Bases of tensor products and geometric Satake correspondence

TL;DR

This paper explores the bases of tensor products in the geometric Satake correspondence, revealing their similarities to dual canonical bases and advancing understanding of representation theory.

Contribution

It generalizes the algebraic cycle construction to tensor products, providing new insights into the structure of bases in representation theory.

Findings

Bases share properties with dual canonical bases

Generalization of algebraic cycles to tensor products

Enhanced understanding of geometric Satake correspondence

Abstract

The geometric Satake correspondence can be regarded as a geometric construction of the rational representations of a complex connected reductive group G. In their study of this correspondence, Mirkovi\'c and Vilonen introduced algebraic cycles that provide a linear basis in each irreducible representation. Generalizing this construction, Goncharov and Shen define a linear basis in each tensor product of irreducible representations. We investigate these bases and show that they share many properties with the dual canonical bases of Lusztig.