  We study the nonnegative part B_{\ge 0} of the flag variety of a reductive
algebraic group G, as defined by Lusztig. Using positivity properties of the
canonical basis it is shown that B_{\ge 0} has an algebraic cell decomposition
indexed by pairs w\le w' of the Weyl group. This result was conjectired by
Lusztig in [Lu; Progress in Math 123].
