论文信息 - Orbits on Linear Algebraic Groups

Orbits on Linear Algebraic Groups

Let G be a linear algebraic group and let p: G GL(V) be a rational representation of G. When G is linearly reductive, D. Mumford has shown that if a point x C V has 0 in the Zariski-closure cl (Gs x) of its orbit, then there exists a one-parameter subgroup X: Gm G such that x(a) x 0 as a 0 (Theorem (4.1)). (See ? 2 for notation and definitions.) Suppose that G and p :are defined over a field k and that x C Vk. It has been conjectured by Mumford (based on a stronger conjecture of J. Tits see [13, p. 64]) that, when k is perfect, X can be chosen to be defined over k. More generally, one can ask when a linear algebraic k-group G has the following property:

D. Birkes | David Spencer Birkes

[1] Lie Group Representations on Polynomial Rings , 1963 .

[2] Armand Borel,et al. Arithmetic subgroups of algebraic groups , 1962 .

[3] S. Helgason. Differential Geometry and Symmetric Spaces , 1964 .

[4] C. Curtis,et al. Representation theory of finite groups and associated algebras , 1962 .

[5] A. Borel,et al. Introduction aux groupes arithmétiques , 1969 .

[6] G. Mostow,et al. On the compactness of arith-metically defined homogeneous spaces , 1962 .

[7] M. Rosenlicht. On quotient varieties and the affine embedding of certain homogeneous spaces , 1961 .

[8] D. Mumford,et al. Geometric Invariant Theory , 2011 .

[9] Robert Steinberg,et al. Regular elements of semisimple algebraic groups , 1965 .