Journal of Lie Theory 31 (2021), No. 2, 367--392
Copyright Heldermann Verlag 2021
Spaces of Bounded Spherical Functions for Irreducible Nilpotent Gelfand Pairs: Part II
Chal Benson
Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
bensonf@ecu.edu
Gail Ratcliff
Dept. of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
ratcliffg@ecu.edu
[Abstract-pdf]
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.
Keywords: Gelfand pairs, spherical functions, nilpotent Lie groups, orbit method.
MSC: 22E30, 43A90.
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