Let 𝓢(Hₙ) be the space of Schwartz functions on the Heisenberg group Hₙ. We define a spherical transform on 𝓢(Hₙ) associated to the action (by automorphisms) of U(p,q) on Hₙ, p + q = n. We determine its kernel and image and obtain an inversion formula analogous to the Godement-Plancherel formula.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-2-5, author = {T. Godoy and L. Saal}, title = {A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q)}, journal = {Colloquium Mathematicae}, volume = {106}, year = {2006}, pages = {231-255}, zbl = {1115.43005}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-2-5} }
T. Godoy; L. Saal. A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q). Colloquium Mathematicae, Tome 106 (2006) pp. 231-255. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm106-2-5/