Universal Capelli identities and quantum immanants for the queer Lie superalgebra

Abstract

We apply the recently introduced idempotents for the Sergeev superalgebra to construct quantum immanants for the queer Lie superalgebra ${\mathfrak{q}}_N$ as central elements of its universal enveloping algebra. We prove universal odd and even Capelli identities for ${\mathfrak{q}}_N$ and use them to calculate the images of the quantum immanants under the action of ${\mathfrak{q}}_N$ in differential operators. We show that the Harish-Chandra images of the quantum immanants coincide with the factorial Schur $Q$-polynomials.

25 pages