This is true iff the admissible representation π belongs to the principal series.
This is true iff the admissible representation π is supercuspidal.
Given a principal series representation π of GL2(Qp), this returns two Dirichlet characters of p-power conductor which represent the restriction to Zp x x Zp x of the character of the split torus of GL2(Qp) associated to π.
Given a minimal supercuspidal representation π of GL2(Qp), this returns a cuspidal inducing datum that gives rise to π.Recall (from Section Supercuspidal Representations) that a cuspidal inducing datum (K, Ξ) consists of a subgroup K of GL2(Qp) and a representation Ξ of K that gives rise to π via induction. Importantly, Ξ factors through some finite quotient K/K1 of K. This function returns such a representation of K/K1. From this one can deduce the representation on K, and hence π.