Spherical function

Remarks

Example: $G = \R$ and $K = \set{0}$, a Gelfand pair since $\R$ is abelian. Clause 1 is empty, and clause 3 reads $\varphi(x + y) = \varphi(x)\varphi(y)$. So the spherical functions are $\varphi(x) = e^{s x}$ with $s \in \C$, and clause 2 holds since $e^0 = 1$. The positive-definite ones are the unitary characters $e^{i \xi x}$ with $\xi \in \R$.

Non-example: $\varphi(x) = 2 e^{x}$ on $\R$ fails clause 2, and also clause 3, since $\varphi(x + y) \ne \varphi(x)\varphi(y)$.

For $(\mathrm{SO}(3), \mathrm{SO}(2))$, the spherical functions are $g \mapsto P_\ell(\cos \theta(g))$, where $\theta(g)$ is the angle between $g e_3$ and $e_3$. Here $P_\ell$ is the Legendre polynomial of degree $\ell \ge 0$.

Equivalent form: a continuous $K$-bi-invariant $\varphi \ne 0$ is spherical exactly when $f \mapsto \int_G f(g) \varphi(g^{-1}) \, dg$ is multiplicative on $C_c(K \backslash G / K)$. If $\pi$ is an irreducible unitary representation with a unit $K$-fixed vector $v$, then $\varphi(g) = \ip{\pi(g) v}{v}$ is a positive-definite spherical function.