---
_id: '11545'
abstract:
- lang: eng
  text: "We classify contravariant pairings between standard Whittaker modules and
    Verma modules over a complex semisimple Lie algebra. These contravariant pairings
    are useful in extending several classical techniques for category O to the Miličić–Soergel
    category N . We introduce a class of costandard modules which generalize dual
    Verma modules, and describe canonical maps from standard to costandard modules
    in terms of contravariant pairings.\r\nWe show that costandard modules have unique
    irreducible submodules and share the same composition factors as the corresponding
    standard Whittaker modules. We show that costandard modules give an algebraic
    characterization of the global sections of costandard twisted Harish-Chandra sheaves
    on the associated flag variety, which are defined using holonomic duality of D-modules.
    We prove that with these costandard modules, blocks of category\r\nN have the
    structure of highest weight categories and we establish a BGG reciprocity theorem
    for N ."
acknowledgement: We thank Catharina Stroppel and Jens Niklas Eberhardt for interesting
  discussions. The first author acknowledges the support of the European Union's Horizon
  2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement
  No. 754411. The second author is supported by the National Science Foundation Award
  No. 1803059 and the Australian Research Council grant DP170101579.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Adam
  full_name: Brown, Adam
  id: 70B7FDF6-608D-11E9-9333-8535E6697425
  last_name: Brown
- first_name: Anna
  full_name: Romanov, Anna
  last_name: Romanov
citation:
  ama: Brown A, Romanov A. Contravariant pairings between standard Whittaker modules
    and Verma modules. <i>Journal of Algebra</i>. 2022;609(11):145-179. doi:<a href="https://doi.org/10.1016/j.jalgebra.2022.06.017">10.1016/j.jalgebra.2022.06.017</a>
  apa: Brown, A., &#38; Romanov, A. (2022). Contravariant pairings between standard
    Whittaker modules and Verma modules. <i>Journal of Algebra</i>. Elsevier. <a href="https://doi.org/10.1016/j.jalgebra.2022.06.017">https://doi.org/10.1016/j.jalgebra.2022.06.017</a>
  chicago: Brown, Adam, and Anna Romanov. “Contravariant Pairings between Standard
    Whittaker Modules and Verma Modules.” <i>Journal of Algebra</i>. Elsevier, 2022.
    <a href="https://doi.org/10.1016/j.jalgebra.2022.06.017">https://doi.org/10.1016/j.jalgebra.2022.06.017</a>.
  ieee: A. Brown and A. Romanov, “Contravariant pairings between standard Whittaker
    modules and Verma modules,” <i>Journal of Algebra</i>, vol. 609, no. 11. Elsevier,
    pp. 145–179, 2022.
  ista: Brown A, Romanov A. 2022. Contravariant pairings between standard Whittaker
    modules and Verma modules. Journal of Algebra. 609(11), 145–179.
  mla: Brown, Adam, and Anna Romanov. “Contravariant Pairings between Standard Whittaker
    Modules and Verma Modules.” <i>Journal of Algebra</i>, vol. 609, no. 11, Elsevier,
    2022, pp. 145–79, doi:<a href="https://doi.org/10.1016/j.jalgebra.2022.06.017">10.1016/j.jalgebra.2022.06.017</a>.
  short: A. Brown, A. Romanov, Journal of Algebra 609 (2022) 145–179.
corr_author: '1'
date_created: 2022-07-08T11:40:07Z
date_published: 2022-11-01T00:00:00Z
date_updated: 2025-04-14T07:43:58Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1016/j.jalgebra.2022.06.017
ec_funded: 1
external_id:
  isi:
  - '000861841100004'
file:
- access_level: open_access
  checksum: 82abaee3d7837f703e499a9ecbb25b7c
  content_type: application/pdf
  creator: dernst
  date_created: 2023-02-02T07:32:48Z
  date_updated: 2023-02-02T07:32:48Z
  file_id: '12473'
  file_name: 2022_JournalAlgebra_Brown.pdf
  file_size: 582962
  relation: main_file
  success: 1
file_date_updated: 2023-02-02T07:32:48Z
has_accepted_license: '1'
intvolume: '       609'
isi: 1
issue: '11'
keyword:
- Algebra and Number Theory
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 145-179
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '754411'
  name: ISTplus - Postdoctoral Fellowships
publication: Journal of Algebra
publication_identifier:
  issn:
  - 0021-8693
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Contravariant pairings between standard Whittaker modules and Verma modules
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 609
year: '2022'
...