Moduli for Spherical Maps and Minimal Immersions of Homogeneous Spaces

The DoCarmo-Wallach theory studies isometric minimal immersions f : G/K → S of a compact Riemannian homogeneous space G/K into Euclidean n -spheres for various n . For a given domain G/K , the moduli space of such immersions is a compact convex body in a representation space for the Lie group G . In 1971 DoCarmo and Wallach gave a lower bound for the (dimension of the) moduli for G/K = S , and conjectured that the lower bound was achieved. In 1997 the author proved that this was true. The DoCarmo-Wallach conjecture has a natural generalization to all compact Riemannian homogeneous domains G/K . The purpose of the present paper is to show that for G/K a nonspherical compact rank 1 symmetric space this generalized conjecture is false. The main technical tool is to consider spherical functions of subrepresentations of C∞(G/K), express them in terms of Jacobi polynomials, and use a recent linearization formula for products of Jacobi polynomials.