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

Let G be the F points of a connected reductive group defined over a local field F of characteristic 0. By Harish-Chandra’s regularity theorem, every 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. 

We explore \e_π, and determine it in the case of a p-adic GL(n). This is done by studying integrability properties of Fourier transforms of nilpotent orbital integrals. As a main technical tool, we use a resolution of singularities algorithm coming from the theory of hyperplane arrangements. As an application, we obtain bounds on the multiplicities of K-types in irreducible representations of G in the p-adic case, where K is an open compact subgroup.

The talk will be accessible to non-representation theorists.

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

================================================