Derivatives of orbital function and an extension of Berezin-Gel’fand’s theorem

Abstract

A generalization of a result of Berezin and Gel’fand in the context of Eaton triples is given. The generalization and its proof are Lie-theoretic free and requires some basic knowledge of nonsmooth analysis. The result is then applied to determine the distance between a point and a G-orbit or its convex hull.We also discuss the derivatives of some orbital functions.

Keywords

• Berezin-Gel’fand’s theorem
• subdifferential
• Clarke generalized gradient
• Lebourg mean value theorem
• Eaton triple
• reduced triple
• finite reflection group