Symmetric and non-symmetric quantum Capelli polynomials (Q1359437)

Summary: 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 (\(q= t^\alpha\) 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.