Riemann's proof of the functional equation of the zeta function relies on the action of the *multiplicative* group of the real numbers on Schwartz functions on the *additive* group. Relative Langlands duality generalizes this paradigm to a correspondence between Schwartz functions on nice spaces, such as spherical varieties, under the action of a (reductive) group G, and automorphic L-functions determined by representations (and more general actions) of its Langlands dual group. For various reasons, it is important to understand the L-function that appears in terms of the geometry of the cotangent bundle of the spherical variety under consideration. I will present a structure theorem, describing the regular locus of the cotangent bundle of a strongly tempered spherical variety as a "toric embedding" of the group scheme of regular centralizers – which is, in some sense, a direct generalization of the underlying geometry of Riemann's proof. This deepens work of Knop (who described this cotangent bundle up to codimension one), confirms an observation of V. Lafforgue, and can be fed into the argument of Hameister–Luo–Morissey to prove a version of the semiclassical/Dolbeault relative Langlands conjecture. This is part of ongoing joint work with David Ben-Zvi and Akshay Venkatesh.
