The $q$-immanants and higher quantum Capelli identities

We construct polynomials ${\mathbb{S}}_{\mu }(z)$ parameterized by Young diagrams $\mu$, whose coefficients are central elements of the quantized enveloping algebra ${\mathrm{U}}_q({\mathfrak{g}\mathfrak{l}}_n)$. 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 ${\mathfrak{g}\mathfrak{l}}_n$. We show that the Harish-Chandra image of ${\mathbb{S}}_{\mu }(z)$ is a factorial Schur polynomial. We also prove quantum analogues of the higher Capelli identities and derive Newton-type identities.