B,TT-, TXoBxm

are proved easily since these operators are reduced to to (2.8~).

The intertwining operator

= BxmTIX (4. le)

the operator B and (4.le) is reduced

(4.2a)

which, exhibits the equivalence between Zx and 2”, [14] is defined by

(Axfk, W, 4 = y(x) j f(gG, W, ZW, (4.2b)

ZxA, = A,%% (4.2~)

(The constant y(x) shall be fixed later.) This operator in the noncompact picture (accounting the equivalence of the general and noncompact pictures we use the same letters):

A,: Cx, + Cx, TxA, = AxTxy (4.3a)

can be represented as follows (cf. [4])

(&0(x, 3 wr, 4 = G&f >GL, 3 W,, zd

(4.3b)


In deriving (4.3b) we have used (2.16) and changed variables x = o(x, -x1). Analogously is defined the inverse operator

Azo: (UC,> --) t&,(CzJ, (4.4a)

(AJ)(g, w, 2) = Y(x&Y&, w, z)dx, Z’@A, = A&XX, (4.4b)

TXmA, = AxmTX.

The constants y(x) shall be fixed (not uniquely) by the condition

verse of A,. We obtain

(A&f)(x, 3 @,z> = Y~x~Y~~~~~~f~~,,wx,,x:,,x:,x,,z~~

(4-k)

that A, is the in-

dx, dx, (- l)m~+m’~4r(%~r(%~)f(x w z)

* (x~2)2+c+Ia(x~J)2-~+II = (/f_C’)[(~2+1)2_c2] 1’ ’ * (4.5)

(Note that for the principal series of unitary representations c = ia, a E R, so c2 = -a2 < 0.) The necessary calculations are given in Appendix A. Setting

&) E (- l)(m1+m,)‘22c n(x) R2 (4.6a)

and requiring that the RHS of (4.5) equals f(xl, W, z), we obtain

n(x)n(xJ = (G-c2)[U2 + I>-c’l, (4.6b)

A,A, = idc .Y’ (4.6~)

Next we de$ne the intertwining operator of the representations TX and @:

GX: C,- -+ C,, (4.7a)

G, = Ax&,,, (4.7b)

(GJ)(x, 3 WI 3 zl>

= Y(X) U

a ml

H a

1 m'

m,!m2! i&X12f&-- - f&z, “2,z2P,

aZ2 aw, ‘EXI2Zl (x~2)2+c+I, * (4.74

Further we note that the operator BzAga : C; --f C, is given by (4.7~) with y(x) replaced

by y&). Thus AxBx, ,and BxAza are proportional and we shall set

r(%) = Y(X), n(L) = n(x). (4-g) In this way

G, E A,Bxa = B,A%,. (4.7b’)

Gfh 3 Gl, ~1) = ’ 5. 5

ml!m2! s %(x12; W1, ~1; W2, z2)$ ~2, - aw 2 E, &a~ dx2,

2 (4.7c’)


with

G&;%,z~;~,,z,) =

(4.7d)

Analogously we define the intertwining operator of the representations TX@ and T;@

G &II’ ’ Cz/ c?&, (4.9a)

G &n z AxWBz= B,..Ag, (4.9b)

and the action of G, is given by (4.7~‘) for f E C;@ and Gx replaced by G,,; also G,(x)

is given by (4.7d) with the substitutions

n(x) + n(j), c + -c, x + x+.

The intertwining properties of GX and G,

TXG, = G,Ti, TXmGxm = Gii,T2~ (4.10)

follow easily from (4.le), (4.3a), (4.4c). It is also trivial to check that G, is the inverse of G,:

GzGi = A,BxuB;A, = A,A,._ = idcx (4.11)

using (4.ld) and (4.6c). Analogously,

G,Gi@ = ide,.,. (4.1 lb)

The operators A,, A,, G, and G, were considered till now for unitary characters

of A, i.e. for c imaginary. Using (1.35) we write

(4.2d)

1 Since k, varies in the compact manifold K we see that -

Y(X) A, is well defined and

analytic in c for Ret < 0. The same applies to 1

- G,, while 1

Y(X) _ A, and _ Y(X)

’ Gy, Y(X)

are well defined and analytic for Ret > 0. All those operators can be extended to meromorphic functions in the entire complex plane c (with poles on the real axis). This analytic continuation will be carried out in the next subsection.

4.B. Harmonic analysis of the intertwining operator G,. The harmonic analysis of G,

on the parabolic subgroup gAM is reduced to the harmonic analysis on 15 and A4 since G,(x) is irreducible with respect to A being homogeneous in X.

ELEMENTARY REPRESEfiTATIONS AND INTERTWINING OPERATORS 253

First one performs the Fourier transform of G,(X) on i z Rzh defining

&p) = 1 G&e-‘P”dx.

We use the integration formula

(4.12)

(4.13)

valid for - 1 < Re a < 0, extending it by analytic continuation to all non-integer a. We introduce the notation

51, = IGpz2, T2,

1/2

We obtain for (4.12)

(4.14)

\ I

~x(P;~1,~l;~,,z*) = (-- 1)2’an(%)

cw2

(~1’~)m~(~2~l)m~~~-i~‘($)2+c*r’dx

= (- l>%(x) ~(-c-z,)r(c+z,+l)

c (z,-z,)!(Z,+Z,)!(5,Tz)‘~-“-’

Jv+c+l,) k (z,-z,-k)!(2z,+k)!k!qc-z,-k+!) -

. (~lp)y~2p)2’l+’ q c-“-k ( 1 (4.15a)

where l-

3ir = - /2

Wf 4rZl (is 1,2), j E-

1, < 0,

II>0 ,

I”( 3132)

x = ‘- (P3l)(P32) ’

(4.15b)

(4.16a)

(4.16b)

(4.17)

Note that in (4.15a) the range of summation in k depends on the sign of I, and is de- termined by the factor l/(Z2-I, -k)!(2Z1 +k)!k!.

In deriving (4.15) we have used the following relation between the C and the 3 variables [17]:

2(P&)(P&) = %‘31)b’32bP2(3132h (4.18)

We have also used an expression for the Jacobi polynomials (8.962.2 of ref. [9]):

P(-a-k)I’(a+@+n+k+l) T(-a-n)P(a+@+n+_l)


To carry out the harmonic analysis of 6,(p) on M for fixed p, we must expandg,(p) in the eigenfunctions of the Casimir operator of the stability group U of p. In order to perform this step we shall use the frame in R4 in which

P = (@,PL = IPI = @-.I. (4.20)

In this frame the group U z SU(2) is generated by (see (1.16) and (2.5))

(rotations in the (j, k)-plane leaving p invariant). Then the Casimir of the stability group is the squared “spin operator”:

(4.22)

We denote for convenience the eigenvalues of S2 by s(s+ 1). For the eigenfunctions of Sz we obtain (the calculations are done as in [17] since the Lie algebra of the stability subgroups of M and SOT (3, 1) may be identified)

F’scp;%,zl; w, ) ZJ = (>~*)“W’yp; %I, z1; wz , z*), (4.23a)

17~s(p;wl,zl;wz,zz) = (- l)+S-‘%k (2(~5,)(ii5,))‘a-“1’40i:lf’“(x) (4.23b)

(s = VA, V11+1, . . ..M.

