Smooth representations and Hecke algebras of $p$-adic ${\mathrm{G}\mathrm{L}}_n(\mathcal{D})$

The main question we are going to address in this paper is: How much does the representation theory of the $p$-adic group ${\mathrm{G}\mathrm{L}}_n(\mathcal{D})$ 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 Sécherre-Stevens decomposition of spherical Hecke algebras associated to types, we obtain that the cuspidal blocks in the Bernstein decomposition of the category $\mathcal{R}({\mathrm{G}\mathrm{L}}_n(\mathcal{D}))$ of smooth complex representations of ${\mathrm{G}\mathrm{L}}_n(\mathcal{D})$ do not depend on the $p$-adic division algebra $\mathcal{D}$. In particular, when $n=1$ or $2$, the category $\mathcal{R}({\mathrm{G}\mathrm{L}}_n(\mathcal{D}))$ does not depend on the $p$-adic division algebra $\mathcal{D}$.