Zonal Spherical Function - Gelfand Pairs

Gelfand Pairs

If G is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, G has a maximal compact subgroup, unique up to conjugation. In this case K is connected and the quotient G/K is diffeomorphic to a Euclidean space. When G is in addition semisimple, this can be seen directly using the Cartan decomposition associated to the symmetric space G/K, a generalisation of the polar decomposition of invertible matrices. Indeed if τ is the associated period two automorphism of G with fixed point subgroup K, then

where

Under the exponential map, P is diffeomorphic to the -1 eigenspace of τ in the Lie algebra of G. Since τ preserves K, it induces an automorphism of the Hecke algebra C_c(K\G/K). On the other hand, if F lies in C_c(K\G/K), then

F(τg) = F(g−1),

so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple,

• the Hecke algebra is commutative

• (G,K) is a Gelfand pair.

More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:

• G is a unimodular locally compact group;
• K is a compact subgroup arising as the fixed points of a period two automorphism τ of G;
• G =K·P (not necessarily a direct product), where P is defined as above.

The two most important examples covered by this are when:

• G is a compact connected semisimple Lie group with τ a period two automorphism;
• G is a semidirect product, with A a locally compact Abelian group without 2-torsion and τ(a· k)= k·a−1 for a in A and k in K.

The three cases cover the three types of symmetric spaces G/K:

1. Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G;
2. Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G;
3. Euclidean type, when A is a finite-dimensional Euclidean space with an orthogonal action of K.