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Applying projective functors to arbitrary holonomic simple modules

Marco Mackaay, Volodymyr Mazorchuk, Vanessa Miemietz

Abstract

We prove that applying a projective functor to a holonomic simple module over a semi-simple finite dimensional complex Lie algebra produces a module that has an essential semi-simple submodule of finite length. This implies that holonomic simple supermodules over certain Lie superalgebras are quotients of modules that are induced from simple modules over the even part. We also provide some further insight into the structure of Lie algebra modules that are obtained by applying projective functors to simple modules.

Applying projective functors to arbitrary holonomic simple modules

Abstract

We prove that applying a projective functor to a holonomic simple module over a semi-simple finite dimensional complex Lie algebra produces a module that has an essential semi-simple submodule of finite length. This implies that holonomic simple supermodules over certain Lie superalgebras are quotients of modules that are induced from simple modules over the even part. We also provide some further insight into the structure of Lie algebra modules that are obtained by applying projective functors to simple modules.
Paper Structure (37 sections, 22 theorems, 26 equations)

This paper contains 37 sections, 22 theorems, 26 equations.

Table of Contents

  1. Motivation and description of the results
  2. Motivation from Lie superalgebras
  3. Main result
  4. Structure of $V\otimes_{\mathbb{C}} L$
  5. Structure of the paper
  6. Preliminaries
  7. Category $\mathscr{O}$
  8. Projective functors
  9. Harish-Chandra bimodules
  10. Soergel's combinatorial description and the integral part
  11. Gelfand-Kirillov dimension
  12. Kazhdan-Lusztig combinatorics
  13. Structure of $\theta L$
  14. Quick recap of birepresentation theory
  15. The bicategory of projective functors
  16. ...and 22 more sections

Key Result

Lemma 1

Let $L$ and $L'$ be two simple $\mathfrak{g}$-modules such that $\mathrm{Hom}_{\mathfrak{g}}(V\otimes_{\mathbb{C}}L,L')\neq0$ or $\mathrm{Hom}_{\mathfrak{g}}(L',V\otimes_{\mathbb{C}}L)\neq0$, for some finite dimensional $\mathfrak{g}$-module $V$. Then $\mathrm{GKdim}(L)=\mathrm{GKdim}(L')$.

Theorems & Definitions (48)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Proposition 4
  • proof
  • Conjecture 5
  • Lemma 6
  • ...and 38 more