caa a22 4500 605496609 CHVBK 20210128105157.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s10231-013-0377-z doi (NATIONALLICENCE)springer-10.1007/s10231-013-0377-z Spaces of bounded spherical functions on Heisenberg groups: part I [Elektronische Daten] [Chal Benson, Gail Ratcliff] Consider a linear multiplicity free action by a compact Lie group $$K$$ K on a finite dimensional hermitian vector space $$V$$ V . Letting $$K$$ K act on the associated Heisenberg group, $$H_V=V\times \mathbb{R }$$ H V = V × R yields a Gelfand pair. In previous work, we have applied the Orbit Method to produce an injective mapping $$\Psi $$ Ψ from the space $$\Delta (K,H_V)$$ Δ ( K , H V ) of bounded $$K$$ K -spherical functions on $$H_V$$ H V to the space $$\mathfrak{h }_V^*/K$$ h V ∗ / K of $$K$$ K -orbits in the dual of the Lie algebra for $$H_V$$ H V . We have shown that $$\Psi $$ Ψ is a homeomorphism onto its image provided that $$K:V$$ K : V is a "well-behaved” multiplicity free action. In this paper, we prove that $$K:V$$ K : V is well-behaved whenever $$K$$ K acts irreducibly on $$V$$ V . Thus, if $$K:V$$ K : V is an irreducible multiplicity free action then $$\Psi :\Delta (K,H_V)\rightarrow \mathfrak{h }_V^*/K$$ Ψ : Δ ( K , H V ) → h V ∗ / K is a homeomorphism onto its image. Our proof involves case-by-case analysis working from the classification of irreducible multiplicity free actions. A sequel to this paper will extend these results to encompass non-irreducible actions. Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg, 2013 Gelfand pairs nationallicence Multiplicity free actions nationallicence Spherical functions nationallicence Nilpotent Lie groups nationallicence Orbit Method nationallicence Benson Chal Department of Mathematics, East Carolina University, 27858, Greenville, NC, USA aut Ratcliff Gail Department of Mathematics, East Carolina University, 27858, Greenville, NC, USA aut Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 321-342 0373-3114 194:2<321 2015 194 10231 https://doi.org/10.1007/s10231-013-0377-z text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10231-013-0377-z text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Benson Chal Department of Mathematics, East Carolina University, 27858, Greenville, NC, USA aut NATIONALLICENCE 700 1- Ratcliff Gail Department of Mathematics, East Carolina University, 27858, Greenville, NC, USA aut NATIONALLICENCE 773 0- Annali di Matematica Pura ed Applicata (1923 -) Springer Berlin Heidelberg 194/2(2015-04-01), 321-342 0373-3114 194:2<321 2015 194 10231 SWISSBIB 605496609