Derived Structures in the Duality of Automorphic Periods

Abstract

Built on top of the global Langlands correspondence – a matching of automorphic forms on a reductive group G and Galois representations into its Langlands dual group Gˇ – is the question of functoriality, i.e., the construction and detection of natural transfers of automorphic forms between varying G, predicted by maps between varying Gˇ. Traditionally, this involved understanding natural numerical invariants attached to both sides of the correspondence: computing automorphic periods on the one hand, and understanding analytic properties of various Langlands L-functions on the other. Recently, Ben-Zvi–Sakellaridis–Venkatesh proposed a duality framework that extends Langlands duality to these numerical invariants, in which automorphic periods and L-functions are extracted from Hamiltonian actions of G and Gˇ, respectively. We expand on this perspective by computing new automorphic periods and by recasting old computations in the context of this emergent duality. In the process, we encounter and introduce the notion of nonabelian L-functions as the Galois-side counterpart to our new automorphic computations, and we suggest that they may be understood as reflecting the presence of singularities or derived structures, on the Hamiltonian Gˇ-action.