On the \(K\)-types and their tensor product for split \(G_2\) (Q1387460)

For a linear connected simple Lie group \(G\) other than split \(G_2\), in order to obtain formulas for the action of intertwining operators on certain subspaces of standard induced representations of \(G\), \textit{M. W. Baldoni-Silva} and \textit{A. W. Knapp} [J. Funct. Anal. 82, 151-236 (1989; Zbl 0683.22010)] discussed the occurrence of \(K\)-types in a tensor product. In this letter, the author has obtained similar results about the occurrence of \(K\)-types in a tensor product for \(G_2\). Clearly, the results obtained are necessary for proving the formulas for the action of intertwining operators for split \(G_2\).