z-Finite distributions on p-adic groups

For a real reductive group G, the center $\mathfrak{z}(\mathcal{U}(\mathfrak{g}))$ of the universal enveloping algebra of the Lie algebra $\mathfrak{g}$ of G acts on the space of distributions on G. This action proved to be very useful (see e.g. [HC63, HC65, Sha74, Bar03]). Over non-Archimedean local fields, one can replace this action by the action of the Bernstein center z of G, i.e. the center of the category of smooth representations. However, this action is not well studied. In this paper we provide some tools to work with this action and prove the following results. 1) The wave-front set of any z-finite distribution on G over any point $g\in G$ lies inside the nilpotent cone of $T_g^*G \cong \mathfrak{g}$. 2) Let $H_1,H_2 \subset G$ be symmetric subgroups. Consider the space J of $H_1\times H_2$-invariant distributions on G. We prove that the z-finite distributions in J form a dense subspace. In fact we prove this result in wider generality, where the groups $H_i$ are spherical groups of certain type and the invariance condition is replaced by equivariance. Further we apply those results to density and regularity of spherical characters. The first result can be viewed as a version of Howe's expansion of characters. The second result can be viewed as a spherical space analog of a classical theorem on density of characters of admissible representations. It can also be viewed as a spectral version of Bernstein's localization principle. In the Archimedean case, the first result is well-known and the second remains open.