## Harmonic analysis on the infinite symmetric group. A deformation of the regular representation.(English. Abridged French version)Zbl 0796.43005

Let $$S(n)$$ be the symmetric group of $$\{1,2,\dots,n\}$$ and $$S = \bigcup S(n)$$ the group of finite permutations of $$\{1,2,\dots\}$$. The authors construct a family of unitary representations $$T_ z$$ of $$S$$, parametrized by complex numbers $$z$$. These representations can be viewed as deformations of the regular $$T_ \infty$$ of $$S$$. They exhibit direct integral decompositions of these representations.

### MSC:

 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 20C32 Representations of infinite symmetric groups