A decomposition for certain real semisimple Lie groups

Abstract:

For a class of real semisimple Lie groups, including those for which G and K have the same rank, Kostant introduced the decomposition $G = K{N_0}K$, where ${N_0}$ is a certain abelian subgroup of N, and conjectured that the Jacobian of the decomposition with respect to Haar measure, as well as the spherical functions, would be polynomial in the canonical coordinates of ${N_0}$. We compute here the Jacobian, which turns out to be polynomial precisely when the equality of ranks is satisfied. We also compute those spherical functions which restrict to polynomials on ${N_0}$.