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Journal of Lie Theory 12 (2002), No. 1, 301--304
Copyright Heldermann Verlag 2002

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

Nazih Nahlus
Dept. of Mathematics, American University of Beirut, c/o New York Office, 850 Third Ave. 18th floor, New York, NY 10022-6297, U.S.A.

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.

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