---
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:10.15479/AT-ISTA-22694
apa: Löwit, J. (2026). Equivariant K-theory of affine Grassmannians in representation
theory and arithmetic. Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-22694
chicago: Löwit, Jakub. “Equivariant K-Theory of Affine Grassmannians in Representation
Theory and Arithmetic.” Institute of Science and Technology Austria, 2026. https://doi.org/10.15479/AT-ISTA-22694.
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. Equivariant K-Theory of Affine Grassmannians in Representation
Theory and Arithmetic. Institute of Science and Technology Austria, 2026,
doi:10.15479/AT-ISTA-22694.
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:
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- '516'
- '512'
- '514'
- '513'
degree_awarded: PhD
department:
- _id: GradSch
- _id: TaHa
doi: 10.15479/AT-ISTA-22694
doi_confirm: '1'
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license: https://creativecommons.org/licenses/by/4.0/
month: '08'
oa: 1
oa_version: Published Version
page: '185'
project:
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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:
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- id: '21751'
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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
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type: dissertation
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abstract:
- lang: eng
text: 'We define a certain class of simple varieties over a field k by a constructive
recipe and show how to control their (equivariant) truncating invariants. Consequently,
we prove that on simple varieties: (i) if k = k and char k = p, the p-adic cyclotomic
trace is an equivalence; (ii) if k = Q, the Goodwillie–Jones trace is an isomorphism
in degree zero; (iii) we can control homotopy invariant K-theory KH, which is
equivariantly formal and determined by its topological counterparts. Simple varieties
are quite special, but encompass important singular examples appearing in geometric
representation theory. We, in particular, show that both finite and affine Schubert
varieties for GLn lie in this class, so all the above results hold for them. '
acknowledgement: "This work was supported by a DOC Fellowship of the Austrian Academy
of Sciences at the Institute of Science and Technology Austria (ISTA) and by an
Erasmus+ staff mobility training. It took place during the author’s visit to Laboratoire
de Mathématiques d’Orsay in the course of his PhD at the Institute of Science and
Technology Austria. First and foremost, I would like to thank Matthew Morrow for
discussions, explanations and ideas without which this work would not have been
carried out. I would further like to thank Brian Conrad for providing an amazing
reference on projective cones in appropriate generality, to Vova Sosnilo for carefully
discussing – among other things – the derived nilinvariance for quotients by any
linearly reductive group, and to Adeel Khan, Timo Richarz, Matthias Wendt and Xinwen
Zhu for helpful conversations\r\nabout the results. I would moreover like to thank
the referee for the very useful comments."
article_number: rnag058
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Jakub
full_name: Löwit, Jakub
id: e3b80ae2-eb8e-11eb-b029-9aef4a9108a0
last_name: Löwit
citation:
ama: Löwit J. Equivariant localizing invariants of simple varieties. International
Mathematics Research Notices. 2026;2026(7). doi:10.1093/imrn/rnag058
apa: Löwit, J. (2026). Equivariant localizing invariants of simple varieties. International
Mathematics Research Notices. Oxford University Press. https://doi.org/10.1093/imrn/rnag058
chicago: Löwit, Jakub. “Equivariant Localizing Invariants of Simple Varieties.”
International Mathematics Research Notices. Oxford University Press, 2026.
https://doi.org/10.1093/imrn/rnag058.
ieee: J. Löwit, “Equivariant localizing invariants of simple varieties,” International
Mathematics Research Notices, vol. 2026, no. 7. Oxford University Press, 2026.
ista: Löwit J. 2026. Equivariant localizing invariants of simple varieties. International
Mathematics Research Notices. 2026(7), rnag058.
mla: Löwit, Jakub. “Equivariant Localizing Invariants of Simple Varieties.” International
Mathematics Research Notices, vol. 2026, no. 7, rnag058, Oxford University
Press, 2026, doi:10.1093/imrn/rnag058.
short: J. Löwit, International Mathematics Research Notices 2026 (2026).
