ИСТИНА

My talk will describe a next step of our joint investigation with Yu.Chernyakov and A.Sorin, see [1, 2, 3]. It is dedicated to the generalizations of the theory of Toda flow, relating it with the theory of Lie groups and algebras. Namely, I will describe the way this system can be defined for arbitrary real form of a semisimple Lie algebra, so that the role of orthogonal group will fall on the maximal compact subgroup K of the corresponding real group G. The dynamical sysem that I will talk about “lives” on the real flag space G/K of the real group; it has been studied in papers [4, 5, 6] and many others. In particular, it was shown there that it is equal to the gradient flow of a Morse function (in generic case). Its constant points correspond to the elements of Weil group W of G and we shall show that (for normal real forms) trajectories of the system are “governed” by the Bruhat order on W; an interesting consequence of this fact is that the real Bruhat cells intersect (transversally) the dual cells if and only if the corresponding elements in W are comparable in Bruhat order. So far this fact has been established only in complex case; and in the case G = SLn(R) we used this property (which can be proved by the virtue of matrix representation) to describe the trajectories of Toda flow. There are many evidences that a similar property should hold for non- split real forms as well, although the size of the intersections should also depend on the dimensions of eigenspaces. Besides this, if time permits, I will also describe the invariants of this system, which will depend on the structure of the representations of G.