The question needs a little more detail, including precise references (for instance to the book by Chriss and Ginzburg). Aside from that, there is a fairly long history of related study in a wider context, for example an old article by V.L. Popov (in a journal translated into English): "Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings" (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322.

Popov works over an arbitrary algebraically closed field, where among other things he can show that the Picard group is trivial for a connected simply connected algebraic group.

ADDED: I should add a 1976 reference which is probably more helpful and which also has numerous references back to the original literature. This is a paper by Birger Iversen, "The geometry of algebraic groups", Adv. in Math. 20 (1976), 57-85. An important source for example is the work of Chevalley in the 1950s.

In any case, what proof of triviality you like best will depend a lot on what you already know about algebraic groups and algebraic geometry. Different approaches are possible.

The question needs a little more detail, including precise references (for instance to the book by Chriss and Ginzburg). Aside from that, there is a fairly long history of related study in a wider context, for example an old article by V.L. Popov (in a journal translated into English): "Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings" (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322.

Popov works over an arbitrary algebraically closed field, where among other things he can show that the Picard group is trivial for a connected simply connected algebraic group.

ADDED: I should add a 1976 reference which is probably more helpful and which also has numerous references back to the original literature. This is a paper by Birger Iversen, "The geometry of algebraic groups", Adv. in Math. 20 (1976), 57-85. An important source for example is the work of Chevalley in the 1950s.

In any case, what proof of triviality you like best will depend a lot on what you already know about algebraic groups and algebraic geometry. Different approaches are possible.

Concerning the use of representation theory by BGG (and others), it should be emphasized that only the most elementary characteristic-free ideas are actually involved (as in Chevalley's seminars). To study line bundles for a connected semisimple group in any characteristic, it's natural to associate them with characters of a maximal torus and related geometrically constructed finite dimensional representations. In turn, being "simply connected" in Chevalley's general sense just involves the position of the root lattice inside the full weight lattice. While the fine details of representation theory in prime characteristic involve questions still not fully answered, none of this is needed for the study of Picard groups.

The question needs a little more detail, including precise references (for instance to the book by Chriss and Ginzburg). Aside from that, there is a fairly long history of related study in a wider context, for example an old article by V.L. Popov (in a journal translated into English): "Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings" (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322.

Popov works over an arbitrary algebraically closed field, where among other things he can show that the Picard group is trivial for a connected simply connected algebraic group.

The question needs a little more detail, including precise references (for instance to the book by Chriss and Ginzburg). Aside from that, there is a fairly long history of related study in a wider context, for example an old article by V.L. Popov (in a journal translated into English): "Picard groups of homogeneous spaces of linear algebraic groups and one-dimensional homogeneous vector fiberings" (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 294–322.

Popov works over an arbitrary algebraically closed field, where among other things he can show that the Picard group is trivial for a connected simply connected algebraic group.

ADDED: I should add a 1976 reference which is probably more helpful and which also has numerous references back to the original literature. This is a paper by Birger Iversen, "The geometry of algebraic groups", Adv. in Math. 20 (1976), 57-85. An important source for example is the work of Chevalley in the 1950s.

In any case, what proof of triviality you like best will depend a lot on what you already know about algebraic groups and algebraic geometry. Different approaches are possible.