Decomposition of representations induced from uniform subgroups and the “Mackey machine”

Let G be a separable locally compact group with a discrete cocompact subgroup Γ; let N be a normal subgroup of G, and suppose that Γ₀ = Γ ∩ N is cocompact in N. Let ? be a finite-dimensional representation of Γ, and let τ = Ind_{Γ → G ?}. We investigate the problem of determining the decomposition of τ, given the decomposition of the corresponding representation and certain other facts (essentially those used in the “Mackey machine”). The last section is devoted to a reciprocity theorem for τ.