On some non-principal locally analytic representations induced by cuspidal Lie algebra representations
Sascha Orlik
TL;DR
The paper constructs locally analytic representations of $G=\mathrm{GL}_{n+1}$ induced from cuspidal Lie algebra modules, which are not in the principal series. It globalizes cuspidal weight modules to produce a functor ${\mathcal F}^G_\kappa$ landing in ind-admissible, compact-type representations and establishes vanishing of parabolic cohomology, a key property for supercuspidality. In the $n=1$ case, it proves topological irreducibility for a central-extension variant under explicit weight conditions, suggesting a path toward a $p$-adic local Langlands perspective for cuspidal data. The work relies on a detailed framework of distribution algebras, pro-coadmissible modules, and Mackey-type criteria to control induction and irreducibility, while a subsidiary analysis shows cuspidality by vanishing of Lie algebra cohomology via Ore localization and Iwahori decompositions.
Abstract
Let $G$ be a split reductive $p$-adic Lie group. This paper is the first in a series on the construction of locally analytic $G$-representations which do not lie in the principal series. Here we consider the case of the general linear group $G=GL_{n+1}$ and locally analytic representations which are induced by cuspidal modules of the Lie algebra. We prove that they are ind-admissible and satisfy the homological vanishing criterion in the definition of supercuspidality in the sense of Kohlhaase. In the case of $n=1$ we give a proof of their topological irreducibility for certain cuspidal modules of degree 1.