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On eigenvectors for semisimple elements in actions of algebraic groups


Type

Thesis

Change log

Authors

Kenneally, Darren John 

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 E is strictly less than the dimension of V provided dimV>dimG+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 dimV>dimG+2, with a short list of possible exceptions. In the majority of cases we consider modules for which dimV>dimG+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 dimG. In more difficult cases, when dimV is only slightly larger than dimG+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 dimVdimG+2, an immediate observation yields the result for dimV<dimB where B is a Borel subgroup of G, while in other cases we argue directly.

Description

Date

Advisors

Keywords

Representation theory, Algebraic groups, Group theory, Eigenvectors

Qualification

Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge