Induced representations of infinite-dimensional groups, I

Induced representations IndHGS were introduced and studied by F.G. Frobenius [8] for finite groups and developed by G.W. Mackey [22,23] for locally compact groups. We generalize the Mackey construction for infinite-dimensional groups. To do this, we construct some G-quasi-invariant measures on an appropriate completion X˜=H˜\G˜ of the initial space X=H\G (since the Haar measure on G does not exist) and extend the representation S of the subgroup H to the representation S˜ of the corresponding completion H˜. Kirillov's orbit method [9] describes all irreducible unitary representations of the finite-dimensional nilpotent group Gn in terms of induced representations associated with orbits in coadjoint action of the group Gn in a dual space gn⁎ of the Lie algebra gn. The induced representation defined in such a way allows us to start to develop an analog of the orbit method for the infinite-dimensional “nilpotent” group B0Z=lim→nG2n−1 of infinite in both directions matrices.