Symmetric weights and $s$-representations

Abstract

We study irreducible representations of compact Lie groups relating an algebraic condition (the highest weight λ is i.e., in any simple factor all non zero ζλ, oC) are equal, for any positive root α and any invariant inner product) with a geometnc one (for all orbits, the d-Xh osculating space coincides with the representation space). We prove that, if d — 2 and λ is symmetric, the irreducible representation with highest weight λ corresponds to the isotropy representation of a symmetric space.