On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities

Kleinian singularities are quotients of $\mathbb{C}^2$ by finite subgroups of $\mathrm{SL}_2(\mathbb{C})$. They are in bijection with the simply-laced Dynkin diagrams via the McKay correspondence. Anti-Poisson involutions and their fixed point loci appear naturally when we want to classify irreducible Harish-Chandra modules over Kleinian singularities. There are three goals of this paper. The first is to classify anti-Poisson involutions of Kleinian singularities up to conjugation by graded Poisson automorphisms. The second is to describe the scheme-theoretic fixed point loci of Kleinian singularities under anti-Poisson involutions. The last and the main goal is to describe the scheme-theoretic preimages of the fixed point loci under minimal resolutions of Kleinian singularities, which are singular Lagrangian subvarieties in the minimal resolutions whose irreducible components are $\mathbb{P}^1$'s and $\mathbb{A}^1$'s.