Repository logo

On eigenvectors for semisimple elements in actions of algebraic groups



Change log


Kenneally, Darren John 


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.





Representation theory, Algebraic groups, Group theory, Eigenvectors


Doctor of Philosophy (PhD)

Awarding Institution

University of Cambridge