Geodesic
Regular elements in spaces of linear representations of reductive algebraic groups
Izvestiya. Mathematics , Tome 24 (1985) no. 2, pp. 383-390.

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A new proof is offered of a differential criterion of regularity for the adjoint representation of a semisimple connected group that does not use the existence of a section in the set of regular elements. Using the ideas of this proof, similar results are obtained for certain linear actions with a Cartan subspace, and, conversely, the existence of a section in the set of regular elements is proved. Bibliography: 7 titles. 
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D. I. Panyushev. Regular elements in spaces of linear representations of reductive algebraic groups. Izvestiya. Mathematics , Tome 24 (1985) no. 2, pp. 383-390. http://geodesic.mathdoc.fr/item/IM2_1985_24_2_a7/

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