Double Bruhat cells and total positivity

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Abstract
We study the totally nonnegative varietyG≥0G_{\ge 0}in a semisimple algebraic groupGG. These varieties were introduced by G. Lusztig, and include as a special case the variety of unimodular matrices of a given order whose all minors are nonnegative. The geometric framework for our study is provided by intersectingG≥0G_{\ge 0}with double Bruhat cells (intersections of cells of the two Bruhat decompositions ofGGwith respect to opposite Borel subgroups).

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