corr_author: '1'
date_created: 2026-04-19T22:07:48Z
date_published: 2026-04-01T00:00:00Z
date_updated: 2026-08-26T06:53:54Z
day: '01'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1093/imrn/rnag058
external_id:
arxiv:
- '2507.09392'
file:
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checksum: 306f4567b7b2dcf38e23f7b55a27514e
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creator: dernst
date_created: 2026-05-06T06:35:05Z
date_updated: 2026-05-06T06:35:05Z
file_id: '21803'
file_name: 2026_IMRN_Loewit.pdf
file_size: 1663246
relation: main_file
success: 1
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fulldoi: https://doi.org/10.1093/imrn/rnag058
has_accepted_license: '1'
intvolume: ' 2026'
issue: '7'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: 901e2a43-16d5-11f0-9cad-9cead34748d6
grant_number: '27004'
name: Arithmetic, geometry, topology and representation theory arising from the
affine Grassmannian
publication: International Mathematics Research Notices
publication_identifier:
eissn:
- 1687-0247
issn:
- 1073-7928
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
related_material:
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relation: dissertation_contains
status: public
scopus_import: '1'
status: public
title: Equivariant localizing invariants of simple varieties
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: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2026
year: '2026'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '22693'
abstract:
- lang: eng
text: We study torus-equivariant algebraic K-theory of affine Schubert varieties
in the perfect affine Grassmannians over Fp. We further compare it to the torus-equivariant
Hochschild homology of perfect complexes, which has a geometric description in
terms of global functions on certain fixed-point schemes. We prove that Fp-linearly,
this comparison is an isomorphism. Our approach is quite constructive, resulting
in new computations of these K-theory rings. We establish various structural results
for equivariant perfect algebraic K-theory on the way; we believe these are of
independent interest.
acknowledgement: "I would like to thank the following people for fruitful discussions,\r\nhelpful
sanity checks or comments on previous drafts: Roman Bezrukavnikov, Jens Niklas Eberhardt,
Mischa Elkner, Tamás Hausel, Andres Fernandez Herrero, Adeel Khan,\r\nBernhard Köck,
Andrei Konovalov, Quoc Ho, Mirko Mauri, Matthew Morrow, Charanya\r\nRavi, Kamil
Rychlewicz, Shyiu Shen, Vladimir Sosnilo, Georg Tamme, Xinwen Zhu. I\r\nwould further
like to thank Marc Hoyois and the anonymous referee for spotting an error\r\nin
a previous version.\r\nThis work was done during author’s PhD at the Institute of
Science and Technology Austria (ISTA). It was funded by a DOC Fellowship of the
Austrian Academy\r\nof Sciences and by the Austrian Science Fund (FWF) 10.55776/P35847.
For open access\r\npurposes, the author has applied a CC BY public copyright license
to any author-accepted\r\nmanuscript version arising from this submission."
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Jakub
full_name: Löwit, Jakub
id: e3b80ae2-eb8e-11eb-b029-9aef4a9108a0
last_name: Löwit
citation:
ama: Löwit J. Equivariant K-theory, affine Grassmannian and perfection. Documenta
Mathematica. 2026. doi:10.4171/dm/1064
apa: Löwit, J. (2026). Equivariant K-theory, affine Grassmannian and perfection.
Documenta Mathematica. EMS Press. https://doi.org/10.4171/dm/1064
chicago: Löwit, Jakub. “Equivariant K-Theory, Affine Grassmannian and Perfection.”
Documenta Mathematica. EMS Press, 2026. https://doi.org/10.4171/dm/1064.
ieee: J. Löwit, “Equivariant K-theory, affine Grassmannian and perfection,” Documenta
Mathematica. EMS Press, 2026.
ista: Löwit J. 2026. Equivariant K-theory, affine Grassmannian and perfection. Documenta
Mathematica.
mla: Löwit, Jakub. “Equivariant K-Theory, Affine Grassmannian and Perfection.” Documenta
Mathematica, EMS Press, 2026, doi:10.4171/dm/1064.
short: J. Löwit, Documenta Mathematica (2026).
