Unicity of types for supercuspidal representations of p-adic SL₂

We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of SL₂ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the SL₂-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in SL₂ is restricted from one in GL₂. From this it follows that any archetype must be induced from a Bushnell-Kutzko type. Given a supercuspidal representation π of SL₂(F), we give an additional explicit description of the number of archetypes admitted by π in terms of its ramification. We also describe a relationship between archetypes for GL₂ and SL₂ in terms of L-packets, and deduce an inertial Langlands correspondence for SL₂.