[Paper Review] Higher Dirac cohomology of modules with generalized infinitesimal character

Pavle Pandžić, Petr Somberg|arXiv (Cornell University)|Oct 14, 2013

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper introduces higher Dirac cohomology as a generalized framework to extend the standard Dirac cohomology to modules with only generalized infinitesimal character, restoring key properties like the six-term exact sequence and the Dirac character formula. The construction preserves standard cohomology for modules with full infinitesimal character while enhancing its applicability to broader classes of representations, particularly in the context of Harish-Chandra modules and highest weight modules.

ABSTRACT

We modify the definition of Dirac cohomology in such a way that the standard properties of the usual Dirac cohomology, valid for modules with infinitesimal character, become valid also for modules with only generalized infinitesimal character.

Motivation & Objective

• To address the failure of standard Dirac cohomology to retain desirable properties—such as the six-term exact sequence and character formula—for modules with only generalized infinitesimal character.
• To generalize the Dirac cohomology framework to include modules that are not necessarily of infinitesimal character, particularly in the context of reductive Lie algebras and their representations.
• To construct a new cohomology theory that reduces to the standard Dirac cohomology for modules with full infinitesimal character but is more robust for generalized cases.
• To enable the application of Dirac cohomology techniques to the study of restrictions of highest weight modules, especially in cases where standard cohomology vanishes or is ill-behaved.

Proposed method

• Introduce a new cohomology functor—higher Dirac cohomology—defined via the Dirac operator acting on $ V \otimes S $, where $ S $ is the spin module for $ \mathfrak{s} $, and $ V $ is a $ \mathfrak{g} $-module with generalized infinitesimal character.
• Define the Dirac operator $ D \in U(\mathfrak{g}) \otimes C(\mathfrak{s}) $ using dual bases of $ \mathfrak{s} $ and the bilinear form $ B $, generalizing Kostant’s cubic Dirac operator for quadratic reductive subalgebras.
• Construct the higher Dirac cohomology as $ H_D^{(n)}(V) = \ker D^n / (\ker D^n \cap \operatorname{im} D^n) $, generalizing the standard $ H_D(V) = \ker D / (\ker D \cap \operatorname{im} D) $.
• Ensure the new cohomology theory respects short exact sequences by proving a six-term exact sequence for $ (\mathfrak{g}, \mathfrak{r}) $-modules with generalized infinitesimal character.
• Use the splitting $ S = S^+ \oplus S^- $ and the odd nature of $ D $ to define $ H_D^{(n)}(V)^\pm $, preserving the $ \tilde{K} $-module structure.
• Verify that the new cohomology agrees with standard Dirac cohomology when $ V $ has full infinitesimal character, ensuring consistency with prior results.

Experimental results

Research questions

• RQ1 Can the standard Dirac cohomology be extended to modules with only generalized infinitesimal character while preserving key structural properties such as the six-term exact sequence?
• RQ2 Does the higher Dirac cohomology recover the standard Dirac cohomology for modules with full infinitesimal character?
• RQ3 How does the higher Dirac cohomology behave under short exact sequences of $ (\mathfrak{g}, \mathfrak{r}) $-modules with generalized infinitesimal character?
• RQ4 Can the higher Dirac cohomology be used to classify or analyze restrictions of highest weight modules, especially in cases where standard cohomology vanishes?
• RQ5 What is the relationship between the generalized infinitesimal character of a module and the highest weights of its higher Dirac cohomology components?

Key findings

• The higher Dirac cohomology construction restores the six-term exact sequence for short exact sequences of $ (\mathfrak{g}, \mathfrak{r}) $-modules with generalized infinitesimal character, extending a key property previously valid only for modules with full infinitesimal character.
• For modules with full infinitesimal character, the higher Dirac cohomology coincides with the standard Dirac cohomology, ensuring consistency with established results.
• The higher Dirac cohomology is non-vanishing for certain modules with generalized infinitesimal character where standard Dirac cohomology vanishes, indicating enhanced detection capability.
• The Dirac character formula, which relates the infinitesimal character to the highest weight of the cohomology, holds in the generalized setting via the higher cohomology framework.
• The construction preserves the $ \tilde{K} $-module structure and allows for a natural extension of the Dirac inequality and unitarity criteria to a broader class of representations.
• The framework enables the application of Dirac cohomology techniques to the study of restrictions of highest weight modules, including cases previously inaccessible due to cohomological obstructions.