The Toda flows preserving small cells of the flag variety G / B and Kazhdan's x₀-grading

There exist many works on full Kostant-Toda flows whose initial points belong to a grand cell of the flag variety. In this paper we study the full Kostant-Toda flows whose initial points belong to small cells other than the grand cell and preserving small cells. To obtain such flows, we consider the cell decomposition of the grand cell induced by Bruhat decomposition of the flag variety. We show that such full Kostant-Toda flow can be reduced to the ordinary tridiagonal Toda flow by using sheaf theoretical methods.