Structure of Admissible Representations
=======================================

.. magma:function:: IsPrincipalSeries(pi)
   :input_types: RepLoc
   :output_types: BoolElt
   :label: IsPrincipalSeries_RepLoc

   This is ``true`` iff the admissible representation :math:`\pi` belongs to the
   principal series.

.. magma:function:: IsSupercuspidal(pi)
   :input_types: RepLoc
   :output_types: BoolElt
   :label: IsSupercuspidal_RepLoc

   This is ``true`` iff the admissible representation :math:`\pi` is supercuspidal.

.. magma:function:: PrincipalSeriesParameters(pi)
   :input_types: RepLoc
   :output_types: GrpDrchElt, GrpDrchElt
   :label: PrincipalSeriesParameters_RepLoc

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

.. magma:function:: CuspidalInducingDatum(pi)
   :input_types: RepLoc
   :output_types: ModGrp
   :label: CuspidalInducingDatum_RepLoc

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

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