Some Results on Gelfand Paires

A Gelfand pair is a pair (G; K) consisting of a group G and a subgroup K (called an Euler subgroup of G) thatsatisfies a certain property on restricted representations. When G is a locally compact topological group and Kis a compact subgroup, (G; K) is a Gelfand pair if and only if the algebra of (K; K)−double invariant compactlysupported continuous functions(measures) on G with multiplication defined by convolution is commutative. Instudying the concept of Gelfand pairs, the identification of spherical functions is of particular importance. In thispaper, the spherical functions of Gelfand pair (G; K) in subspace E۱ of L۱(G) containing functions of form f ∗ f~ isintroduced, where f belonges to Cc(G)(The convolution algebra of continuous, complex-valued functions on G withcompact support). Also the characters of E۱# have been identified. Finally, by introducing the space Gb# includingthe bi-K-invariant unitary characters and the space Gd# including bounded spherical functions, the locally compactgroups G relatively to Gb#=Gd#, are characterized.

Keywords:

Gelfand pair , spherical function , locally compact group , character , irreducible representation