Dimension of Spaces of Automorphic Forms

I will first formulate a problem in the theory of group representations and show how to solve it; then I will discuss the relation of this problem to the theory of automorphic forms. Since there is no point in striving for maximum generality, I start with a connected semisimple group G with finite center. An irreducible unitary representaiton π of G on the Hilbert space H is said to be square-integrable if for one and hence, as one can show, every pair u and v of nonzero vectors in H the function (π(g)u, v) is square-integrable on G. It is said to be integrable if for one such pair (π(g)u, v) is integrable. Suppose Γ is a discrete subgroup ofG and Γ\G is compact. As was shown by Godement in an earlier lecture the representation π of the previous paragraph occurs a finite number of times, sayN(π), in the regular representation onL(Γ\G). The problem is first to find a closed formula forN(π). The method which I will now describe of obtaining such a formula is valid only when π is actually integrable.