Arbeitsgemeinschaft: Relative Langlands Duality

Abstract

One of the fundamental properties of automorphic forms is that their periods - integrals against certain distinguished cycles or distributions - give special values of $L$-functions. The Langlands program posits that automorphic representations for a reductive group $G$ correspond to (generalizations of) Galois representations into its Langlands dual group $\check G$. Periods and $L$-functions are specific ways to extract numerical invariants from the two sides of the Langlands program; in interesting cases, they match with one another. Relative Langlands Duality is the systematic study of the manifestations of this matching at all "tiers" of the Langlands program (global, local, geometric, arithmetic, etc.). A key point is a symmetric conceptualization of both sides: periods arise from suitable Hamiltonian $G$-actions $G\circlearrowright M$ and $L$-functions from suitable Hamiltonian ${\check G}$-actions ${\check G}\circlearrowright \check M$.