Relative Langlands Duality of Toric Periods

The relative Langlands program introduced by Ben-Zvi--Sakellaridis--Venkatesh posits a duality structure exchanging automorphic periods and L-functions, which can be encoded by pairs of dual Hamiltonian actions. In work of the author and Venkatesh, an extension of the definitions to certain singular spaces was made with the objective of restoring duality in some well-known automorphic integrals. In this companion article we apply these definitions to establish duality in the context of affine toric varieties, and study finer structures regarding regularization and stabilizers that are instructive for the general case.