Contents

** The quasi-spherical transformations **

** Cornelie MITCHA MALANDA **

Université Marien NGOUABI BP : 69, Brazzaville, Congo

** Kinvi KANGNI **

Université Félix Houphouët Boigny,

22 BP 582 Abidjan 22 Abidjan, Côte d'Ivoire

*Mathematics Subject Classification:* **(MSC 2010)** 22E25, 22E27, 22E60

*Key words *: Quasi-Gelfand pair, quasi-spherical function

Abstract:

Let G be a locally compact group, K a compact subgroup of G. is a
Gelfand pair if the convolution subalgebra L^{1}^{♮} of L^{1} that are biinvariant under
the action K, is commutative.

Let L_{K}^{1} be the algebra of K-invariant L^{1}-functions on G. L^{1}^{♮} is isomorphic
to L_{K}^{1}(G), then L^{1}^{♮} is nilpotent is equivalent to the fact that L_{K}^{1}(G) is
nilpotent.

Let H_{n} be the (2n + 1)-dimensional Heisenberg group. The action of the compact
subgroup SO of the group of automorphisms Aut of H_{n}, is not of multiplicity
free [1] then L_{SO}^{1} is not commutative and the couple (H_{n},SO(n, ℝ) is not a
Gelfand pair.

In this work, we will interest in this case where the associated algebra is nilpotent but not commutative and we will construct and study the corresponding spherical transformation which will be called a quasi-spherical transformation.