Spherical Fourier transform of type \(\delta\). (Transformation de Fourier sphérique de type \(\delta\).) (French) Zbl 0945.43004

Let \(K\) be a compact subgroup of a locally compact unimodular group \(G\) and let \(\delta\in\widehat K\). A Banach space valued function on \(G\) is quasibounded if it is bounded by a seminorm (= a positive lower semicontinuous submultiplicative function) on \(G\). A spherical function of type \(\delta\) is a continuous quasibounded function \(\phi:G\to\operatorname{Hom} (E,E)\) \((E\) a finite-dimensional Banach space) such that (i) \(\phi(k\times k^{-1})=\phi(x)\), \(x\in G\), \(k\in K\); (ii) \(\chi_\delta*\phi=\phi=\phi*\chi_\delta\); and (iii) the map \(u_\phi\) defined by \(u_\phi(f)=\int_Gf(x)\phi(x^{-1})dx\) gives an irreducible representation of the algebra \({\mathcal K}^\#_\delta(G)\) of \(K\)-central continuous compactly supported functions \(f\) on \(G\) such that \(\overline \chi_\delta*f=f=f*\overline\chi_\delta\). The main result gives a one-to-one correspondence between spherical functions of type \(\delta\) and height \(m\) and \(m\)-dimensional irreducible representations of \({\mathcal K}^\#_\delta (G)\). This result and a notion of generalised Gelfand transform on the noncommutative algebra \({\mathcal K}^\#_\delta(G)\) combine to give a definition of spherical Fourier transform of type \(\delta\). When \((G,K)\) is a Gelfand pair and \(\delta\) is the trivial one-dimensional representation one gets the usual spherical Fourier transform.


43A90 Harmonic analysis and spherical functions
22D10 Unitary representations of locally compact groups
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