Structure of Admissible Representations

IsPrincipalSeries(pi) : RepLoc -> BoolElt

This is true iff the admissible representation π belongs to the principal series.

IsSupercuspidal(pi) : RepLoc -> BoolElt

This is true iff the admissible representation π is supercuspidal.

PrincipalSeriesParameters(pi) : RepLoc -> GrpDrchElt, GrpDrchElt

Given a principal series representation π of GL₂(Q_p), this returns two Dirichlet characters of p-power conductor which represent the restriction to Z_p ^x x Z_p ^x of the character of the split torus of GL₂(Q_p) associated to π.

CuspidalInducingDatum(pi) : RepLoc -> ModGrp

Given a minimal supercuspidal representation π of GL₂(Q_p), 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(Q_p) 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.