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Let $F$ be a non-archimedean local field, $A$ be a central simple$F$-algebra, and $G$ be the multiplicative group of $A$. To construct types forsupercuspidal representations of $G$, simple types by S\'echerre and Yu'sconstruction are already known. In this paper, we compare these constructions.In particular, we show essentially tame supercuspidal representations of $G$defined by Bushnell-Henniart are nothing but tame supercuspidal representationsdefined by Yu.

Comparing Bushnell-Kutzko and S\'echerre's constructions of types for $\mathrm{GL}_{N}$ and its inner forms with Yu's construction