  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.
