The space of bounded spherical functions on the free 2-step nilpotent Lie group

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.