On the types for supercuspidal representations of inner forms of GLN

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 π, there exists a [G,π]G-type (J,λ), called a (maximal) simple type. We will show that [G,π]G-types defined over some maximal compact subgroup are unique up to G-conjugations under some unramifiedness assumption on a simple stratum.