The $q$-immanants and higher quantum Capelli identities

Abstract

We construct polynomials ${\mathbb{S}}_{\hat{I}}¼(z)$ parameterized by Young diagrams $\hat{I}¼$, whose coefficients are central elements of the quantized enveloping algebra ${\mathrm{U}}_q({\mathfrak{gl}}_n)$. Their constant terms coincide with the central elements provided by the general construction of Drinfeld and Reshetikhin. For another special value of $z$, we get $q$-analogues of Okounkov's quantum immanants for ${\mathfrak{gl}}_n$. We show that the Harish-Chandra image of ${\mathbb{S}}_{\hat{I}}¼(z)$ is a factorial Schur polynomial. We also prove quantum analogues of the higher Capelli identities and derive Newton-type identities.

19 pages, more detailed proofs are given, references extended