The Space of Bounded Spherical Functions on the Free 2-Step Nilpotent Lie Group

Abstract

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 Δ(K, N) of bounded spherical functions for such a Gelfand pair and a set \( {\user1{\mathcal{A}}}{\left( {K,N} \right)} \) of K-orbits in the dual \( \mathfrak{n}* \) of the Lie algebra for N. The construction involves an application of the Orbit Method to spherical representations of K ⋉ N. We conjecture that the correspondence \( \Delta {\left( {K,N} \right)} \leftrightarrow {\user1{\mathcal{A}}}{\left( {K,N} \right)} \) 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 Δ(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.