Geometry of Multiplicity-Free Representations of SO ( N ) $\mathrm {SO}(N)$ and Visible Actions

Abstract

For compact simple groups G$G$ of type B or D, we find the pairs (V1,V2)$(V_{1},V_{2})$ of irreducible representations of G$G$ such that the tensor product representation V1?V2$V_{1}\otimes V_{2}$ is multiplicity-free by a geometric consideration based on the notion of visible actions on complex manifolds, introduced by T. Kobayashi. The pairs that we find exhaust all the multiplicity-free pairs (V1,V2)$(V_{1},V_{2})$ in comparison with the classification that was obtained earlier by Stembridge by combinatorial methods.