Journal of Lie Theory 30 (2020), No. 3, 779--810
Copyright Heldermann Verlag 2020
Spaces of Bounded Spherical Functions for Irreducible Nilpotent Gelfand Pairs: Part I
Chal Benson
Department of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
bensonf@ecu.edu
Gail Ratcliff
Department of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.
ratcliffg@ecu.edu
[Abstract-pdf]
\newcommand{\fn}{\mathfrak n} In prior work an orbit method, due to Pukanszky and Lipsman, was used to produce an injective mapping $\Psi\colon \Delta(K,N)\rightarrow \fn^*/K$ 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 $\fn$ of $N$. We have conjectured that $\Psi$ is a topological embedding. This has been proved for all pairs $(K,N)$ with $N$ a Heisenberg group. A nilpotent Gelfand pair $(K,N)$ is said to be {\em irreducible} if $K$ acts irreducibly on $\fn/[\fn,\fn]$. In this paper and its sequel we will prove that $\Psi$ is an embedding for all such irreducible pairs. Our proof involves careful study of the non-Heisenberg entries in Vinberg's classification of irreducible nilpotent Gelfand pairs. Part I concerns generalities and six related families of examples from Vinberg's list in which the center for $\fn$ can have arbitrarily large dimension.
Keywords: Gelfand pairs, spherical functions, nilpotent Lie groups, orbit method.
MSC: 22E30, 43A90.
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