Cohomology of symmetric stacks

We construct decompositions of: (1) the cohomology of smooth stacks, (2) the Borel--Moore homology of $0$-shifted symplectic stacks, and (3) the vanishing cycle cohomology of $(-1)$-shifted symplectic stacks, assuming a good moduli space exists and the tangent space has a pointwise orthogonal structure. These conditions are satisfied by many stacks of interest, including moduli stacks of semistable $G$-bundles and (twisted) $G$-Higgs bundles on curves, $G$-character stacks of oriented closed 2-manifolds and various 3-manifolds, and moduli stacks of semistable coherent sheaves on Calabi--Yau threefolds and K3 surfaces with generic polarization. As a special case, we prove a PBW-type theorem for cohomological Hall algebras of $3$-Calabi--Yau categories with commutative orientation data, a strong form of the cohomological integrality conjecture for such categories. We define the BPS cohomology as the primary summand of the decomposition. When the stack is smooth, the BPS cohomology coincides with the intersection cohomology of the good moduli space, generalizing a theorem of Meinhardt--Reineke. Using the BPS cohomology for singular spaces, we propose a formulation of the topological mirror symmetry conjecture for the stack of $G$-Higgs bundles generalizing the work of Hausel and Thaddeus for type A groups, and a version of Langlands duality for character stacks of compact oriented 3-manifolds, following Ben-Zvi--Gunningham--Jordan--Safronov.