Zonal Spherical Function - Harish-Chandra's Formula

Harish-Chandra's Formula

If G is a non-compact semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in P, then A is isomorphic to its Lie algebra under the exponential map and, as a further generalisation of the polar decomposition of matrices, every element of P is conjugate under K to an element of A, so that

G =KAK.

There is also an associated Iwasawa decomposition

G =KAN,

where N is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by A. Thus S=AN is a closed solvable subgroup of G, the semidirect product of N by A, and G = KS.

If α in Hom(A,T) is a character of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding unitary induced representation σ of G on L2(G/S) = L2(K), a so-called (spherical) principal series representation.

This representation can be described explicitly as follows. Unlike G and K, the solvable Lie group S is not unimodular. Let dx denote left invariant Haar measure on S and Δ_S the modular function of S. Then

The principal series representation σ is realised on L2(K) as

where

is the Iwasawa decomposition of g with U(g) in K and X(g) in S and

for k in K and x in S.

The representation σ is irreducible, so that if v denotes the constant function 1 on K, fixed by K,

defines a zonal spherical function of G.

Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function

as an integral over K.

Harish-Chandra proved that these zonal spherical functions exhaust the characters of the C* algebra generated by the C_c(K \ G / K) acting by right convolution on L2(G / K). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, where s is in the Weyl group of A

the quotient of the normaliser of A in K by its centraliser, a finite reflection group.

It can also be verified directly that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of G-invariant differential operators on G/K and their simultaneous eigenfunctions (see below). In the case of complex semisimple groups, Harish-Chandra and Felix Berezin realised independently that the formula simplified considerably and could be proved more directly.

The remaining positive-definite zonal spherical functions are given by Harish-Chandra's formula with α in Hom(A,C*) instead of Hom(A,T). Only certain α are permitted and the corresponding irreducible representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied by Bargmann (1947) for G = SL(2,R) and by Harish-Chandra (1947) and Gelfand & Naimark (1947) for G = SL(2,C). Subsequently in the 1960s, the construction of a complementary series by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, Elias Stein and Bertram Kostant. Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on G (or G / K).