[Paper Review] Generalized Jacquet modules of parabolic induction

Noriyuki Abe|ArXiv.org|Oct 17, 2007

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper generalizes the Jacquet module for parabolic inductions in reductive Lie groups by introducing a Bruhat filtration on the generalized Jacquet module, whose successive quotients are described via Arkhipov's twisting functor. The key contribution is a structural decomposition of Whittaker models through these quotients, enabling a systematic approach to determining such models via extension data and twisted induction.

ABSTRACT

In this paper we study a generalization of the Jacquet module of a parabolic induction and construct a filtration on it. The successive quotient of the filtration is written by using the twisting functor.

Motivation & Objective

• To generalize the Jacquet module construction for parabolic inductions in reductive Lie groups beyond the classical case.
• To define a Bruhat filtration on the generalized Jacquet module and describe its successive quotients using the twisting functor.
• To provide a strategy for determining Whittaker models of parabolic inductions by analyzing the filtration quotients and their extensions.
• To extend Casselman's generalized Jacquet module framework to include character twists and module structure changes via the twisting functor.

Proposed method

• Define the generalized Jacquet modules $ J'_{ u}(V) $ and $ J^*_{ u}(V) $ as joint kernels of $ (X - \eta(X))^k $ on dual spaces for $ X \in \mathfrak{n}_0 $, capturing $ \eta $-twisted nilpotent conditions.
• Construct the twisting functor $ T_{w,\eta} $ using localization of the universal enveloping algebra $ U(\mathfrak{g}) $ at elements $ e_i - \eta(e_i) $, forming a $ \mathfrak{g} $-bimodule $ S_{w,\eta} $.
• Define the Bruhat filtration $ \{I_i\} $ on $ J'_{\eta}(\operatorname{Ind}_P^G(\sigma \otimes e^{\lambda+\rho})) $ by support on $ N_0 $-orbits $ \bigcup_{j\leq i} N_0 w_j P/P $.
• Use the Weyl group action to define twisted characters $ w_i^{-1}\eta $ on $ \mathfrak{m} \cap \mathfrak{n}_0 $, enabling the construction of $ J'_{w_i^{-1}\eta}(\sigma \otimes e^{\lambda+\rho}) $ as the successive quotient.
• Apply the generalized twisting functor $ T_{w_i,\eta} $ to relate the successive quotient of the filtration to twisted induction from the Levi subgroup $ M $.
• Use distribution theory on nilpotent groups to analyze $ \mathcal{D}'(N, \mathcal{L}) $, showing that distributions annihilated by powers of $ \ker \eta $ lie in the span of polynomial distributions with coefficients in $ V' $.

Experimental results

Research questions

• RQ1 How can the classical Jacquet module construction be generalized to accommodate non-unitary characters and non-principal series representations?
• RQ2 What is the structure of the generalized Jacquet module of a parabolic induction, and how can it be filtered meaningfully?
• RQ3 How do the successive quotients of the Bruhat filtration on the generalized Jacquet module relate to the representation theory of the Levi subgroup $ M $?
• RQ4 Can the twisting functor of Arkhipov be used to describe these quotients, and what is the nature of the resulting 'twisted' induction?
• RQ5 What is the role of the generalized Jacquet module in determining Whittaker models of parabolic inductions?

Key findings

• If the character $ \eta $ is not unitary, then the generalized Jacquet module $ J'_{\eta}(\operatorname{Ind}_P^G(\sigma \otimes e^{\lambda+\rho})) $ vanishes.
• The successive quotients of the Bruhat filtration on the generalized Jacquet module are isomorphic to the twisted induction $ T_{w_i,\eta}(J'_{w_i^{-1}\eta}(\sigma \otimes e^{\lambda+\rho})) $, which has the same character as standard induction but a different module structure.
• In the case where the parabolic induction has a unique Langlands quotient, the successive quotients reduce to standard induction from the Jacquet module of the Levi subgroup.
• The filtration structure allows a systematic approach to computing Whittaker models: it suffices to analyze the successive quotients and their extensions.
• Distributions on a nilpotent Lie group $ N $ annihilated by a power of $ \ker \eta $ are finite linear combinations of monomials times $ \delta $-distributions with coefficients in $ V' $, establishing a polynomial model for such distributions.
• The generalized Jacquet module is isomorphic to a direct sum of twisted modules over the nilpotent radical, with the structure governed by the action of the Weyl group and the character $ \eta $.