  In this work we are concerned with the multiplicity of the eigenvalues of the
Neumann Laplacian in regions of Rn which are invariant under the natural action
of a compact subgroup G of O(n). We give a partial positive answer (in the
Neumann case) to a conjecture of V. Arnold [1] on the transversality of the
transformation given by the Dirichlet integral to the stratification in the
space of quadratic forms according to the multiplicities of the eigenvalues. We
show, for some classes of subgroups of O(n) that, generically in the set of
G-invariant, C2-regions, the action is irreducible in each eigenspace. These
classes include finite subgroups with irreducible representations of dimension
not greater than 2 and, in the case n=2, any compact subgroup of O(2). We also
obtain some partial results for general compact subgroups of O(n).
