The Cascade of Orthogonal Roots and the Coadjoint Structure of the Nilradical of a Borel Subgroup of a Semisimple Lie Group

Let G be a semisimple Lie group and let g = n− + h+ n be a triangular decomposition of g = LieG. Let b = h+ n and let H, N, B be Lie subgroups of G corresponding respectively to h, n and b. We may identify n− with the dual space to n. The coadjoint action of N on n− extends to an action of B on n−. There exists a unique nonempty Zariski open orbit X of B on n−. Any N -orbit in X is a maximal coadjoint orbit of N in n−. The cascade of orthogonal roots defines a cross-section r×− of the set of such orbits leading to a decomposition X = N/R× r×−. This decomposition, among other things, establishes the structure of S(n)n as a polynomial ring generated by the prime polynomials of Hweight vectors in S(n)n. It also leads to the multiplicity 1 of H weights in S(n)n. 2010 Math. Subj. Class. 20C, 14L24.