Let N be a connected and simply connected 2-step nilpotent Lie group and let K be a compact subgroup of Aut(N). We say that (K, N) is a Gelfand pair when the set of integrable K-invariant functions on N forms an abelian algebra under convolution. In this paper we construct a one-to-one correspondence between the set Delta(K, N) of bounded spherical functions for such a Gelfand pair and a set A(K, N) of K-orbits in the dual n* of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K x N. We conjecture that the correspondence Delta(K, N) <-> A(K, N) is a homeomorphism. Our main result shows that this is the case for the Gelfand pair given by the action of the orthogonal group on the free 2-step nilpotent Lie group. In addition, we show how to embed the space Delta(K; N) for this example in a Euclidean space by taking eigenvalues for an explicit set of invariant differential operators. These results provide geometric models for the space of bounded spherical functions on the free 2-step group.
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机译：令N为一个连通且简单连通的两步幂等李群，令K为Aut（N）的紧凑子群。我们说（K，N）是N上的可积K不变函数集在卷积下形成阿贝尔代数的Gelfand对。在本文中，我们构造了此类Gelfand对的有界球面函数集Delta（K，N）与该对偶n *中K轨道的集合A（K，N）的一对一对应关系。 N的李代数。该构造涉及将轨道方法应用于K x N的球面表示。我们推测对应关系Delta（K，N）<-> A（K，N）是同胚的。我们的主要结果表明，正交基对自由两步幂立李群的作用给出的Gelfand对就是这种情况。另外，我们展示了如何通过获取明确的不变微分算子的特征值，将示例Delta（K; N）嵌入到欧几里得空间中。这些结果为自由两步组上有界球面函数的空间提供了几何模型。
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