Bases of tensor products and geometric Satake correspondence

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.