Wick theorem and matrix Capelli identity for quantum differential operators on Reflection Equation Algebras

Abstract: Quantum differential operators on Reflection Equation Algebras, corresponding to Hecke symmetries R were introduced in previous publications. In the present paper we are mainly interested in quantum analogs of the Laplace and Casimir operators, which are invariant with respect to the action of the Quantum Groups U_q(sl(N)), provided R is the Drinfeld-Jimbo matrix. We prove that any such operator maps the characteristic subalgebra of a Reflection Equation algebra into itself. Also, we define the notion of normal ordering for the quantum differential operators and prove an analog of the Wick theorem. As an important corollary we find a universal matrix Capelli identity generalizing the results of [Ok2] and [JLM].