An explicit integral representation of Whittaker functions for the ...

519

J. Math. Kyoto U n iv . (JMKYAZ)

37 - 3 (1997) 519-530

An explicit integral representation of Whittaker functions for the representations of the discrete series —the case of SU(2 ,2) — By Takahiro

1.

HAYATA 1 and

T akayuki

ODA

Introduction

In a se n se , th is p a p e r is a supplem ent to th e p a p e r [1 2 ] o f Yamashita. A lso it is an analogue of a re su lt in [6]. W e c o n sid e r t h e L ie g ro u p G = S U ( 2 ,2 ) . T h e la r g e discrete series representation o f G h a s a W hittaker m odel w ith re sp ec t t o a nondegenerate c h a ra c te r o f th e m a x im a l u n ip o te n t subgroup N o f G . U sin g t h e Schmid o p e ra to r, Y a m a s h ita [1 2 ] e x p lic itly c o m p u te d t h e differential equations satisfied by the minimal K type vectors in the W hittaker m odel of the discrete series representations. T he purpose of this paper is to push this com putation one step further to o b t a in a n e x p lic it in te g r a l r e p r e s e n ta tio n o f t h e W hittaker functions representing these vectors belonging to th e m in im a l K -ty p e (Theorem s 4.4 a n d 4 . 5 ) . T h e r e i s a g e n e ra l in te g ra l re p re se n ta tio n d u e t o Jacquet for W h itta k e r fu n c tio n s. B u t th is re p re se n ta tio n is so m e tim e s in tra c ta b le for h ig h e r ra n k g ro u p s. W e h o p e o u r form ula is u se fu l for the investigation of L factors of autom orphic representations o f t h e d is c re te s e rie s a t t h e real places. T h e c o n te n t o f t h is p a p e r is a s follow s: I n § 2 , w e b rie fly re v ie w the s tr u c tu r e o f S U ( 2 ,2 ) , th e d is c re te s e rie s a n d t h e representations o f th e maximal compact subgroup K . B asic notations and definitions a re found in [1 ,2 ] , w h ic h w e fo llo w . In § 3 , w e review th e re su lts o f Y am ashita and of K o s ta n t o n th e d im e n s io n o f th e s p a c e o f W h itta k e r v e c to rs . W e a ls o calculate th e r a d ia l A -p a rt o f t h e S c h m id o p e ra to r e x p lic itly . I n § 4 , we describe the holonomic system of differential equations of W hittaker functions w h ic h h a s a p p e a r e d i n [ 1 2 ] . F u rth e rm o re , w e s h o w t h a t t h e integral representation o f th e a n aly tic W h itta k er fu n ctio n c an b e o b ta in ed fro m the differential equations under th e parity condition of a nondegenerate character of N. B e c a u se w e b e lo n g to th e c u ltu re o f a u to m o rp h ic fo rm s, the m axim al -

-

Communicated by Prof. T. Hirai, September 27,1996

JSPS Research Fellow

520

Takahiro Hayata and Takayuki Oda

compact subgroup K appears in the right hand of G and the unipotent group N in the left s id e . T h is is different from [12]. The authors thank Hiroshi Y am ashita for constant communications.

2.

The group SU(2,2) and its discrete series 2 .1 . Structure of Lie group and Lie algebra.

Let G be the special

unitary group SU(2,2) realized as, G = { g SL 4 (C) Ig*12,2 g

=

I,,,= diag (1, 1,

12,2}

—

1,

,

where g*= to denotes the adjoint of a matrix g. W e fix some notation for this group and its discrete series representations, used throughout this paper. L et U (4 ) b e th e unitary group o f degree 4 in S L 4 (C ). T a k e a maximal compact subgroup K = G n u (4) = s (u ( 2 ) x ( 2 ) . W e denote by g, f the Lie -

algebra of G, K, respectively. Let 0 ( x ) = tx be a Cartan involution and g-=t+p is the C artan decomposition of g. W e set, a = + RH2 with, 1/1 = X 2 3 ± X 3 2 , H2 = X14 ± X41, w here the Xii's are elementary matrices given by, X i•=

(

5

5

1P id ) W , 4 5 4

with Kronecker's delta öi p.

