Representations of compact groups and minimal immersions into spheres.
Manfredo P. do Carmo, Nolan R. Wallach · Journal of Differential Geometry · 1970
Let G be a compact group, K a closed subgroup of G, and C(M) the space of all real-valued continuous functions on the homogeneous space M = G/K.Then G has a natural action on C{M) given by g-f(p) = f(g~ιp), fζC(M), geG, peM.Let V be a (necessarily finite-dimensional) invariant irreducible subspace of C(M).Then V may be given an inner product = I fgdμ, where the homogeneous measure dμ normalized in such M a way that I dμ -dim V; relative to , G acts orthogonally on V., f r are r linearly independent in V and Σ f\(p) -1 for all p e M. i = lIn this paper, we are concerned with the following question: For which homogeneous spaces M is condition A satisfied for all invariant irreducible subspaces of C(M)ΊWe shall restrict ourselves to the simplest homogeneous spaces, namely, the simply connected homogeneous spaces G/K, where (G, K) is a symmetric pair of compact type.We recall that for such a pair, G is a compact, semisimple Lie group with an involutive automorphism s:G -> G which is such that K is left fixed by s, and K contains the component of the identity of the fixed point set of s.To ensure the simply connectedness of G/K, we assume further that G is connected, simply connected and that K is connected.In this situation, condition A is strangely rare.In fact, we prove the following:Theorem 1.Let M = G/K be a homogeneous space such that (G, K) is a symmetric pair of compact type, G is connected and simply connected, and K is connected.Then condition A is satisfied for all invariant, irreducible subspaces of C{M) if and only if M is the 2-dimensional sphere S 2 = SC/(2)/ί/(l).In §2, we prove Proposition 1, which says that the invariant, irreducible subspaces of C(S 2 ) satisfy condition A. In § 3, we prove Proposition 2, which Communicated by S. S.