Little group method for smooth representations of finite length (Q1913354)

Let \(G=H \times_sA\) be the semidirect product Lie group formed by a real Lie group \(H\) acting linearly on a real vector group \(A\), so that the orbits of the dual action of \(H\) on \(A^*\) are locally closed. The author uses the Mackey little group method to obtain a functorial bijection between smooth representations of \(G\), admitting a finite topologically split Jordan-Hölder composition series of subrepresentations and a certain category \(\overline {\mathcal C}_{SA}\) of representations of the inhomogeneous stabilizer \(SA\) of the orbit \({\mathcal O}\) of \(H\) in \(A^*\). The starting point is Theorem 2.2, which shows that even when representations of finite length are not induced they do act locally.