Motivic Wave Front Sets

Abstract

Abstract The concept of wave front set was introduced in 1969–1970 by Sato in the hyperfunctions context [1, 34] and by Hörmander [23] in the $\mathcal C^{\infty }$ context. Howe in [25] used the theory of wave front sets in the study of Lie groups representations. Heifetz in [22] 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 $k\mathopen{(\!(} t \mathopen{)\!)}$-setting with $k$ a Characteristic 0 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 framework of [13] and [14] 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 behavior under natural operations like pullback, tensor product, and products of distributions.