Some Conjectures for Macdonald Polynomials of Type B, C, D

We present conjectures giving formulas for the Macdonald polynomials of type B, C, D which are indexed by a multiple of the first fundamental weight. The transition matrices between two different types are explicitly given. Introduction Among symmetric functions, the special importance of Schur functions comes from their intimate connection with representation theory. Actually the irreducible polynomial representations of GLn(C) are indexed by partitions λ = (λ1, . . . , λn) of length ≤ n, and their characters are the Schur functions sλ. In the eighties, I. G. Macdonald introduced a new family of symmetric functions Pλ(q, t). These orthogonal polynomials depend rationally on two parameters q, t and generalize Schur functions, which are obtained for t = q [9,10]. When the indexing partition is reduced to a row (k) (i.e. has length one), the Macdonald polynomial gk(q, t) of n variables x = (x1, . . . , xn) are given by their generating function n ∏ i=1 (tuxi; q)∞ (uxi; q)∞ = ∑ r≥0 ugr(x; q, t), with the standard notation (a; q)∞ = ∏∞ i=0(1 − aq ). Of course for t = q the complete functions s(r) = hr are recovered. A few years later, generalizing his previous work, Macdonald introduced another class of orthogonal polynomials, which are Laurent polynomials in several variables, and generalize the Weyl characters of compact simple Lie groups [11,12]. In the most simple situation of this new framework, a family P (R) λ (q, t) of polynomials, depending rationally on two parameters q, t, is attached to each root system R. Typeset by AMS-TEX 1 2 MICHEL LASSALLE These orthogonal polynomials are elements of the group algebra of the weight lattice of R, invariant under the action of the Weyl group. They are indexed by the dominant weights of R. When R is of type A, the orthogonal polynomials P (R) λ (q, t) correspond to the symmetric functions Pλ(q, t) previously studied in [9,10]. For t = q, they correspond to the Weyl characters χ (R) λ of compact simple Lie groups. This paper is only devoted to the Macdonald polynomials which are indexed by a multiple of the first fundamental weight ω1. Since H. Weyl [15], it is well known that χ (R) rω1 is given by (i) hr(X) + hr−1(X), when R = Bn, (ii) hr(X), when R = Cn, (iii) hr(X)− hr−2(X), when R = Dn, with X = (x1, . . . , xn, 1/x1, . . . , 1/xn). However, as far as the author is aware, no such result is known when t 6= q, and no explicit expansion is available for the Macdonald polynomials P (R) rω1 (q, t). The purpose of this paper is to present some conjectures generalizing the previous formulas. Actually this problem can be considered in a more general setting, allowing two distinct parameters t, T , each of which is attached to a length of roots. We give an explicit formula for P (R) rω1 (q, t, T ) when R is of type B,C,D, together with an explicit formula for the transition matrices between different types. The entries of these transition matrices appear to be fully factorized and reveal deep connections with basic hypergeometric series. We provide some support for these conjectures by showing that they are verified upon principal specialization. On the other hand, computer calculations show a very strong empirical evidence in their favor. 1. Macdonald polynomials In this section we introduce our notations, and recall some general facts about Macdonald polynomials. For more details the reader is referred to [11,12,13]. The most general class of Macdonald polynomials is associated with a pair of root systems (R, S), spanning the same vector space and having the same Weyl group, with R reduced. Here we shall only consider the case of a pair (R,R), with R of type B,C,D. Let V be a finite-dimensional real vector space endowed with a positive definite symmetric bilinear form 〈u, v〉. For all v ∈ V , we write |v| = 〈v, v〉 , and v = 2v/|v|. MACDONALD POLYNOMIALS 3 Let R ⊂ V be a reduced irreducible root system, W the Weyl group of R, R the set of positive roots, {α1, . . . , αn} the basis of simple roots, and R = {α | α ∈ R} the dual root system. Let Q = ∑n i=1 Z αi and Q + = ∑n i=1 N αi be the root lattice of R and its positive octant. Let P = {λ ∈ V | 〈λ, α〉 ∈ Z ∀α ∈ R} and P = {λ ∈ V | 〈λ, α〉 ∈ N ∀α ∈ R} be the weight lattice of R and the cone of dominant weights. A basis of Q is formed by the simple roots αi. A basis of P is formed by the fundamental weights ωi defined by 〈ωi, αj 〉 = δij . A partial order is defined on P by λ ≥ μ if and only λ− μ ∈ Q. Let A denote the group algebra over R of the free Abelian group P . For each λ ∈ P let e denote the corresponding element of A, subject to the multiplication rule ee = e. The set {e, λ ∈ P} forms an R-basis of A. The Weyl group W acts on P and on A. Let A denote the subspace of W invariants in A. Such elements are called “symmetric polynomials”. There are two important examples of a basis of A . The first one is given by the orbit-sums mλ = ∑