Locally algebraic representations and integral structures on the cohomology of arithmetic groups

Abstract

This paper introduces the notion of locally algebraic representations and corresponding sheaves in the context of the cohomology of arithmetic groups. These representations are of relevance for the study of integral structures and special values of cohomological automorphic representations, as well as corresponding period relations. We introduce and investigate related concepts such as locally algebraic $({\mathfrak g},K)$-modules and cohomological types of automorphic representations. Applying the recently developed theory of tdos and twisted $\mathcal D$-modules over schemes by Hayashi and the author, we establish the existence of canonical global $1/N$-integral structures on spaces of automorphic cusp forms. As an application, we define canonical periods attached to regular algebraic automorphic representations, potentially related to the action of Venkatesh's derived Hecke algebra on cuspidal cohomology.