# A proof of the refined PRV conjecture via the cyclic convolution variety

## TL;DR

This paper uses geometric Satake correspondence and cyclic convolution varieties to provide a simple proof of the PRV conjecture and its refinement, connecting MV-cycles with orbit closures.

## Contribution

It offers a novel, geometric proof of the PRV conjecture and its refinement using the cyclic convolution variety and MV-cycles, clarifying their orbit structure.

## Key findings

- Proof of the PRV conjecture using geometric methods
- Refinement of the PRV conjecture established
- Explicit characterization of MV-cycles as orbit closures

## Abstract

In this brief note we illustrate the utility of the geometric Satake correspondence by employing the cyclic convolution variety to give a simple proof of the Parthasarathy-Ranga Rao-Varadarajan conjecture, along with Kumar's refinement. The proof involves recognizing certain MV-cycles as orbit closures of a group action, which we make explicit by unique characterization. In an appendix, joint with P. Belkale, we discuss how this work fits in a more general framework.