Principal series representation - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive Lie group and let $P_0=MAN$ be a minimal parabolic subgroup. Given an irreducible unitary representation $\sigma$ of $M$ and a parameter $\lambda\in\mathfrak a_{\mathbb C}^*$ , extend $\sigma\otimes e^\lambda$ , where $e^\lambda(a)=e^{\lambda(\log a)}$ , to $P_0$ by making $N$ act trivially. The principal series representation with parameters $(\sigma,\lambda)$ is

I(\sigma,\lambda) =i_{P_0}^G(\sigma\otimes e^\lambda),

where $i_{P_0}^G$ denotes normalized parabolic induction. If $\lambda\in i\mathfrak a^*$ , the inducing character is unitary and $I(\sigma,\lambda)$ is a unitary representation.

Compact picture

Choose a maximal compact subgroup $K$ with $G=KP_0$ . Restriction to $K$ realizes every $I(\sigma,\lambda)$ on the same space of functions $f:K\to V_\sigma$ satisfying the appropriate $M\cap K$ -equivariance. Only the $G$ -action depends on $\lambda$ . This fixed-space realization makes the analytic dependence on the parameter visible and is the standard compact picture of the principal series Knapp, Chapter VII.

Spherical and unitary cases

When $\sigma$ is trivial, the family is the spherical principal series; it contains a nonzero $K$ -fixed vector. Parameters on the imaginary axis give the unitary principal series. At exceptional parameters the induced representation can be reducible, and complementary series may remain unitarizable for certain nonimaginary parameters. Thus “principal series” does not by itself mean irreducible or unitary.

Role in noncompact representation theory

Principal series supply continuous families of representations and are the starting point for harmonic analysis on many noncompact groups. Their irreducible subquotients include important tempered and nontempered representations. Induction from arbitrary parabolics, rather than only minimal ones, produces generalized principal series and the standard modules appearing in the Langlands classification.

Example

For $G=\operatorname{SL}(2,\mathbb R)$ , a minimal parabolic is the subgroup of upper triangular matrices. Its compact $M$ -factor has two characters, and its one-dimensional split factor $A$ contributes a complex parameter. The resulting even and odd principal-series families are unitary on the imaginary parameter axis and become reducible at a discrete set of parameters.

Conventions and scope

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on minimal principal series, compact pictures, and intertwining operators.
Nolan R. Wallach, Real Reductive Groups I, Pure and Applied Mathematics 132, Academic Press, 1988. Publisher record. Relevant: chapters on real reductive groups, principal series, and the Langlands classification.