T hen a is a maximally R-split abelian subalgebra of g contained in p. the restricted root system zl =d (g, a) is expressed as,

Then

A= A (g, a) = {±21± 22, ± 22i, ±222). -

-

w h e r e 2 ; i s t h e d u a l of H . W e c h o o s e a p o s it iv e s y s te m d + a n d a fundamental system —fund A of A as follows:

A+={2 ± 22, 221, 222), 1

- -

22, 222).

fund —

W e also denote the corresponding nilpotent subalgebra by n=EBEA,- gB. Here gs is th e root subspace o f g corresponding to [3E d + . T h e n one obtains an Iwasawa decomposition of g and G:

g = n + a + t,

G = NAK ,

with A =exp a, N = e x p r t. Now let Ei A / —

1

X13 +

1 X31, E2 — H24 A/

1 -K24 +1/ 1 X42,

E 3

1/2

(X 1 2 X 2 1 X 1 4 + X 2 3 ± X 3 2 X 4 1 X 3 4 + X 4 3 )

E4

1/

1 /2

— -

(X

12

-

1X -

2 i

X 14 X 23 ±X 32 ±X 41 X 34 X 43) ,

E 5 1 / 2 (X 1 2 X 2 1 ±X 1 4 ± X 2 3 + X 3 2 ±X 4 1 + X 3 4 X 4 3 ) , E6

— 1/

1 /2

(X 12

X 2 1 + X 1 4 X 2 3 ±X 3 2 X 4 1 ±X 3 4 + X 4 3 ) ,

521

W hittaker functions for the representations of discrete series where H o =

—

1 (X ii — X ) fo r 1

Then it is easily seen that

g22; = R E; (j =1,2) , gÂ1l-2—RE3+RE4, gAi_22=RE5 - FRE6. L e t u s now parametrize 2 . 2 . Parametrization of discrete series. subalgebra t defined a com pact C artan T ake of S U ( 2 ,2 ) . th e discrete series by t = R,/ — 1 hi ± Ri/ — 1 h 2 -ER,/ — 1/2,2,

with h X 1

11

X22, h

2

= X33 X44

and let tc b e its complexification. Then th e absolute root system , of type A 3, is expressed as,

A = A (g c, tc) = { [± 2 , 0 ; 0], [0 , ± 2 ; 0 ], [± 1 , ± 1 ; ± 2 ]). where 13= [r, s; u] means r= [3 (h IF =

s =13 (h 2 ) and u = /3(/2,2) .

Let

1[2, 0; 0], [0 ,2 ; 0 ], [± 1 , ± 1 ; 2 ]). - +21 " = [2 , 0 ; 0 ] , [0 , 2 ; 0 ])

W e w rite the set of com pact positive roots by w e fix it h e re a fte r. T he W eyl group where,

c

IN=1-47 (gc, tc)

and

is generated by s , s , 5 i

2

3

= [— r, s; u],

siEr, s;

s2 [r, s; it] -= [(r — s +u ) / 2 , ( — r+s+u) /2; r+s ], S3

— [r, — s; u ].

[r, s;

of degree 4 by the map:

W e identify 1/1-7 and the symmetric group

j + 1 ) . T h e c o m p a ct W e y l g r o u p is g iv e n b y W = ( s , 5 ) , also s identified canonically with th e subgroup S2 X S2. '4 containing There are exactly six positive system s Ai', L1ji ..... 21 i

c

.

-

defined by W/ =

1

Aï=tv i d , where the elements w +

1, w

1

=

S3, Win = S2S3, Wry = S2S3,

3

W a re given as,

j

Wv

= 525351, WVI S 2 5 1

5 5

3 2.

W e denote by Lt ; the noncompact positive roots in By definition, the space of the Harish-Chandra param eters E , is given by, .

E c = {AE MA is A -regular, K-analytically integral and L1 -dominant}. Put

Ev= {A E

I

dominant).

W e also p u t p G j = 2 E B E3 1,3 (resp. p K =2 E s . 3:1(3) , th e half sum of positive r o o ts ( r e s p . th e half sum of com pact positive roots.) T hen th e space E c c t t a r e d iv id e d in to s ix p a r t s : E = U isfsvE.r. W e n o te t h a t E (re sp . E v i) - 1

- 1

-

c

i

522

Takahiro Hayata a n d Takayuki Oda

corresponds t o t h e holomorphic (resp. anti holomorphic) discrete series. For A E UisfsviEj, we denote the corresponding discrete series by rn and call th is A the Harish Chandra param eter of r n . A s determ ined i n [H, §10.4] the Gelfand-Kirillov dimensions of the discrete series representations n"A are given as follows: -

-

4

e

GK dim (zit) = { 6

(A

-

5

UENO,

.2-71

U

),

E Ent U Ely) .

Therefore the representations belonging to Ell U Ev is a large representation in the sense of Vogan [8, T h . 6.2, f)], hence has an algebraic W hittaker model. (Harish-Chandra param eters of discrete series of SU(2,2) are described as in Figure 1.)

