Spherical principal series representation - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive group, $K$ a maximal compact subgroup, and $P=MAN$ a minimal parabolic in Langlands form. For $\lambda\in\mathfrak a_{\mathbb C}^{*}$ , the spherical principal series is the normalized induction

I(\lambda)=\operatorname{Ind}_{P}^{G} \bigl(1_M\otimes e^\lambda\otimes 1_N\bigr).

Its restriction to $K$ contains the trivial $K$ -type with multiplicity one; a nonzero vector on this line is called spherical. Thus this is precisely the principal-series family induced from trivial $M$ -data, rather than an arbitrary principal series that happens to have a special parameter.

Spherical vector and functions

In the compact picture the spherical vector is represented by the constant function on $K/M$ . Its normalized matrix coefficient is an elementary spherical function on $G/K$ , so the family connects principal-series representation theory with harmonic analysis on the Riemannian symmetric space. The normalized induction and compact-picture realization are described in Knapp–Trapa, Lecture 3, pp. 42–43.

Example and near-miss

For $G=\mathrm{SL}(2,\mathbb R)$ , normalized induction from a character of the positive diagonal subgroup, trivial on the compact factor of the minimal parabolic, gives the spherical principal series and an even $K$ -type spectrum. Induction using the nontrivial character of the finite group $M$ gives a nonspherical principal series: it has no $K$ -fixed vector and is not part of this family.

Conventions and scope

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Mathematical Series 36, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on nonunitary and spherical principal series.
Anthony W. Knapp and Peter E. Trapa, Representations of Semisimple Lie Groups, Park City Mathematics Institute lecture notes, 2000. Author PDF. Relevant: Lecture 3, pp. 42–43 on normalized induction and the compact picture of principal series.