Generalized Jacquet modules of parabolically induced representations
type: Item
instance of: scholarly article
title: Generalized Jacquet modules of parabolically induced representations
zbMATH Open document ID: 1245.22009
DOI: 10.2977/PRIMS/75
author: Noriyuki Abe
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2012-07-11T00:00:00Z
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6054397
zbMATH Keywords: Jacquet module
zbMATH Keywords: parabolically induced representation
zbMATH Keywords: Bruhat filtration
zbMATH Keywords: twisting functor
zbMATH Keywords: Fréchet representation
zbMATH Keywords: Whittaker vector
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/75
OpenAlex ID: W2028878631
Recommended article: Q4012132
Recommended article: Jacquet modules of principal series generated by the trivial \(K\)-type
Recommended article: The asymptotic behavior of holomorphic representations
Recommended article: Q3530287
Recommended article: Weight filtrations for induced representations of real reductive Lie groups
Recommended article: A Strong Generalization of Helgason's Theorem
Recommended article: Jacquet modules and local Langlands correspondence
Recommended article: On certain Iwahori invariants in the unramified principal series
Recommended article: Jacquet modules of 𝑝-adic general linear groups
Recommended article: Q4796251
review text: Let \(G\) be a connected semisimple linear Lie group with Lie algebra \(\mathfrak g\). In 1978, \textit{W. Casselman} [Proc. int. Congr. Math., Helsinki 1978, Vol. 2, 557--563 (1980; Zbl 0425.22019)] introduced the Jacquet module of a representation of \(G\). One of the tools for investigating this module is the Bruhat filtration defined by the Bruhat decomposition, see [\textit{W. Casselman, H. Hecht} and \textit{D. Miličić}, Proc. Symp. Pure Math. 68, 151--190 (2000; Zbl 0959.22010)].NEWLINENEWLINE Let \(G=KA_0N_0\) be an Iwasawa decomposition and \(P_0 = M_0A_0N_0\) a minimal parabolic subgroup together with its Langlands decomposition. Fix some parabolic subgroup \(P\) with \(P_0 \subseteq P\) and Langlands decomposition \(P=MAN\), \(A_0 \subseteq A\). Let \(\mathfrak a\) be the Lie algebra of \(A\) and \({\mathfrak a}^\ast = {\Hom}_{\mathbb C}({\mathfrak a}, {\mathbb C})\). For \(\lambda \in {\mathfrak a}^\ast\) and some irreducible representation \(\sigma\) of \(M\) set \(I(\sigma,\lambda) := {\text{ Ind}}_P^G(\sigma \otimes e^{\lambda + \rho})\), where \(\rho \in {\mathfrak a}^\ast\) is half the sum of all positive roots, as the space of \(C^\infty\)-sections of a certain vector bundle on the quotient \(G/P\). Finally, fix some character \(\eta \in {\mathfrak n}_0\) of the Lie algebra of \(N_0\) and define NEWLINE\[NEWLINE J'_\eta(I(\sigma,\lambda)) := \{ v \in I(\sigma,\lambda)'\,| \, \exists k \in {\mathbb N}\;\forall X \in {\mathfrak N}_0\;: \;(X-\eta(X))^kv=0\,\}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe author uses the Bruhat filtration on \(J'_\eta(I(\sigma,\lambda))\) for studying this very module. The investigation of the structure of the successive quotients of the Bruhat filtration are one of the main parts of the paper.NEWLINENEWLINENEWLINEConditions are stated under which these quotients become zero eventually. In the second half of the paper, inspired by the work of \textit{S. Arkhipov} [Adv. Stud. Pure Math. 40, 27--68 (2004; Zbl 1096.17003)], …
description: scientific article; zbMATH DE number 6054397