{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/224782"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/224782","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"On eigenvectors for semisimple elements in actions of algebraic groups","abstract":"Let $G$ be a simple simply connected algebraic group defined over an algebraically closed field $K$ and $V$ an irreducible module defined over $K$ on which $G$ acts. Let $E$ denote the set of vectors in $V$ which are eigenvectors for some non-central semisimple element of $G$ and some eigenvalue in $K^∗$. We prove, with a short list of possible exceptions, that the dimension of $\\overline{E}$ is strictly less than the dimension of $V$ provided $\\dim V > \\dim G + 2$ and that there is equality otherwise. In particular, by considering only the eigenvalue $1$, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of $V$ provided $\\dim V > \\dim G + 2$, with a short list of possible exceptions. In the majority of cases we consider modules for which $\\dim V > \\dim G + 2$ where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds $\\dim G$. In more difficult cases, when $\\dim V$ is only slightly larger than $\\dim G + 2$, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying $\\dim V \\leq \\dim G + 2$, an immediate observation yields the result for $\\dim V < \\dim B$ where $B$ is a Borel subgroup of $G$, while in other cases we argue directly.","abstract_html":"Let $G$ be a simple simply connected algebraic group defined over an algebraically closed field $K$ and $V$ an irreducible module defined over $K$ on which $G$ acts. Let $E$ denote the set of vectors in $V$ which are eigenvectors for some non-central semisimple element of $G$ and some eigenvalue in <span class=\"etd-inline-math\">K<sup>∗</sup></span>. We prove, with a short list of possible exceptions, that the dimension of $\\overline{E}$ is strictly less than the dimension of $V$ provided $\\dim V &gt; \\dim G + 2$ and that there is equality otherwise. In particular, by considering only the eigenvalue $1$, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of $V$ provided $\\dim V &gt; \\dim G + 2$, with a short list of possible exceptions. In the majority of cases we consider modules for which $\\dim V &gt; \\dim G + 2$ where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds $\\dim G$. In more difficult cases, when $\\dim V$ is only slightly larger than $\\dim G + 2$, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying $\\dim V \\leq \\dim G + 2$, an immediate observation yields the result for $\\dim V &lt; \\dim B$ where $B$ is a Borel subgroup of $G$, while in other cases we argue directly.","abstract_has_math":true,"creators":["Kenneally, Darren John"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-02-09","date_published":"2010-02-09","updated_at":"2026-07-22T22:24:18Z","subjects":["Representation theory","Algebraic groups","Group theory","Eigenvectors"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/5ce608cc-6d85-4750-8a9a-7bddefd9c7d8/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.16210","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Kenneally, Darren John"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2010-02-09"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["http://www.dspace.cam.ac.uk/handle/1810/224782","https://www.repository.cam.ac.uk/handle/1810/224782"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Representation theory","Algebraic groups","Group theory","Eigenvectors"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/5ce608cc-6d85-4750-8a9a-7bddefd9c7d8/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.16210"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/7e166b73-a2a6-4cef-9ca6-4efda8a435d3/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let $G$ be a simple simply connected algebraic group defined over an algebraically closed field $K$ and $V$ an irreducible module defined over $K$ on which $G$ acts. Let $E$ denote the set of vectors in $V$ which are eigenvectors for some non-central semisimple element of $G$ and some eigenvalue in $K^∗$. We prove, with a short list of possible exceptions, that the dimension of $\\overline{E}$ is strictly less than the dimension of $V$ provided $\\dim V > \\dim G + 2$ and that there is equality otherwise. In particular, by considering only the eigenvalue $1$, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of $V$ provided $\\dim V > \\dim G + 2$, with a short list of possible exceptions. In the majority of cases we consider modules for which $\\dim V > \\dim G + 2$ where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds $\\dim G$. In more difficult cases, when $\\dim V$ is only slightly larger than $\\dim G + 2$, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying $\\dim V \\leq \\dim G + 2$, an immediate observation yields the result for $\\dim V < \\dim B$ where $B$ is a Borel subgroup of $G$, while in other cases we argue directly."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["cc83912d67194bb8ee306973ef9c75c6","6abbd0216aa1e6f05b424a6b867d8cd2"]},{"key":"dc:title","label":"Title","values":["On eigenvectors for semisimple elements in actions of algebraic groups"]}]}],"canonical_facts":{"dc:creator":["Kenneally, Darren John"],"dc:date.issued":["2010-02-09"],"dc:description.abstract":["Let $G$ be a simple simply connected algebraic group defined over an algebraically closed field $K$ and $V$ an irreducible module defined over $K$ on which $G$ acts. Let $E$ denote the set of vectors in $V$ which are eigenvectors for some non-central semisimple element of $G$ and some eigenvalue in $K^∗$. We prove, with a short list of possible exceptions, that the dimension of $\\overline{E}$ is strictly less than the dimension of $V$ provided $\\dim V > \\dim G + 2$ and that there is equality otherwise. In particular, by considering only the eigenvalue $1$, it follows that the closure of the union of fixed point spaces of non-central semisimple elements has dimension strictly less than the dimension of $V$ provided $\\dim V > \\dim G + 2$, with a short list of possible exceptions. In the majority of cases we consider modules for which $\\dim V > \\dim G + 2$ where we perform an analysis of weights. In many of these cases we prove that, for any non-central semisimple element and any eigenvalue, the codimension of the eigenspace exceeds $\\dim G$. In more difficult cases, when $\\dim V$ is only slightly larger than $\\dim G + 2$, we subdivide the analysis according to the type of the centraliser of the semisimple element. Here we prove for each type a slightly weaker inequality which still suffices to establish the main result. Finally, for the relatively few modules satisfying $\\dim V \\leq \\dim G + 2$, an immediate observation yields the result for $\\dim V < \\dim B$ where $B$ is a Borel subgroup of $G$, while in other cases we argue directly."],"dc:format.checksum.md5":["cc83912d67194bb8ee306973ef9c75c6","6abbd0216aa1e6f05b424a6b867d8cd2"],"dc:identifier.doi":["10.17863/CAM.16210"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/7e166b73-a2a6-4cef-9ca6-4efda8a435d3/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["http://www.dspace.cam.ac.uk/handle/1810/224782","https://www.repository.cam.ac.uk/handle/1810/224782"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/5ce608cc-6d85-4750-8a9a-7bddefd9c7d8/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Representation theory","Algebraic groups","Group theory","Eigenvectors"],"dc:title":["On eigenvectors for semisimple elements in actions of algebraic groups"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:18Z"}