Motivic wave front sets

Abstract

The concept of wave front set was introduced in 1969-1970 by M. Sato in the hyperfunctions context and by L. H\"ormander in the $\mathcal C^{\infty}$ context. Howe used the theory of wave front sets in the study of Lie groups representations. Heifetz defined a notion of wave front set for distributions in the $p$-adic setting and used it to study some representations of $p$-adic Lie groups. In this article, we work in the $k((t))$-setting with $k$ a characteristic zero field. In that setting, balls are no longer compact but working in a definable context provides good substitutes for finiteness and compactness properties. We develop a notion of definable distributions in the motivic integration framework of Cluckers--Loeser for which we define notions of singular support and $\Lambda$-wave front sets (relative to some multiplicative subgroups $\Lambda$ of the valued field) and we investigate their behaviour under natural operations like pull-back, tensor product, and products of distributions.