This is true iff the admissible representation πbelongs to the principal series.
This is true iff the admissible representation πis supercuspidal.
Given a principal series representation πof 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 πof GL2(Qp), this returns a cuspidal inducing datum that gives rise to π.[Next][Prev] [Right] [Left] [Up] [Index] [Root]Recall (from Section Supercuspidal Representations) that a cuspidal inducing datum (K, Ξ) consists of a subgroup K of GL2(Qp) and a representation Ξof K that gives rise to πvia induction. Importantly, Ξfactors 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.