Automorphism invariant Cartan subgroups and power maps of disconnected groups

作者：Chatterjee Pralay^*

来源：Mathematische Zeitschrift, 2011, 269(1-2): 221-233.

DOI：10.1007/s00209-010-0723-4

• Cartan subgroups of algebraic groups
• Power maps
• Surjectivity
• Weak-exponentiality

We extend a well-known result of R. Steinberg on the existence of an invariant maximal torus under a semisimple automorphism of an algebraic group over an algebraically closed field. We show that the same result holds when the underlying field is of characteristic zero, but not necessarily algebraically closed. We next study surjectivity of the power maps g -> g(n) of disconnected algebraic groups of characteristic zero. In the case of disconnected real algebraic groups we apply our generalisation of Steinberg's result to obtain results on the surjectivity of the power maps. We also extend a result of A. Borel on weak exponentiality in real Lie groups by relating it with the surjectivity of the square map.

• 出版日期2011-10
• 单位常州工学院