Two Permanents in the Universal Enveloping Algebras of the Symplectic Lie Algebras

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent, and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the columndeterminant by A. Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra. Introduction. In this paper we give new generators for the center of the universal enveloping algebra of the symplectic Lie algebra spN . These generators Dk(u) are expressed in terms of the “column-permanent,” and similar to the Capelli determinants, i.e., well-known central elements of the universal enveloping algebra of the general linear Lie algebra glN . As the key of the Capelli identity, these Capelli determinants are used to analyze the representations of glN acting via the polarization operators (see [Ca1], [Ca2], [HU]). One of the remarkable properties of the Capelli determinants is that we can easily calculate their eigenvalues on irreducible representations. It is also easy to calculate the eigenvalues of our central elements Dk(u). On the other hand, it is not so obvious that Dk(u) is actually central in the universal enveloping algebra. This fact can be proved as follows. In addition toDk(u), we consider another central element D k(u) expressed in terms of the “symmetrized permanent.” We can easily check that this D k(u) is central, but its eigenvalue is difficult to calculate. In spite of this difference, these Dk(u) and D ′ k(u) are actually equal. We will prove this 2000 Mathematics Subject Classification. Primary 17B35; Secondary 15A15.