Derived Hecke algebra and cohomology of arithmetic groups

TL;DR

This paper introduces a graded extension of the Hecke algebra that acts on the cohomology of arithmetic groups, revealing new algebraic structures and conjectures related to Galois cohomology and motivic lattices.

Contribution

It constructs a graded Hecke algebra acting on arithmetic group cohomology and links it to Galois cohomology, proposing a new conjecture about motivic lattices.

Findings

Cohomology is freely generated over the graded Hecke algebra in certain cases.

An action of $p$-adic Galois cohomology groups on cohomology is established.

Formulation of a conjecture relating motivic lattices to Galois cohomology.

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 $H^*(\Gamma,\mathbb{Q})$.