Harmonic analysis of the space of Sa×Sb×Sc-invariant vectors in the irreducible representations of the symmetric group

In this paper, we find an orthogonal basis for the Sa×Sb×Sc-invariant vectors in the irreducible representations S(α,β,γ) of the symmetric group. The basis chosen is part of a Gel'fand basis (or adapted basis) coming from the chain of subgroups Sa+b+c>Sa+b×Sc>Sa×Sb×Sc. This is a generalization and a completion of the work of Dunkl [Pacific J. Math. 92 (1981) 57-71], who considered the Sa×Sb×Sc-invariant vectors in S(N−k,k).