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 below only in terms of the group G, and calculate it in the case of a p-adic GL(n). 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.

Based on a joint work with Itay Glazer and Julia Gordon.