Michel Brion's [Lectures on the Geometry of Flag Varieties][1] answers all of your questions in the special case $G = SL_n$ and $P = B$ (the Borel subgroup of upper triangular matrices).  See Section 1.4.  If you are not so familiar with this particular field, you may find the entire first section quite helpful.

  [1]: http://arxiv.org/abs/math/0410240