The operators FfS has the following orthogonality property

F’S o + : VI --, v’ 3

I;3”oFtr = 0 , s # d.

Indeed

(4.24a)

F’” o F%’ = $+ 1) F’” 0 ( S2FLf ) = ,,(,f+ 1) (S’F*‘) 0 FL’

._ :‘:

‘@+ ‘1 F” o Fh’ _ 0

F s’(s’+ 1) - , s # d.

The crucial- point in deriving the last formula is that if 1 = [II, 121, then I” = i-h, bl and F’” depends on j11l (not on II).

Let us denote

7P: v’ --, v’, ,tE =_ F’” o Fii. (4.24b)

The 7~‘~ are orthogonal:.


We shall fix the constant AI, in (4.23) by requiring

12

c 7clS = i&l. S=llll

(4.25)

For this we first decompose zx in F” using the formula

p:“.B’(x) = rQ?+z+ 1) + (2s+/9+ l)r(a+B+l+s+ l)P(a+Z-4 -____ r(cx+/f?+I+ l)r(cx) s (I-s)!P(B+l+s+2) _ *

* P$O*f‘yx) .

We obtain using (4.15b) and (4.26)

(4.26)

2 (p) _ (12+l~1I)~(~2-I~ll)~~(x)~(l--+~2) P2 c x - (1h1--41‘(2+c+~,) ( 1 2’

12

22 (- l)+““lr(c+sf l)lr(c--s)(2s+ 1) --__ S=,,I’i r(c+II,l+ I)P(c- I11,)(1,+s+ 1)!(12-s)!A[, FrS(p)*

Then we use

&$)(?&I) = idYZ = (I,+ 1Z,[)!2(1,-1111)!2 2 (2s+ 1)2F’“(p)F’“S(p)

s=,,l, (b+s+ ~)!2(~2-s)!2&s

where we have accounted for (4.6b), (4.9, (4.24) and

I’(u+k) = (-l)k+n

T(l-a-n)

r(cc+n) T(l-a-k)’

We obtain (4.25) by setting

(4.27)

(we fix the phase factor consistently with the case of symmetric tensor representations, cf. [4]).

Finally we obtain

z;c(P) = (4.28a)

where

cY,Il(C) = (-I)+ qc+s+ l)F(c--S)

_. m+I4l+wk--I4I)

(4.28b)

Note that in the symmetric tensor case (I, = 0) F” reduces to 17’” which are orthogonal

projections if we identify the spaces V’ and VT (I = 1); also xl’ reduces to 17’” since zrs = F’“$ = n’“fl’” = fl” in this case.


The normalization condition (46b) (using (4.8))

n(x)n(i) = (I:-c2)[(z,+ 1)2-c2] (4.29)

does not fix n(x) uniquely and there seems to be no universal choice of n(x) equally suited for all purposes. Following [14] we can choose n(x) to be a meromorphic function of c with all zeroes in the right half-plane and poles in the left-hand plane. The simplest choice of this type is given by

n+(x) = (l+c+Z,) m +c+ IU) WA-C) *

Another choice of normalization

%(X) = (Iu-w(2+cf~2)

F(l-c+Z2)

(4.30)

(4.31)

is particularly convenient in writing some differential identities between sesquilinear forms for exceptional representations (see Section 5.E below).

A third choice is appropriate in the study of exceptional points 2;; (cf. (3.1)):

n_(x) = (Iz,I-c)(z~+l-c). (4.32)

We shall use the notation G+, Go and G- for the intertwining operators G with normalization (4.30), (4.31) and (4.32), respectively.

It is easily seen in the p-space picture that the operator G” is a meromorphic function of c and is not ‘defined for xl;n (3.Ia) and &;; (3.1~). Analogously the operator Go is not defined for x = 2;” and xii’; (3.ld) and G- is not defined for x = x$” (3.lb) and x;$,. (For G- it is appropriate to use the x-space picture.)

Thus the intertwining operator G, is defined for every x (using the appropriate nor-

malization) and we can conclude that the representations TX and Tx”are partially equivalent.

In fact unless x is some exceptional representation of the type (3.1), TXand Tx”are equivalent and for unitary irreducible TX they are unitarily equivalent for normalizations obeying (4.29).

4.C. Invariant sesquilinear forms involving G, . We saw in Section 3.B that the scalar product (2.13) is G-invariant for imaginary c. It is easy to check that (2.13) is not G-invariant for c other than imaginary. However, there may be other unitary representations given by a hermitian form on C, x Cr:

UflYf2) = S<fdx), $Xx,Y)f26wx&. (4.33)

The invariance condition

%@“2(g)f~ 9 T*(g)f2) = M_f~ ,f2)

forg=ii,,g=m,g=a,andg=wgives

$&,Y) = &(x--Y),

S,(Ax)fY(m) = D’(m)S,(x),

(4.34)

(4.35a)

(4.35b)


M4x) = la12~“c-4Sx(x), (4.35c)

1 (x2)2-‘(y2)2-’

@(m:)S,(+Y)D’(%) = SAX-Y), (4.35d)

where

If we consider (4.35d) for x = @, x4 > 0), Y = (0, Y4 > 0) we obtain using (4.35~)

x4 ( 1 c-c

-_ S&-y) = S&-y) or C = c. Y4

(4.36)

Further we notice that properties (4.35) (with C = c) are those of an intertwining operator of the representations TX and TX1 with x = [I, c] and x1 = [I, -cl. But we know

that TX is equivalent to T% and T:. Thus we see that the hermitian form S exists iff

i= I. (4.37)

Then i = [i, -cl = [f, -cl = x1 and identifying V’ and VT (when r= r) we obtain:

S,(x) = G,- (x,. (4.38)

We shall also use the standard approach to the problem [14]. One starts with the identity

(X2y’,)2_, ~b(~:)~x,(~x-~Y)~‘(~,) = A&-Y) (4.39)

using D:(m) = BxaDi(m)B;; (4.40)

to obtain (multiplying (4.39) on the left with B,-)

d2~ D’im:)G,-(Wx--y)D1(my) = G,-(X-Y) (4.41)

and so S, exists iff r” = I and S, = G,.

Identifying Vi with V’ (for r”= I), so that Bx, and B,- are operators in V’ which Will

be denoted in compliance with (2.8) by B and B-l respectively, we may rewrite (4.4) as

D;(m) = BD’(m) B-l. (4.42) On the other hand

D:(m) = D’(omco-‘) = D’(co)D’(m)D*(o)-l (4.43)

where we have introduced D’(o), because in the case r” = I it can be defined [14]. Compar- ing (4.42) and (4.43) we see that up to some unitary transformation commuting with D’ we must define

D’(o) = B, (444a)

(D’(co)+i; z) = y(ze, 8%). (444b)


Returning to the question of unitary representations besides the principal series we reach the following (well known) conclusion that necessary conditions for the existence of such representations are (for x = [l, c])

id, c=c. (4.45)

The necessary and sufficient conditions are found in [4] and the complementary series of unitary representations is singled out.

