Lagrangian Intersections, Symplectic Reduction and Kirwan Surjectivity

Given a smooth holomorphic symplectic variety $X$ with a Hamiltonian $G$-action, $G$-invariant Lagrangians $C^{\mathrm{\prime }}s$ induce Lagrangians in the symplectic quotient $X\mathrm{/}\mathrm{/}G$. Given clean intersections $B=C_1\cap C_2$ whose conormal sequence splits, we show that $$C_1\mathrm{/}G{\times }_{X\mathrm{/}\mathrm{/}G}{C}_2\mathrm{/}G\cong T^{\vee }[-1](B\mathrm{/}G)\mathrm{.}$$ When $det(N_{B\mathrm{/}{C}_2})$ is torsion, we have $Ex{t}_{X\mathrm{/}\mathrm{/}G}^{\bullet }({\mathcal{O}}_{C_1\mathrm{/}G},{\mathcal{O}}_{C_2\mathrm{/}G})\cong H_G^{\bullet }(B,det(N_{B\mathrm{/}{C}_2}{)}_{\delta })$ provided that the Hodge-to-de Rham degeneracy holds. Furthermore, we have a generalized version of Kirwan surjectivity $Ex{t}_{X\mathrm{/}\mathrm{/}G}^{\bullet }({\mathcal{O}}_{C_1\mathrm{/}G},{\mathcal{O}}_{C_2\mathrm{/}G})\twoheadrightarrow Ex{t}_{X^{ss}\mathrm{/}\mathrm{/}G}^{\bullet }({\mathcal{O}}_{C_1^{ss}\mathrm{/}G},{\mathcal{O}}_{C_2^{ss}\mathrm{/}G})$ if $B$ is proper. When $C_1=C_2$, this is the Kirwan surjectivity, which is now interpreted as the symmetry commutes with reduction problem in 3d B-model. We also obtain similar results for $K_{C_1\mathrm{/}G}^{1\mathrm{/}2}$ and $K_{C_2\mathrm{/}G}^{1\mathrm{/}2}$.