Jacquet tensors

Let G be a split reductive p-adic group. The category of admissible p-adic Banach space representations of G is equivalent to the corresponding category of finitely generated Iwasawa modules, via the duality map V ↦ V\u27. In this paper, we define certain tensors on Iwasawa modules, which are intended to play the role of Jacquet modules. We describe some properties of Jacquet tensors and show how they can be applied to the study of principal series representations

Deligne-Lusztig Constructions for Division Algebras and the Local Langlands Correspondence

Let $K$ be a local non-Archimedean field of positive characteristic and let $L$ be the degree-$n$ unramified extension of $K$. Via the local Langlands and Jacquet-Langlands correspondences, to each sufficiently generic multiplicative character of $L$, one can associate an irreducible representation of the multiplicative group of the central division algebra $D$ of invariant $1/n$ over $K$. In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive $p$-adic groups analogous to Deligne-Lusztig theory for finite reductive groups. In this paper we prove that when $n=2$, the $p$-adic Deligne-Lusztig (ind-)scheme $X$ induces a correspondence between smooth one-dimensional representations of $L^\times$ and representations of $D^\times$ that matches the correspondence given by the LLC and JLC.Comment: 61 pages. Version 2: minor revision

The Bernstein Center of a p-adic Unipotent Group

Francois Rodier proved that it is possible to view smooth representations of certain totally disconnected abelian groups (the underlying additive group of a finite-dimensional p-adic vector space, for example) as sheaves on the Pontryagin dual group. For nonabelian totally disconnected groups, the appropriate dual space necessarily includes representations which are not one-dimensional, and does not carry a group structure. The general definition of the topology on the dual space is technically unwieldy, so we provide three different characterizations of this topology for a large class of totally disconnected groups (which includes, for example, p-adic unipotent groups), each with a somewhat different flavor. We then use these results to demonstrate some formal similarities between smooth representations and sheaves on the dual space, including a concrete description of the Bernstein center of the category of smooth representations

Jacquet modules of principal series

We study the Jacquet module of principal series representations. Using the Bruhat filtration, we can introduce a filtration {Vi} of the Jacquet module. In a special case, it is proved that Vi/Vi−1 is isomorphic to the generalized Verma module