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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On Gel′fand pairs associated with solvable Lie groups
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by Chal Benson, Joe Jenkins and Gail Ratcliff PDF
Trans. Amer. Math. Soc. 321 (1990), 85-116 Request permission

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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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 321 (1990), 85-116
  • MSC: Primary 22E25; Secondary 22D25, 43A20
  • DOI: https://doi.org/10.1090/S0002-9947-1990-1000329-0
  • MathSciNet review: 1000329