Spaces of bounded spherical functions on Heisenberg groups: part I

作者：Benson Chal^*; Ratcliff Gail

来源：Annali di Matematica Pura ed Applicata, 2015, 194(2): 321-342.

DOI：10.1007/s10231-013-0377-z

• Gelfand pairs
• Multiplicity free actions
• Spherical functions
• Nilpotent Lie groups
• Orbit Method

Consider a linear multiplicity free action by a compact Lie group K on a finite dimensional hermitian vector space V. Letting K act on the associated Heisenberg group, H-V = V x R yields a Gelfand pair. In previous work, we have applied the Orbit Method to produce an injectivemapping Psi from the space Delta( K, H-V) of bounded K-spherical functions on HV to the space h(V)K of K-orbits in the dual of the Lie algebra for H-V. We have shown that Psi is a homeomorphism onto its image provided that K : V is a "well-behaved" multiplicity free action. In this paper, we prove that K : V is well-behaved whenever K acts irreducibly on V. Thus, if K : V is an irreducible multiplicity free action then Psi : Delta( 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.

• 出版日期2015-4