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Many phenomena in geometry and analysis can be explained via the theory of D-modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of D-modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely Cexp-class distributions from Cluckers et al. ['Distributions and wave front sets in the uniform nonarchimedean setting', Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld ['The wave front set of the Fourier transform of algebraic measures', Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld ['The wave front set of the Fourier transform of algebraic measures', Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the Cexp-class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the Cexp-class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld ['The wave front set of the Fourier transform of algebraic measures', Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the Cexp-class. We strengthen a link from Cluckers et al. ['Distributions and wave front sets in the uniform nonarchimedean setting', Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the Cexp-class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory. [ABSTRACT FROM AUTHOR]

We prove that any relative character (a.k.a. spherical character) of any admissible representation of a real reductive group with respect to any pair of spherical subgroups is a holonomic distribution on the group. This implies that the restriction of the relative character to an open dense subset is given by an analytic function. The proof is based on an argument from algebraic geometry and thus implies also analogous results in the p-adic case. As an application, we give a short proof of some results of Kobayashi-Oshima and Kroetz-Schlichtkrull on boundedness and finiteness of multiplicities of irreducible representations in the space of functions on a spherical space. To deduce this application we prove the relative and quantitative analogs of the Bernstein-Kashiwara theorem, which states that the space of solutions of a holonomic system of differential equations in the space of distributions is finite-dimensional. We also deduce that, for every algebraic group $${{G}}$$ defined over $$\mathbb {R}$$ , the space of $${{G(\mathbb {R})}}$$ -equivariant distributions on the manifold of real points of any algebraic $${{G}}$$ -manifold $${{X}}$$ is finite-dimensional if $${{G}}$$ has finitely many orbits on $${{X}}$$ . [ABSTRACT FROM AUTHOR]

For a real reductive group G , the center z ( U ( g ) ) of the universal enveloping algebra of the Lie algebra g of G acts on the space of distributions on G . This action proved to be very useful (see e.g. [17,18,32,8] ). Over non-Archimedean local fields, one can replace the action of z ( U ( g ) ) 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. • The wavefront set of any z -finite distribution ξ on G over any point g ∈ G lies inside the nilpotent cone of T g ⁎ G ≅ g . • Let H 1 , H 2 ⊂ G be symmetric subgroups. Consider the space J of H 1 × 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 subgroups of certain type and the invariance condition is replaced by equivariance. Further we apply those results to density and regularity of relative 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 finite length 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. [ABSTRACT FROM AUTHOR]