Virtual Jeffrey--Kirwan localisation

We express integrals over virtual cycles of GIT quotients $X\mathrm{/}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}G$ 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.