{"degree_awarded":"PhD","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"}],"doi_confirm":"1","publication_status":"published","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.","tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"related_material":{"record":[{"id":"21751","relation":"part_of_dissertation","status":"public"},{"id":"22693","relation":"part_of_dissertation","status":"public"}]},"file":[{"content_type":"application/pdf","date_updated":"2026-08-14T11:42:31Z","date_created":"2026-08-14T11:42:31Z","creator":"jloewit","file_id":"22709","checksum":"2d0be77791dc296621c6c0f76be9d0d7","file_name":"2026_Löwit_Jakub_Thesis.pdf","file_size":1574709,"relation":"main_file","access_level":"open_access"},{"file_size":1085118,"relation":"source_file","access_level":"closed","file_id":"22710","checksum":"61bde4b58c1e6c7baeb561e41f82659b","file_name":"2026_Löwit_Jakub_Source_files.zip","creator":"jloewit","content_type":"application/zip","date_created":"2026-08-14T11:42:42Z","date_updated":"2026-08-14T11:42:42Z"}],"ddc":["510","516","512","514","513"],"date_created":"2026-08-12T14:05:36Z","type":"dissertation","supervisor":[{"id":"4A0666D8-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9582-2634","last_name":"Hausel","full_name":"Hausel, Tamás","first_name":"Tamás"}],"year":"2026","title":"Equivariant K-theory of affine Grassmannians in representation theory and arithmetic","publisher":"Institute of Science and Technology Austria","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","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>","ieee":"J. Löwit, “Equivariant K-theory of affine Grassmannians in representation theory and arithmetic,” Institute of Science and Technology Austria, 2026.","short":"J. Löwit, Equivariant K-Theory of Affine Grassmannians in Representation Theory and Arithmetic, Institute of Science and Technology Austria, 2026.","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>.","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>.","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>"},"publication_identifier":{"issn":["2663-337X"]},"date_updated":"2026-08-26T06:53:55Z","_id":"22694","date_published":"2026-08-05T00:00:00Z","project":[{"grant_number":"27004","name":"Arithmetic, geometry, topology and representation theory arising from the affine Grassmannian","_id":"901e2a43-16d5-11f0-9cad-9cead34748d6"},{"grant_number":"P35847","name":"Geometry of the tip of the global nilpotent cone","_id":"34b2c9cb-11ca-11ed-8bc3-a50ba74ca4a3"}],"oa":1,"status":"public","department":[{"_id":"GradSch"},{"_id":"TaHa"}],"oa_version":"Published Version","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","corr_author":"1","author":[{"last_name":"Löwit","full_name":"Löwit, Jakub","first_name":"Jakub","id":"e3b80ae2-eb8e-11eb-b029-9aef4a9108a0"}],"page":"185","alternative_title":["ISTA Thesis"],"doi":"10.15479/AT-ISTA-22694","file_date_updated":"2026-08-14T11:42:42Z","month":"08","language":[{"iso":"eng"}],"article_processing_charge":"No","day":"05","has_accepted_license":"1","OA_place":"publisher"}