This research reveals the commutativity of Gelfand pairs in weakly symmetric spaces, suggesting significant extensions of prior findings.
Let (M, G, ) be a Riemannian weakly symmetric space.Fix a base point x 0 M and denote by H to be the compact isotropy subgroup of G at x 0 .It is proven that L 1 (H \ G/H) is commutative, i.e. (G, H) is a Gelfand pair.This extends . Cartan's result for Riemannian symmetric spaces.Conversely, if (G, H) is a Riemannian weakly symmetric pair,
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H. Nguyen (1999) studied this question.
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