The motivic Satake equivalence

We refine the geometric Satake equivalence due to Ginzburg, Beilinson-Drinfeld, and Mirkovi'c-Vilonen to an equivalence between mixed Tate motives on the double quotient $L^{+}G\mathrm{\backslash }LG\mathrm{/}{L}^{+}G$ and representations of Deligne’s modification of the Langlands dual group $\widehat{G}$.