Some singular examples of relative Langlands duality

Abstract Relative Langlands duality structures the study of automorphic periods around a putative duality between certain G -spaces and $$\\check{G}$$ G ˇ -spaces, where $$G, \\check{G}$$ G , G ˇ are 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 $$3 \\times 3$$ 3 × 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.