Structure of Admissible Representations

IsPrincipalSeries(pi): RepLoc → BoolElt

This is true iff the admissible representation \(\pi\) belongs to the principal series.

IsSupercuspidal(pi): RepLoc → BoolElt

This is true iff the admissible representation \(\pi\) is supercuspidal.

PrincipalSeriesParameters(pi): RepLoc → GrpDrchElt, GrpDrchElt

Given a principal series representation \(\pi\) of \({\operatorname{GL}}_2({\mathbb{Q}}_p),\) this returns two Dirichlet characters of \(p\)-power conductor which represent the restriction to \({\mathbb{Z}}_p^{\times} \times {\mathbb{Z}}_p^{\times}\) of the character of the split torus of \({\operatorname{GL}}_2({\mathbb{Q}}_p)\) associated to \(\pi.\)

CuspidalInducingDatum(pi): RepLoc → ModGrp

Given a minimal supercuspidal representation \(\pi\) of \({\operatorname{GL}}_2({\mathbb{Q}}_p),\) this returns a cuspidal inducing datum that gives rise to \(\pi.\)

Recall (from Section Supercuspidal Representations) that a cuspidal inducing datum \((K,\Xi)\) consists of a subgroup \(K\) of \({\operatorname{GL}}_2({\mathbb{Q}}_p)\) and a representation \(\Xi\) of \(K\) that gives rise to \(\pi\) via induction. Importantly, \(\Xi\) factors through some finite quotient \(K/K_1\) of \(K.\) This function returns such a representation of \(K/K_1\). From this one can deduce the representation on \(K\), and hence \(\pi\).