Mackey theory for \(p\)-adic Lie groups (Q2508565)

Let \(H\) be a finite group acting on a finite abelian group \(A\). Let \(G\) denote the semidirect product of \(H\) by \(A\). The method of little groups of Wigner and Mackey classifies all the irreducible representations of \(G\) as certain induced representations. The method also works in the situation of Hilbert space representations of separable locally compact groups and this is due to \textit{G. W. Mackey} [Ann. Math. (2) 58, 193--221 (1953; Zbl 0051.01901)]. In the special case of the Poincaré group, Mackey's result goes back to \textit{E. P. Wigner} [Ann. Math. (2) 40, 149--204 (1939; Zbl 0020.29601)]. Closely following Mackey, the paper under review gives a \(p\)-adic analogue of Wigner-Mackey theory. The precise result is as follows: Let \(A\) be a locally compact totally disconnected abelian group such that its dual \(\widehat A\) has the same property. Let \(H\) be a locally compact totally disconnected group with a continuous action \(t\) on \(A\), and a dual action \(\widehat t\) on \(\widehat A\). Let \(G=H\times_t A\). For each orbit of \(t'\), select out a point \(\chi\) on it. Every irreducible smooth representation \(\pi_0\) of the stabilizer \(H_\chi\) of \(\chi\) gives an irreducible smooth representation \(\text{ Ind}_{H_\chi\times_t A}^G(\pi_0\cdot\chi)\) of \(G\). Every irreducible smooth representation of \(G\) is equivalent to the one obtained in this way. Furthermore, if \(t'\) is smooth, then the representations obtained in such a way are not equivalent to each other.