[{"has_accepted_license":"1","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","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.","type":"dissertation","year":"2026","oa_version":"Published Version","publisher":"Institute of Science and Technology Austria","day":"05","fulldoi":"https://doi.org/10.15479/AT-ISTA-22694","status":"public","doi":"10.15479/AT-ISTA-22694","abstract":[{"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.","lang":"eng"}],"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","name":"Geometry of the tip of the global nilpotent cone","grant_number":"P35847"}],"publication_status":"published","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"_id":"22694","file":[{"creator":"jloewit","date_updated":"2026-08-14T11:42:31Z","relation":"main_file","checksum":"2d0be77791dc296621c6c0f76be9d0d7","content_type":"application/pdf","file_id":"22709","date_created":"2026-08-14T11:42:31Z","file_name":"2026_Löwit_Jakub_Thesis.pdf","access_level":"open_access","file_size":1574709},{"date_created":"2026-08-14T11:42:42Z","file_size":1085118,"access_level":"closed","file_name":"2026_Löwit_Jakub_Source_files.zip","date_updated":"2026-08-14T11:42:42Z","relation":"source_file","checksum":"61bde4b58c1e6c7baeb561e41f82659b","creator":"jloewit","content_type":"application/zip","file_id":"22710"}],"oa":1,"alternative_title":["ISTA Thesis"],"date_created":"2026-08-12T14:05:36Z","supervisor":[{"orcid":"0000-0002-9582-2634","first_name":"Tamás","id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","full_name":"Hausel, Tamás","last_name":"Hausel"}],"language":[{"iso":"eng"}],"file_date_updated":"2026-08-14T11:42:42Z","department":[{"_id":"GradSch"},{"_id":"TaHa"}],"article_processing_charge":"No","title":"Equivariant K-theory of affine Grassmannians in representation theory and arithmetic","publication_identifier":{"issn":["2663-337X"]},"corr_author":"1","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","date_published":"2026-08-05T00:00:00Z","ddc":["510","516","512","514","513"],"degree_awarded":"PhD","related_material":{"record":[{"status":"public","id":"21751","relation":"part_of_dissertation"},{"status":"public","id":"22693","relation":"part_of_dissertation"}]},"month":"08","page":"185","author":[{"id":"e3b80ae2-eb8e-11eb-b029-9aef4a9108a0","full_name":"Löwit, Jakub","first_name":"Jakub","last_name":"Löwit"}],"date_updated":"2026-08-26T06:53:55Z","doi_confirm":"1","OA_place":"publisher","citation":{"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>.","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>","short":"J. Löwit, Equivariant K-Theory of Affine Grassmannians in Representation Theory and Arithmetic, Institute of Science and Technology Austria, 2026.","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.","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>."}},{"has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"      2026","status":"public","OA_type":"hybrid","article_number":"rnag058","fulldoi":"https://doi.org/10.1093/imrn/rnag058","publisher":"Oxford University Press","oa_version":"Published Version","type":"journal_article","year":"2026","day":"01","project":[{"_id":"901e2a43-16d5-11f0-9cad-9cead34748d6","grant_number":"27004","name":"Arithmetic, geometry, topology and representation theory arising from the affine Grassmannian"}],"doi":"10.1093/imrn/rnag058","abstract":[{"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. ","lang":"eng"}],"tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"publication":"International Mathematics Research Notices","_id":"21751","publication_status":"published","oa":1,"file":[{"file_size":1663246,"access_level":"open_access","file_name":"2026_IMRN_Loewit.pdf","date_created":"2026-05-06T06:35:05Z","success":1,"content_type":"application/pdf","file_id":"21803","checksum":"306f4567b7b2dcf38e23f7b55a27514e","date_updated":"2026-05-06T06:35:05Z","relation":"main_file","creator":"dernst"}],"quality_controlled":"1","volume":2026,"article_processing_charge":"Yes (via OA deal)","title":"Equivariant localizing invariants of simple varieties","department":[{"_id":"TaHa"}],"file_date_updated":"2026-05-06T06:35:05Z","scopus_import":"1","publication_identifier":{"eissn":["1687-0247"],"issn":["1073-7928"]},"language":[{"iso":"eng"}],"date_created":"2026-04-19T22:07:48Z","article_type":"original","corr_author":"1","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.","external_id":{"arxiv":["2507.09392"]},"month":"04","PlanS_conform":"1","related_material":{"record":[{"relation":"dissertation_contains","status":"public","id":"22694"}]},"ddc":["510"],"date_published":"2026-04-01T00:00:00Z","arxiv":1,"issue":"7","OA_place":"publisher","citation":{"short":"J. Löwit, International Mathematics Research Notices 2026 (2026).","ieee":"J. Löwit, “Equivariant localizing invariants of simple varieties,” <i>International Mathematics Research Notices</i>, 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.","chicago":"Löwit, Jakub. “Equivariant Localizing Invariants of Simple Varieties.” <i>International Mathematics Research Notices</i>. Oxford University Press, 2026. <a href=\"https://doi.org/10.1093/imrn/rnag058\">https://doi.org/10.1093/imrn/rnag058</a>.","mla":"Löwit, Jakub. “Equivariant Localizing Invariants of Simple Varieties.” <i>International Mathematics Research Notices</i>, vol. 2026, no. 7, rnag058, Oxford University Press, 2026, doi:<a href=\"https://doi.org/10.1093/imrn/rnag058\">10.1093/imrn/rnag058</a>.","ama":"Löwit J. Equivariant localizing invariants of simple varieties. <i>International Mathematics Research Notices</i>. 2026;2026(7). doi:<a href=\"https://doi.org/10.1093/imrn/rnag058\">10.1093/imrn/rnag058</a>","apa":"Löwit, J. (2026). Equivariant localizing invariants of simple varieties. <i>International Mathematics Research Notices</i>. Oxford University Press. <a href=\"https://doi.org/10.1093/imrn/rnag058\">https://doi.org/10.1093/imrn/rnag058</a>"},"author":[{"last_name":"Löwit","first_name":"Jakub","id":"e3b80ae2-eb8e-11eb-b029-9aef4a9108a0","full_name":"Löwit, Jakub"}],"date_updated":"2026-08-26T06:53:54Z"},{"author":[{"id":"e3b80ae2-eb8e-11eb-b029-9aef4a9108a0","full_name":"Löwit, Jakub","first_name":"Jakub","last_name":"Löwit"}],"supplementarymaterial":"no","date_updated":"2026-08-26T06:53:54Z","OA_place":"publisher","mathsc":["19E08","19L47","20G44","14G17","19D55","14F43","14L30","14D24","14M25"],"arxiv":1,"citation":{"mla":"Löwit, Jakub. “Equivariant K-Theory, Affine Grassmannian and Perfection.” <i>Documenta Mathematica</i>, EMS Press, 2026, doi:<a href=\"https://doi.org/10.4171/dm/1064\">10.4171/dm/1064</a>.","ama":"Löwit J. Equivariant K-theory, affine Grassmannian and perfection. <i>Documenta Mathematica</i>. 2026. doi:<a href=\"https://doi.org/10.4171/dm/1064\">10.4171/dm/1064</a>","apa":"Löwit, J. (2026). Equivariant K-theory, affine Grassmannian and perfection. <i>Documenta Mathematica</i>. EMS Press. <a href=\"https://doi.org/10.4171/dm/1064\">https://doi.org/10.4171/dm/1064</a>","short":"J. Löwit, Documenta Mathematica (2026).","ieee":"J. Löwit, “Equivariant K-theory, affine Grassmannian and perfection,” <i>Documenta Mathematica</i>. EMS Press, 2026.","ista":"Löwit J. 2026. Equivariant K-theory, affine Grassmannian and perfection. Documenta Mathematica.","chicago":"Löwit, Jakub. “Equivariant K-Theory, Affine Grassmannian and Perfection.” <i>Documenta Mathematica</i>. EMS Press, 2026. <a href=\"https://doi.org/10.4171/dm/1064\">https://doi.org/10.4171/dm/1064</a>."