  Let k be a field, G a smooth connected linear algebraic group and X a
homogeneous space of G over k, such that the geometric stabilizers are
extensions of a smooth group of multiplicative type by a smooth connected
characterfree group. If k has characteristic zero and if X^c is a smooth
compactification of X over k, we obtain a formula for the algebraic Brauer
group of X^c. Several variants are obtained in positive characteristic p,
including the finite field case and the global field case, where the formulae
describe the prime-to-p part of the algebraic unramified Brauer group of X,
without assuming the existence of a smooth compactification of X. Moreover,
assuming that stabilizers are connected, then our formulae hold for the
prime-to-p part of the whole unramified Brauer group.
