On G-invariant norms

A result of R. Mathias and Horn [cf. Linear Algebra Appl. 142 (1990) 63] on the representation of the unitarily invariant norm is extended in the context of Eaton triples and of real semisimple Lie algebras. The representation is related to a function • α. Criteria for • α being a norm is given in terms of α and the longest element of the underlying finite reflection group. In particular, for the real simple Lie algebras, • α is a norm on its Cartan subspace for a given nonzero α if and only if ω0α=−α, where ω0 is the longest element of the Weyl group (this is the case if the Weyl group contains −id). Some related results are obtained.