[Paper Review] A simple proof of the algebraic version of a conjecture by Vogan

Tim Bratten, Sergio Corti|ArXiv.org|Aug 27, 2007

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper provides a concise algebraic proof of an algebraic version of Vogan's conjecture, showing that the maximal globalization of a Harish-Chandra module commutes with $ $-cohomology if and only if the minimal globalization does. Using duality and the Hochschild-Serre spectral sequence, the authors establish that the conjecture holds for the maximal globalization, leveraging the known truth for the minimal case.

ABSTRACT

In a recent manuscript, D.Vogan conjectures that four canonical globalizations of Harish-Chandra modules commute with certain n-cohomology groups. In this article we prove that Vogan's conjecture holds for one of the globalizations if and only if it holds for the dual. Using a previously published result of one of the authors, which establishes the conjecture for the minimal globalization, we can therefore deduce Vogan's conjecture for the maximal globalization.

Motivation & Objective

• To resolve the algebraic version of Vogan's conjecture concerning the compatibility of $ $-cohomology with globalizations of Harish-Chandra modules.
• To establish a duality between the conjecture's validity for a globalization and its dual, reducing the problem to known results.
• To prove that the conjecture holds for the maximal globalization by leveraging the known truth for the minimal globalization.
• To use the Hochschild-Serre spectral sequence as a central tool to relate cohomology and homology structures under duality.
• To provide a formal, algebraic proof independent of classification theorems, avoiding reliance on Beilinson-Bernstein for broader applicability.

Proposed method

• Define $ $-homology and cohomology groups via standard resolutions of the trivial module over $U( )$, using the tensor and Hom functors.
• Apply the Hochschild-Serre spectral sequence to relate the cohomology of $ $ with the cohomology of a larger algebraic structure, particularly in the context of parabolic subalgebras.
• Use the duality isomorphism $H_p( , M^*) o H^p( , M)^*$ to relate homology and cohomology of dual modules.
• Establish a canonical isomorphism between $H^p( , M_{ ext{max}})$ and the maximal globalization of $H^p( , M)$, using spectral sequence filtrations.
• Apply a key lemma on filtered $(rak{l}, K_0 imes L_0)$-modules to conclude that an isomorphism on associated graded pieces implies an isomorphism at the level of $K_0$-finite vectors.
• Leverage the fact that the minimal globalization satisfies the conjecture (from prior work) to deduce the result for the maximal globalization via duality and spectral sequence arguments.

Experimental results

Research questions

• RQ1 Does the maximal globalization of a Harish-Chandra module commute with $ $-cohomology?
• RQ2 Is the validity of Vogan's conjecture for a globalization equivalent to its validity for the dual globalization?
• RQ3 Can the conjecture be proven algebraically using spectral sequences and duality, without relying on classification theorems?
• RQ4 How do the $K_0$-finite vectors in cohomology groups behave under duality and spectral sequence filtrations?
• RQ5 What is the precise relationship between the cohomology of a Harish-Chandra module and its globalizations via the Hochschild-Serre spectral sequence?

Key findings

• The conjecture holds for the maximal globalization of a Harish-Chandra module if and only if it holds for the minimal globalization.
• The duality between $H_p( , M^*)$ and $H^p( , M)^*$ establishes a canonical isomorphism that allows transferring the conjecture between dual modules.
• The Hochschild-Serre spectral sequence provides a formal framework to compare cohomology and homology structures, enabling the proof via induction on spectral sequence terms.
• The $K_0$-finite dual functor is exact on good $(rak{l}, K_0 imes L_0)$-modules, which allows the use of spectral sequence filtrations to deduce global isomorphisms.
• The final result confirms that $H^p( , M_{ ext{max}}) o H^p( , M)_{ ext{max}}$ is a canonical isomorphism, establishing the conjecture for the maximal globalization.
• The proof is algebraic and self-contained, avoiding dependence on Beilinson-Bernstein classification, making it applicable to general reductive Lie groups of Harish-Chandra class.