Relative Langlands duality for linear periods

Abstract

We prove relative Langlands duality for $X=G{L}_{p+q}/(G{L}_p\times G{L}_q)$. These symmetric spaces arise in the setting of the linear period of Friedberg-Jacquet when $p=q$, and exhibit non-tempered behavior when $q>p+1$. Via the real-symmetric correspondence, we establish a derived Satake equivalence for the real group $G_{\mathbb{R}}=U(p,q)$, and use it to obtain a constructible realization of coherent sheaves on Kato's exotic nilpotent cones. Along the way, we describe the dual variety of $X$ in terms of the Whittaker cotangent bundle and regular centralizers, viewed as Coulomb branches.