Hecke operators and Q-groups associated to self-adjoint homogeneous cones

Let G be a reductive algebraic group associated to a self-adjoint homogeneous cone defined over Q, and let Gamma subset of G be an appropriate neat arithmetic subgroup. We present two algorithms to compute the action of the Hecke operators on H-i (Gamma; Z) for all i. This simultaneously generalizes the modular symbol algorithm of Ash-Rudolph (Invent. Math. 55 (1979) 241) to a larger class of groups, and proposes techniques to compute the Hecke-module structure of previously inaccessible cohomology groups. (C) 2003 Elsevier Science (USA). All rights reserved.