FIGURE 1. Harish-Chandra param eters in tt

2 .3 . Representations of the maximal compact subgroup. d , d EZ

and d EZ. For d = [d1, d2; d E t , define 1

2

Let

3

k 0

rd

(hi )A dik2

=

'rd (4) r d (e- ) f e k 2 =

rd

( 2k

(d ) r ( [ 2 ,2 ) fk i k2

k by the following rule:

dj) t k ik d

(j = 1, 2) ,

j)— )_1_,,S11,,.2+,•2/9

f(tedi)k2 =

j

E

,

k

fe-

1,,k2-52,,

(d ) d 3f k i k2.

(1)

523

W hittaker functions for the representations of discrete series Here, Va = {f(kdi1210

j} C is the standard b a sis (se e [1 , § 3 ]) and

hl ,

7

2

, e 1+ = X 12 , e2+

= X 34,

ei_ = i ei+

Then according to [1, Prop. 3.1], I? is exhausted by

are the generators of tc.

("cd•Vd)ld = [di, d2; d3], d1±d2±d3 is even}.

T h e a d jo in t representation Ad = Ad pc o f K o n p c is decom posed into a direct sum of two irreducible subrepresentations: pc= p+ +p_, where, P+=CX13±CX14 FCX23+CX24,

P--=`P+.

-

In fact, Ad ± = Adlp+ is isomorphic to r11,1, ±21, respectively. F o r late r use, we fix the K-isomorphisms (±: p,--*v [ ,,,, + 2 , as follows: C

(

-(11) . 11)

f io ,

(X 2 3 , X 1 3 , X 2 4 , X 1 4 )

(AT4 1 , X 3 1 , X 4 2 , X 3 2 )

1 -4

r(11) (fÔ 101),.110

AP ,

—

)

—fan

( Ll, 1, Prop. 3.10]). [

The irreducible decomposition of tc module V a OPC is given a s follows: -

V a

pc = Va o p + V,-

V d op±

0

V E r+ E I,S i-E 2 . U ± 2 ]•

£1,62E I ± 1)

The projectors ®P+— )1/

P

€1 62)

r.3 '

:

Vd e p _

—

,

u+21,

v[r+Ei,s+E2, u-2],

are explicitly given by [1, Lemma 3.12].

3 . Radial A part of the differential operator Di, ,d -

Let ri be a unitary character of N = e x p n. F o r F (g ) cC7,,„(AAG/K), let

n,aF (9) =ER x ,F(g) ®Xk, {Xk}: orthonormal basis of p be the S c h m id o p e ra to r. Put

) ( f) ;,/,d= ° 7, 7,d w ith the projectors Pd

PY : V

V .

=

)

fie 3 /

L e t r A b e t h e d is c re te s e rie s re p re s e n ta tio n o f G w h e r e A E ,E j i s a Harish C handra p a ra m e te r. T h e n d = [r , s ; u ] = A ± PG ,, — 2pK, called the B lattner param eter o f r n , is th e highest w eight of the m inim al K type of ityi. For a W hittaker vector Hom(g,K) (7r1, (A G )) , define (N\G/K) by the following: -

-

(v*, O

(g)) =0 (1) * ) (g )

T h e algebraic W hittaker function

(1)* E V g

E G).

fo r th e representation o f th e discrete

524

Takahiro Hayata and Takayuki Oda

se rie s is u n iq u e ly d e te rm in e d a t th e v a lu e a EA =exp ( a ) b y v ir t u e of the Iwasawa decomposition.

Theorem 3.1 ( [11] ) . Let AE E j . H om o

Then,

C77(N \G )) 7-=-ker (O M ,

w here d is the Blattner param eter of rn. According to the re su lt of Kostant [4, T h. 6.8.1], the dimension formula can be obtained:

Theorem 3.2. dimc (Homwo (7rt C (N \ G ))) =

(j=

4 0

,

V),

otherwise.

P r o o f. It is su ffic ie n t to conside th e c a se o f A E Ell U Ev because of the largeness of 7rA. In order to specify the dimension, consider F # = exp (ac) = (diag (1,

— 1 ,1 , —

1) , a = i/- 1 ( 1 2

12) )

Define another representation 7r " b y 7r( " ) (g) i t (a g a ) . Then, we see from (

r[r,s u l

( a .Xa) 1

)

l

r[s,r, -id (X)

-= T w v w [ r , „ ]

(X)

th a t th e Harish - Chandra param eter o f 7rna) b e lo n g s to E v . U sin g th e same argument in the proof o f [2, Th. 4.2], we obtain the theorem. It is also know n that the rapidly-decreasing solution is unique if exists. For a = (ai, a2)=exp( (loga i )111 - 1- (loga2)H2) E A , le t 0 a5a/aa; and

L1= 1/2a, /1= 1/2

(a2± 722a), -0= (ai/a2) 2 , J'='(ai/a2) 2 .

