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

We study the number $N_{\mathrm{sd}}^K(\lambda)$ of self-dual cuspidal automorphic representations of $GL_N(\mathbb{A_Q})$ which are $K$-spherical with respect to a fixed compact subgroup $K$ and whose Laplacian eigenvalue is $\leq \lambda$. 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 $c_1\lambda^{d/2}\leq N_{\mathrm{sd}}^K(\lambda)\leq c_2\lambda^{d/2}$ 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.