We note a useful relation between the forms (4.33) and (3.11). Observing that for

i- = I, c = c, *x = [I, --Cl = [r; -c] E 2

and thus f E C, = C*-,, Gi = G*, and G*JE C+* we obtain

&cfi, fi) = UJI 3 h.fJ <r” = 0 - (4.46)

Eq. (4.46) enables us to define an invariant sesquilinear form on C, x CL,- for all x. Actually we shall define a form on C*, x C;i (fr E C*,,fi E C:)

Lxcfi J-i) = Uf, 3 Wd = ~(G*zf~ J’z). (4.47)

For exceptional x (cf. (3.1)) the form (4.47) is degenerate. However it shall be used in Section 5.C to define invariant sesquilinear forms on the invariant subspaces. Then

(again in the case i = 1) additional unitary representations will emerge.

5. Properties of exceptional elementary induced representations

5.A. Subrepresentations of reducible exceptional representations. The present section will be devoted to the study of the six families of exceptional representations, x:Wr, $, and xi:.” (see (3.1) and (3.3)). We shall use the shorthand notation CL C$,, C;i$ for the corresponding (noncompact picture) representation spaces, and *Cl&“)* for the representation space of the respective *x (3.14). Note that *C$:j,* = C:l,b’ b = 21+2-n), *Cii: = C;:i.

As was pointed out in Section 3.A the space Ci;, contains a finite-dimensional invariant subspace Er,. which is defined as follows

E ,vn = (P E C;;,) P polynomial, (-jt?‘P(x, W, 2.) = O>

( a 3/r = $qpz, v = -

8X 1 .

The general expression for the elements of I& can be written as follows

v-1

P(x, w, 2) = c

(X2)‘-k--Iht(~; fi, 2, x)),

k4

where @, z, x) stands for the 4-tuple

p, z, x} = (W, z, W-X, xz)

(5.1)

(5.2a)

(5.2b)


and Irk is a harmonic polynomial of degree k in the first argument and homogeneous polynomial of degrees 22-n+ 1 and n- 1 in W and z respectively.

The invariance of E,,, follows directly from (5.2a). The invariance of the subspaces introduced below will be proved in Section 5.B.

Note that the four linear forms (5.2b) satisfy the first order differential equation

(3V@, 2, x> = 0. (5.2~) To see this we use [17]

(&&&a = 2%&a. (5.2d)

The space Cz, contains an infinite-dimensional invariant subspace

fi, = i fE C&l 3g E c;;: f(x, w,z> = $ x g(x7 w, 4 - ( a + a )’ - ) (5.3)

The subspaces *FL,,,, = Elvp and Fl,. (p = 2Z+ 2-n) are orthogonal with respect to the invariant sesquilinear form (3.11) on C;;, x Clt, . Indeed if P E Elvp, f E FI,, (or P E El,, ,

f E FI,,) then

L(P,f) =s(P(x),f(x):dx =~~(x,&,~&)f(x,W,z)dx

= p x&,$ s ( )(

&V+& ‘g(x,W,z)dx )

= ,_1~“&$-6~)‘~(x,&- ,$)]g(x,ii,z)dx = 0. (5.4)

Conversely, if for a fixed f E CL,, Eq. (5.4) holds for any P E Ervp, then f E FI,,. Thus one can use (5.4) as alternative definition of Flrp. Sim.ilarly, definmg Ft, by (5.3), we can use (5.4) to define Elvp.

The space C$, contains an infinite dimensional invariant subspace

F;, E {f s C;,i 3g E CI,,: (3V)“g(x, M’, z) = f(x, ii, z)}.

Finally, the space C’$ contains an infinite-dimensional invariant subspace

(5.5)

Dbn f (fEciil (++ -+jfcx,w.4 = 0). (5.6)

The subspaces *Dim = F&, and Dlvn are orthogonal with respect to the imm-iant sesquihnear form (3.11) on C$ x C;$, i.e. for fI E F&,, fi E Dlvn

Qfi sfi> = s_& (xc&-, &) f&, W, z)dx = 0. (5.7)

(The proof is analogous to that of (5.4).) It is also true that (5.7) can be used as a definition of either F;,,” or D,, depending on which of them we have defined independently.

We shall use in general the notation I!$* c Cl:!* for the invariant subspaces. Note that *II;:* =I;::’ (p = 21+2-n).

260 V. K. DOEREV and V. B. PETKOVA

5.B. Intertwining d$erential operators. Partial equivalence relations among the exceptional representations. We shall consider the following differential operators (Y = 1,2, . . . ;

k = 1,2, .*.; p = 1,2, . ..).

fl: G, -+ c,,, (5.8)

(dW(x, w, 2) = @yY(G w, z),

Xl = (ml, m,; c), x2 = (ml+v, mz+v; c+v);

d”: C,, --f C,,, (5.9

Xl = (ml, mz; 4, x2 = (m,-v,m2-v; c+y), v < m,, v f m2;

(5.10)

Xl = (m,,m2;c), x2 = (m,+k,m,-k;c+k), k< m,;

alp: c Xl -+ G,, (5.11)

( 1 P

(ay-)(x,w,z) s +yz f(x,W,z),

XI = h,m2,4, x2 = (ml-p,m,+p;c+p), p < ml.

The relations between x1 and x2 in the above formulae follow from

QT”l(g) = Pa(g)Q (Q = d”, d’“, ak, VP) (5.12)

for g = mEMandg= a E A. Eqs. (5.12) are true (without restrictions on xr) for g

= i E*. They are established most easily in infinitesimal form. They are trivial for the

generators of i? and M (see (2.18a) and (2.18b)) and for the generator of A (2.18~) it sufZices to observe that

[(rlv)r, XVI = WV, (5.13)

7 = Wqz ( a_ a a or $4+x’ W9.s XY aw EC&r

1 (q2 = O),

to obtain (r193’DI = ~z(W, (5.14)

DI = -2-q-XV (i = 1,2).

The intertwining property (5.12) is not valid for g = n EN for general x1 in (5.8-5.11). We shall find the restrictions on x1 brought by imposing (5.12).

First we establish the commutation formulae:

ELEMENTARY REPRESENTATIONS AND INTJ?RTWINING OPERATORS 261

KW, &I = rrl&?v-‘7 WV, x21 = 2@?)(7m-‘,

rw)r9 x&v)1 = ~Ex,hv+ (r- 1 +xmplwrl. Using (5.15) and (2.18d) we derive

(7#qrc:= = C,=(gV)‘+2r[(l+r+c,+xV)rl,-(x~)v~](77~-’.

