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 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.
Cite this article
F. Knop, Symmetric and non-symmetric quantum Capelli polynomials. Comment. Math. Helv. 72 (1997), no. 1, pp. 84–100