DERIVED HECKE ALGEBRA AND COHOMOLOGY OF ARITHMETIC GROUPS

Abstract

We describe a graded extension of the usual Hecke algebra: it acts in a graded fashion on the cohomology of an arithmetic group Γ. Under favorable conditions, the cohomology is freely generated in a single degree over this graded Hecke algebra. From this construction we extract an action of certain p-adic Galois cohomology groups on H*pΓ,Qpq, and formulate the central conjecture: the motivic Q-lattice inside these Galois cohomology groups preserves H*pΓ,Qq.