Induced representations of infinite-dimensional groups, I

Induced representations Ind(II)(G) S 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) over tilde = (H) over tilde\(G) over tilde of the initial space X = H\G (since the Haar measure on G does not exist) and extend the representation (S) over tilde of the subgroup (H) over tilde 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 in terms of induced representations associated with orbits in coadjoint action of the group G(n) in a dual space g(n)* of the Lie algebra g(n). 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 [GRAPHICS] infinite in both directions matrices. (C) 2014 Elsevier Inc. All rights reserved.