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 Ē 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 ≤ 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:541715 |
| Date | January 2010 |
| Creators | Kenneally, Darren John |
| Contributors | Lawther, Ross |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.repository.cam.ac.uk/handle/1810/224782 |
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