THE DIMENSION OF SPACES OF AUTOMORPHIC FORMS 3
CHAPTER I
SELBERGfS DIMENSION FORMULA
Selberg's method of finding the dimension of spaces of automorphic forms
on a bounded complex domain D, outlined in [8],consists essentially inthe
following. Suppose that D admits a transitive group of analytic automor-
phisms G, and let j(g,P) be the jacobian of the mapping g e G at P e D.
It turns out that the space of holomorphic functions on D which are square
integrable on D with respect to ordinary Lebesque measure is a Hilbert space
^ds and that the evaluation maps f - f (P) (P e D) are continuous linear
functionals on this Hilbert space (seeHelgason, [3]);hence, there exists a
unique function k(P,Q) on D x D (the Bergman kernel function) such that,
as a function of Q, k(P,Q) is in %f and
f(P) = /Dk(P,Q)f(Q)dQ (1)
for all f zjf. Expanding k(P,Q) in terms of an orthonormal basis of _%f-,
one finds that k(P,Q) = k(Q,P); and checking that j(g,P)d(gP,gQ)j(g,Q) also
satisfies (1)for all g e G, one see that
j(g,P)k(gP,gQ)TQTQ) = k(P,Q) (2)
for all g e G. In fact, the transitivity of G implies that (2)determines
k(P,Q) uniquely up to a constant. Moreover, da)(P) = k(P,P)dP is clearly
the G-invariant volume element on D.
If onenow considers the space Jzf* of holomorphic functions f on D
such that
/ ,f(P)|2JaJEi . .,
U k(P,P)m
one again gets a Hilbert space, which is non-empty if m _ 1 (it contains the
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