On the Jacobi differential operators associated to minimal isometric immersions of symmetric spaces into spheres. II
This is a continuation of our first paper [ibid. 18, 115--145 (1981; Zbl 0482.53046)] concerning the linear mappings \(S_{\sigma}\) from that paper. In section 6 we study subrepresentations of a representation of a compact Lie group \(G\) with respect to its closed subgroup \(K\). We introduce certain constants associated to a representation of \(G\) which describe some intertwining homomorphisms. And under certain conditions, called \((P_ 1)\) and \((P_ 2)\), we prove some properties of these constants, which play important roles for the study of the linear mappings \(S_{\sigma}\). In section 7 we consider an orthogonal symmetric Lie algebra \((\mathfrak g,\sigma)\), and study a 3-dimensional subalgebra of \(\mathfrak g\) as well as its representation induced from that of \(\mathfrak g\). The results in sections 6 and 7 will be used in the later computations. In section 8 we study minimally imbedded symmetric \(R\)-spaces into spheres. It is shown that in these cases the Jacobi differential operator \(\tilde S\) reduces to Casimir operators. In section 9 we recall some basic results on representations of the special orthogonal group \(\mathrm{SO}(n+1)\), and study in detail certain representations of the group \(\mathrm{SO}(n+1)\). It is shown that the properties \((P_ 1)\) and \((P_ 2)\) are satisfied in the cases where immersed manifolds are spheres.
scientific article; zbMATH DE number 3899689