  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.
