Total positivity in the De Concini-Procesi Compactification

We study the nonnegative part $\overline{G_{>0}}$ of the De Concini-Procesi compactification of a semisimple algebraic group $G$, as defined by Lusztig. Using positivity properties of the canonical basis and parametrization of flag varieties, we will give an explicit description of $\overline{G_{>0}}$. This answers the question of Lusztig in Total positivity and canonical bases, Algebraic groups and Lie groups (ed. G.I. Lehrer), Cambridge Univ. Press, 1997, pp. 281-295. We will also prove that $\overline{G_{>0}}$ has a cell decomposition which was conjectured by Lusztig.