An Algebraic Cell Decomposition of the Nonnegative Part of a Flag Variety

We study the nonnegative part ≥ 0of the flag variety of a reductive algebraic group, as defined by Lusztig. Using positivity properties of the canonical basis it is shown that ≥ 0has an algebraic cell decomposition indexed by pairs of elementsw ≤ w′ of the Weyl group. This result was conjectured by G. Lusztig (1994,in“Progr. Math.,” Vol. 123, pp. 531–568, Birkhäuser, Basel).