Then we obtain

[ ($Y, wqk Y&+z’qi & 1 = 2t(%vk-~kV&~v)r-‘,

[

a a (7jV)r, wqk --=--zzIqk - at4 aZ 1 = 2rs~,@#.‘1,V&v)V

Using (5.17) and (2.18d) we find

(t3Yz)q = c;‘(KJyzy + 2v I

m1;m2 -(xv)) VP+ (W)V, WY1 1 which summed with (5.16) (for q = Gqz and r = v) gives

SC; = c;(Iy+2v 1

1+v4-Cf mr;m2 &-‘(Gq,z). 1 If

we obtain

l+v+c+-- = m-f-m2 o

d’C; = C;d’.

(5.15)

(5.16)

(5.17a)

(5.17b)

(5.18a)

(5.18b)

(5.18~)

(5.18d)

For fixed v (5.18~) means that we have two free parameters, c being real, negative and integer or half-integer. Thus we set

c= -l-y-l, I= q!L?- = o,;, 1, . . . . (5.18e)

m2 =n- 1, n = I,2 )...) 21+1, ml = 21-n-I-l.

Thus (5.18d) is fulfilled iff (cf. (3.1)) ,-

Xl = Xlvn 9 x2 = XIvn (5.18f)

Analogously we find that d”‘Cf; = c;d” (5.19)

holds iff x1 = x$, and x2 = & ; ant; = c;an (5.20)

holds iff either x1 = xi;, x2 = xi;; (cf. (3.3a)), or xl = xi:; (cf. (3.3b)), x2 = x;y’, or

XI = (m,m+k; -k/2), x2 = 2,; avC1 = C2p cc Ir (5.21)


holds iffeither x1 = x;$, xz = *xi:;, or x1 = * x;:p’, x2 = *J&Y, or x1 = (m +k, m; -k/2), xZ = iI. In (5.19), (5.20), (5.21) the 3-tuple (Z,Y,~) has the same ranges as in (5.18e); m = 0, 1, . . . . k = 1, 2, . . . . p = 1,2, . . . . 21+1.

Later on we shall need all equivalence relations between the representations in (3.1) and (3.3). For this we note that making the change p = 21+2-n, n = 1,2, . . . ,21+ 1 in (5.21) we obtain

a r21+2-nc1 cc

= C2(3f21+2-n u (5.22)

for x1 = xi;, x2 = x;Z, or x1 = x2, x2 = x2. Summarizing the results obtained so far in this subsection we have proved:

PROPOSITION 5.1. Let d’, d”, an, alp (p = 21+2-n) are the dlflerential operators (5.8-5.11). Then all partial equivalence relations realized through these operators are the

following d’Ti& = T;,d”, (5.23a)

,,,,;A = T&d”, (5.23b)

Z’T;y, = T,‘;,,? G”, (5.23~)

a”T;:; = T;;: S”, (5.23d)

arpT;v; = T;;,- a’p 9 (5.23e)

a’PTi:z = T;;; alp, (5.23f)

akT%l = T:+k 3 (5.24a)

a!kTXi = T<@. (5.24b)

(In(5.23)2=0,$,1,... ;v=1,2 ,... ;n= 1,2 ,..., 21+1.In(5.24)~,=(m,m+k;-k/2), xi = (m+k, m; -k/2), m = 0, 1, . . . . k = 1,2, . ..).

Remark. The representations TX1 and TXi must be equivalent to Tiii and T% respectively in accord with our previous statement. In order to verify it we must show that the operators G-,, and G-,; reduce to ak and Yk respectively. The proof is straightforward.

Another useful fact is:

PROPOSITION 5.2. Let i3” and afp be the operators (5.10-5.11) and F;,, Dlvn be as in (5.5-5.6). Then

anFiVn = 0, (5.25a)

arPFiyll = 0

ant;:; = d,,. ,

(5.25b)

(5.25~) arpcI+ _ D lvn - Ivn . (5.25d)

Proof: Let f E F;,. . Then there is g E Ci;,,: f = d’g and so

ay = o”Vg = 2 (5) (a”-kd’)akg. k=O


Noting that Pg = 0 (mz = n - 1) and ad = 0 (%ye &) (Gyz) = ~vetv~ = -iiy+&W

-- z-- WEWVPV~ = 0) we conclude that every term in the above sum equals zero which gives (525a). For (5.25b) we use a’2’+2-ng = 0 (ml = 21-n+ 1) and a’d = 0. For (5.25~) let f E C;;;. We must show d”‘d”f = 0 which follows from d”f = 0 (ml = Y- 1) and d’a = 0. Analogously (5.25d) follows using d’y = 0 (mz = Y- l,fs C;$+) and d’a’ = 0. For (5.25c,d) we use also invertibility of d” and Yp which is established in Proposition

5.7 below. Next using Propositions 5.1 and 5.2 we shall show that If:!* are indeed invariant sub-

spaces of Cl,,.

PROPOSITION 5.3. Let El,“, F,,,, Fi’“, and Dtvn be as in (5.1), (5.3), (5.5) and (5.6). Then

Ti;, EI,, = El,, , C’ii FL, = I;;:, , (5.26)

T&F,,, = FI,,, T’+D = D,,.,,. lvn Ivn

Proof: The invariance of El,, was shown in Section 5.A. To obtain the rest of (5.26) we apply (5.23a) to Cc,, (5.23b) to C;,f; and (5.23d) to Cj:; . For the last equation besides the definitions we use also (5.25~).

For exceptional reducible representations TX and TZ are only partially equivalent because the intertwining operators G, annihilate the invariant subspaces in their domains and map onto the invariant subspaces of their images. This is the content of the following

PROPOSITION 5.4. Let G,f is G, with normalization n*(X), set

G;;;” = G*, - XI ‘.” - (5.27) Y

Then Ker GiLj,* = Ii;!,‘, ImGjij,* = Iii!* (5.28)

where KerA is the set of vectors mapped to the zero vector, ImA is the image of the map-

ping A.

Proof Note that ImG;;, = El,,, + d”G& = 0, (5.29a)

Ker GG, = F,, SS- G&d’” = 0, (5.29b)

ImGZ, = 4, o Gi;nG,&, = 0, (5.29~)

KerG& = EIvn cs G$,,Gi;n = 0, (5.29d)

Im G$, = Dlvn - d”G$ = 0, (5.29e)

KerG;; = F;, oG$d’ = 0, (5.29f)

Im G& = Fi,,” * G$G& = 0, (5.298)

Ker G;; = D,,, - G;;G$ = 0. (5.29h)


The proof of (5.29) in the “a” direction is trivial. To prove it in the “+” direction

we use the fact that Z\Lj,* are invariant subspaces and G$,* are intertwining operators. Note that we prove (5.29c,d) using (5.29a,b), and (5.29g,h) using (5.29e,f). Now to prove Proposition 5.4 we shall establish the validity of the equations in the right-hand sides of (5.29). (We refer to them by the respective number of the relations.)

