Sur le spectre des opérateurs de Markov de designs sphériques

Let X be a nonempty finite subset of the sphere of dimension n−1, where n⩾3. For every nonnegative integer k, we define a Markov operator MX(k) acting on the space H(k) of the restrictions to the sphere of polynomial functions on Rn which are homogeneous of degree k and harmonic as follows: MX(k)=1|X|∑x∈Xπ(k)(rx), where π(k) is the natural representation of O(n) on H(k), and rx is the reflection fixing the hyperplane orthogonal to x. We show that, when X is antipodal, MX(s) is a homothety if and only if X is a spherical (2s+1)-design. In the case of tight spherical designs, we get more information on the spectrum of these operators.