Relative Langlands Duality of Toric Periods

TL;DR

This paper explores the duality in the relative Langlands program for affine toric varieties, extending previous definitions to singular spaces and analyzing structures relevant for automorphic periods and L-functions.

Contribution

It applies the extended duality framework to affine toric varieties and investigates regularization and stabilizer structures for broader automorphic applications.

Findings

Established duality for affine toric varieties.

Analyzed regularization techniques for singular spaces.

Studied stabilizer structures relevant to automorphic integrals.

Abstract

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.