Groupes engendrés par des réflexions, designs sphériques et réseau de Leech∗

Abstract

For a finite subset $\text{X⊂}\text{R}{}^{\mathrm{n}}$ of unit vectors, G_X denotes the group generated by reflections r_x fixing hyperplanes orthogonal to x∈X. When X is a spherical t-design and π^(k)_har the representation of G_X in the harmonic polynomials in n variables of degree k, the spectrum of the Markov operator $\text{1}\text{|X|}\text{∑}{}_{\mathrm{x\in X}}\text{π}{}^{\mathrm{(k)}}{}_{\mathrm{<mtext>har</mtext>}}\text{(r}{}_{\mathrm{x}}\text{)}$ is analyzed. If k is small enough, this operator is a scalar multiple of the identity. When X is the spherical 11-design of short vectors in a Leech lattice, it is shown that the infinite group G_X contains the finite Conway group Co and is a quotient of a remarkable Coxeter group.