W e also put, for a character )7 of N, 772= 1/ — 17 (E2)

(E5) +1/ — 1 7) (Eo),

(E5) —A/-172 (E6),

no=

T h e follow ing lem m a is obtained by d ire c t c a lc u la tio n sim ila r to [1, Lemma 6.5].

525

Whittaker functions for the representations of discrete series

Lemma 3 .3 .

For

F (g) = E lock a (g).fi jEC7,2- (N\G/K) , d = [r, s; u] , we d

d

have,

(k—r) (s — /).0 (k r) (1+1) (L1 (2k 21-1 r-F3s —

—

P ' ' V +c (

-

--

-

u+6) /4)

—

-

(k + l) (s

-

1) (L i (2k 21 r+s (k+1) (1+1) J23' —

(k

—

(s

—

—

u 12) /4)

—

-

-

1) s3

k) (1+ 1 ) ( I I + (2k - 21 — r+ s — u +2)/4) (k + 1 ) (l — s) (L + (2k — 21 3r s u 6) /4) (k+1) (l+1),23' (r

—

1

4.

—

—

—

—

Integral representation of discrete series Whittaker function

I n t h e follow ing, 77 i s a s s u m e d t o b e nondegenerate. A ccording to Theorem 3.2, w e treat the cases of J= H a n d V . M o re o v er, sin c e w e have ( E 2 ) , th e W hittaker functions o f 7 r4 (A E E v) can be obtained E ) , b y taking th e p a ra m e te rs (s, r, — u, — 1) ) in from TE (A ' = c p la c e o f (r, s , u , 17 ). T h u s w e o n ly tre a t th e c a s e o f A c Ell. Then, the Blattner param eter of rrn is cl = A + [0, 0; 2].

r) ( " ) (E 2 ) = A

—

i

,

ll

2

2

Lemma 4.1.

The projector P d( 1 1 ) decomposes into four projectors as follows:

= p -,-) (

Proof.

p(-,-) e p(-,+) p(-F,-)

.

(2)

W e find that

di+ir= 1[1, —1; 2], [1, 1; ±2], [-1, 1; 2], [0 , 2; 0] , [ 2 ,0 ; 0]

and that

A

11

=

{ [ i ,

—1; 2], [ 1 ,1 ; ±2], H 1, 1; 2]).

T hus the lemma follows. A ccording to Theorem 3.1, W hittaker functions a re characterized by the differential equations derived by the com position of the Schm id operator and projectors w hich appears in the decomposition of P P . Let (a) = E rack,, (a)fka. For notation, w e w rite ( 0) k a = ck ,i. Then, ck,i's )

satisfy the following system which is equivalent to 0;71,140=0:

Takahiro Hayata and Takayuki Oda

526 ( r'+.

(C2 :

kJ

1

(b i :

2

=

) kl

-

-

(P - - - ) . 17 0) k,i

0,

) kl

(P -

-

)

.

—

1)

°

0) k,1-1-1

0) k,1 + ( 1 + 1 )

°

0) k,i+i

(s

—

0, o,

( k +1 ) ( P +- ) . v ) k +i,i=o ,

v

(p( - ) .V J4 k,l — (r — k ) (P ± ' ) °

(P) ki-id= 0,

-

where we write V =V d for simplicity. Given a d = [r, s; 14], we put, ,

bo=b o ( d ) = ( r ± s + u ) / 2 , b l = b i ( d ) = ( b2=b 2 (d ) = (r s + u )/ 2 , b 3 = b 3 (d ) = ( —

—

—

r+ s+ u )/2 , r — s +u) /2.

(3

)

B y L em m a 3.3 as w ell as [1, L em m a 6.5] , the coefficients {ek,/} satisfy the following concrete equations: (k r) (1+1) (L 1+ (b3— s— k+1 - 3) /2)c k,14-1 —

1(L i+ (62 — k+ 1 - 1) / 2) ck-FLI + (k+1) (1+1) sYc k+1,1+1+ (k — r) (s — 1) s23ck,i = 0,

(Ci : 1) kJ F

+ (k + 1 ) (s

—

(62—k+1+1) /2)C k,I+1

(r—k) (C i. 1 ) k l

(C 2 :

2

) kl

1) kl (C i:

2

) kl

+ (k+ 1) C k+ 1,I+ 1+ (1? + 1) (.3 1) C k+ 1,1

(k+1)

(1+1) (LI (s

(60+r-1?-1+1)12)(4+1d— (k

1) (L1

—

r).Ock,I=0,

(s —1) s23ck ,i + (k-1- 1) (s

(b 2 — k+1+1) /2)c

—

—

=

(b i +k+1+3) / 2)c ki-1,1

—

-

1)ck-H.,1=0,

(1 +1) sYc k+1,1+1=0.

