Symmetric and non-symmetric quantum Capelli polynomials

Abstract

We introduce families of symmetric and non-symmetric polynomials (the quantum Capelli polynomials) which depend on two parameters q and t. They are defined in terms of vanishing conditions. In the differential limit $(q=t^{\alpha }\mathrm{and}t\to 1)$ they are related to Capelli identities. It is shown that the quantum Capelli polynomials form an eigenbasis for certain q-difference operators. As a corollary, we obtain that the top homogeneous part is a symmetric/non-symmetric Macdonald polynomial. Furthermore, we study the vanishing and integrality properties of the quantum Capelli polynomials.