Intertwining operators and polynomials associated with the symmetric group

There is an algebra of commutative differential-difference operators which is very useful in studying analytic structures invariant under permutation of coordinates. This algebra is generated by the Dunkl operators , (i=1, ...,N, where (ij) denotes the transposition of the variablesx _i x _j andk is a fixed parameter). We introduce a family of functions {p }, indexed bym-tuples of non-negative integers = (₁, ..., _m ) formN, which allow a workable treatment of important constructions such as the intertwining operatorV. This is a linear map on polynomials, preserving the degree of homogeneity, for which ,i = 1, ...,N, normalized byV1=1 (seeDunkl, Canadian J. Math.43 (1991), 1213–1227). We show thatT _i p =0 fori>m, and where (₁, ₂, ..., _m ) is the partition whose parts are the entries of (That is, _{1 2} ... _m 0), = (₁, ..., _m ), _i=1 ^m _i = _i=1 ^m _m and the sorting of is a partition strictly larger than in the dominance order. This triangular matrix representation ofV allows a detailed study. There is an inner product structure on span {p } and a convenient set of self-adjoint operators, namelyT _ii , whereip p(₁, ...., _i + 1, ..., _m ). This structure has a bi-orthogonal relationship with the Jack polynomials inm variables. Values ofk for whichV fails to exist are called singular values and were studied byDe Jeu, Opdam, andDunkl in Trans. Amer. Math. Soc.346 (1994), 237–256. As a partial verification of a conjecture made in that paper, we construct, for anya=1,2,3,... such that gcd(N–m+1,a)<(N–m+1)/m andmN/2, a space of polynomials annihilated by eachT _i fork=–a/(N–m+1) and on which the symmetric groupS _N acts according to the representation (N–m, m).During the research for this paper, the author was partially supported by NSF grant DMS-9401429, and also held a Sesquicentennial Research Associateship at the University of Virginia