  This paper introduces the notion of locally algebraic representations and
corresponding sheaves in the context of the cohomology of arithmetic groups.
These representations are particularly relevant for studying 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, 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.
