Bases of tensor products and geometric Satake correspondence

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