Let N be a connected and simply connected nilpotent Lie group, and let K be a subgroup of the automorphism group of N. We say that the pair (K, N) is a nilpotent Gelfand pair if \( {L}_K^1(N) \) is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs (K, N) where the K-orbits in the center of N have a one-parameter cross section and satisfy a certain non-degeneracy condition. More specifically, we show that the one-to-one correspondence between the set Δ(K, N) of bounded K-spherical functions on N and the set \( \mathcal{A} \)(K, N) of K-orbits in the dual 𝔫* of the Lie algebra for N established in [BR08] is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for N a free group and N a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs in [BR08].

令N为一个连通且简单连通的幂等李群，令K为N的自同构群的子群。我们说，如果\（{L} _K ^ 1（N）\）是卷积下的阿贝尔代数，则对（K，N）是幂幂Gelfand对。在本文中，我们为零能Gelfand对（K，N）的Gelfand谱建立了几何模型，其中N中心的K轨道具有一个单参数横截面并满足一定的非简并性条件。更具体地说，我们表明集合Δ（K，N）的有界ķ上-spherical功能Ñ和设定\（\ mathcal {A} \） （ķ，Ñ）的ķ在双𝔫*李代数为-orbits Ñ成立[BR08]为同胚这类零能Gelfand对。先前已针对N个自由基团和N个海森堡基团显示了该结果，并推测[BR08]中的所有幂等Gelfand对都成立。