INTERTWINING-OPERATORS AND POLYNOMIALS ASSOCIATED WITH THE SYMMETRICAL GROUP

There is an algebra of commutative differential-difference operators w hich is very useful in studying analytic structures invariant under pe rmutation of coordinates. This algebra is generated by the Dunkl opera tors T-i:= partial derivative/partial derivative x(i) + k Sigma(j not equal i) 1-(ij)/x(i)x(j), (i = 1,...,N, where (ij) denotes the transpo sition of the variables x(i) x(j) and k is a fixed parameter). We intr oduce a family of functions {p(alpha)}, indexed by m-tuples of non-neg ative integers alpha = (alpha(1),...,alpha(m)) for m less than or equa l to N, which allow a workable treatment of important constructions su ch as the intertwining operator V: This is a linear map on polynomials , preserving the degree of homogeneity, for which TiV = V partial deri vative/partial derivative x(i), i = 1,...,N, normalized by V1 = 1 (see DUNKL, Canadian J. Math. 43 (1991), 1213-1227). We show that T-ip alp ha = 0 for i > m, and V(X-1(alpha 1) ... x(m)(alpha m)) = lambda(1)!la mbda(2)!...lambda(m)!/(Nk+1)(lambda 1) (Nk-k+1)(lambda 2) ... (Nk-(M-1 )k+1)(lambda m beta)p alpha+Sigma A beta alpha p beta, where (lambda(1 ),lambda(2),...,lambda(m)) is the partition whose parts are the entrie s of alpha (That is, lambda(1) greater than or equal to lambda(2) grea ter than or equal to ... lambda(m) greater than or equal to 0), beta = (beta(1),..., beta(m)), Sigma(i=1)(m) beta(i) = Sigma(i=1)(m) alpha(i ) and the sorting of beta is a partition strictly larger than lambda i n the dominance order. This triangular matrix representation of V allo ws a detailed study. There is an inner product structure on span{p(alp ha)} and a convenient set of self-adjoint operators, namely T-i rho i, where rho(i)p(alpha) := p((alpha i,...,alpha i+1,...,alpha m)). his s tructure has a bi-orthogonal relationship with the Jack polynomials in m variables. Values of k for which V fails to exist are called singul ar values and were studied by DE JEU, OPDAM, and DUNKL in Trans. Amer. Math. Sec. 346 (1994), 237-256. As a partial verification of a conjec ture made in that paper, we construct, for any a = 1,2, 3,... such tha t gcd(N - m + 1, a) < <(N-m+1)/m and m less than or equal to N/2, a sp ace of polynomials annihilated by each T-i for k = = -a/(N - m + 1) an d on which the symmetric group S-N acts according to the representatio n (N - m, m).