---
OA_place: publisher
_id: '22694'
abstract:
- lang: eng
  text: "We develop and employ techniques from equivariant algebraic K-theory and
    related invariants\r\nin the context of geometric representation theory, in both
    arithmetic and topological situations.\r\nWe showcase the use of such techniques
    on the affine Grassmannian Gr, a space of fundamental\r\ninterest in the geometric
    Langlands program.\r\n\r\nIt is a deep development of mathematics of the last
    century that many concrete, yet combina-\r\ntorially complex algebraic problems
    may be effectively studied through the lens of algebraic\r\ngeometry. The objects
    of interest can be often realized as cohomological invariants of algebraic\r\nvarieties,
    and good understanding of their geometry sheds light into the original questions.\r\nSuch
    techniques have seen immense applications in the Langlands program, where they
    go\r\nunder the label of geometric representation theory.\r\n\r\nOne source of
    powerful invariants in algebraic geometry comes from algebraic K-theory,\r\nHochschild
    homology, and their relatives. These localizing invariants contain large amount\r\nof
    information, but are quite hard to compute. For this reason, their usage in geometric\r\nrepresentation
    theory has been limited.\r\n\r\nThe aim of this thesis is to showcase how to control
    such invariants in the situations of\r\ninterest and use them to obtain new insights.
    We start by reinterpreting equivariant Hochschild\r\nhomology in terms of functions
    on certain fixed-point schemes, which are of independent\r\ninterest. We compare
    it to equivariant K-theory via the trace map. We give new computations\r\nand
    comparisons of such invariants of affine Schubert varieties in Gr, including arithmetic\r\nsituations.
    We show that they behave much better than expected.\r\n\r\nWe finally utilize
    this circle of ideas in a purely topological setting. We describe the varying\r\nfixed
    points of the extended torus action on the affine Grassmannian, and use it to
    compute\r\nits equivariant topological K-theory ring. The answer is nontrivial
    and verifies an outstanding\r\nconjecture in the subject.\r\n\r\nWe compare, partly
    conjecturally, the resulting K-theory ring to the completed center of an\r\nintegral
    even hybrid quantum group and its deformed quantum category O. This gives a\r\ngenuine
    application of our computations in pure representation theory."
acknowledgement: "It was funded by a DOC Fellowship of the Austrian Academy of Sciences
  entitled Arithmetic,\r\ngeometry, topology and representation theory arising from
  the affine Grassmannian. It was\r\nfurther funded by the Austrian Science Fund FWF
  10.55776/P35847, and an Erasmus+ staff\r\nmobility training. \r\n"
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Jakub
  full_name: Löwit, Jakub
  id: e3b80ae2-eb8e-11eb-b029-9aef4a9108a0
  last_name: Löwit
citation:
  ama: Löwit J. Equivariant K-theory of affine Grassmannians in representation theory
    and arithmetic. 2026. doi:<a href="https://doi.org/10.15479/AT-ISTA-22694">10.15479/AT-ISTA-22694</a>
  apa: Löwit, J. (2026). <i>Equivariant K-theory of affine Grassmannians in representation
    theory and arithmetic</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT-ISTA-22694">https://doi.org/10.15479/AT-ISTA-22694</a>
  chicago: Löwit, Jakub. “Equivariant K-Theory of Affine Grassmannians in Representation
    Theory and Arithmetic.” Institute of Science and Technology Austria, 2026. <a
    href="https://doi.org/10.15479/AT-ISTA-22694">https://doi.org/10.15479/AT-ISTA-22694</a>.
  ieee: J. Löwit, “Equivariant K-theory of affine Grassmannians in representation
    theory and arithmetic,” Institute of Science and Technology Austria, 2026.
  ista: Löwit J. 2026. Equivariant K-theory of affine Grassmannians in representation
    theory and arithmetic. Institute of Science and Technology Austria.
  mla: Löwit, Jakub. <i>Equivariant K-Theory of Affine Grassmannians in Representation
    Theory and Arithmetic</i>. Institute of Science and Technology Austria, 2026,
    doi:<a href="https://doi.org/10.15479/AT-ISTA-22694">10.15479/AT-ISTA-22694</a>.
  short: J. Löwit, Equivariant K-Theory of Affine Grassmannians in Representation
    Theory and Arithmetic, Institute of Science and Technology Austria, 2026.
corr_author: '1'
date_created: 2026-08-12T14:05:36Z
date_published: 2026-08-05T00:00:00Z
date_updated: 2026-08-26T06:53:55Z
day: '05'
ddc:
- '510'
- '516'
- '512'
- '514'
- '513'
degree_awarded: PhD
department:
- _id: GradSch
- _id: TaHa
doi: 10.15479/AT-ISTA-22694
doi_confirm: '1'
file:
- access_level: open_access
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  content_type: application/pdf
  creator: jloewit
  date_created: 2026-08-14T11:42:31Z
  date_updated: 2026-08-14T11:42:31Z
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  file_size: 1085118
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file_date_updated: 2026-08-14T11:42:42Z
has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '08'
oa: 1
oa_version: Published Version
page: '185'
project:
- _id: 901e2a43-16d5-11f0-9cad-9cead34748d6
  grant_number: '27004'
  name: Arithmetic, geometry, topology and representation theory arising from the
    affine Grassmannian
- _id: 34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3
  grant_number: P35847
  name: Geometry of the tip of the global nilpotent cone
publication_identifier:
  issn:
  - 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
publisher_comment: "For open access purposes, the author has applied a CC BY public
  copyright\r\nlicense to any author-accepted manuscript version arising from this
  submission."
related_material:
  record:
  - id: '21751'
    relation: part_of_dissertation
    status: public
  - id: '22693'
    relation: part_of_dissertation
    status: public
status: public
supervisor:
- first_name: Tamás
  full_name: Hausel, Tamás
  id: 4A0666D8-F248-11E8-B48F-1D18A9856A87
  last_name: Hausel
  orcid: 0000-0002-9582-2634
title: Equivariant K-theory of affine Grassmannians in representation theory and arithmetic
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: dissertation
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
year: '2026'
...