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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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Generalization of the standard construction

Generalization of the standard construction

9 Generalization of the standard construction
Classification of Pseudo-reductive Groups (AM-191)

Brian Conrad

Gopal Prasad

Princeton University Press

This chapter deals with “generalized standard” pseudo-reductive group over a field k. Because the notion of root field has only been defined for absolutely pseudo-simple G whose root system over ks is reduced, the first step is to remove that restriction. Thus, in the non-reduced case the focus is on the longest (equivalently, divisible) roots when defining the root field of G. For absolutely pseudo-simple k-groups of minimal type with a non-reduced root system over ks the condition that the root field coincides with the ground field is a reasonable one to impose for the purpose of a structure theorem. The chapter proves a rigidity property of generalized standard presentations involving the notion of “pseudo-isogeny.” It concludes with a detailed discussion of the structure theorem.

Keywords:   generalized standard, pseudo-reductive group, root field, root system, pseudo-simple k-group, minimal type, structure theorem, rigidity property, generalized standard presentation, pseudo-isogeny

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