A representation-theoretic proof of the branching rule for Macdonald polynomials

We give a new representation-theoretic proof of the branching rule for Macdonald polynomials using the Etingof–Kirillov Jr. expression for Macdonald polynomials as traces of intertwiners of $U_q (\mathfrak{gl}_n)$ given in [11]. In the Gelfand–Tsetlin basis, we show that diagonal matrix elements of such intertwiners are given by application of Macdonald’s operators to a simple kernel. An essential ingredient in the proof is a map between spherical parts of double affine Hecke algebras of different ranks based upon the Dunkl–Kasatani conjecture of [8, 9, 13, 20].

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Accepted 17 September 2015

Published 8 July 2016