A generalized Macdonald operator

Details A generalized Macdonald operator A generalized Macdonald operator

2010-09-22

arxiv.org/abs/1009.4486v1

International Mathematics Research Notices, Vol. 2011, No. 15, pp. 3560-3574

Mathematics

Representation Theory

11 pages. To appear in Int. Math. Res. Not. IMRN

Scientific paper

10.1093/imrn/rnq233

We present an explicit difference operator diagonalized by the Macdonald polynomials associated with an (arbitrary) admissible pair of irreducible reduced crystallographic root systems. By the duality symmetry, this gives rise to an explicit Pieri formula for the Macdonald polynomials in question. The simplest examples of our construction recover Macdonald's celebrated difference operators and associated Pieri formulas pertaining to the minuscule and quasi-minuscule weights. As further by-products, explicit expansions and Littlewood-Richardson type formulas are obtained for the Macdonald polynomials associated with a special class of small weights.

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