Although perhaps the question was directed more at finite groups: for reductive or semi-simple real Lie groups, (serious) results of Harish-Chandra (starting in the early 1950s) show that among regular semi-simple elements of the group, the supports of characters of principal series are the smallest, being confined to (maximally) *split* conjugacy classes, while at the opposite end the characters of discrete series have non-trivial support on *all* semi-simple conjugacy classes. The in-between repns, i.e., induced from discrete series on Levis, have in-between supports, in terms of split-ness.

I have the impression that suitable analogues hold for p-adic reductive groups, tho' the corresponding results are relatively much-newer (by a displacement of 15-20 years?), not to mention the complication in description of discrete series (supercuspidal repns) as induced from compact-open subgroups (Kutzko-Bushnell-et-alia). Early computations for the p-adic case go back to MacDonald (SL2) in the 1950s, and Shalika, Sally in the 1960s (SL2), Jacquet in the late 1960s, so far as I know.