Multiplicity in root components via Geometric Satake

In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands $V(\lambda+\mu-N\beta) \subset V(\lambda) \otimes V(\mu)$ for ${\bf G}=SL_{n+1}$, where $N\ge 1$ and $\beta$ is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when $\beta$ is not a simple root. In an appendix, we contrast this approach with a combinatorial proof of the same results using Littelmann paths.