The Space of Bounded Spherical Functions on the Free Two Step Nilpotent Lie Group

Let N be a connected and simply connected 2-step nilpotent Lie group and 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 ∆(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 nN . We conjecture that the correspondence ∆(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 ∆(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.