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.


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