Types and stable wave front set of theta representations

We compute the depth 0 K -types and stable wave front set of theta representations for certain tame Brylinski–Deligne covers of a connected split reductive p -adic group. To determine the K -types we utilize the structure theory of the pro- p Iwahori-Hecke algebra developed by Gao–Gurevich–Karasiewicz and Wang. We find that the K -types are inflated minimal principal series representations of the reductive quotient. To compute the stable wave front set we extend Barbasch–Moy's test functions to the universal tame cover. The passage to the universal tame cover ensures that the quotients of parahoric subgroups by their pro-p radical are reductive, and this allows us to utilize Lusztig's formulas for wave front sets over finite fields and our knowledge of the K -types to reduce the computation to a novel question about the ℤ / 𝑛 -values of affine roots on the apartment that we solve. This resolves a conjecture of Gao–Tsai under mild hypotheses on the residual characteristic and as a corollary generalizes Friedberg–Ginzburg's upper bound on wavefront sets of global theta representations.