Multiplicity in root components via Geometric Satake

Abstract

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 $G^{\vee }=S{L}_{n+1}$, 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. Finally, we demonstrate that not all such top-dimensional components can be realized as closures of orbits.

17 pages, fixed minor errors and enhanced exposition