Extended Picard complexes and homogeneous spaces

Let k be an algebraically closed field of characteristic 0. Let G be a connected linear k-group acting on a smooth irreducible k-variety X. In works of Popov, Kraft, Knop, Vust, and others the equivariant Picard group Pic_G(X) of G-line bundles on X was investigated. We introduce the extended equivariant Picard complex UPic_G(X) in degrees 0 and 1, whose first cohomology is Pic_G(X). In other words, we compute Pic_G(X) in terms of divisors and rational functions on X and X\times G. This extended equivariant Picard complex UPic_G(X) is an equivariant version of the extended Picard complex UPic(X), introduced in our previous paper. When Pic(G)=0 and X is a homogeneous space of G with stabilizer H, we compute UPic(X) (up to an isomorphism in the derived category) in terms of the character groups of G and H. Using this result, we compute the ``algebraic part'' Br_a(X) of the Brauer group Br(X), when X is a homogeneous space (maybe without rational points) defined over a number field. This is a joint work with the late Joost van Hamel.