Multiplicity in root components via Geometric Satake

TL;DR

This paper explicitly constructs certain top-dimensional components of cyclic convolution varieties related to the geometric Satake correspondence, providing new insights into the multiplicities of subrepresentations in tensor products for SL(n+1).

Contribution

It offers explicit constructions of top-dimensional components corresponding to specific irreducible summands and establishes lower bounds on their multiplicities, revealing limitations of orbit closures.

Findings

Constructed explicit top-dimensional components for cyclic convolution varieties.

Derived lower bounds on multiplicities of certain subrepresentations.

Showed not all components are realizable as orbit closures.

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