A family of generalized Gelfand pairs attached to $3$-step nilpotent Lie groups

Abstract

It is well known that if $(K\ltimes N, K)$ is a Gelfand pair with $N$ a nilpotent Lie group and $K$ a compact subgroup of the automorphism group of $N$, then $N$ is at most $2$-step. The notion of Gelfand pairs is extended to the notion of generalized Gelfand pairs, where $K$ is not necessarily compact. Gallo and Saal constructed an example of a generalized Gelfand pair $(K\ltimes N, K)$ with $N$ $3$-step and $K$ non-compact. In this work, we construct a family of generalized Gelfand pairs $(K_d \ltimes N_d, K_d)_{d\ge 1}$, where $N_d$ is $3$-step and $K_d$ is isomorphic to $\mathbb{R}^{d+1}$; the case of $d=1$ recovers the example of Gallo and Saal.