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Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras

Simone Noja, Steffen Schmidt, Raphael Senghaas

TL;DR

The paper develops a comprehensive framework for Dirac cohomology in basic classical Lie superalgebras using cubic Dirac operators associated with parabolic subalgebras. It proves a super Casselman–Osborne lemma for modules with infinitesimal character, shows that the Dirac cohomology of highest-weight modules is nontrivial, and provides explicit DC formulas for finite-dimensional type-1 modules with typical highest weight as well as for simple modules in parabolic BGG category O. It further establishes embeddings of Dirac cohomology into Kostant’s ${ rak u}$-cohomology and ${ar{ rak u}}$-homology, with unitarizable modules yielding isomorphisms and a Kostant-Hodge-type decomposition. The results illuminate how Dirac cohomology can determine or constrain the structure of representations, connect to Kostant theory, and extend to broader superalgebra settings, offering a path toward explicit computations and potential reconstruction of modules from DC data.

Abstract

We study the Dirac cohomology of supermodules over basic classical Lie superalgebras, formulated in terms of cubic Dirac operators associated with parabolic subalgebras. Specifically, we establish a super-analog of the Casselman-Osborne theorem for supermodules with an infinitesimal character and use it to show that the Dirac cohomology of highest-weight supermodules is always non-trivial. In particular, we explicitly compute the Dirac cohomology of finite-dimensional simple supermodules for basic Lie superalgebras of type 1 with a typical highest weight, as well as of simple supermodules in the parabolic BGG category. We further investigate the relationship between Dirac cohomology and Kostant (co)homology, proving that, under suitable conditions, Dirac cohomology embeds into Kostant (co)homology. Moreover, we show that this embedding lifts to an isomorphism when the supermodule is unitarizable.

Cubic Dirac Operators and Dirac Cohomology for Basic Classical Lie Superalgebras

TL;DR

The paper develops a comprehensive framework for Dirac cohomology in basic classical Lie superalgebras using cubic Dirac operators associated with parabolic subalgebras. It proves a super Casselman–Osborne lemma for modules with infinitesimal character, shows that the Dirac cohomology of highest-weight modules is nontrivial, and provides explicit DC formulas for finite-dimensional type-1 modules with typical highest weight as well as for simple modules in parabolic BGG category O. It further establishes embeddings of Dirac cohomology into Kostant’s -cohomology and -homology, with unitarizable modules yielding isomorphisms and a Kostant-Hodge-type decomposition. The results illuminate how Dirac cohomology can determine or constrain the structure of representations, connect to Kostant theory, and extend to broader superalgebra settings, offering a path toward explicit computations and potential reconstruction of modules from DC data.

Abstract

We study the Dirac cohomology of supermodules over basic classical Lie superalgebras, formulated in terms of cubic Dirac operators associated with parabolic subalgebras. Specifically, we establish a super-analog of the Casselman-Osborne theorem for supermodules with an infinitesimal character and use it to show that the Dirac cohomology of highest-weight supermodules is always non-trivial. In particular, we explicitly compute the Dirac cohomology of finite-dimensional simple supermodules for basic Lie superalgebras of type 1 with a typical highest weight, as well as of simple supermodules in the parabolic BGG category. We further investigate the relationship between Dirac cohomology and Kostant (co)homology, proving that, under suitable conditions, Dirac cohomology embeds into Kostant (co)homology. Moreover, we show that this embedding lifts to an isomorphism when the supermodule is unitarizable.

Paper Structure

This paper contains 38 sections, 78 theorems, 246 equations, 1 table.

Table of Contents

  1. Introduction
  2. Vue d'ensemble
  3. Main results
  4. Leitfaden
  5. Conventions
  6. Preliminaries
  7. Parabolic subalgebras, parabolic induction, highest weight supermodules
  8. Parabolic subalgebras
  9. Parabolic induction
  10. Parabolic category $\mathcal{O}^{{\mathfrak{p}}}$
  11. Highest weight supermodules
  12. Unitarizable supermodules and real forms
  13. Clifford superalgebras
  14. Clifford and exterior superalgebras.
  15. Embedding of ${\mathfrak l}$ into $C({\mathfrak{s}})$.
  16. ...and 23 more sections

Key Result

Theorem 1.2.1

Let $M$ be a ${\mathfrak{g}}$-supermodule with infinitesimal character $\chi_\lambda$ for $\lambda \in {\mathfrak h}^\ast$. Then $z \in {\mathfrak{Z}({\mathfrak{g}})}$ acts on the Dirac cohomology $\operatorname{H}_{\operatorname{D}({\mathfrak{g}},{\mathfrak l})}(M)$ as $\eta_{\, {\mathfrak l}} (z)

Theorems & Definitions (124)

  • Theorem 1.2.1
  • Theorem 1.2.2
  • Theorem 1.2.3
  • Theorem 1.2.4
  • Theorem 1.2.5
  • Proposition 2.0.1: Kac
  • Definition 2.1.1
  • Lemma 2.1.2: DFG
  • Definition 2.1.3
  • Remark 2.1.4
  • ...and 114 more