To prove (5.29a) we note that the intertwining kernel

Gi&;~1,z1;~z,zJ = (5.30)

(C1 and C2 are given in (4.14)) is a polynomial in x of the type (5.2) and is therefore an- nihilated by 6. The RHS of (5.29b) is reduced to (5.29a): Let frs C;$

(Gi;nd”l)(xr , zl, ZI)

Gin XI,; w,&- a

s aw, t&, ‘E -

az2 = (21-n+ l)!(?z- l)! -aj V2+ & ‘f(x2 ; z2,Z2)h2

2 2

( l)S(

a I---.-- 1 = -

aw 2 (21--n+ l)!(n- l)! *

* Gm

a x12; ~1,z1;---

aii f&, r&--

2 af

2 )f(x,; ~2,zz)dxz = 0.

Using the p-space picture we see that

G”(P) = I+r+i .

I

. c

(-l)‘-~(r+v+1+S)!(z+v--S)!(S-lFz-z-ll)! n,:,s, (5 31)

s=-in-l-11 (s+ In-Z- II)!

(ZiX s 17p-1.2l-n+l)s - cf. (4.23)))

is a homogeneous function of p of degree 2(Zfv+ l), which annihilates the Fourier transform of polynomials P E El, since (FP)(p, w, z) = P(I’V,, i?, z) S(p) and the degree of P(x, %, z) does not exceed 2(1 -I-V- 1). This proves (5.29c) and (5.29d).

To prove (5.29e) we shall use the expression of Gz in (4.15a) with normalization n,:

MO, ,z1;W2,z2) = (-1)2’~r(-c-z~)r(l+c+1z11)

~WlI--4

* g (;I)(;) rcc+ &_z2j (~I'&~;)L(z,&z2)k~,hZt)ll-k(~2~z1)m1-k. (5.32)

w

ELEMENTARY REPRESENTATIONS AND INTERTWININ G OPERATORS 265

Thus we obtain for d”G: in the p-space picture:

(~p&~g+(p;zl,zl;w2,z2) = (-l)“~~(-c-~2)~(l+c+l~ll) (I.$ W1I-4

* (F2iz2)

m,!m,!r(c-Z,+v) c (w,‘&W2)k(Z1EZ2)k(W1~Z2)m’-v-k F(c-I,) k=.O k!(mI-v-k)!(m,-v-k)!(l+c-l,+k+v) *

* (wj2zl)m~--k.

Observing that

while for x = xi.+.

(5.33)

(5.34a)

q-c-l,) = (_ I)l+“+$_n+l~ (l+l--b-l- II)!

WA-4 (22-w+ l)! (5.34b)

we conclude that (5.33) equals zero for x = x$, which gives (5.29e). Eq. (5.29f) is reduced

to Eq. (5.29e). In order to prove (5.29g) and (5.29h) we need the explicit form of G’& in x space,

which cannot be obtained directly by setting x =x$, in (4.8) (using (4.30)) because of

undefined expressions of the type

1 2 2+k

( ) T(O) xz (k > 0, integer).

We define G;:(x) = !j; G&+(X) (5.35a)

with xi+ Z (Y+n--l,221+V-n+l;I+1-6). (5.35b)

Using (5.32) we obtain

G;:(x) = $(&&+)(_x) = (- l)‘+‘+“+‘-“‘(l +I+ II+ l-n])! .

2’+‘(21+Y+ I)!

. (1 +I- II+ 1 -nl>! T (‘+;-I) (2z+v;n+1) k! (k+l-v)!

(Wl’&W2)yZ1&Z2)f.

. ($2yZ1)2~+v-n+l-k (W1 yZ2).+“-l-kAk+1-vd(%)

where the distribution-theoretic identity was used [8]

(5.36a)

Zik-6 = (2x)24 k9(X)

2k!(k+ l)! (5.36b)

and the sum in k is over Y- 1 d k d min(v+n- 1, ~Z+Y-n+ 1).

266 V. K: DOBBEV and V. B. PETKOVA

Next we derive using (4.7b’), (4.ld), (4.30) and (4.32):

n+(X;+>n-(iA+) = [n(n-21-2)+6][V(21+Vi-2)+6]

(21+2+v- _ 6)2(1+1+Il-n+ll-6)r(l+21+Iz-n+lI-~), (537) r(ll--n+ 11 -I-- 1+ 6)[n(n-21-2)+ 8][v(2z+Y+2)+ 4 *

where AZ is (analogously to G$) A, with normalization n*(x). We note

G;,(x) = ',';G;;+(x)

and using (5.35a) and (5.37), we obtain

G’+G’- = limGt+GZt+ = 0 Ivn ivn a&o%% *

Noting that G&+Gl;+ equals the same expression (5.37) we obtain also

G;,G;,f. = 0.

This completes the proof of Proposition 5.4. The final result of this subsection is

PROPOSITION 5.5. The following isomorphisms (and equivalences of the corresponding IR’s of G) among the invariant subspaces I, and the factor spaces Gx.Iz for all exceptional x (cf. (3.1), (3.3)) hold

4,. = CG,I&, = FL = C;,t,/&n , (5.38a)

E Ivn CL: CL,, IFI,, , (5.38b)

D,“n N C;,/F;,, N Cl,,” 2: C;;;. ‘I+ (5.38~)

Proof: (5.38a) follows from Propositions 5.1 and 5.4; (5.38b) follows from Propo- sition 5.4; (5.38~) follows from Propositions 5.1, 5.2 and 5.4 assuming that C$z have, no invariant subspaces, which will be shown in Section 5.E.

We remind that discarding the spaces C;iz and the operators a”, a’p, considering integer 1 and n = I+ 1 we obtain from Propositions 5.1-5.5 results established in [4] for symmetric tensor representations.

Our results can be compared also with those obtained in [6] for the representations of Spin(N, 1) (double covering of SOt (N, 1)) for the case N = 5. The authors work in the compact picture (which we do not consider) of the elementary induced representations. They introduce in the representation space the (infinite) basis compounded by,the Gel’fand- Zeitlin bases of each (finite-dimensional) representation of K contained in the elementary representation ‘XX. The subspaces and operators are defined in terms of this basis. For that reason the comparison with such quantities in our approach is done through the labels of the representations. We find in [6] the following results: (a) the full list ot fhe

ELEMENTARY REPRESENT.ATrONS AND INTERTWINING OPERATORS 267

reducible representations (which coincide with (3.1)); (b) the matrix elements of some of the operators in Eqs. (5.23); (c) Eqs. (5.25a, c) of Proposition 5.2; (d) Proposition 5.4. We also point out some errors in ref. [6]: (a) the inverse of alp: C$, + C;i.’ is defined by composing G& 8” which in fact is equal to zero (G& Z?‘C&- = G$Dl,. = 0). As we shall show in the next subsection for this inverse we need an operator B$: DJ,, -+ C,‘“;/ Fj’,.,; (b) Scheme 1, which would contain our Proposition 5.5 is somewhat incorrect. The true scheme for the group G (and SO.? (5, 1)) is as follows

G+ h”

CG * - c+ I””

Gil A

d' d”

Fig. 1. Scheme of the intertwining operators at exceptional points

5.C. Sesquilinear forms on invariant subspaces. In this subsection we shall define invariant sesquilinear forms (ISF) on the invariant subspaces of the reducible exceptional representations (3.1). The ISF’s (4.47) are actually forms on the factor spaces Cx/Ix because of Proposition 5.4.

