Symmetric and Non-Symmetric Quantum Capelli Polynomials

Generalizing the classical Capelli identity has recently attracted a lot of interest ([HU], [Ok], [Ol], [Sa], [WUN]). In several of these papers it was realized, in various degrees of generality, that Capelli identities are connected with certain symmetric polynomials which are characterized by their vanishing at certain points. From this point of view, these polynomials have been constructed by Sahi [Sa] and were studied in [KS]. The purpose of this paper is twofold: we quantize the vanishing condition in a rather straightforward manner and obtain a family of symmetric polynomials which is indexed by partitions and which depends on two parameters q, t. As in [KS], their main feature is that they are non-homogeneous and one of our principal results states that the top degree terms are the Macdonald polynomials. It is an interesting problem whether these quantized Capelli polynomials are indeed connected with quantized Capelli identities (see [WUN]) as it is in the classical case. But the main progress over [KS] is the introduction of a family of non-symmetric polynomials which are also defined by vanishing conditions. They are non-homogeneous and their top degree terms turns out to be the non-symmetric Macdonald polynomials. To prove this, we introduce certain difference operators of Cherednik type of which our polynomials are a simultaneous eigenbasis. Because of these operators, the non-symmetric functions are much easier to handle than the symmetric ones. Moreover, the latter can be obtained by a simple symmetrization process. More specifically, the non-symmetric vanishing conditions are as follows: For λ ∈ Λ := N let |λ| := ∑ λi and let wλ be the shortest permutation such that w −1 λ (λ) is a partition (i.e., a non-increasing sequence). Let q and t be two formal parameters and