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Classification of Pseudo-reductive Groups (AM-191)$
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Brian Conrad and Gopal Prasad

Print publication date: 2015

Print ISBN-13: 9780691167923

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691167923.001.0001

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Constructions when Φ‎ has a double bond

Constructions when Φ‎ has a double bond

8 Constructions when Φ‎ has a double bond
Classification of Pseudo-reductive Groups (AM-191)

Brian Conrad

Gopal Prasad

Princeton University Press

This chapter describes constructions when Φ‎ has a double bond. In particular, it considers a construction that goes beyond SO(q)'s and provides the right generalization of the basic exotic construction for type-Bn when n is not equal to 2. A type-B generalized basic exotic k-group is the universal smooth k-tame central extension of a type-B adjoint generalized basic exotic k-group. The root-field hypothesis in the rank-1 adjoint type-B case is automatic in the higher-rank case. The chapter also builds a large class of absolutely pseudo-simple k-groups of type C via fiber products using type-B generalized basic exotic groups. Finally, it discusses exceptional construction for rank-2 and introduces auxiliary Weil restrictions to explain how the generalized exotic groups of types B and C, as well as the rank-2 basic exceptional groups, underlie a construction beyond the standard case that is exhaustive under a locally of minimal type hypothesis.

Keywords:   double bond, exotic construction, k-tame central extension, rank-1, generalized exotic group, rank-2, Weil restriction, minimal type, pseudo-simple k-group

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