INTRODUCTION
xv
instead o f symplecti c o r orthogona l groups : (i ) Unitar y group s ca n b e spli t o r
nonsplit, an d therefor e the y hav e th e characteristic s o f general reductiv e algebrai c
groups, whil e symplecti c group s ar e spli t an d specia l i n tha t sense , (ii ) Besides ,
the symplecti c cas e ha s bee n treate d i n detai l i n a serie s o f recen t paper s b y th e
author, an d s o i t seem s desirabl e t o presen t th e "nonspli t aspect " o f th e theory .
(iii) Holomorphi c form s ca n be considered als o on orthogonal group s of a restricte d
type. A unifor m treatmen t o f al l thes e classica l group s i s no t impossible , bu t a t
some poin t th e tas k wil l becom e cumbersome . Fo r example , i t require s a carefu l
analysis of certain Eisenstei n serie s in the orthogonal case , which would hav e mad e
the book much longer. Fo r this reason, we discuss only the arithmetic aspec t o f the
orthogonal case , but no t it s analyti c aspect .
Now the function Z i s closely connected with an Eisenstein series of the followin g
type. Give n G^ a s abov e wit h p = f (pp, w e consider G^ an d G v wit h
where m i s a positiv e integer . Le t P b e th e paraboli c subgrou p o f G^ consistin g
of al l th e matrice s whos e lowe r lef t m x ( m + n)-bloc k i s 0 . Give n a congruenc e
subgroup A o f G^ , a holomorphi c cus p for m / wit h respec t t o a congruenc e sub -
group o f G^ define d a s a functio n o n 3^ , an d a Heck e characte r \ o f K, w e ca n
define an infinite serie s E(z, s; /, \) r ( zi s ) 3 ^ x C unde r a natural consistenc y
condition o n 4 , / , an d \. I n the simples t cas e i n which F = Q , i t ca n b e give n i n
the for m
E(z, s; /, x ) = X ) sgn(A 0(c*)) V(Ao(a)Z)£(*, s; f)\\ ka,
R = P\G*, 6(z, s; /) = f(p(z)) [Sn
(z)/6v{p(z))]'-"/2
.
Here Xo(a) i s the determinant o f the lowe r right m x m-block o f a , p i s a certai n
projection ma p o f 3 ^ int o 3^ , 8^ i s th e functio n o n 3 ^ suc h tha t 8^{^w) =
\j1{w)\~28ip{w)
fo r ever y 7 G G^ wit h th e standar d scala r facto r o f automorph y
j1{w)^ 8^ i s similarl y define d o n 3^ , k i s th e weigh t o f / , an d (p||^a)(z ) =
j1(z)~kg(az)
fo r a function g o n 3^-
In orde r t o stud y th e analyti c natur e o f thi s series , w e conside r anothe r grou p
Gu wit h u = diag[^ , —/?] . Now i t ca n easil y b e see n tha t G" i s isomorphi c t o
G(rym+n), an d w e can defin e a n Eisenstei n serie s £ o n G(rj rn^-n) wit h respec t t o it s
standard paraboli c subgroup. Sinc e G^ x G^ ca n be embedded i n G" i n an obviou s
fashion, w e obtain a function H(z, w; s) o f (z , w; s) G 3^ x 3^ x C b y pulling bac k
a suitabl e transfor m £' o f £ t o 3 ^ x 3^ - The n w e prov e a formul a whic h i n th e
simplest cas e can b e writte n
(7) / H(z, w; s)f(w)8
ip(w)kdw
= c(s)Z(s,
X
)E{z, s; /, *) ,
where c i s a product o f explicitly give n gamm a factors .
To defin e E(z, s; /, x) w e assume d m 0 . However , w e ca n mak e H an d th e
integral meaningfu l eve n whe n m = 0 by takin g ip = (p. Then w e obtai n
(8) / H(z, w; s)f(w)8^(w) kdw = c'(s)Z(s, *)/(*) ,
1/,:
p 0
0 7]
and7 7
= r]m
=
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