corr_author: '1'
das_tickbox: '0'
date_created: 2026-08-12T13:29:17Z
date_published: 2026-03-26T00:00:00Z
date_updated: 2026-08-26T06:53:54Z
day: '26'
ddc:
- '500'
department:
- _id: GradSch
- _id: TaHa
doi: 10.4171/dm/1064
external_id:
arxiv:
- '2409.18925'
fulldoi: https://doi.org/10.4171/dm/1064
has_accepted_license: '1'
keyword:
- equivariant algebraic K-theory
- perfection in positive characteristic
- affine Grassmannian
- affine Schubert varieties
- Dennis trace map
- equivariant Hochschild homology
- fixed-point schemes
- toric varieties
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.4171/DM/1064
mathsc:
- '19E08'
- 19L47
- 20G44
- 14G17
- 19D55
- 14F43
- 14L30
- 14D24
- 14M25
month: '03'
oa: 1
oa_version: Published Version
project:
- _id: 34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3
grant_number: P35847
name: Geometry of the tip of the global nilpotent cone
publication: Documenta Mathematica
publication_identifier:
eissn:
- 1431-0643
issn:
- 1431-0635
publication_status: epub_ahead
publisher: EMS Press
quality_controlled: '1'
related_material:
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relation: dissertation_contains
status: public
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Equivariant K-theory, affine Grassmannian and perfection
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)
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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...
---
OA_place: publisher
OA_type: hybrid
_id: '18154'
abstract:
- lang: eng
text: 'In 1976, Deligne and Lusztig realized the representation theory of finite
groups of Lie type inside étale cohomology of certain algebraic varieties. Recently,
a p-adic version of this theory started to emerge: there are p-adic Deligne–Lusztig
spaces, whose cohomology encodes representation theoretic information for p-adic
groups – for instance, it partially realizes the local Langlands correspondence
with characteristic zero coefficients. However, the parallel case of coefficients
of positive characteristic ℓ≠p has not been inspected so far. The purpose of
this article is to initiate such an inspection. In particular, we relate cohomology
of certain p-adic Deligne–Lusztig spaces to Vignéras''s modular local Langlands
correspondence for GLn.'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Jakub
full_name: Löwit, Jakub
id: e3b80ae2-eb8e-11eb-b029-9aef4a9108a0
last_name: Löwit
citation:
ama: Löwit J. On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn.
Journal of Algebra. 2025;663(2):81-118. doi:10.1016/j.jalgebra.2024.08.033
apa: Löwit, J. (2025). On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties
for GLn. Journal of Algebra. Elsevier. https://doi.org/10.1016/j.jalgebra.2024.08.033
chicago: Löwit, Jakub. “On modulo ℓ Cohomology of P-Adic Deligne–Lusztig Varieties
for GLn.” Journal of Algebra. Elsevier, 2025. https://doi.org/10.1016/j.jalgebra.2024.08.033.
ieee: J. Löwit, “On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for
GLn,” Journal of Algebra, vol. 663, no. 2. Elsevier, pp. 81–118, 2025.
ista: Löwit J. 2025. On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties
for GLn. Journal of Algebra. 663(2), 81–118.
mla: Löwit, Jakub. “On modulo ℓ Cohomology of P-Adic Deligne–Lusztig Varieties for
GLn.” Journal of Algebra, vol. 663, no. 2, Elsevier, 2025, pp. 81–118,
doi:10.1016/j.jalgebra.2024.08.033.
short: J. Löwit, Journal of Algebra 663 (2025) 81–118.
corr_author: '1'
date_created: 2024-09-29T22:01:37Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-02-27T12:32:40Z
day: '01'
ddc:
- '510'
department:
- _id: TaHa
doi: 10.1016/j.jalgebra.2024.08.033
external_id:
arxiv:
- '2404.11176'
isi:
- '001325207800001'
file:
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checksum: eb240e93c178e48429ad918c9058f1fe
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creator: dernst
date_created: 2025-01-13T08:57:57Z
date_updated: 2025-01-13T08:57:57Z
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file_size: 731175
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intvolume: ' 663'
isi: 1
issue: '2'
language:
- iso: eng
month: '02'
oa: 1
oa_version: Published Version
page: 81-118
publication: Journal of Algebra
publication_identifier:
eissn:
- 1090-266X
issn:
- 0021-8693
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
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title: On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn
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user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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year: '2025'
...