A Note on Observable Subgroups of Linear Algebraic Groups and a Theorem of Chevalley

N. Nahlus

doi:10.5802/jolt.264

N. Nahlus

Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 301-304

Abstract

Let H be an algebraic subgroup of a linear algebraic group G over an algebraically closed field K. We show that H is observable in G if and only if there exists a finite-dimensional rational G-module V and an element v of V such that H is the isotropy subgroup of v as well as the isotropy subgroup of the line Kv. Moreover, we give a similar result in the case where H contains a normal algebraic subgroup A which is observable in G. In this case, we deduce that H is observable in G whenever H/A has non non-trivial rational characters. We also give an example from complex analytic groups.

Zbl

DOI: 10.5802/jolt.264

Classification: 20G05, 20G15, 22E45, 20E07
Keywords: linear algebraic groups, finite-dimensional rational modules, isotropy subgroups, normal algebraic subgroups, complex analytic groups

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N. Nahlus. A Note on Observable Subgroups of Linear Algebraic Groups and a Theorem of Chevalley. Journal of Lie Theory, Volume 12 (2002) no. 1, pp. 301-304. doi: 10.5802/jolt.264

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