Journal of Lie Theory 12 (2002), No. 2, 551--570
Copyright Heldermann Verlag 2002
Moduli for Spherical Maps and Minimal Immersions of Homogeneous Spaces
Gabor Toth
Dept. of Mathematical Sciences, Rutgers University, Camden, NJ 08102, U.S.A.
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 = Sm, 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 Cinfinity(G/K), express them in terms of Jacobi polynomials, and use a recent linearization formula for products of Jacobi polynomials.
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