ИСТИНА

Schubert cells in standard flag manifolds and their generalizations to arbitrary Lie groups, Bruhat cells, play important role in algebraic geometry and representation theory. These cells are open dense subsets in algebraic subvarieties in flag spaces (Schubert varieties), enumerated by the elements of the corresponding Weyl group. Namely, the Schubert cell XwC, corresponding to w ∈ W(gC) is equal to the BC+ orbit of the element [w] ∈ Fl(gC) = GC/BC+. Here [w] is the point in Fl(gC), represented by the element w ̃ in the normalizer of Cartan subalgebra, corresponding to w ∈ W(gC). Similarly dual Schubert cell YwC, corresponding to w is the orbit BC− · [w]. Similarly, one defines real Bruhat cells Xw and dual real Bruhat cells Yw in real flag manifold Fl(G) = G/B+ = K/ZK(h) (where B+ is the real Borel subgroup) as the orbits of [w] with respect to B+ and B−. In complex case the structure of these cells is relatively well understood and has been extensively studied in the last 40 years. For example, using complex geometry and algebraic groups theory one can show that complex Bruhat cells intersect dual cells in accordance with Bruhat order on the corresonding Weil group: YwC 􏰚 XwC′ ̸= ∅ iff w′ ≺ w in Bruhat order. Moreover, in the latter case the intersection is always transversal and the (complex) dimension of this variety is equal to the difference of lengths of the corresponding Weil elements: l(w) − l(w′). It turned out that the similar properties for real Bruhat cells, though mutely implied by many authors, have not been accurately proved to this moment. In my talk I shall show that the following is true: Proposition 1 Suppose the real Lie algebra g of G is non-split (normal). Then the intersection Xw 􏰚 Yw′ is nonempty, iff w′ ≺ w in Bruhat order; moreover, if it is not empty, its (real) dimension is equal to the difference l(w) − l(w′). In addition to filling the existing gap (this statement used to be accurately proved only in the case of G = SLn(R), where it followed from the geometric description of Schubert cells in terms of matrix ranks), the purpose of this talk is to demonstrate the connection of this theory with the full symmetric Toda flow. Namely, the result follows from the general properties of the Toda flow on real groups. In this sense the talk paper is a continuation of the research, we began in the papers 1, 2 and 3.