Principal series representation

Summary

A principal series representation of a real reductive group is induced from a minimal parabolic subgroup. The inducing data are a character of its split part and an irreducible representation of its compact part. For $\mathrm{SL}_2(\R)$, it acts on homogeneous functions on the plane, or equivalently on functions on the circle. The unitary principal series supply the continuous part of the Plancherel decomposition of $L^2(G)$. By a theorem of Casselman, every irreducible admissible representation embeds into some principal series, which makes these representations a universal source.