The Structure of Root Data and Smooth Regular Embeddings of Reductive Groups
AffiliationUniversity of Arizona
MetadataShow full item record
PublisherCambridge University Press
CitationThe Structure of Root Data and Smooth Regular Embeddings of Reductive Groups, Proc. Edinb. Math. Soc. (2) 62 (2019), no. 2, 523-552.
Rights© Edinburgh Mathematical Society 2018
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AbstractWe 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 parametrization 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 In an unpublished manuscript, Asai proved three key reduction techniques that are used for reducing statements about arbitrary connected reductive algebraic groups, equipped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. Using our investigations into root data we give new proofs of Asai's results and generalize 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.
Note6 month embargo; published online: 29 November 2018
VersionFinal accepted manuscript
SponsorsINdAM (Istituto Nazionale di Alta Matematica Francesco Severi); European Commission via an INdAM Marie Curie Fellowship; University of Padova [CPDA125818/12, 60A01-4222/15]