INTRODUCTION
The object of this paper is to compute the dimension of certain spaces
of automorphic forms on the complex domain D = {(z,u) e ( C 2Imzu 0},
which is of interest in the theory of automorphic forms as the simplest ex
ample of a bounded symmetric domain which is not a "tube domain". Hopefully,
the information obtained on the dimension of these spaces will facilitate the
further study of automorphic forms on such domains, which differ from forms
on the domains so far studied in that they have Fourier coefficients which are
not constants, but are theta functions.
The main result which we derive is the following dimension formula, ob
tained by means of Selberg's trace formula:
j * * rr%\ (3m1) (3m2) 1
r
» . ,.3nu f ^3 m ,
n
d l m
V
r ) =
3x64  TT^'QIil/Q^
+ 3 (

1 } + 1 ]
+ 3*T28 [ 1 0 t r Q [ i ] / Q { ( 3 m  1 ) i 3 m ~ 1 " (3m2)i3m"2} + 17(l) 3 m (6m3)]
+
JL
+
±tT (L^
+
J_
t r
fC3"1
,
+
l^r f n3«H
128 32 t r Q[i]/Q l 'li' 16trQ[5]/Q j T ^ ' 36CrQ[P]/QlP J
+ T2 t r Q[i]/Q ( i m ) t r Q [ p ] / Q ( ^
1
f
.3m+l) ,p ,
" 12trQ[i]/QU trQ[p]/CT 1PJ
f
T N
3m 3m1 . .
12 trQ[p]/Ql 1P
J
9 *
Here r is a certain discrete group of analytic automorphisms of the domain
D; A (r) is the space of "cusp forms for r of weight m" (seebelow for
the definition); p and c denote primitive cube and 8th roots of unity;
Received by the editor November 16, 1973.
1