Cohomological degree-shifting operators on Shimura varieties

An automorphic form can appear in multiple degrees of the cohomology of arithmetic manifolds, and this happens mostly when the arithmetic manifolds are not algebraic. This phenomenon is a part of the “derived” structures of the Langlands program, suggested by Venkatesh. However, even over algebraic arithmetic manifolds, certain automorphic forms like weight-one elliptic modular forms possess a derived structure. In this talk, we discuss this idea over Shimura varieties. A part of the story is the construction of archimedean/p-adic “derived” operators on the cohomology of Shimura varieties, using complex/p-adic Hodge theory.