The Dimension of Spaces of Automorphic Forms

Abstract

1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in iarish-Chandra's investigations of the representations of semi-simple groups. These integrals have been evaluated. In this paper, after clarifying the relation between the two types of integrals, we go on to complete the evaluation of the integrals appearing in the trace formula. Before the formula for the dimension that results is described let us review iarish-Chandra's construction of bounded symmetric domains and introduce the automorphic forms to be considered. If (7 is the connected component of the identity in the group of pseudoconformal mappings of a bounded symmetric domain then (7 has a trivial centre and a maximal compact subgroup of any simple component has nondiscrete centre. Conversely if (7 is a connected semi-simple group with these two properties then (7 is the connected component of the identity in the group of pseudo-conformal mappings of a bounded symmetric domain [2(d)]. Let g be the Lie algebra of (7 and gc its complexification. Let GC be the simplyconnected complex Lie group with Lie algebra gc; replace G by the connected subgroup G of GC with Lie algebra g. Let K be a maximal compact sub