# 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.

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