A Central Element in the Universal Enveloping Algebra of Type Dn via Minor Summation Formula of Pfaffians

It is well known that the universal enveloping algebra U (o 2n ) of the orthogonal Lie algebra o 2n of size even has a central element given explicitly in terms of Pfaffian of a certain matrix which is alternating along the antidiagonal (we call such a matrix anti-alternating for short) whose entries are in U (o 2n ).In this paper, we establish a minor summation formula of Pfaffian of the noncommutative anti-alternating matrix, as well as of commutative antialternating matrix.As an application, we show that the eigenvalues of the Pfaffian-type central element on the highest weight representations of o 2n can be easily computed.