Some spherical analysis related to the pairs (U(n), H n ) and (U(p,q), H n ), p + q = n

Abstract

In this paper, we define the normalized spherical transform associated with the generalized Gelfand pair (U(p,q),H n ), where H n is the Heisenberg group 2n + 1-dimensional and p+q=n. We show that the normalized spherical transform $\mathcal {F}(f)$ of a Schwartz function f on H n restricted to the spectrum of the Gelfand pair (U(n),H n ) is the Gelfand transform $\hat {g}$ of a radial Schwartz function g on H n . Moreover, by the Godement–Plancherel inversion formula the function g can be recovered explicitly from $\mathcal {F}(f)$ .