On the Selberg trace formula in the case of compact quotient

Introduction.Let G be a connected unimodular Lie group.Let T be a discrete subgroup of G so that T\G is compact.We fix a Haar measure, dg, on G. Then dg induces a G-invariant measure on T\G.We can then form a unitary representation (rrr, L 2 (F\G)) where (rrr(g)f)(x)= : f(xg) for feL 2 (T\G), xeT\G, geG.If eC7(G) (the space of all C 00 compactly supported complex valued functions on G) we can formIt is a standard fact (see §2) that 7r r (4>) is of trace class.In particular, 7Tr(4>) is completely continuous for 4>eC7(G).This implies that L 2 (r\G) decomposes into an orthogonal direct sum of irreducible invariant subspaces, {HjK°=i and for each i there are only a finite number of k so that H, is equivalent with H k as a representation of G (cf. Gelfand, Graev, Pyateckiï-Shapiro [9]).Let G denote the set of equivalence classes of irreducible representations of G. Then we have observed that 7Tr= ^Nr( eC7(G), / 7r(c/)) = jG</)(g) / n'(g) dg is a trace class operator on H.If coeG is of trace class, then set 0 w (</))=tr 7r(c/>) for (IT, H) G co.The above observations imply that if co G G and N r ( G C7(G), then tr7Tr(</>)= I"Nr(a>)a.to>). weGThe numbers N r (co) have been the subject of a great deal of investigation in the last few years.In this article we will give a short survey of various techniques that have been used to study these integers.We will concentrate our attention on semisimple Lie groups, G.We will also, for most of the article, look at the easiest groups T. These groups have no elements of finite order other than the identity.Without this assumption many (interesting)