Wreath Macdonald polynomials at 𝑞=𝑡 as characters of rational Cherednik algebras

Abstract

Using the theory of Macdonald [Symmetric functions and Hall polynomials, The Clarendon Press, Oxford University Press, New York, 1995], Gordon [Bull. London Math. Soc. 35 (2003), pp. 321–336] showed that the graded characters of the simple modules for the restricted rational Cherednik algebra by Etingof and Ginzburg [Invent. Math. 147 (2002), pp. 243–348] associated to the symmetric group S n \mathfrak {S}_n are given by plethystically transformed Macdonald polynomials specialized at q = t q=t . We generalize this to restricted rational Cherednik algebras of wreath product groups C ℓ ≀ S n C_\ell \wr \mathfrak {S}_n and prove that the corresponding characters are given by a specialization of the wreath Macdonald polynomials defined by Haiman in [Combinatorics, symmetric functions, and Hilbert schemes, Int. Press, Somerville, MA, 2003, pp. 39–111].