Given a smooth quasiprojective variety $X$ with a coherent sheaf $E$, if $X$ is not projective, then the sheaf cohomology of $E$ may not be finitedimensional. However, if we also have the action of a torus $T$ on $X$ so that $E$ is equivariant, then the cohomology groups are naturally representations of $T$, and it could be the case that the weight spaces are finitedimensional. Under what hypotheses will this be true? For instance, if $X$ has proper fixed locus, is that enough?
