Friedlander, Holley, William Grodzicki, Wayne Johnson, Gail Radcliff, Anna Romanov, Benjamin Strasser, and Brent Wessel. An Orbit Model for the Spectra of Nilpotent Gelfand Pairs. Transformation Groups 25 (2020): 859-886. https://link.springer.com/article/10.1007/s00031-019-09541-8

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 L1K(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 A(K, N) of K-orbits in the dual n* of the Lie algebra for N established in C. Benson, G. Ratcliff, The space of bounded spherical functions on the free 2-step nilpotent Lie group, Transform. Groups 13 (2008), no. 2, 243–281. 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 C. Benson, G. Ratcliff, The space of bounded spherical functions on the free 2-step nilpotent Lie group, Transform. Groups 13 (2008), no. 2, 243–281.

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