Two determinants in the universal enveloping algebras of the orthogonal Lie algebras

This paper gives a direct proof for the coincidence of the following two central elements in the universal enveloping algebra of the orthogonal Lie algebra: an element recently given by A. Wachi in terms of the column-determinant in a way similar to the Capelli determinant, and an element given by T. Umeda and the author in terms of the symmetrized determinant. The fact that these two elements actually coincide was shown by A. Wachi, but his observation was based on the following two non-trivial results: (i) the centrality of the first element, and (ii) the calculation of the eigenvalue of the second element. The purpose of this paper is to prove this coincidence of two central elements directly without using these (i) and (ii). Conversely this approach provides us new proofs of (i) and (ii). A similar discussion can be applied to the symplectic Lie algebras.