The result

Let k be a number field and G the split simple group over k of type G2. There is only one such group: it is both simply connected and adjoint. Set G = G(). Denote by {M0, 1} the equivalence class of the pair consisting of the split torus M0 and of the trivial representation of M0. Denote by the direct sum of irreducible subspaces of. Langlands determined the subspace of K-invariant vectors of. It is of dimension 2. Besides the constants, it contains an element whose cuspidal exponents are short roots. We are interested here in what happens when we suppress the hypothesis of invariance under K. A complete study shows that decomposes into two subspaces. The first is of dimension 1 and is reduced to the constants. The K-finite elements of the other all have short roots as cuspidal exponents. We propose to determine the representation of the group G in this last space. A complete study would necessitate a local study at the archimedean places which has not been done. We will study the space V consisting of K-finite elements of whose cuspidal exponents are short roots and which are invariant under K∞. The group Gf operates on V. Denote by Σ the set of finite places of k.