Quantum immanants and higher Capelli identities

We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants. We express them in terms of generatorsE _ij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants. The author is supported by the International Science Foundation and the Russian Fundamental Research Foundation.