Zonal Spherical Function - Definitions

Definitions

Let G be a locally compact unimodular topological group and K a compact subgroup and let H₁ = L2(G/K). Thus H₁ admits a unitary representation π of G by left translation. This is a subrepresentation of the regular representation, since if H= L2(G) with left and right regular representations λ and ρ of G and P is the orthogonal projection

from H to H₁ then H₁ can naturally be identified with PH with the action of G given by the restriction of λ.

On the other hand by von Neumann's commutation theorem

where S' denotes the commutant of a set of operators S, so that

Thus the commutant of π is generated as a von Neumann algebra by operators

where f is a continuous function of compact support on G.

However Pρ(f) P is just the restriction of ρ(F) to H₁, where

is the K-biinvariant continuous function of compact support obtained by averaging f by K on both sides.

Thus the commutant of π is generated by the restriction of the operators ρ(F) with F in C_c(K\G/K), the K-biinvariant continuous functions of compact support on G.

These functions form a * algebra under convolution with involution

often called the Hecke algebra for the pair (G, K).

Let A(K\G/K) denote the C* algebra generated by the operators ρ(F) on H₁.

The pair (G, K) is said to be a Gelfand pair if one, and hence all, of the following algebras are commutative:

Since A(K\G/K) is a commutative C* algebra, by the Gelfand–Naimark theorem it has the form C₀(X), where X is the locally compact space of norm continuous * homomorphisms of A(K\G/K) into C.

A concrete realization of the * homomorphisms in X as K-biinvariant uniformly bounded functions on G is obtained as follows.

Because of the estimate

the representation π of C_c(K\G/K) in A(K\G/K) extends by continuity to L1(K\G/K), the * algebra of K-biinvariant integrable functions. The image forms a dense * subalgebra of A(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is also continuous for the norm ||·||₁. Since the Banach space dual of L1 is L∞, it follows that

for some unique uniformly bounded K-biinvariant function h on G. These functions h are exactly the zonal spherical functions for the pair (G, K).