Abstract
We study irreducible representations of compact Lie groups relating an algebraic condition (the highest weight λ is “symmetric”, i.e., in any simple factor all non zero <λ, α> are equal, for any positive root α and any invariant inner product) with a geometric one (for all orbits, the d-th 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.
