  Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of
$G:=H(\mathbb{R})$, and $\Gamma\subset H(\mathbb{Q})$ a congruence arithmetic
subgroup. In this paper, we generalize existing subconvex bounds for
Hecke-Maass forms on the locally symmetric space $\Gamma \backslash G/K$ to
corresponding bounds on the arithmetic quotient $\Gamma \backslash G$ for
cocompact lattices using the spectral function of an elliptic operator. The
bounds obtained extend known subconvex bounds for automorphic forms to
non-trivial $K$-types, yielding subconvex bounds for new classes of automorphic
representations, and constitute subconvex bounds for eigenfunctions on compact
manifolds with both positive and negative sectional curvature. We also obtain
new subconvex bounds for holomorphic modular forms in the weight aspect.
