On the types for supercuspidal representations of inner forms of $\mathrm{GL}_{N}$

Abstract

Let $F$ be a non-Archimedean local field, $A$ be a central simple $F$-algebra, and $G$ be the multiplicative group of $A$. It is known that for every irreducible supercuspidal representation $\pi$, there exists a $[G, \pi]_{G}$-type $(J, \lambda)$, called a (maximal) simple type. We will show that $[G, \pi]_{G}$-types defined over some maximal compact subgroup are unique up to $G$-conjugations under some unramifiedness assumption on a simple stratum.