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Transactions of the American Mathematical Society

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On Gelfand pairs associated with solvable Lie groups

Authors: Chal Benson, Joe Jenkins and Gail Ratcliff
Journal: Trans. Amer. Math. Soc. 321 (1990), 85-116
MSC: Primary 22E25; Secondary 22D25, 43A20
MathSciNet review: 1000329
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Abstract: Let $ G$ be a locally compact group, and let $ K$ be a compact subgroup of $ {\operatorname{Aut}}(G)$, the group of automorphisms of $ G$. There is a natural action of $ K$ on the convolution algebra $ {L^1}(G)$, and we denote by $ L_K^1(G)$ the subalgebra of those elements in $ {L^1}(G)$ that are invariant under this action. The pair $ (K,G)$ is called a Gelfand pair if $ L_K^1(G)$ is commutative. In this paper we consider the case where $ G$ is a connected, simply connected solvable Lie group and $ K \subseteq {\operatorname{Aut}}(G)$ is a compact, connected group. We characterize such Gelfand pairs $ (K,G)$, and determine a moduli space for the associated $ K$-spherical functions.

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