On Jacquet modules of discrete series: the first inductive step (Q2810924)

The author determines certain parts of Jacquet modules of the discrete series representations of \(p\)-adic classical groups. Let us explain the results in more detail: let \(G_n\) be a tower of odd orthogonal, full even-orthogonal, symplectic or unitary classical \(p\)-adic groups (in this explanation, we may turn our attention to symplectic groups, so that \(n\) denotes the split rank of \(G_n\)). Then, the Mœglin-Tadić classification of the discrete series representations of \(\{G_n\}\) classifies them in terms of so-called admissible triples which consist of a Jordan block (which corresponds to the Langlands parameter in Langlands parametrization), partial supercuspidal support and an \(\epsilon\) function on the Jordan block (these two components correspond, roughly, to a character of the component group of the centralizer of the Langlands parameter). These admissible triples are built in sort of inductive way, starting from the so-called strongly positive discrete series representations by adding more elements in their Jordan block (in a certain way). The author analyzes those discrete series representations which are obtained in the first step of this inductive procedure, those obtained by adding two elements in the Jordan block of a strongly positive representation. The author describes, for those discrete series, all the representations occurring in their Jacquet modules with respect to the maximal parabolic subgroups for which the part on the general linear group (of that Jacquet-module representation) is an essentially square-integrable representation. He describes which essentially square-integrable representations of the general linear group occur, and, for a fixed such representation, he describes which representations occur on the classical part of the Jacquet module.