Duality in the space of tempered virtual characters of a reductive \(p\)-group (Q884982)

Let \(G\) be a connected reductive group defined over a non-archimedean local field \(F\) of characteristic 0. The paper is concerned with a certain involution in the space of tempered virtual characters of \(G(F)\). The involution is given by the formula \[ D=\sum_{M}(-1)^{\text{dim}A_M}\, i^G_M \circ r^G_M, \] where \(i^G_M\) denotes parabolic induction and \(r^G_M\) parabolic restriction followed by taking the maximal tempered quotient. \(M\) runs through a set of standard Levi groups (in the paper \(M\) is supposed to run through a set of semi-standard Levi groups, but then the sum does not have the wanted properties). \(D\) commutes with induction and tempered restriction and \(D^2 = \text{id}\). From the symmetry of \(D\) (Lemma 4) it follows immediately that \(D\) maps each irreducible tempered character to \(\pm\) an irreducible tempered character (Theorem 5.1; the argument in the paper is obscure). The behaviour of \(D\) on elliptic characters is also treated.