Zonal Spherical Function - Properties

Properties

A zonal spherical function h has the following properties:

1. h is uniformly continuous on G
2. h(1) =1 (normalisation)
3. h is a positive definite function on G
4. f * h is proportional to h for all f in C_c(K\G/K).

These are easy consequences of the fact that the bounded linear functional χ defined by h is a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection with unitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions of invariant differential operators on G/K (see below).

In fact, as a special case of the Gelfand–Naimark–Segal construction, there is one-one correspondence between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by

Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the induced representation π on H₁. Each representation σ extends uniquely by continuity to A(K\G/K), so that each zonal spherical function satisfies

for f in A(K\G/K). Moreover, since the commutant π(G)' is commutative, there is a unique probability measure μ on the space of * homomorphisms X such that

μ is called the Plancherel measure. Since π(G)' is the centre of the von Neumann algebra generated by G, it also gives the measure associated with the direct integral decomposition of H₁ in terms of the irreducible representations σ_χ.