Invent. math. 97, 257-290 (1981) 1Springer-Verlag 1989 Summary: Invent. math. 97, 257-290 (1981) 1Springer-Verlag 1989 The L2-Lefschetznumbers of Hecke operators James Arthur* Department of Mathematics, University of Toronto, Toronto, Canada Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1. Hecke operators and L2-cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .52. The spectral decomposition of cohomology . . . . . . . . . . . . . . . . . . . . . . . . .6 3. Application of the trace formula 54. The functions 'PA,T). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5. Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. The main formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction Suppose that G is a semisimple Lie group and that F is a discrete subgroup of G. We assume that F is an arithmetic subgroup defined by congruence condi- tions, and for simplicity, suppose also that G is contained in a simply connected complex group. A fundamental problem is to decompose the regular representa- Collections: Mathematics