Arithmetic Wavefront Set and Microlocal Structure of Harish-Chandra Character

Abstract

In this paper, we establish in Theorem 1.2 the reciprocity of wavefront sets for irreducible admissible representations $\pi$ of classical groups $G$ over any local field $F$ of characteristic zero if $\pi$ has a generic local $L$-parameter. Over archimedean local fields, based on the progress made in our previous work (arXiv:2207.04700), we prove in Theorem 1.5 that for an irreducible Casselman--Wallach representation $\pi$ with a generic local $L$-parameter, the Wavefront Set Conjecture (arXiv:2207.04700, Conjecture 1.2) and its refinement (Conjecture 1.1) hold for the arithmetic wavefront set ${\mathrm{WF}}_{\mathrm{ari}}(\pi)$ as defined by the associated enhanced local $L$-parameter of $\pi$ and the wavefront set ${\mathrm{WF}}_{\mathrm{tr}}(\pi)$ defined by the Harish--Chandra distribution character $\Theta_\pi$ of $\pi$. Hence the microlocal structure of $\Theta_\pi$ is completely determined by the arithmetic information carried by the enhanced local $L$-parameter of $\pi$. The relations with the algebraic wavefront set ${\mathrm{WF}}_{\mathrm{wm}}(\pi)$ defined by the degenerate Whittaker models are extensively discussed by means of the composition law (Theorem 3.4) over all local fields of characteristic zero. Under Conjecture 1.3, the Wavefront Set Conjecture is fully established over archimedean local fields. As a consequence, we prove a refinement of Vogan's maximal-orbit principle (Theorem 1.7).