Groupe de Picard et groupe de Brauer des compactifications lisses d'espaces homogènes

Let k be a field of characteristic zero, G a connected linear algebraic group over k and H a connected closed k-subgroup of G. Let X be a smooth k-compactification of Y = G/H. We prove that the Galois lattice given by the geometric Picard group of X is flasque. The result was known in the case H = 1. We compute this Galois lattice up to addition of a permutation module. When G is semisimple and simply connected, the result shows that the Brauer group of X is determined by the maximal toric quotient of H.