The fundamental representation of the affine Lie algebra \(A_{n-1}^{(1)}\) and the Feynman path integral (Q1917229)

In a series of previous works carried out by the authors' group, some Feynman path integrals on the coadjoint orbits of noncompact Lie groups are computed [e.g., \textit{T. Hashimoto}, Hiroshima Math. J. 23, 607-627 (1993; Zbl 0838.22011)]. They showed that the path integrals give the kernel functions of the irreducible unitary representations. Their next problem is to carry out this idea to infinite dimensional Lie groups, where one encounters a new difficulty of nonexistence of the quasi-invariant measure on the coadjoint orbits. However, it is known that the irreducible representations of certain Kac-Moody Lie groups are realized by using the complex white noise on a coadjoint orbit. In this paper the authors utilize this fact for constructing the fundamental representation of the affine Kac-Moody Lie algebra \(A^{(1)}_{n-1}\) by means of the path integral.