Semisimple Subgroups of Semisimple Groups

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Abstract
The projection theorem for weights of a representation of a semisimple group G on restriction to a semisimple subgroup is derived, and the existence of a subgroup corresponding to a given projection is discussed. Dynkin's definition of the index of a simple subgroup is extended to the case of G being only semisimple, and the geometrical meaning of the index is given. A method is developed for finding branching rules for both regular and nonregular subgroups. Explicit general formulas for the branching multiplicities are obtained for all cases when G is of rank 2 and for B3(R7) → G2. Applications to the construction of weight diagrams and the ``state‐labeling'' problems for B2 and G2 are mentioned.