Arithmetic Wavefront Sets and Generic $L$-packets

Abstract

Let $G$ be a classical group defined over a local field $F$ of characteristic zero. Let $\pi$ be an irreducible admissible representation $\pi$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean. If $\pi$ has a generic local $L$-parameter, we define the arithmetic wavefront set ${\rm WF_{ari}}(\pi)$ of $\pi$, which is a subset of $F$-rational nilpotent orbits of the Lie algebra $\mathfrak{g}(F)$ of $G(F)$, by means of the arithmetic structures of the enhanced $L$-parameter $(\varphi,\chi)$ of $\pi$. Those arithmetic structures are discovered by using our method of consecutive descents of enhanced $L$-parameters, based on the rationality of the local Langlands correspondence and the local Gan-Gross-Prasad conjecture. We study the basic structure of ${\rm WF_{ari}(\pi)}$ and prove that it is an invariant of $\pi$ (Theorem 5.10). Furthermore, those basic structures ${\rm WF_{ari}(\pi)^{max}}$ are expected to yield the precise $F$-rational structure of ${\rm WF_{ari}(\pi)^{max}}$, which has been realized, when $F$ is archimedean, in Theorems 1.3 and 1.4 (Theorems 7.10, 7.14, and 9.2). Based on the local Langlands reciprocity, the Wavefront Set Conjecture (Conjecture 1.2 and Conjecture 5.14) asserts that the wavefront sets on the L-parameter side should be closed related to those on the representation side, namely, \[ {\rm WF_{wm}(\pi)^{max}}={\rm WF_{ari}(\pi)^{max}}={\rm WF_{tr}(\pi)^{max}} \] when $\pi$ has a generic local $L$-parameter, where the algebraic wavefront set ${\rm WF_{wm}}(\pi)$ is defined by Moeglin and Waldspurger in [MW87], using generalized Whittaker models and the analytic wavefront set ${\rm WF_{tr}}(\pi)$ is defined by Howe [H81, Hd85] using distribution characters, and also by [H74, HC78, BV80]. Conjecture 1.2 is verified for families of interesting cases.