Lagrangian Subvarieties of Hyperspherical Varieties

Given a hyperspherical G-variety 𝒳 we consider the zero moment level Λ_𝒳⊂𝒳 of the action of a Borel subgroup B⊂G. We conjecture that Λ_𝒳 is Lagrangian. For the dual G^∨-variety 𝒳^∨, we conjecture that that there is a bijection between the sets of irreducible components \(\operatorname {Irr}\Lambda _{{\mathscr{X}}}\) and \(\operatorname {Irr}\Lambda _{{\mathscr{X}}^{\vee }}\). We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.