Unicity of types for supercuspidal representations of p-adic SL(2)

We consider the question of unicity of types on maximal compact subgroups for supercuspidal representations of $\mathbf{SL}_2$ over a nonarchimedean local field of odd residual characteristic. We introduce the notion of an archetype as the $\mathbf{SL}_2$-conjugacy class of a typical representation of a maximal compact subgroup, and go on to show that any archetype in $\mathbf{SL}_2$ is restricted from one in $\mathbf{GL}_2$. From this it follows that any archetype must be induced from a Bushnell--Kutzko type. Given a supercuspidal representation $π$, 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 $\mathbf{GL}_2$ and $\mathbf{SL}_2$ in terms of $L$-packets, and deduce an inertial Langlands correspondence for $\mathbf{SL}_2$.

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