Coherent cohomology of Shimura varieties, motivic cohomology, and archimedean $L$-packets

We formulate an analogue of the archimedean motivic action conjecture of Prasanna–Venkatesh for irregular cohomological automorphic forms on Shimura varieties, which appear on multiple degrees of coherent cohomology of Shimura varieties. Such multiple appearances are due to many infinity types in a single $L$-packet with equal minimal $K$-types. Accordingly, we formulate the conjecture comparing periods of forms of different automorphic representations. We provide evidences for the conjecture by showing its compatibility with existing conjectures on periods of automorphic forms. The conjectures suggest the existence of certain operations which move between different infinity types in an $L$-packet.