},"date_published":"2026-03-26T00:00:00Z","ddc":["500"],"das_tickbox":"0","month":"03","related_material":{"record":[{"relation":"dissertation_contains","id":"22694","status":"public"}]},"PlanS_conform":"1","corr_author":"1","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.","external_id":{"arxiv":["2409.18925"]},"article_type":"original","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"],"date_created":"2026-08-12T13:29:17Z","language":[{"iso":"eng"}],"department":[{"_id":"GradSch"},{"_id":"TaHa"}],"title":"Equivariant K-theory, affine Grassmannian and perfection","article_processing_charge":"Yes (in subscription journal)","publication_identifier":{"issn":["1431-0635"],"eissn":["1431-0643"]},"scopus_import":"1","quality_controlled":"1","oa":1,"doi":"10.4171/dm/1064","abstract":[{"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.","lang":"eng"}],"project":[{"_id":"34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3","name":"Geometry of the tip of the global nilpotent cone","grant_number":"P35847"}],"tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"_id":"22693","publication_status":"epub_ahead","publication":"Documenta Mathematica","year":"2026","type":"journal_article","publisher":"EMS Press","oa_version":"Published Version","main_file_link":[{"open_access":"1","url":"https://doi.org/10.4171/DM/1064"}],"researchdata_availability":"no","day":"26","fulldoi":"https://doi.org/10.4171/dm/1064","OA_type":"hybrid","status":"public","has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"isi":1,"has_accepted_license":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","intvolume":"       663","fulldoi":"https://doi.org/10.1016/j.jalgebra.2024.08.033","status":"public","OA_type":"hybrid","type":"journal_article","year":"2025","oa_version":"Published Version","publisher":"Elsevier","day":"01","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."}],"doi":"10.1016/j.jalgebra.2024.08.033","publication":"Journal of Algebra","tmp":{"image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)"},"_id":"18154","publication_status":"published","quality_controlled":"1","file":[{"file_name":"2024_JourAlgebra_Loewit.pdf","file_size":731175,"access_level":"open_access","date_created":"2025-01-13T08:57:57Z","success":1,"content_type":"application/pdf","file_id":"18830","creator":"dernst","relation":"main_file","checksum":"eb240e93c178e48429ad918c9058f1fe","date_updated":"2025-01-13T08:57:57Z"}],"oa":1,"volume":663,"file_date_updated":"2025-01-13T08:57:57Z","department":[{"_id":"TaHa"}],"title":"On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn","article_processing_charge":"Yes (via OA deal)","publication_identifier":{"eissn":["1090-266X"],"issn":["0021-8693"]},"scopus_import":"1","date_created":"2024-09-29T22:01:37Z","language":[{"iso":"eng"}],"article_type":"original","corr_author":"1","external_id":{"isi":["001325207800001"],"arxiv":["2404.11176"]},"month":"02","page":"81-118","date_published":"2025-02-01T00:00:00Z","ddc":["510"],"OA_place":"publisher","issue":"2","arxiv":1,"citation":{"mla":"Löwit, Jakub. “On modulo ℓ Cohomology of P-Adic Deligne–Lusztig Varieties for GLn.” <i>Journal of Algebra</i>, vol. 663, no. 2, Elsevier, 2025, pp. 81–118, doi:<a href=\"https://doi.org/10.1016/j.jalgebra.2024.08.033\">10.1016/j.jalgebra.2024.08.033</a>.","ama":"Löwit J. On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn. <i>Journal of Algebra</i>. 2025;663(2):81-118. doi:<a href=\"https://doi.org/10.1016/j.jalgebra.2024.08.033\">10.1016/j.jalgebra.2024.08.033</a>","apa":"Löwit, J. (2025). On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn. <i>Journal of Algebra</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.jalgebra.2024.08.033\">https://doi.org/10.1016/j.jalgebra.2024.08.033</a>","short":"J. Löwit, Journal of Algebra 663 (2025) 81–118.","ista":"Löwit J. 2025. On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn. Journal of Algebra. 663(2), 81–118.","chicago":"Löwit, Jakub. “On modulo ℓ Cohomology of P-Adic Deligne–Lusztig Varieties for GLn.” <i>Journal of Algebra</i>. Elsevier, 2025. <a href=\"https://doi.org/10.1016/j.jalgebra.2024.08.033\">https://doi.org/10.1016/j.jalgebra.2024.08.033</a>.","ieee":"J. Löwit, “On modulo ℓ cohomology of p-adic Deligne–Lusztig varieties for GLn,” <i>Journal of Algebra</i>, vol. 663, no. 2. Elsevier, pp. 81–118, 2025."},"author":[{"first_name":"Jakub","full_name":"Löwit, Jakub","id":"e3b80ae2-eb8e-11eb-b029-9aef4a9108a0","last_name":"Löwit"}],"date_updated":"2025-02-27T12:32:40Z"}]