  The main question we are going to address in this paper is: How much does the
representation theory of the $p$-adic group $\mathrm{GL}_n(\mathcal{D})$ depend
on the $p$-adic division algebra $\mathcal{D}$? Let $\mathcal{D}$ be a central
division algebra defined over some locally compact non-archimedean local field.
Using Bushnell-Kutzko theory of types and S\'echerre-Stevens decomposition of
spherical Hecke algebras associated to types, we obtain that the cuspidal
blocks in the Bernstein decomposition of the category $\mathcal{R} \left(
\mathrm{GL}_n(\mathcal{D}) \right)$ of smooth complex representations of
$\mathrm{GL}_n(\mathcal{D})$ do not depend on the $p$-adic division algebra
$\mathcal{D}$. In particular, when $n=1$ or $2$, the category $\mathcal{R}
\left( \mathrm{GL}_n(\mathcal{D}) \right)$ does not depend on the $p$-adic
division algebra $\mathcal{D}$.
