Unitary Representations of Heisenberg Groups

초록

The aim of this paper is to investigate the unitary representations of the Heisenberg group $H_R^{(g,h)}$ explicitly. In particular, we compute the coadjoint orbits of the Heisenberg group (due to A.A. Kirilov). Recently we realized that we need to investigate the group $H_R^{(g,h)}$ for the study of Jacobi forms, degeneration of abelian varieties and so on. I would like to mention that recently a Heisenberg group was paid to an attention by some differential geometers, e.g., M.L. Gromov, in the sense of a parabolic geometry. A Heisenberg group is regarded as a principal fibre bundle over an Euclidean space with a vector space or a circle as fibres and may be also regarded as the boundary of a complex ball. The geometry of this group is quite different from that of an Euclidean space.