On a quantitative version of Harish-Chandra's regularity theorem and singularities of representations

Let G be a reductive group defined over a local field of characteristic
0 (real or p-adic). By Harish-Chandra’s regularity theorem, the
character Θ_π of an irreducible representation π of G is given by a
locally integrable function f_π on G. It turns out that f_π has even
better integrability properties, namely, it is locally
L^{1+r}-integrable for some r>0. This gives rise to a new singularity
invariant of representations \e_π by considering the largest such r.

We explore \e_π, show it is bounded from below by a constant only
depending on the group G, and calculate it in the case of a p-adic
GL(n). In order to do so, we relate \e_π to the integrability of Fourier
transforms of nilpotent orbital integrals appearing in the local
character expansion of Θ_π. As a main technical tool, we use explicit
resolutions of singularities of certain hyperplane arrangements. We
obtain bounds on the multiplicities of K-types in irreducible
representations of G for a p-adic G and a compact open subgroup K.