We saw in Section 4.B that the operators G.$ (with normalization factor n, (4.30)) are not defined on the spaces CjlL+ (for x = xj$), and G; (with normalization n_ (4.32)) are not defined on C$i- (for x = &A+).

On the other hand G,+(X) differs from G;(x) by the numerical factor

n+(x) _ (l+c+z~)r(l+c+lfll)

n- (xl (1-c+r,>r(l-c+lI,I)’ (5.39)

Note that G; vanishes on the invariant subspaces of CC,::+ (for x = xii’.-). We have to show that their product G,+ defines a non-vanishing (and finite) ISF on fiyp x Fl,, for

x = xi;. and on 4, x 01,. for x = & (p = 22 +2-n). To do that we shall use a limiting procedure.

Let, for example, JY; = (21-n + 1, n- 1; -Z-Y- 1 + s)(lirnx; = xi;.). Consider the d10


sesquilinear form

which is defined for 6 > 0. In order to find the limit of (5.40) for 6 10 we use the Laurent expansion of G&(x) around the point 6 = 0:

G&(x) = (_ l)(l-n+11-l-v

(2+21+a)(l+y+ I+ II+ I-nl)!(l+V+ 1r+ l-nl)! L GI;A(x) i- 6

The crucial point is that the pole term (N l/6) in (5.41) vanishes since it is proportional to G& and fi E F,,,. Thus we can go to the limit 6 $0, obtaining

= <fib,), Blt,(x,-x,lf,(x,))dx,dx, ss (5.42) where

(- l)“_ n+11-b-v+1 &W) = (2+21+V)(z+Y+1+~z+1-nl)!(z+~+~z+l-nl)! Gi;n(x)x

lnxz-ry(l+l+~-lZ+l-nj)-y(l+Z+~+lZ+l-~~)+~ 1 (5.43)

where y(a) is the logarithmic derivative of r(a):

v(a) = -&- lnP(a).

Note that the last three terms in the parenthesis in (5.43) give no contribution to the form (5.42) (as the l/6 term of G&).

The kernel B;t,(x) is not covariant under the representation T& of G. Under dilatations and special conformal transformations it acquires’non-homogeneous terms (because of the ln(xz/2) term). However it is not difficult to see that the form (5.42) is invariant, since the inhomogeneous terms can be split into a sum of polynomials of El,, (and El,,) in x2 (respectively x1) (multiplied by more complicated functions of x1 (resp. x,)) and therefore vanishing for fi E Flyp (resp. fi E FJ,,).

In general we define the ISF BILL* on I$:‘,’ x fii!* by

%‘cfi ,fi) = y; -g 6qdfl ,fi) = WI, &l’fJ

= <fiW, 42*(x,- ss ~,)f~Cd> dx, dxz 9 (5.44)

ELEMENTARY REPRESENTATIONS AND INTERTWINING OPERATORS

where %$‘I* 3 (HZ,, m2 ; CT 8) (l&x$“* = #,*) and

269

B!:!*(x) = lim Lc~G.&(x). dJ0 &

One finds by inspection that the forms (5.44) are finite (non-vanishing) and indeed invariant under the corresponding representations of G.

The explicit form of the kernels (5.45) look simpler in the x-space picture for c > 0 and in the p-space picture for c < 0. We have (discarding the terms that do not contribute to the corresponding form):

BZ(& WI, z1; w2, z2)

(_1)Y+l~+l-~l+ly(2/~2)v+l

= (2n)2(f+ /r+ 1 -nl)!(I--II+ 1 -nl)! In ~ts,~)*I+Y-“+1(52X)Y+“--l, (5.46)

&&;~,,z1;~2,z,) = (- 1)"(1+ II+ 1 -nl)! (~~C*)2lZ+i-*‘(p2/2)I+r+l

(I- ir+ I-nl)!@-- 1)!(2Z+%J+2)! x

(-1)“-~‘+‘-“‘(1+l+VfS)!(z+Y--S)!(S-~II+1-?znj) !

(1+z+Y+(I+1--n~)!(~+V-_Iz+l--nl)!(s+~l+l-nl)! 17Gi3 (5.47)

S=Il+l-n/

B;,-,(p; W, , ~1; w,, ~2) = (I+v+ II+ 1 -nl)!Y! (Jzjr*)2”+1-n’ 2

(Ii- Y- II+ l-n/) (21+ +2)fP ‘2) I+1

! x v . I

x c

(- l)“- l+l-n’(z+ 1 +.$!(I--s)!(S-- IZf l--n/)! W_“. -. SEII+I_n, (1+I+~I+1-nl)!(z-If+1-nl)!(s+II+1-nl)! Ivn

(5*4*)

The definition of B,$‘,j* implies the following

PROPOSI~ON 5.6. The ISF Bjii* on I&?* x Iii;* is related to the ISF L& on Cj;jV/I$ji x 1vn

x Cj;iT Jfi;iT by

BXA*V; ,f2> = L&d& F2h (5.49)

where _fl ~Ijii*, f2 EI$*, Fl E C$~‘/$~~, F2 E Cj~,jTfIj~~T (in (5.49) we take arbitrary representations of the respective cosets)

fl = G::;* Fl , f2 = Gj;;” F2.

Proof: Using definitions (5.45) and (4.47) we obtain

Bj;;*cfi ,fi) = tCf, , B::!*f,) = L(Gj;p Fl , Bj;;’ G::;* F2)

= L(F, , Gj:j,* B::i* G:;i* F2) = L(F, , Gj;;’ F2) z L&V,, F2).


In the derivation of (5.49) we used that on Z$j,*

6G&rr = lim -!!- 6id,c<r* = srodd d

id crf. %)I

(5.50a)

(Note that (&?G.&+)Z~~b* - G’,;,j,TZjCi* = 0; cf. (5.29) and, e.g. (5.41).)

Analogously we can prove

B$S*Gj~~* = id %n %n *

‘jr (#jr (5.50b)

Turning to the question of unitary representations in the irreducible components of the exceptional representations &j,* we note that such representations can only arise in the case 1 integer and n = I+ 1, when the ISF Bit)* z B$bI becomes a hermitian form on Zji)* = Z’$$ i = Zj$* (p = 21+2-n = 1+ 1 = n). This case is discussed in detail in [4].

