In the theory of reductive algebraic groups, there is the following map (notation: $G$ reductive over an algebraically closed field, $T$ a maximal torus, $B$ a Borel, $X(T)$ the characters of $T$, $Pic^G$ denotes the group of isomorphism classes of $G$-equivariant line bundles):

$$X(T) \to Pic^G(G/B), \lambda \mapsto \mathcal O(-\lambda) := G \times^B {k_\lambda}$$

My questions:

- Is (analogously to $Pic(S) = H^1(S, \mathcal O^\times)$) $Pic^G$ expressible as some cohomology group?
- Is the above map some kind of connection homomorphism or does it have another cohomological interpretation?

Thank you!