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 GL(2). From this it follows that any archetype must be induced from a Bushnell-Kutzko type. Given a supercuspidal representation n of SL2(F), we give an additional explicit description of the number of archetypes admitted by pi in terms of its ramification. We also describe a relationship between archetypes for GL(2) and SL2 in terms of L-packets, and deduce an inertial Langlands correspondence for SL2. (C) 2015 The Author. Published by Elsevier Inc.