Some Singular Examples of Relative Langlands Duality

TL;DR

This paper explores specific instances of relative Langlands duality involving singular spaces related to nilpotent matrices and tensors, connecting them to known Rankin--Selberg integrals.

Contribution

It provides a detailed exposition of relative Langlands duality and applies it to novel singular examples involving nilpotent cones, linking to established integral formulas.

Findings

Identification of duality structures in singular nilpotent spaces

Connections established with Rankin--Selberg integrals

Enhanced understanding of duality in complex geometric contexts

Abstract

Relative Langlands duality structures the study of automorphic periods around a putative duality between certain group actions of Langlands dual reductive groups. In this article, after giving a self-contained exposition of the relevant ingredients from relative Langlands duality, we examine this proposal for some interesting pairs of singular spaces: one pair arising from the cone of nilpotent (3 x 3)-matrices, and the other pair arising from the nilpotent cone of (2,2,2)-tensors. These relate, respectively, to Rankin--Selberg integrals discovered by Ginzburg and Garrett.