Each equation (C t: q ) is derived from (C t: q). Next, define hk,/ =ktl! (r — k)!(s

—

1)! e'dnz/zaTb1-k+1-2aib2+k-Ick,i.

(4)

Then the system satisfied by hkd's is given as follows: ( a 1 + 2 b 3 - 2 ) h k ,i + 1 — ( a d a 2 ) 2 h k + 1 ,1 + 1 + ( a i / a 2 ) 2 h k ,i

(Hi: 1) kl

(H i: 1)1.a

—

2172A + 2b2— 2k + 21) (ada2) 2 hk+1,1=0

(al/a2) hk+1,1+1 1 2 (1± 1 ) (ai/a2) hki-1,/ — 2

2

- -

s

(,) (I12 : 1) k0

(Hi: 2)

k,

a 2 h k ,O + (ai/a2)

2

hk+1,0 —

(al-2 r-f -2 k +2 )h k +1 ,1 +

,

0 (O

k

—1) ,

527

Whittaker functions for the representations of discrete series

7

(11 : 1 ) k l

( ai/ a2 ) 2 h k ,,+

2 (r— k ) (ai/a2) hk-F1d —0 2

(0

(ai/a2) h , = 0

(H i: 1) ri

2

s

s-1 ). (Hi: 2) k l (0 ( ai-21)hk+1,, titk+1,,±1—o (HP q) k i is a d ir e c t consequence o f (CI,: q) Ia. C oncluding, w e obtain the differential equations satisfied by the W hittaker functions.

Proposition 4.2. h , satisfies the following: k

Let

= Ek ,ick atk j. d

Define h c i b y (4).

Then

i

2n2 ai 2b 3 — 2) (ai/a2) k k +id = 0 ( 0 - k r 1, 0 a2hk,,,i+ (a i/a2) 2 (a 1 + 2 )h k + i,,-0 (a1 2 r+ 2 k + 2 ) h k i - 1 , i + h k , i = 0 ( 0 k r - 1) , (ô 21) h k+1,1 1) (0 - 1. ,3 a 2 h k , O + (a da2) hk+1,0= 0

(a1 I 2b3 -

-

—

2) hk,i+1 —(2 a rk az

2

—

-

(n) (iii)

(iv)

(V) (vi)

-

1

-

2

a2hr,i+i

(ai/a2) hr,, = 0 2

E q u a tio n (i) can be obtained from E quations (H P: 1) k l, (H t: 1 ) k l a n d (115 : 1) E quation (ii) is fro m (Hi: 2) k ! a n d (H i: 1) The others are clear. Proof.

E quations (iii) and (iv) in P roposition 4.2 tell us that hr,o determines the system {hk,i}. By the proposition above, w e see that from (iii) a n d (v), (5!,32—

(ada2) ) hr,o = 0,

(5)

2

and from (i) , (iii) a n d (iv) , (ai ±2b 3 — 2) aihr,o + (2 a , + a 2 -2 )7 2 a Z + 2 0 -2 ) (ai/a2) 7ohr,o= O. 2

Operating a t o (6), we obtain the following: 2

(6)

528

Takahiro Hayata and Takayuki Oda

Corollary 4.3. (a1a2—

((a, +a2)

(2b3

2

no(cii/a2)

2

) hr,0 = 0,

2) (0 1 + a 2 )

-

(7)

272 aZa ) h , = O.

—

2

2

E quation (6 ) can be recovered from ( 8 ) by operating check that this system becomes holonomic of rank 4. To get an integral representation, first we consider -

a,.

W e can also

u du

q5(u)exp(71°`d

*/(a i a2 ) =

(8)

r o

(9

4aZ ) u

)

f o r OcC - (R , ). T h e n "W form ally satisfies t h e differential equation (7). Suppose s a t is f y ( 8 ) , t h e n 0 s h o u ld b e th e s o lu tio n o f th e following differential equation, 0

42

d 20

4

+46 du d u

77 u0= 0.

3

Putting v=.,/Fs and 0(u) =vd

b3+1/20

v

,

( ) w e see that E quation (10) becomes

\ + 1/4—(b 3 — 1) 2 )

± ( 1

2

0.

2

4

dV2

(10)

2

2

V

If 72 >0 (i.e. Tm (12 (E2))