Classical Invariant Theory and the Virasoro Algebra

Abstract

The Virasoro algebra (a certain central extension of the Lie algebra of vector fields on the circle) has come to play a more important role in such fields as quantum field theory and number theory. In [3], Wakimoto and Yamada gave a new relationship with classical invariant theory. The purpose of this note is to give a more direct relationship between highest weight vectors for the Virasoro algebra and characters of the unitary groups U(n), n⩾1. In particular, our result gives (in principle) a method of calculating ail highest weight vectors for all Verma modules for the Virasoro algebra. However, the combinatorial problems in the general case are very difficult. In the special case studied in [3], we give a simple proof of the main result announced in [3].