Spherical Function of Type \(\delta \) on Groupoids

Let \(\vartheta \) be a locally compact groupoid and K a compact subgroup of \( \vartheta .\) In this chapter, we give a generalization of Fourier transform on the convolution algebra \(I_{c,\delta }(\vartheta )\) of all continuous functions f with compact support on \(\vartheta \) such that \(\chi _{\delta }\ast f=_{\delta }f=f,\forall \delta \in \widehat {K}\) ; here, \(\chi _{\delta }\) denotes the character of a unitary irreducible representation of K times its dimension. We obtain an inversion formula for the spherical transform by using the Fourier inversion formula in \(\vartheta .\)