5.D. Irreducibility of the exceptional representations xi::. We start the study of the representations xii,” writing down the explicit expressions for the intertwining operators

Gx;l: - This needs some consideration because substituting in (4.7d) we obtain (for &n+)

G x;‘,;: N (Lx) 21+v+l(~2x)v-1

which is not a priori well defined expression (a+~+- 1 > 3 and the distribution (2/x2)“+” is singular for c1 = 0, 1, . . . [S]). However, using the identity

a+k = (- l>kr(a> r(a+k)

(5.51)

(derived for nonsingular a) for k = v+n, CI = 1, we obtain

Gx;;; (x;t3,,~1;~2,zz) = n(x;i,*)(-- 0””

(27c)2(n + Y)! (5.52)

Analogously we find

G,;~,(x;W~,Z~;W~,Z~) = - fl(x&)(- l>‘- I v (2n)‘(2l+2+v-n)!

(5.53)

We do not specify the choice of the normalization constant n(x) because all three choices (4.30), (4.31) and (4.32) are well defined for x = xi::. Thus the intertwining operators GX;;,’ are well defined (as derivatives of well defined distributions) and

GX;: G,,,$ = id,;;;, Y Ivn

(5.54)


which means that the representations Tj:: and T;i; are equivalent (and not just partially

equivalent). However, they may still turn out to be reducible if some of the operators which intertwine Ti:z with T;,f, is not invertible. We start with the following

PROPOSITION 5.7. Consider the operators

Then

iYPBiY+. 8 - . C ii, + CiZ, (5.55a)

anBi+,a’pm C;‘: Y * Y + Ci$. (5.55b)

C-1) a’P&$ a* = - 11+1-nl-n-2i-121+1(n- l)! (21-n+ l)! (21+y+ 1)!2

(I+ll-n+ll)!(l-Il-n+ll)! id

cXyn9 (5.W Y

(_ I)l’+‘-“l-“-2’+‘-‘n!(21-nf2)!2 .

a”BiAa’P = (Y-1)!(21+y+1)!221+1(2z_n+1)! 1dcx;:;:* (5.56b)

The proof is given in Appendix B. Thus we conclude (see also [ 121):

PROPOSITION 5.8. There are no invariant subspaces in Ci$ .

Proof: Eq. (5.56a) means that

and that the mapping

is onto. Similarly, Eq.

and that the mapping

is onto.

KerP 3 (f~ c;;;] a"f = 0} = {0}

alp: C;,-./Fj’V. + Ci$-

(5.56b) implies that

Ker B’P = {J-EC;:,+] a'+ o} ={o)

8”: C;G/F;~. --) Ci$

(5.57a)

(5.57b)

(5.58a)

(5.58b)

5.E. D@erential identities between sesquilinear forms for exceptional representations.

The relations (5.49) between the forms BjiL* and LGiik and relations (5.56) are not

the only ones between sesquilinear forms on the irreducible components of the exceptional elementary representations. The objective of this subsection is to establish six other sets of identities of that type. In order to write these identities in a simple and symmetric form, we shall use different normalizations, specific for each identity. For instance, we note that the bilinear forms B& and Bi, are proportional to the appropriate limits of the

intertwining operator G,O normalized according to (4.31) for x + &, and x + xi,%, respectively. This follows from (5.44) and the proportionality of the intertwining operators G,f and G,O (for non-exceptional x).

PROPOSITION 5.9. The foIlowing identities hold:

d”GP,,_ d” = ac.G$- , (5.59)

272

where

V. K. DOBREV and V. B. PETKOVA

a’PB;$ a’p = b Go,,_ ‘9” XIP” ’

a’P@ a’P = b

2;:;:

G’+ tvn Im,

d”G$; am = ctvn G;,+, ,

ang;y+, aa = CIV” G& , Y

Gs:‘R, s limL6G$,,a, 810 dd d

a~, = (21+Y-n+1)!(n+~-1)!(I+v+1-!-(I+-1-4)! Y (21-n+l)!(n-l)!(l+l+lI+l--nl)! ’

b (- 1y’-“+“-‘f1(21-n+2)!y21+v+ l)!

lvn = n(l+ jl-n+ I/)!(/- I/-n+ ll)!(v-- l)! ’

Clm = - (- 1)21+“+‘+1’-“+‘ln!2(2z+y+ l)!

(v-l)!(l+E+II-n+l)l!(l+l--ll-n+ll)! *

Proof: Note that

GE,_ = (-‘) ‘~+l-~‘-yl+v- II-I- 1 -nl)!(l+v+ 1 + II+ 1 -nj)!

Y!(2lfVf l)! B;t, ,

G;:_ = G;;_. = (21-t-v+2)! G,_

Y. I Ivn 9 v

G z”,+ = v!

(21+2+v)! Bk

GP,., = Go (- l)‘-l’+‘-“y2l+v+ l)! Y!

%&I = (I+~-_II+l-nl)!(l+v+l+lz+l-nl~!G:”~

(5.60)

(5.61)

(5.62)

(5.63)

(5.64)

(5.65)

(5.66a)

(5.66b)

(5.66~)

(5.67a)

(5.67b)

(5.67~)

(5.67d)

We start with the proof of (5.59). Let f E C;,$,, then

(d’Gp,- O-)(x, ; ~1,211

xf(x2 ; w2, z&x, =

ELEMENTARY RJZPRJSENT ATIONS AND INTERTWINING OPERATORS 273

Consider

($VZJ(W~~Z&%- (x; WI, zi ; wz , .G)

= K&“(31 V)1(32V)Y(T1~)21-“+1(tZ~)“-L (-$-)“-I In f , (5.69)

where K, ~ (- l)‘+“(l+Y+ II+ 1 -nl)

v =

(2x)+- 1)!(2Z+yi- l)!

and we have used (4.14), (4.16a) and (5.43). The non-log terms in (5.43) do not contribute to (5.68) and (5.69) since they are pro-

portional to GI;, and by Proposition 5.4 d”G,;.d’” = 0. Accounting for the fact that (3i t) = 0 (i, j = 1,2) we must only apply the formula

(31V)“(32V)Y ; ( 1 v-1

In< = (-~)‘Y!(Y-l)! f ( 1

‘+1(5,xy([2xy (5.70)

to obtain (5.59). Further we prove (5.61) and (5.63) using G.&- and G,$- instead of B;; and G;*-

1Wll

and going to the limit 6 $0 after all differentiations are done. Then we see that (5.60), (5.62) and (5.64) are corrolaries of (5.59), (5.61) and (5.63) respectively. The necessary calculations are presented in Appendix B.

Acknowledgments

We want to express our gratitude to Professor I. T. Todorov for illuminating dis- cussions and many useful remarks on the first draft of the manuscript.

Appendix A. Proof of formula (4.5)

Noting that

we shall evaluate the integral

I(x1J;W,,z,;i?2,z2) = lim&, (A.2) 810

& = s

dxz(Tijlx12~~~&W2)m’(Z2&~~~X12Zl)m’ (_L)“c+l=-d (-L)‘-=+l*.

