## 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.

### MSC:

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

 [1] Barut, A.O. and Raczka, R.. Theory of group representations and applications. Polish Scientific PublishersWarszawa1980. · Zbl 0471.22021 [2] Dieudonne, J. : Eléments d’Analyse Tome VIGauthier-Villars, Paris 6è , 1971. [3] Kangni, K. : Transformation de Fourier et Représentation unitaire sphériques de type d. Thèse de Doctorat de 3è cycle, FAST, Université d’Abidjan. Nov. 1994. [4] Mackey, G.W. : Infinite-dimensional group representationsAnnals of Mathematics Vol. 54, N°1, November 1951. [5] Mautner, F.I. : Unitary representations of locally compact groups II. Annals of Mathematics, Vol. 52, n°3, November 1950. · Zbl 0039.02201 [6] Sugiura, M. : Unitary représentation and HarmonicAnalysis Kodansha Scientific Books1928. · Zbl 0344.22001 [7] Warner, G. : Harmonie analysis on semi-simple Lie Group. Tome ISpringer-VerlagBerlinHeidelberg New-York1972. · Zbl 0265.22020 [8] Warner, G. : Harmonic Analysis on semi-simple Lie Groups. Tome II. Springer-VerlagBerlinHeidelberg New-York1972. · Zbl 0265.22021 [9] Wers, G.L. -Boothby, W.H. : Symmetric Spaces. Department of Mathematics, Washington UniversityST. Louis Missouri1972.
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