Spherical representation

Let $G$ be a real reductive Lie group and let $K$ be a maximal compact subgroup. An irreducible unitary representation $(\pi,H)$ of $G$ is $K$ -spherical, or of class one, if

H^K=\{v\in H:\pi(k)v=v\text{ for every }k\in K\}

is nonzero. A nonzero member of $H^K$ is a spherical vector. Because $(G,K)$ is a Gelfand pair in the standard reductive setting, $\dim H^K=1$ , so a unit spherical vector is unique up to a scalar of absolute value one.

Spherical matrix coefficient

If $v$ is a unit spherical vector, then

\varphi_\pi(g)=\langle\pi(g)v,v\rangle

is $K$ -bi-invariant, positive definite, and normalized by $\varphi_\pi(e)=1$ . This coefficient function is the spherical function attached to $\pi$ . Conversely, normalized positive-definite elementary spherical functions recover spherical unitary representations through the GNS construction for a positive-definite function.

Examples and representation theory

The trivial representation is spherical. Spherical principal series provide the basic nontrivial family and are treated separately as spherical principal series. Spherical representations are precisely the irreducible constituents relevant to harmonic analysis of $K$ -bi-invariant functions and of the symmetric space $G/K$ .

Conventions and scope

Some literature defines a spherical representation for a general pair $(G,K)$ , or does not require unitarity. In that broader usage, the one-dimensionality of $H^K$ requires the Gelfand-pair hypothesis and is not automatic. The present definition fixes the unitary real-reductive setting and should not be confused with a representation of a sphere.

References

Sigurdur Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions, American Mathematical Society, 2000. AMS record. Relevant: Chapter IV, §§2–3 on class-one representations and spherical functions.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author record. Relevant: Chapter VII on spherical representations and principal series.