Journal of Lie Theory

no. 2
Spaces of Bounded Spherical Functions for Irreducible Nilpotent Gelfand Pairs: Part II
Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 367-392
In prior work an orbit method, due to Pukanszky and Lipsman, was used to produce an injective mapping $\Psi\colon \Delta(K,N)\rightarrow\mathfrak{n}^*/K$ from the space of bounded $K$-spherical functions for a nilpotent Gelfand pair $(K,N)$ into the space of $K$-orbits in the dual for the Lie algebra $\mathfrak{n}$ of $N$. We have conjectured that $\Psi$ is a topological embedding. In this paper we complete the proof of this conjecture under the hypothesis that $(K,N)$ is an {\it irreducible} nilpotent Gelfand pair. Following Part I of this work it remains to verify the conjecture in six exceptional cases from Vinberg's classification of irreducible nilpotent Gelfand pairs.
DOI: 10.5802/jolt.1176
Classification: 22E30, 43A90
Keywords: Gelfand pairs, spherical functions, nilpotent Lie groups, orbit method 
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     author = {C. Benson and G. Ratcliff},
     title = {Spaces of {Bounded} {Spherical} {Functions} for {Irreducible} {Nilpotent} {Gelfand} {Pairs:} {Part} {II}},
     journal = {Journal of Lie Theory},
     pages = {367--392},
     year = {2021},
     volume = {31},
     number = {2},
     doi = {10.5802/jolt.1176},
     zbl = {1493.22006},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1176/}
}
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C. Benson; G. Ratcliff. Spaces of Bounded Spherical Functions for Irreducible Nilpotent Gelfand Pairs: Part II. Journal of Lie Theory, Volume 31 (2021) no. 2, pp. 367-392. doi: 10.5802/jolt.1176

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