On the cotangent space of a strongly tempered spherical variety

Let $X=H\backslash G$ be a homogeneous spherical variety for a reductive group $G$, such that the dual group of $X$ is equal to the dual group of $G$. We prove a structure theorem for the cotangent space $T^*X$ ``up to codimension two,'' which strengthens some results of Knop. The theorem confirms observations of V.\ Lafforgue and Gaiotto, and a conjecture of Hameister, Luo, and Morrissey, and by their work implies a ``classical'' or ``Dolbeault'' limit of the global geometric relative Langlands conjecture. Conditionally on the local relative Langlands conjecture of [BZSV], the theorem also amounts to a description of the Braverman--Finkelberg--Nakajima ``Coulomb branch'' of hyperspherical symplectic vector spaces. Finally, we generalize the classical Chevalley--Richardson restriction theorem to this setting, giving a description of the invariant-theoretic double quotient $H\backslash G/H$ in terms of root data and invariants of the spherical variety.