ORTHOGONAL POLYNOMIALS OF TYPE-A AND TYPE-B AND RELATED CALOGERO MODELS

There are examples of Calogero-Sutherland models associated to the Wey l groups of type A and B. When exchange terms are added to the Hamilto nians the systems have non-symmetric eigenfunctions, which can be expr essed as products of the ground state with members of a family of orth ogonal polynomials. These polynomials can be defined and studied by us ing the differential-difference operators introduced by the author in Trans. Am. Math. Sec. 311, 167-183 (1989). After a description of know n results, particularly from the works of Baker and Forrester, and Sah i; there is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms mon omially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group ther e is a skew operator on polynomials which leads to evaluation at (1, 1 ,..., 1) of the quotient of the unique skew polynomial in a given irre ducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polyno mials for the type B Weyl group with an emphasis on the Hermite-type p olynomials. These can be expressed by using the generalized binomial c oefficients. A complete basis of eigenfunctions of Yamamoto's B-N spin Calogero model is obtained by multiplying these polynomials by the gr ound state.