Replacing x1, = xX3 -x23 in the first bracket and x23 = x13-xl2 in the secondand using x+x = x21 (cf. (1.6)) we obtain


x (Z2EZ1)~ri( -2)%-k-I (&)“c-ll-d (-gC”’ dx2

64.3)

where the differential operators p3 , + Q act only inside the brackets. We have used (cf. [4]) the convolution integral in %h-dimensional space

for h = 2, a = 2+c-II-d, /I = 2-c+l,. Performing the differentiation in (A.3) we obtain

Id = z (;I) (72) (!) (3 22r+(27c)2(- 1)21,-k-j(~1&tS*)rnI-sX

x (Z2&ZJm’-s s!2r(c-l,)r(l--c-l-6)r(2-6+j+k-s) r(2-c+z,+k)~(2+c-I,+j-6)r(l-t6) x

2+8-d

(%zx13)(3A3))s

3i SiGiCjZ(, i =

1/2 1,2

where we have used (5.2d). Using that

z+s--6

(2n)‘(3 I 32)S@&) S-l-l

we obtain

where I = 22’q27r)46(x13)(- f - ml w1 EWJ (ZIEZJrn’ K

To evaluate K we use twice the formula [9]

“1 T(u +j)P(b +j)r(c) F(a9by ” ‘) = &I r(u)I@)IYc+i)j! l-yc)P(c-a-b)

= r(c-u)r(c-b)

(A-4)

(A.5)

(A-6)

(A.7)

(A.0


and

Thus we have

c * Qa+k) = T(a+mf 1) k!

kc0 am! * (A-9)

K= 1

(I: - c”) [(12 + 1y - c”] * (A. 10)

Noting that

(b.&J)(%; WI 3 4

= YwYho~(- l)m,+m, I(x,3; w 3

m,!m2!24+2[2 s 1s 1,

z *w,, Z2)f

( X3;r z dx

3 2 &?q 1

we end up with (4.5).

Appendix B. Proof of formulae (5.56), (5.60)-(5.64)

We start with the proof of (5.61). We reduce the evaluation of d’P&$P to calcu- lating

(B.1)

1 = _- ( 1 I+” n+(&-) (2b-n+2)!r(21-n+2-6) 2 (29V r(l+Y+r))2r(-Q

,(,,_:,,)X i

X I-‘(j- 6)P(21-n+3+~-j+ 6)

j!(Y- 1 -j)! X

2

( 1

2Z-n+3+v-j+d

X- X2

[‘@,E~2)(Zl&Z2)]j(i?1~Z2)Y-1-j(i?2~Zl)2~+Y+1-i

(we have changed B;$, with G,f;-).

For (B.1) and (B.2) we have used twice (A.8) and the formula

(B-2)

n!r(2fn+a) = ‘--‘)’ (n_k)!r(n+2+a_k) (~lE’2)k~~~Z2)n-kgg(ZZ), (B.3)


where g(zz) is a homogeneous function of z2 of degree c(. Taking into account that (cf. (4.30))

*im n+(x’-)r(j~ = L 0, .i# 0,

r( - fJ2 (_ I)'-w-11+1

610 (Z+[l+l-nl)!(l-IZ+l-nl)! ’ j= O9

we recover (5.61) using the definitions (5.45) and (5.53). The proof of (5.63) is similar to that of (5.61) and we omit it. Eqs. (5.60), (5.62) and (5.64) are direct consequences of (5.59), (5.61) and (5.63)

respectively, using the intertwining properties of the operators G, and property (5.50). Following the lines of the derivation of (5.61) we calculate (for (5.56a))

taking the limit 8 JO after the differentiations and using Eq. (A.8). The same procedure is applied for (5.56b).

REFERENCES

[l] Bargmann, V., and I. T. Todorov: J. Math. Phys. (to be published) (1977). [2] Bruhat, F.: Bull. Sot. Math. France 84 (1956), 97. [3] Casselman, W.: private communication (to be published). [4] Dobrev, V. K., G. Mack, V. B. Petkova, S. G. Petrova, and I. T. Todorov: Harmonic Analysis on

the n-Dimensional Lorentz Group and its Application to Conformal Quantum Field Theory, Springer Lecture Notes in Physics NO. 63, Springer-Verlag, Berlin 1977; see also: Elementary Represen- tations and Intertwining Operators for the Generalized Lorentz Group, IAS preprint, May 1975, which is the tist part of the above book.

[S] (a) Dobrev, V., V. Petkova, S. Petrova, and I. T. Todorov: On the Exceptional Integer Points in the Representation Space of the Pseudo-Orthogonal Group J0(2h+ 1, l), ICTP preprint IC/75/1, Trieste 1975. (b) Dobrev, V. K., V. B. Petkova, S. G. Petrova, and I. T. Todorov: Phys Rev. D13 (1976). 887.

[6] Gavrilik, A. M., and A. U. Klimyk: Analysis of the Representations of the Lorentz and Euclidean Groups of n-th Order, ITP preprint, ITP-75-18E, Kiev, 1975.

[7] Gel’fand, I. M., and M. A. Naimark: Zzv. AN SSSR, Math. 11 (1947), 411.

[S] Gel’fand, I. M., and G. E. Shilov: Generalized Functions, Vol. 1, Academic Press, N. Y., 1964. 191 Gradshteyn, I. S., and I. M. Ryzhik: Tables of Integrals, Series and Products, Academic Press, N.Y.

1965. [lo] Grensing, G.: J. Math. Phys. 16 (1975), 312. [ll] Helgason, S.: Differential Geometry and Symmetric Spaces, Academic Press, N.Y. 1962. [12] Hirai, T.: Proc. Japan. Acad. 38 (1962), 83 and 258. [13] Iwasawa, K.: Ann. Math. 50 (1949), 507. [14] Knapp, A. W., and E. M. Stein: Ann. Math. 93 (1971), 489. [15] Ltischer, M., and G. Mack: Comm. Math. Phys. 41 (1975), 203. [16] Thieleker, E.: Trans. Amer. Math. Sot. 199 (1974), 327. [17] Todorov, I. T., and R. P. Zaikov: J, Math. Phys. 10 (1969), 2014. 1181 Wallach, N.: Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, Inc., N.Y., 1973.


[19] Warner, G.: Harmonic AnaIysis on Semi-Simple Lie Groups, I and 11, Springer-Verlag, Berlin, 1972.

[20] Zaikov, R. P.: Bulg. J. Phys. 2 (1973, 89; see also G. M. Sotkov and R. P. Zaikov: Conformal In- variant Two- and Three-Point Functions for Fields with Arbitrary Spin, JINR preprint, E2-10250,

Dubna, 1976.

[21] Zhelobenko, D. P.: Compact Lie Groups and Their Representations, AMS, Providence, Rhode Island,

1973.

[U] Zhelobenko, D. P. Zm. AN SSSR, Math., 40 (1976), 1055.

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