A spherical transform on Schwartz functions on the Heisenberg group associated to the action of U(p,q)

Abstract

Abstract. Let S(H n ) be the space of Schwartz functions on the Heisenberg group H n . We define a spherical transform on S(H n ) associated to the action (by automorphisms) of U (p, q) on H n , p + q = n. We determine its kernel and image and obtain an inversion formula analogous to the Godement-Plancherel formula.1. Introduction. Let n ≥ 2 and let p, q be natural numbers such that p + q = n. Let H n be the Heisenberg group defined bywhereFor x = (x 1 , . . . , x n ) ∈ R n , we write x = (x ′ , x ′′ ) with x ′ ∈ R p , x ′′ ∈ R q . So, R 2n can be identified with C n via the mapIn this setting, the form − Im B(z, w) agrees with the standard symplectic form on R 2(p+q) , and the vector fieldsform a standard basis for the Lie algebra h n of H n . Thus H n can be viewed as R n × R n × R via the map (x, y, t) → (ϕ(x, y), t). From now on, we will use freely this identification. Let S(H n ) be the Schwartz space on H n and let S ′ (H n ) be the space of corresponding tempered distributions. Consider the action of U (p, q) on H n given by g · (z, t) = (gz, t) (note that since we have assumed that p, q ≥ 1, U (p, q) is noncompact). So U (p, q) acts on L 2 (H n ), S(H n ) and S ′ (H n ) in 2000 Mathematics Subject Classification: Primary 43A80; Secondary 22E25.