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 $\Gamma$. 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 $H^*(\Gamma, \mathbb{Q}_p)$, and formulate the central conjecture: the motivic $\mathbb{Q}$-lattice inside these Galois cohomology groups preserves $H^*(\Gamma,\mathbb{Q})$.