Virtual Jeffrey--Kirwan localisation

We express integrals over virtual cycles of GIT quotients $X/\!\!/G$ in terms of integrals over virtual cycles of fixed loci $X^T$.

The results hold for both perfect obstruction theories and $(-2)$-shifted symplectic structures, in cohomology and in $K$-theory, and for noncompact Deligne-Mumford stacks acted on by a reductive group with compact quotient.