Spherical function

Summary

A spherical function for a Gelfand pair $(G, K)$ is a continuous function on $G$, invariant under $K$ on both sides. It satisfies a product formula with an average over $K$. Spherical functions are exactly the multiplicative functionals on the commutative algebra of $K$-bi-invariant functions. They give the spherical transform, the analogue of the Fourier transform on the homogeneous space $G/K$. Positive-definite spherical functions correspond to the irreducible unitary representations of $G$ with a nonzero $K$-fixed vector.