Induced representations of infinite-dimensional groups

Induced representations  were introduced and studied by F.G. Frobenius in 1898  for finite groups
and developed by G.W. Mackey (1949)  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 of the initial space X=H\G.  A.A. Kirillov's orbit method (1962) describes all irreducible unitary representations of the finite-dimensional nilpotent  group G in terms of induced representations  associated  with orbits in coadjoint action of the group G in a
dual  space g* of the Lie algebra  g. As the illustration we  start to develop an analog of the
orbit method for  infinite-dimensional  ``nilpotent'' group of upper triangular matrices.