  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.
