Moduli for spherical maps and minimal immersions of homogeneous spaces

The DoCarmo-Wallach theory studies isometric minimal immersions f: G/K → Sⁿ 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^m, 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.