On some non-principal locally analytic representations induced by Whittaker modules
Sascha Orlik
TL;DR
The work addresses constructing non-principal locally analytic $G$-representations induced by Whittaker modules attached to the Lie algebra, defining the bifunctor ${\mathcal F}^G_{\hat{\eta}}$ via $D(G_0)$-modules and compact induction. It proves exactness and ind-admissibility, and shows that simple Whittaker modules yield topologically irreducible outputs, while the naive Jacquet functor vanishes for all parabolics yet Kohlhaase’s homological vanishing criterion fails in general. The analysis leverages Milic\u{c}i\u{c}-Soergel Whittaker theory and Agrawal–Strauch techniques, with Kostant’s results providing the simple Whittaker quotients and a Mackey-type criterion establishing irreducibility. Collectively, the paper broadens the landscape of locally analytic, non-principal representations and clarifies their cuspidal-like properties within the $p$-adic analytic framework.
Abstract
Let G be a connected split adjoint semi simple p-adic Lie group. This paper can be seen as a continuation of [12] and is about the construction of locally analytic G-representations which do not lie in the principal series. Here we consider locally analytic representations which are induced by Whittaker modules of the attached Lie algebra. We prove that they are inadmissible and topologically irreducible in case the Whittaker module is simple. On the other hand, we show that the naive Jacquet functor of these representations vanishes for all parabolic subgroups. However, they do not satisfy the definition of supercuspidality in the sense of Kohlhaase.