McGovern applies Borel-Weil-Bott to obtain a branching rule from G=Spin(7,C) to H=G2. The main point here is that there are parabolics P \subset G and Q \subset H such that G\P=H\Q. Then, via Borel-Weil-Bott, he reduces the branching problem to one between the Levi factors of these parabolics. The same idea applies to branching from SL(2n,C) to Sp(2n,C). The relevant paper is "McGovern, William M. A branching law for ${\rm Spin}(7,C)\to G\sb 2$ Spin(7,C)→G2 and its applications to unipotent representations. J. Algebra 130 (1990), no. 1, 166--175."
McGovern applies Borel-Weil-Bott to obtain a branching rule from G=Spin(7,C) to H=G2. The main point here is that there are parabolics P \subset G and Q \subset H such that G\P=H\Q. Then, via Borel-Weil-Bott, he reduces the branching problem to one between the Levi factors of these parabolics. The same idea applies to branching from SL(2n,C) to Sp(2n,C). The relevant paper is "McGovern, William M. A branching law for ${\rm Spin}(7,C)\to G\sb 2$ and its applications to unipotent representations. J. Algebra 130 (1990), no. 1, 166--175."