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.

💡 Research Summary

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The paper “Relative Langlands Duality of Toric Periods” develops the relative Langlands program of Ben‑Zvi, Sakellaridis, and Venkatesh in the concrete setting of affine toric varieties, extending the framework introduced in the author’s earlier work with Venkatesh to singular spaces. The central goal is to exhibit a precise duality between automorphic periods on a toric variety (X) and “spectral” L‑functions attached to its Langlands dual toric variety (\check X).

The authors begin by recalling the notion of relative Langlands duality: a pair of Langlands dual reductive groups ((G,\check G)) should give rise to a duality of Hamiltonian actions on symplectic manifolds (M=T^{}X) and (\check M=T^{}\check X). In the simplified polarized setting, the duality is expressed through two equations (1) and (2) linking (L^{2}) pairings of Poincaré series with Hecke eigenforms on one side to nonlinear L‑functions on the other. The paper stresses that these equations are only schematic; a rigorous statement requires regularization of divergent contributions, removal of small orbits, and cancellation of (\zeta(1)) factors.

Section 2 sets up the combinatorial language of toric geometry. A split torus (T) with character lattice (X^{}(T)) and its Langlands dual (\check T) are fixed. For a strongly convex rational cone (\sigma\subset X^{}(T){\mathbb R}) the affine toric variety (U{\sigma}=\operatorname{Spec}\mathbb Z