  In this thesis, two $\bar{\mathbb{Q}}_\ell$-local systems,
$\vphantom{\mathcal{E}}^\circ \mathcal{E}$ and $\vphantom{E}^\circ
\mathcal{E}^\prime$ on the regular unipotent subvariety $\mathcal{U}_{0,K}$ of
$p$-adic $\operatorname{SL}_2(K)$ are constructed. Making use of the
equivalence between $\bar{\mathbb{Q}}_\ell$-local systems and $\ell$-adic
representations of the \'etale fundamental group, we prove that these local
systems are equivariant with respect to conjugation by $\operatorname{SL}_2(K)$
and that their nearby cycles, when taken with respect to appropriate integral
models, descend to local systems on the regular unipotent subvariety of
$\operatorname{SL}_2(k)$, $k$ the residue field of $K$. Distributions on
$\operatorname{SL}_2(K)$ are then associated to $\vphantom{E}^\circ
\mathcal{E}$ and $\vphantom{E}^\circ \mathcal{E}^\prime$ and we prove
properties of these distributions. Specifically, they are admissible
distributions in the sense of Harish-Chandra and, after being transferred to
the Lie algebra, are linearly independent eigendistributions of the Fourier
transform. Together, this gives a geometrization of important admissible
invariant distributions on a nonabelian $p$-adic group in the context of the
Local Langlands program.
