On G-invariant norms

Abstract

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 ω 0 α=− α, where ω 0 is the longest element of the Weyl group (this is the case if the Weyl group contains − id). Some related results are obtained.