Note on Observable Subgroups of Linear Algebraic Groups

Nazih Nahlus

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.

Classification (MSC2000): Primary 20G05, 20G15, 22E45

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