On eigenvectors for semisimple elements in actions of algebraic groups
On eigenvectors for semisimple elements in actions of algebraic groups
复制标题
关于代数群作用中半单元的特征向量
DOI：
10.17863/cam.16210
复制
发表时间：
2010
期刊：
影响因子：
--
通讯作者：
Darren John Kenneally
中科院分区：
文献类型：
--
作者：
Darren John Kenneally
设G是定义在代数闭域K上的单连通代数群，V是定义在K上的G作用的不可约模。设E表示V中的一组向量，它们是G的某个非中心半单元的特征向量，K中的某个特征值.我们证明了，如果V和G+2为dim V>；dim G+2，则E的维度严格小于V的维度，否则存在相等.特别地，通过仅考虑本征值1，得出了非中心半单元素的不动点空间的并的闭包的维度严格小于给定dim V>；dim G+2的V的维度，并列出了一系列可能的例外。在大多数情况下，我们考虑对其执行权重分析的模进行dim V>；dim G+2。在许多情况下，我们证明了对于任何非中心半单元和任何本征值，特征空间的余维超过dim G。在更困难的情况下，当dim V仅略大于DIMG+2时，我们根据半单元素的中心子的类型来细分分析。在这里，我们证明了每种类型的一个稍弱的不等式，它仍然足以建立主要结果。最后，对于满足Dimv 6 dim G+2的相对较少的模，直接观察就得到了Dimv<；dim B的结果，其中B是G的Borel子群，而在其他情况下我们直接证明.
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 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 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 dimV 6 dim G+2, an immediate observation yields the result for dimV < dim B where B is a Borel subgroup of G, while in other cases we argue directly.