Component Groups of Stabilizers in Algebraic Groups: Computational Methods and Applications

The component groups of the stabilizers of nilpotent elements in complex semisimple Lie algebras have been a relevant topic of research over the last century. The works of Alekseevskii (1979) and Sommers (1998) contributed significantly to the determination of such groups for exceptional Lie algebras, while Jantzen (2004) provided similar results for Lie algebras of classical types. However, the cited papers primarily compute the isomorphism types of these component groups. In this thesis, we investigate methods to compute explicit generators of the component groups for simple Lie algebras and for two θ-groups. In the latter case, we use the framework of Galois cohomology to derive the real orbits of the action of the θ-groups on certain vector spaces. For this purpose, the computation of the explicit generators of the component groups of stabilizers of nilpotent and semisimple elements is essential. We develop ad hoc algorithmic methods to perform these calculations across a wide range of situations. In this setting, computer algebra systems such as GAP and Magma play a fundamental role, and most of our algorithms have been implemented within these systems.