The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups

Abstract

We investigate the structure of root data by considering their decomposition as a product of a semisimple root datum and a torus. Using this decomposition we obtain a parameterisation of the isomorphism classes of all root data. By working at the level of root data we introduce the notion of a smooth regular embedding of a connected reductive algebraic group, which is a refinement of the commonly used regular embeddings introduced by Lusztig. In the absence of Steinberg endomorphisms such embeddings were constructed by Benjamin Martin. In an unpublished manuscript Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped with a Frobenius endomorphism, to those whose derived subgroup is simple and simply connected. By using our investigations into root data we give new proofs of Asai's results and generalise them so that they are compatible with Steinberg endomorphisms. As an illustration of these ideas, we answer a question posed to us by Olivier Dudas concerning unipotent supports.