Density of Self-Dual Automorphic Representations of GL_n(A_Q)

Abstract

We study the number $N_{\mathrm{sd}}^K(\lambda )$ of self-dual cuspidal automorphic representations of $G{L}_N({\mathbb{A}}_{\mathbb{Q}})$ which are $K$-spherical with respect to a fixed compact subgroup $K$ and whose Laplacian eigenvalue is $\le \lambda$. We prove Weak Weyl's Law for $N_{\mathrm{sd}}^K(\lambda )$ in the form that there are positive constants $c_1,c_2$ (depending on $K$) and $d$ such that $c_1{\lambda }^{d/2}\le N_{\mathrm{sd}}^K(\lambda )\le c_2{\lambda }^{d/2}$ for all sufficiently large $\lambda$. When $N=2n$ is even and $K$ is a maximal compact subgroup at all places, we prove Weyl's Law for the number of self-dual representations, i.e., $N_{\mathrm{sd}}^K(\lambda )=c{\lambda }^{d/2}+o({\lambda }^{d/2})$. These results are based on considering functorial descents of self-dual representations $\Pi$ to quasisplit classical groups $\mathbf{G}$. In order to relate the properties of representations under functoriality, we discuss the infinitesimal character of the real component ${\Pi }_{\infty }$, which determines the Laplacian eigenvalue. To relate the existence of $K$-fixed vectors, we study the depth of $p$-adic representations, proving a weak version of depth preservation. We also consider the explicit construction of local descent, which allows us to improve the results towards depth preservation for generic representations.

PhD thesis at Purdue University. Advisor: Freydoon Shahidi