A geometric approach to Feigin-Loktev fusion product and cluster relations in coherent Satake category

TL;DR

This paper introduces a geometric method to compute the Feigin-Loktev fusion product of modules over current algebras, linking it to convolution of sheaves on the affine Grassmannian and exploring cluster relations in the Satake category.

Contribution

It provides a new geometric realization of the fusion product, enabling calculations in new cases and connecting it to cluster relations in the coherent Satake category.

Findings

Computed the fusion product in new cases using geometric methods

Established a relation between fusion product and convolution of sheaves

Proposed conjectural cluster relations in the Grothendieck ring

Abstract

We propose a geometric realization of the Feigin-Loktev fusion product of graded cyclic modules over the current algebra. This allows us to compute it in several new cases. We also relate the Feigin-Loktev fusion product to the convolution of perverse coherent sheaves on the affine Grassmannian of the adjoint group. This relation allows us to establish the existence of exact triples, conjecturally corresponding to cluster relations in the Grothendieck ring of coherent Satake category.