  The primary goal of this paper is to study Spanier-Whitehead duality in the
$K(n)$-local category. One of the key players in the $K(n)$-local category is
the Lubin-Tate spectrum $E_n$, whose homotopy groups classify deformations of a
formal group law of height $n$, in the implicit characteristic $p$. It is known
that $E_n$ is self-dual up to a shift; however, that does not fully take into
account the action of the Morava stabilizer group $\mathbb{G}_n$, or even its
subgroup of automorphisms of the formal group in question. In this paper we
find that the $\mathbb{G}_n$-equivariant dual of $E_n$ is in fact $E_n$ twisted
by a sphere with a non-trivial (when $n>1$) action by $\mathbb{G}_n$. This
sphere is a dualizing module for the group $\mathbb{G}_n$, and we construct and
study such an object $I_{\mathcal{G}}$ for any compact $p$-adic analytic group
$\mathcal{G}$. If we restrict the action of $\mathcal{G}$ on $I_{\mathcal{G}}$
to certain type of small subgroups, we identify $I_{\mathcal{G}}$ with a
specific representation sphere coming from the Lie algebra of $\mathcal{G}$.
This is done by a classification of $p$-complete sphere spectra with an action
by an elementary abelian $p$-group in terms of characteristic classes, and then
a specific comparison of the characteristic classes in question. The setup
makes the theory quite accessible for computations, as we demonstrate in the
later sections of this paper, determining the $K(n)$-local Spanier-Whitehead
duals of $E_n^{hH}$ for select choices of $p$ and $n$ and finite subgroups $H$
of $\mathbb{G}_n$.
