Geometric wavefront sets of genuine Iwahori-spherical representations

For Iwahori-spherical genuine representations of central covers with positive real Satake parameters, we prove the upper bound inequality for their geometric wavefront sets, formulated for general genuine representations in an earlier work by Gao–Liu–Lo–Shahidi. Meanwhile, we show the equality is attained for covers of type A groups and for some representations of covers of the exceptional groups. We also verify the equality for certain Iwahori-spherical representations occurring in regular unramified principal series; this uses and generalizes the earlier work of Karasiewicz–Okada–Wang on theta representations. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of a theta representation when its geometric wavefront set is of a special type.