# 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](introduction.md#sec-supercuspidal)) 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 $\pi$.