The spherical transform of any \(k\)-type in a locally compact group (Q2892859)

The paper is devoted to the development and study of a spherical transform on the convolution algebra \(\mathcal{C}_{c,\delta}(G)\) of all continuous functions \(f\) with compact support on \(G\) such that \(\overline{\chi}_{\delta}\ast f= f \ast \overline{\chi}_{\delta}=f\), where \(G\) is a locally compact group and \(K\) is a compact subgroup of \(G\). Here, \(\chi_{\delta}\) denotes the character of a unitary irreducible representation of \(K\) times its dimension.NEWLINENEWLINEOne of the main results of the work is an inversion formula for the spherical transform by using the Fourier inversion formula in \(G\). The case of the group \(G=SU(2,1)\) and the compact subgroup \(K=U(2)\) is discussed in detail. The last part of the paper is devoted to the spherical transform in the complex hyperbolic plane. The authors give explicit expressions for the spherical transform and they compute all irreducible unitary spherical functions. Moreover, they determine all irreducible positive definite spherical functions associated with unitary principal series representations of \(SU(2,1)\) and with discrete series representations. Their description is used to give an explicit expression for the inversion formula for the spherical transform in terms of the matrix hypergeometric function \(_2H_1\).